Spin-polarized electrons in atomic layer materials formed on solid surfaces

Abstract

In this review, we summarize the recent progress in the understanding of the spin-polarized electronic states in two-dimensional (2D) atomic layer materials (ALMs) formed on solid surfaces. The spin-polarized electronic states caused by the combination of spin-orbit coupling (SOC) with broken spatial inversion symmetry along the surface normal direction is one of the most exotic phenomena that appears on ALMs formed on solid surfaces as well as clean solid surfaces. The so-called Rashba-Bychkov (RB) effect that arises from the potential gradient induced by broken inversion symmetry was believed to be the main origin of these spin-polarized electronic states. However, the spin texture of most ALMs are different from that caused by the ideal RB effect. Due to the high impact of the spin-polarized electronic states of 2D materials in not only spin-related fundamental science but also in applications since they are the key concepts to realize future semiconductor spintronics devices, much efforts have been made to elucidate the origin of these peculiar spin textures. So far, the deviations in spin texture from the ideal one have been attributed to be induced by perturbation, such as entanglement of spin and orbital momenta. In this review, we first illustrate how the symmetry of the ALM’s atomic structure can affect the spin texture, and then introduce that various spin textures, ranging from the RB-type and symmetry-induced type to spin textures that cannot be explained based on the origins proposed so far, can be simply induced by the orbital angular momentum. This review aims to provide an overview on the insights gained on the spin-polarized electronic states of ALMs and to point out opportunities for exploring exotic physical properties when combining spin and other physics, e.g. superconductivity, and to realize future spintronics-based quantum devices.

Introduction

Two-dimensional (2D) atomic layer materials (ALMs) formed on solid surfaces show exotic quantum phenomena that do not appear in 3D bulk systems, and thus have attracted an incredible interest in the past decade. The formation of ALMs on solid surfaces also allows to create 2D materials that do not exist in nature. The graphene family silicene [1], germanene [2], [3], borophene [4], [5], which can not survive as free standing ALM but can be formed on Ag(111) or Al(111) surfaces, have been reported to host massless Dirac Fermions, and the atomic layer FeSe formed on ${\text{SrTiO}}_3$ shows a 10 times higher superconducting transition temperature than that of bulk FeSe [6]. The combination of broken spatial inversion symmetry along the surface normal direction [i.e. one side of the ALM is the substrate and the other side is vacuum or air or liquid as shown in Fig. 1(a)] and spin-orbit coupling (SOC) leads to further exotic phenomena for these ALMs, which are related to electron spins that are of interest for not only fundamental science but also applications since they contain key concepts needed to realize future semiconductor spintronics devices [7], [8], [9], [10], [11], [12], [13].

In the case of a nonmagnetic “ideal” 2D electron gas, the bands of electrons with opposite spin direction are degenerate and show a parabolic dispersion following the equation,

$$E_B\left({\overrightarrow{k}}_{\left|\right|}\right)=\frac{{\hslash }^2}{2m^{\ast }}{\overrightarrow{k}}_{\left|\right|}^2$$

as shown in Fig. 1(b), where ${\overrightarrow{k}}_{\left|\right|}=(k_x,k_y,0)$ ($k_{\left|\right|}=\sqrt{k_x^2+k_y^2}$) and $m^{\ast }$ are the momentum parallel to the 2D plane and the effective mass, respectively. The spin degeneracy in this 2D gas results from the presence of both the time-reversal and the spatial inversion symmetries, which can be expressed as $E_B({\overrightarrow{k}}_{\left|\right|},\uparrow )=E_B(-{\overrightarrow{k}}_{\left|\right|},\downarrow )$ and $E_B({\overrightarrow{k}}_{\left|\right|},\uparrow )=E_B(-{\overrightarrow{k}}_{\left|\right|},\uparrow )$ ($\uparrow$ and $\downarrow$ indicate up-spin and down-spin). That is, the only solution to satisfy the two symmetries simultaneously is $E_B({\overrightarrow{k}}_{\left|\right|},\uparrow \downarrow ))=E_B(-{\overrightarrow{k}}_{\left|\right|},\uparrow \downarrow )$, which means that the band of the up-spin has the same dispersion as that of the down-spin. However, in case of having one of the two symmetries broken, this spin degeneracy is lifted. When the spatial inversion symmetry is broken, i.e. $E_B({\overrightarrow{k}}_{\left|\right|},\uparrow )\ne E_B(-{\overrightarrow{k}}_{\left|\right|},\uparrow )$, the effect of SOC can basically be explained by the so-called Rashba-Bychkov (RB or simply Rashba) effect [14]. The RB Hamiltonian can be expressed as

