Dirac-cone-like surface state in W(110): dispersion, spin texture and photoemission from first principles

Rashba-split surface states are well described by model Hamiltonians (see [3, 17] for Au(111) and bichalkogenide topological insulators, respectively). For the W(110) surface, we derived an analogous Hamiltonian from k·p perturbation theory [18], taking into account the twofold symmetry of this surface (point group 2mm [19]). This Hamiltonian agrees with that derived by Miyamoto et al [11]. The model Hamiltonian

$$\begin{equation} H = H_{0}(\boldsymbol{k}) + H_{\mathrm{soc}}^{(1)}(\boldsymbol{k}) + H_{\mathrm{soc}}^{(3)}(\boldsymbol{k}), \quad \boldsymbol{k} = (k_{x}, k_{y}) \end{equation} \tag{ 1 }$$

comprises three terms. (i) H₀(k) describes the basic dispersion of an anisotropic two-dimensional electron gas without spin–orbit coupling,

$$\begin{equation} H_{0}(\boldsymbol{k}) = \frac{\hbar^{2} k_{x}^{2}}{2 m_{x}^{\star}} + \frac{\hbar^{2} k_{y}^{2}}{2 m_{y}^{\star}}. \end{equation} \tag{ 2 }$$

The effective masses m^☆_y and m^☆_x are −4.7m_e and 3.3m_e, respectively (electron mass m_e; all parameters in this section have been taken from [11]). (ii) H⁽¹⁾_soc(k) is the Rashba-type spin–orbit Hamiltonian in first order in the wavenumbers k_x and k_y,

$$\begin{equation} H_{\mathrm{soc}}^{(1)}(\boldsymbol{k}) = \alpha_{x} k_{x} \sigma_{y} + \alpha_{y} k_{y} \sigma_{x} \end{equation} \tag{ 3 }$$

with Pauli matrices σ_x and σ_y (α_x = −0.08 eV Å and α_y = 1.05 eV Å). (iii) The contribution of third order to the spin–orbit coupling is given by

$$\begin{equation} H_{\mathrm{soc}}^{(3)}(\boldsymbol{k}) = \alpha_{x^{3}} k_{x}^{3} \sigma_{y} + \alpha_{x^{2}y} k_{x}^{2} k_{y} \sigma_{x} + \alpha_{xy^{2}} k_{x} k_{y}^{2} \sigma_{y} + \alpha_{y^{3}} k_{y}^{3} \sigma_{x} \end{equation} \tag{ 4 }$$

(α_x³ = 5.57 eV Å³, α_x²y = 12.3 eV Å³, α_xy² = −25 eV Å³ and α_y³ = 1.13 eV Å³).

The above Hamiltonian reveals considerable differences with respect to those for systems with (111) surfaces [20, 21]. The latter surfaces show threefold rotational symmetry; their point group 3m (cf Au(111), Bi/Ag(111) and Bi₂Te₃(111)) implies α_x = −α_y = α_R (Rashba parameter). Also the warping introduced by the third-order terms depends only on a single parameter [3]. Further, it tilts the spin polarization of the surface states out of the surface plane, as is described by the Pauli matrix σ_z. For a system with point group 2mm, the two mirror planes that are perpendicular to each other allow, in principle, for an out-of-plane tilt. However, time-reversal symmetry forbids this because E(k,s) = E(−k,−s); compare the actions of C_2z and T on the spin vector s in table 1 which dictates s_z ≡ 0. Hence, the spin polarization vector s has to be in-plane for all wavevectors and H⁽³⁾_soc(k) comprises only terms with σ_x and σ_y. Furthermore, for a wavevector parallel to the k_x-axis, the table yields s_x ≡ 0; thus, on the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$ line of the two-dimensional Brillouin zone (figure 1), only s_y is nonzero. Analogously, for k along the k_y-axis, one obtains s_y ≡ 0, which implies a nonzero s_x along the $\overline {\mathrm {N}}$–$\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$–$\overline {\mathrm {N}}$. These impositions are fully supported by our first-principles calculations.

