Toward surface orbitronics: giant orbital magnetism from the orbital Rashba effect at the surface of sp-metals

As the inversion symmetry is broken at a surface, spin-orbit interaction gives rise to spin-dependent energy shifts – a phenomenon which is known as the spin Rashba effect. Recently, it has been recognized that an orbital counterpart of the spin Rashba effect – the orbital Rashba effect – can be realized at surfaces even without spin-orbit coupling. Here, we propose a mechanism for the orbital Rashba effect based on sp orbital hybridization, which ultimately leads to the electric polarization of surface states. For the experimentally well-studied system of a BiAg2 monolayer, as a proof of principle, we show from first principles that this effect leads to chiral orbital textures in k-space. In predicting the magnitude of the orbital moment arising from the orbital Rashba effect, we demonstrate the crucial role played by the Berry phase theory for the magnitude and variation of the orbital textures. As a result, we predict a pronounced manifestation of various orbital effects at surfaces, and proclaim the orbital Rashba effect to be a key platform for surface orbitronics.

where n 1 and n 2 are integers and are unit lattice vectors, where a is the lattice constant. For convenience, we also define a 3 = − (a 1 + a 2 ). Within each unit cell, there are one Bi and two Ag atoms, the locations of which are We first consider a spinless model. We will return to a spin-full model when we consider the effect of spin-orbit coupling. Near the Fermi energy, the most relevant atomic orbitals are the p x , p y , p z orbitals of Bi atoms, and s orbitals of Ag atoms, which are sufficient to capture the essential features of the orbital Rashba effect in BiAg 2 . We define φ pα (r) (α = x, y, z) and φ s (r) as p α -and s-like Wannier functions localized around r = 0, respectively. Considering up to the first-and second-nearest hoppings, we define the following hopping amplitudes: V z(2) = ± drφ * px (r)Ĥφ pz (r ± ax) = ± drφ * py (r)Ĥφ pz (r ± aŷ).
Note that V z(1) and V z(2) are zero if z-inversion symmetry is preserved. The effect of V z(2) was studied by Petersen et al. [S1].
In sp alloys such as BiAg 2 , the quantity V z(1) arising due to the buckling of Bi atoms becomes important.
In order to find a k-space representation of the Hamiltonian, let us define basis states where n = p x , p y , p z , s 1 , s 2 , and r s 1(2) = r Ag1(2) , and N is the number of lattice sites. We compute a matrix representation of the Hamiltonian as where The above equations define a spin-less model Hamiltonian. A spin-full model Hamiltonian can be constructed by simply adding spin-orbit coupling as where is the spin-orbit coupling Hamiltonian at the intra-atomic level.

Supplementary Note 2: Parameters of the tight-binding model
The parameters occuring in the above tight-binding model were chosen as E s = 1, E p = 0, t pσ = 0.7, t pπ = 0, t s(1) = 1.5, t s(2) = 0, γ sp = 1.3, V z(1) = 0.6, V z(2) = 0, λ soc = 1.2, in order to reproduce the band structure which resembles the result obtained from the first principles calculation (see Supplementary Figure 1). The Fermi energy is introduced as a parameter, and we set E F = 0.6 and E F = 0.75 for the case without and with spin-orbit coupling, respectively, such that the location of the spin-split bands near the Gamma-point is similar to that of the first-principles calculation. All parameters are given in electronvolt.

Supplementary Note 3: Derivation of Eq. (5)
In order to derive the orbital Rashba Hamiltonian from a tight-binding model of the sp-alloy, let us consider a two-dimensional square lattice in the xy-plane with three p orbitals and one s orbital at each site. A generalization to different lattice systems of sp-alloys is straightforward. The tight-binding Hamiltonian is written as where the basis states are |ϕ nk = (1/ √ N ) R e ik·R |φ nR with |φ nR as the n-th (n = p x , p y , p z , s) Wannier function localized around the Bravais lattice vector R, and N is the total number of the lattice sites. Here, H p(s) is the Hamiltonian spanned by p(s) orbitals, i.e., H p (k) = diag E px (k), E py (k), E pz (k) , and H s (k) = E s (k), where E n (k) where E n (k) refers to the energy dispersion of the n-th isolated basis state. Most importantly, describes the hybridization between s and p orbitals, where γ sp is the nearest-neighbor hopping amplitude between s and p x(y) orbitals, and V z (k) is the hopping amplitude between p z and s orbitals, which is induced by the surface potential gradient. Near up to the first order in k. Considering k and V z (0) as perturbative parameters, the eigenstates of Eq. (S35) are given by Using these basis states to obtain a matrix representation of the Hamiltonian Eq.(S35), we arrive at an effective Hamiltonian decoupling the manifolds of s and p orbitals: In particular for the effective Hamiltonian describing the manifold of p orbitals [Eq. (S42)], we note the occurence of the so-called orbital Rashba Hamiltonian: where The orbital Rashba constant α OR is given by where we assumed E px (0) = E py (0) ≡ E pxy (0). The orbital angular momentum operator in a given basis is defined aŝ