$$H_{\mathit{RB}}={\alpha }_R\left(\right|\epsilon \left|\right)\overrightarrow{\sigma }·({\overrightarrow{k}}_{\left|\right|}\times {\widehat{e}}_z),$$

where, ${\widehat{e}}_z=(0,0,1),{\alpha }_R$ is the Rashba parameter (${\alpha }_R={\hslash }^2{k}_0/m^{\ast }\text{;}{k}_0$ is the offset by which the $E\left({\overrightarrow{k}}_{\left|\right|}\right)$ parabola is shifted away from the time-reversal invariant momenta such as the $\bar{\mathrm{\Gamma }}$ point of the Brillouin zone), and $\epsilon$ is an electric field determined by the potential gradient normal to the surface. This RB effect splits the band of a nonmagnetic electron 2D gas into two parabolas along the momentum direction, whose dispersion follows the equation

$$E_B\left({\overrightarrow{k}}_{\left|\right|}\right)=\frac{{\hslash }^2}{2m^{\ast }}{\overrightarrow{k}}_{\left|\right|}^2\pm {\alpha }_R{\overrightarrow{k}}_{\left|\right|},$$

as shown in Fig. 1(c). Furthermore, as can be understood by the RB Hamiltonian, the split band is spin-polarized with the spin polarization vector ($\overrightarrow{P}$) parallel to the 2D plane and perpendicular to the momentum, and the directions of $\overrightarrow{P}$ of the two bands are opposite, i.e. $\overrightarrow{P}$(${\overrightarrow{k}}_{\left|\right|}$)$=-\overrightarrow{P}$($-{\overrightarrow{k}}_{\left|\right|}$) (Fig. 1(c)). This ideal RB-type spin splitting was observed in the L gap of the (111) surfaces of Au, Ag, and Cu [15], [16], [17], [18], [19], [20], [21], [22], but with energy splittings much larger than expected for a 2D electron gas. Taking into account the average potential gradient ∼1 eV/Å on metal surfaces, the Rashba parameter ${\alpha }_R$ is in the order of $\sim {10}^{-6}$ eVÅ for a 2D electron gas. This means that the splitting would be only a few $\mu$eV even at the boundary of the Brillouin zone, where the splitting would be largest, while the splitting observed at the Fermi wave vectors are 110 meV on Au(111) [15], 5 meV on Ag(111) [22] and 16 meV on Cu(111) [22]. Further large Rashba-type spin splittings were observed on ALMs formed on solid surfaces [23], [24], [25], [26], and it has been argued that they are originating from a local asymmetric charge distribution along the direction normal to the 2D layer in close proximity to the nuclei of the surface atoms [23], [27], where the essential role of inversion symmetry breaking is to mix different parity states. In contrast to these reports, in which the spin-textures are similar to that of an ideal RB effect, spin textures that differ from the ideal case have been reported for several ALMs in e.g. Refs. [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40]. The origins of these peculiar spin textures were attributed to, e.g. the perturbation induced by the entanglement of spin and orbital momenta, but they are still explained within the framework of the RB effect [26], [38], [41], [42].

This review aims to provide an overview on the insights gained on the spin-polarized electronic states of ALMs and to point out opportunities for exploring exotic physical properties when combining spin and other physics, e.g. superconductivity, and to realize future spintronics-based quantum devices. Here, we present two novel effects that go beyond the framework of the RB effect for creating spin-polarized bands in ALMs formed on solid surfaces. The first effect is the geometric symmetry of the atomic structure [28], [29], [32], [33], [35], [43] and the second one is the orbital angular momentum (OAM) of valence electrons [40]. The geometric symmetry was not considered to affect the“RB spins” in the early studies (for example Refs. [24], [15], [16], [20], [44], [45], [46], [47], [48], [49], [50]), but it is now considered to have influence on the spin texture, and is expected to be a necessary condition to gain a proper understanding on the spin texture in recent studies (for example Refs. [30], [39], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60]). Below, we focus on the effects of three different symmetries $C_3,C_{3v}$ and $C_{1h}$ (the relation between the symmetry of the crystal and that of the k point is shown in Table 1), and show the peculiar spin textures originated from them. Regarding the OAM, the so-called chiral OAM has been reported to play a role in defining the size of the spin splitting [26], [41], [42], but not so much the spin texture. (Here we use “chiral OAM” as an effect that produces a spin texture similar to that of the RB effect.) In this review we will show that various spin textures, ranging from the RB-type and symmetry-induced type, can be induced by the OAM. A fundamental understanding of the spin physics arising from the SOC in ALMs is necessary for the realization of not only semiconductor spintronics and/or quantum devices [7], [10], [12] but also for possible novel physical properties such as topological superconductors formed by 2D materials with spin-polarized states [61], [62], [63].