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This single-band approach leads to agreement of model and experiment in the energy region close to the Dirac point (band crossing at k_x = k_y = 0, $\overline {\Gamma }$). However, it does not hold at higher energies where—as we shall show below—the Dirac state hybridizes with another surface state (see also [15, 16]).

We have performed ab initio calculations within the framework of the local density approximation to density functional theory. The electronic properties have been obtained by multiple-scattering computations with a relativistic layer Korringa–Kohn–Rostoker (KKR) code [22, 23]. Spin–orbit coupling is fully accounted for by solving the Dirac equation. The exchange-correlation functional is taken from [24].

The layer-resolved Green function G_ll(E + iη,k) provides detailed information on the electronic structure. l is the layer index, E the energy, η a small offset from the real energy axis and k = (k_x,k_y) is the in-plane wavevector. Both orbital composition and spin texture of the electronic states have been obtained from the spectral density [25]

$$\begin{equation} N_{l}(E, \boldsymbol{k}) = -\frac{1}{\pi} \lim_{\eta \to 0^{+}}{\rm Tr}\,G_{ll}(E + {\mathrm{i}} \eta, \boldsymbol{k}) \end{equation} \tag{ 5 }$$

by taking appropriate partial traces. A comparison of N_l(E,k) for different layers l at fixed (E,k) gives the localization of the surface states in the vacuum–surface region.

The spin textures of the electronic states are addressed by means of spin differences s^↑_μ − s^↓_μ that are obtained from the spectral density in (5). The spin projections ↑ and ↓ are given with respect to the chosen spin quantization axis μ (e.g. μ = y for k on the k_x-axis).

The entire system comprises three regions: (i) semi-infinite bulk, (ii) the surface region that consists of six W layers and four layers of vacuum and (iii) semi-infinite vacuum. The image potential barrier is mimicked by the so-called empty muffin-tin spheres, rather than by a smooth interpolating function [26]. In a KKR calculation of the surface electronic structure of W(110) by Giebels et al [27], a smooth interpolating function is used as an image-potential barrier. We find no significant changes with respect to our calculation that uses empty muffin-tin spheres. The only exception is a slightly better binding energy of the Dirac surface state in that calculation because of the adjustment of the barrier parameters. All other properties, in particular the spin polarization, agree very well. Bulk layers are denoted ${\mathsf {B}}$, whereas surface layers are named ${\mathsf {S}}$, ${\mathsf {S-1}}$, ${\mathsf {S-2}}$, etc, starting with the topmost surface layer ${\mathsf {S}}$. Similarly, vacuum layers are ${\mathsf {Vac}}$, ${\mathsf {Vac+1}}$, etc. The surface layer is relaxed toward the bulk by δd₁₂ = −2.75% of the bulk interlayer distance, as has been obtained from the surface x-ray diffraction [28]. Low-energy electron diffraction (LEED) gives comparable contractions (e.g. −2.2% in [29]).

The layer KKR method relies on two representations for the electronic states: plane waves and spin-angular functions. The number of plane waves, used in the interlayer scattering, was at least 50. The maximum angular momentum, used in the single-site scattering and the intralayer scattering, was l_max = 3. Both numbers guarantee converged results. The offset η from the real energy axis in (5) is 5 meV.

We have also computed spin- and angle-resolved photoemission intensities within the relativistic one-step model of photoelectron spectroscopy [22, 30], using the potentials of the first-principles calculations as input. These theoretical spectra provide a direct link between the spectral density maps of the initial states (occupied states) and the measured intensity maps of the photoelectrons.

The spin-density matrix of the photoelectron is calculated from

$$\begin{equation} \fl \rho_{\sigma \sigma'}(E, \boldsymbol{k}) \propto \left\langle \Phi_{\sigma}(E, \boldsymbol{k}) | \Delta(\omega) G(E - \hbar\omega, \boldsymbol{k}) \Delta^{\dagger}(\omega) | \Phi_{\sigma'} (E, \boldsymbol{k}) \right\rangle, \quad \sigma, \sigma'\!=\uparrow, \downarrow, \end{equation} \tag{ 6 }$$

which represents the triangular Feynman diagram of the one-step model of photoemission [31, 32]. The initial (occupied) electronic states are represented by the Green function G(E − ℏω,k). The dipole operator Δ(ω)∝α·A(ω) mediates the transition from the initial state to the outgoing photoelectron state, that is the time-reversed LEED state Φ_σ(E,k) with spin projection σ. α is a vector of Dirac matrices [33], while A(ω) is the vector potential of the incident radiation with photon energy ℏω.

The spin-averaged photocurrent is given by I(E,k) = tr ρ(E,k), while the spin polarization of the photoelectrons reads

$$\begin{equation} \boldsymbol{s}(E, \boldsymbol{k}) = \frac{{\rm tr}[\boldsymbol{\sigma} \rho(E, \boldsymbol{k})]}{I(E, \boldsymbol{k})} \end{equation} \tag{ 7 }$$

(σ is the vector of Pauli matrices). Due to spin–orbit coupling, the photocurrent is, in general, spin polarized even for nonmagnetic samples, depending on the specific setup (see [34] and references therein).

The above model takes into account the correct boundary conditions [35], the electronic structure above the vacuum level (via the time-reversed LEED state), dipole selection rules (via the dipole operators) and the electronic structure of the occupied states. The in-plane wavevector k is conserved due to translational invariance within the layers. Hence, it allows for quantitative comparison with experiments (see e.g. [17, 36]).

The photoemission calculations use the same setup as the electronic structure calculations. The Green function in (6) introduces a sum over all layers [37] which is approximated by a finite sum over the outermost layers. This is justified by the finite lifetime of the photoelectrons that introduces the surface sensitivity of photoelectron spectroscopy in the vacuum ultraviolet (VUV) range [38]. Here, the topmost 30 layers contribute to the photocurrent. The lifetime broadening is mimicked by an imaginary part of the energy, taken as −0.025 eV for the occupied and −1.0 eV for the unoccupied states. These values are chosen to clearly show the spin texture of the Dirac-like surface states. However, for reproducing experimental spectra these values have to be optimized and have to be taken as energy dependent. The quality of our approach may be judged from the data shown in [36].

The setup is adopted from [9, 11] (see figure 1 in these publications). The optical plane is spanned by the surface normal, the direction of light incidence and the direction of photoelectron detection. The sample is rotated about the surface normal by an angle ϕ_e and the photoelectrons are detected at a polar angle θ_e. The polar angle θ_ph of light incidence is 50°. We use both completely s- and p-polarized light with a photon energy of 22.5 eV (as in the experiment).

In the energy range that is relevant for the Dirac-like surface state, the W band structure shows a pocket-shaped partial band gap that is induced by spin–orbit coupling (top row in figure 2). This band gap is occupied by a strongly dispersive bulk band in the vicinity of the Brillouin zone center $\overline {\Gamma }$. As we will see below, also a strongly dispersive surface state follows its band edge.

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At an energy of −1.25 eV, an M-shaped region with a high spectral density shows up around $\overline {\Gamma }$. Along the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$ the dispersion is concave (bent upward, figure 2(a)) whereas along the $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$ there is, in addition to the concave shape, a very small convex region (bent downward, figure 2(b)) very close to $\overline {\Gamma }$. This anisotropy has consequences for the dispersion of the Dirac-like surface state that is split off this bulk band.

Considering the spectral density of the topmost surface layer ${\mathsf {S}}$, we find a surface state at −1.15 eV (experiment: about −1.2 eV), whose concave and weak dispersion follows that of the associated bulk band edge (this Dirac state is marked by an arrow in figure 2(c)). Along the $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$, its dispersion becomes linear and much stronger than along the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$. It is this finding that is clearly reminiscent of the Dirac state in a topological insulator, say Bi₂Se₃. Due to the strong anisotropy of the dispersion, however, we can hardly speak of an isotropic Dirac cone.

When deliberately changing the inward relaxation δd₁₂ of the top W layer ${\mathsf {S}}$ to zero, the Dirac point is shifted into the bulk band region at lower energies. This behavior is typical for a Shockley surface state that occurs due to a change of the crystal potential at the surface (rather than to a simple truncation of the infinite bulk system). From this finding we conclude that the surface relaxation is not essential for the appearance of the Dirac surface state. However, the inward relaxation 'pushes' the Dirac point into the bulk band gap. Hence, other heavy bcc(110) surfaces may show Dirac surfaces states as well.

As has been found in the experiment by Miyamoto et al, the Dirac state does not cross the band gap but displays a sharp maximum at k = ± 0.2 a⁻¹₀ and E = −0.86 eV (figure 2(d)). This change from an upward to a downward dispersion is explained by the strongly dispersive surface state that is associated with the bulk band edge mentioned above (marked 'surface state' in the figure). Both states belong to the same irreducible representation of the small group of k and, therefore, cannot cross. Their small distance in (E,k) results in a sizable hybridization which we will discuss by means of surface localization now.

The spectral densities of the surface states become blurred when they appear in a region of bulk states, an indication of hybridization of surface and bulk states [39]. Another manifestation of hybridization but among surface states is their surface localization. Close to the Brillouin zone center $\overline {\Gamma }$, the Dirac state shows largest spectral weight in the topmost layer (filled triangles at layer ${\mathsf {S}}$ in figure 3) and decays monotonously toward the bulk without modulation. At larger wavenumbers, the Dirac state and the strongly dispersive surface state hybridize (figure 2(d)). As a consequence, they show similar properties: the spectral densities of these states, for example, are largest at layer ${\mathsf {S}}$ with almost identical size. However, their decay toward the bulk is slightly modulated, in contrast to the Dirac state at $\overline {\Gamma }$.

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The linear dispersion of the Dirac state along $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$ mimics that of a massless fermion. Along the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$ this state becomes 'massive', that is, it exhibits parabolic dispersion. Hence, this strong anisotropy could be used to gradually increase or decrease the effective electron mass by changing the azimuth of the wavevector k.

To elucidate the anisotropy of the dispersion in more detail, we present constant energy contours (CECs) of the spectral density in the surface layer ${\mathsf {S}}$ at selected energies (figure 4). At −1.0 eV, the Dirac state shows up as the inner rhombus that is elongated along k_x (the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$; figure 4(a)). This finding illustrates the weak dispersion in k_x and the strong dispersion in k_y. Due to the maximum in its dispersion (figure 2(d)), the Dirac state manifests itself in the CEC also as the innermost oval (indicated by an arrow as well). The strongly dispersive surface state shows up as the second innermost oval whose curvature follows closely that of the Dirac state. As a consequence, these states hybridize sizably in all k directions.

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The anisotropy of the Dirac state's dispersion shows the importance of the third-order terms in the model Hamiltonian. It becomes even enhanced at lower energies, as is evident from the extremely elongated rhombus at −1.1 eV (figure 4(d)). Below the Dirac point but still above the bulk band region, for example at −1.2 eV, it appears that the spectral density shows weight only along k_x; see the almost parallel lines in figure 4(c). At even smaller energies, the hybridization of the Dirac state and bulk states results in a blurred spectral density (figure 4(b)).

At higher energies, a band gap opens up in the Dirac state along the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$ (cf −0.9 eV in figure 2(c)). This opening has the consequence that the Dirac state's rhombus and oval transform into two chevron-shaped features (figure 4(e)) that lie mirror symmetric with respect to the k_x-axis.

A comparison with experimental ARPES data, figure 3 in [11], establishes agreement with our calculations concerning the shape of the Dirac state's rhombohedral structure in the CECs. However, the ARPES measurements do not 'illuminate' the complete shape of the rhombus which is attributed to dipole selection rules in the photoemission process. Similar findings are reported by Winkelmann et al [36]. We will address this observation in section 3.3.

The Rashba spin–orbit coupling implies a specific spin texture of the surface states. For an isotropic two-dimensional electron gas, the Rashba Hamiltonian yields a spin alignment within the confinement plane (here: xy or surface plane) and normal to the wavevector k. For systems with threefold rotational symmetry, a tilt of the electron spin out-of-plane has been found: small for Au(111) [17] but as large as 45° for Bi₂Te₃ [40]. For W(110), however, the two orthogonal mirror planes in conjunction with time reversal forbid such an out-of-plane tilt (see section 2.1). Consequently, we focus on spin differences s^↑_μ − s^↓_μ for an in-plane spin quantization axis normal to k (figure 5).

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The Dirac state exhibits the expected spin texture that fully agrees with that derived from the model Hamiltonian. Interestingly, the strongly dispersive surface state that hybridizes with the Dirac state has a spin orientation opposite to that of the Dirac state (figure 5(b)). The degrees of spin polarization of the Dirac state are as large as 87.5% at k = 0.05 a⁻¹₀ and 95.9% at k = 0.28 a⁻¹₀ at E = −1.07 eV (i.e. at the (E,k) points used in 3); the surface state's spin polarization equals 86.8% at k = 0.31 a⁻¹₀.

A similar observation has been made for the surface states in Bi/Cu(111) [15] and Bi/Ag(111) [16]. In these surface alloys, the Bi atoms induce two sets of surface states, one at lower energy of mostly sp_z orbital composition and another one at higher energy with p_xp_y orbital composition. The outer branch of the lower set hybridizes with the inner branch of the upper set, both showing opposite spin polarization. For W(110), the strongly dispersive surface states play the role of the upper branch. The other electronic states at the surface are as well spin polarized due to spin–orbit coupling [41, 42].

Our findings reported so far corroborate the experimental observations and the conclusions drawn by Miyamoto et al [9, 11]. In particular, they support the surface origin of the spin-resolved properties. In turn, they contrast with a recent finding by Rybkin et al [43]: in their study on W(110), the spin-polarized features were attributed to bulk origin.

Having analyzed the Dirac state in the previous sections, we now address how its properties manifest themselves in the photoemission intensities. Without spin–orbit coupling, electronic states are either even or odd with respect to mirror operations. With spin–orbit coupling, however, these components become mixed in the respective double group [19]. Furthermore, even spin-up orbitals can hybridize with odd spin-down orbitals and vice versa, as has been found in Bi/Cu(111) [15]. As a result, the spin polarization is less than 100%. Since the spatial character of the initial state in the photoemission process can be selected by the polarization of the incident light, one could observe a reversal of the spin texture upon changing from s-polarized light to p-polarized light.

We will now address this effect for the $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$ azimuth. For s-polarized light, the vector potential A of the incident light lies parallel to the surface plane and normal to the yz plane. Hence, only A_x is nonzero and A is odd under m_yz (table 1). Therefore, only odd orbitals under m_yz can be probed. For p-polarized light, A is even under m_yz and even orbitals are probed.

The spectral-density calculations yield that the Dirac state near $\overline {\Gamma }$ is mainly of d_z² orbital character which is even under m_yz. Consequently, its photoemission intensity is large for p-polarization (figure 6(a)) and small for s-polarization (c). As suggested before, we find the expected spin reversal: the Dirac state at k_y > 0 is spin-up (blue in figure 6(e)) for p-polarization but spin-down (red in figure 6(g)) for s-polarization. As found already in figure 5(g), the highly dispersive surface state shows a spin polarization opposite to that of the Dirac state.

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Analogous considerations hold for the $\overline {\mathrm {N}}$–$\overline {\Gamma }$–$\overline {\mathrm {N}}$ azimuth (figures 6(b) and (d)). Similar to the $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$ azimuth, the strongly dispersive surface state is observed for p-polarized light but is absent for s-polarized light, which implies that this state is even with respect to the mirror operation m_xz. The flattened states shown along k_x consist of even and odd states. For p-polarized light, the flattened states show a larger intensities near the center of the Brillouin zone and fade out for larger k. The situation is reversed for s-polarized light: intensities are weaker close to the center of the Brillouin zone and become stronger for larger momenta.

The experiments by Miyamoto et al [11] reveal a rather strong sensitivity of the Dirac state's intensity on the electron detection direction: for negative k (along $\overline {\mathrm {H}}$–$\overline {\Gamma }$–$\overline {\mathrm {H}}$), the Dirac state shows up clearly but is suppressed for positive k (cf state S₁ in figure 2(a) of their publication). Our computations do not show such a strong asymmetry, a finding that is easily understood from the Dirac state's orbital character (mostly d_z²). To resolve this issue, we would like to suggest detailed experimental and theoretical investigations that include a spin analysis of the photocurrent.

The spin polarization of the Dirac state lies within the surface plane of all k, as is shown in the previous section. The complete in-plane spin polarization of the (initial) Dirac surface state is due to the joint effect of rotation and time-reversal invariance. The photoemission process, however, breaks the time-reversal symmetry; as a consequence, the restriction induced by time reversal on the spin polarization (operation T in table 1) is no longer valid, and the out-of-plane component of the photoelectron's spin polarization can be nonzero for k that do not lie within a mirror plane.

To investigate the out-of-plane spin component that is induced by the photoemission process itself, we have computed spin-resolved ARPES maps at constant energies (figure 7). The intensities have been obtained for polar-angle scans, with the azimuth running from −90 ° to +90 °; this is the setup as sketched in figure 1 of [9].

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For both light polarizations, s_z vanishes in a mirror plane of the surface (see table 1). However, the spin-resolved intensity patterns of s- and p-polarized light differ in their symmetry. For s-polarized light, s_z shows twofold rotational symmetry (figures 7(b) and (d)); by crossing a mirror plane, it changes sign (the same holds for the in-plane spin polarization (not shown)). However, for off-normally incident p-polarized light (figures 7(a) and (c)), the twofold symmetry is broken but s_z changes sign when turning k_y into −k_y for fixed k_x. This result is fully in line with table 1 (cf the effect of the operation m_xz on s_z). Also the in-plane components s_x and s_y comply with table 1. The out-of-plane spin polarization is as large as 92% at k = (0.004,0.008) a⁻¹₀ in figure 7(b) and 44% at k = (0.098,0.044) a⁻¹₀ in figure 7(c). Note that these numbers depend on the chosen lifetime broadening; a larger lifetime broadening would reduce these values. We conclude that if a spin polarization component is not forbidden by symmetry it is actually nonzero. Evidently, its degree and sign depend on the transition matrix elements that are calculated from realistic initial and final states and consequently depend on the specific setup.

We would like to emphasize that the purpose of the present SARPES calculations is to show the major effects concerning the spin polarization of photoelectrons excited from the Dirac surface state. For this reason we used the setup detailed in the publications by Miyamoto et al [9, 11]. Nevertheless, we performed additional photoemission calculations using different setups: with fixed incidence angles of 20° and 50° with respect to the sample. To check the effect of the final states, we performed calculations for different photon energies as well. As a result in both cases, the photoemission intensities from the Dirac surface state and its out-of-plane spin polarization change.