Spin-polarized electronic surface states of Re(0001): an ab-initio investigation
Andrea Urru Andrea Dal Corso





International School for Advanced Studies (SISSA), 

Via Bonomea 265, 34136 Trieste (Italy).

International School for Advanced Studies (SISSA), 

Via Bonomea 265, 34136 Trieste (Italy).
IOM-CNR Trieste (Italy).


We study the electronic structure of the Re(0001) surface by means of ab-initio
techniques based on the Fully Relativistic (FR) Density Functional Theory (DFT)
and the Projector Augmented-Wave (PAW) method.
We identify the main surface states and resonances and study in detail their
energy dispersion along the main symmetry lines of the SBZ. Moreover, we discuss
the effect of spin-orbit coupling on the energy splittings and the spin-polarization 
of the main surface states and resonances. Whenever possible, we compare the results with
previously studied heavy metals surfaces.
We find empty resonances, located below a gap similar to the L-gap of the
(111) fcc surfaces, that have a downward dispersion and cross the Fermi level,
similarly to the recently studied Os(0001) surface. Their spin polarization at the
Fermi level is similar to that predicted by the Rashba model, but the usual level
crossing at Γ̅ is not found with our slab thickness.
Moreover, for selected states, we follow the spin polarization along the high 
symmetry lines, discussing its behavior with respect to k_∥, the 
wave-vector parallel to the surface.


Spin-polarized electronic surface states of Re(0001): an ab-initio investigation
    Andrea Dal Corso
    Received: date / Revised version: date
================================================================================





§ INTRODUCTION


The presence of electronic states localized in the last few atomic layers of a solid, namely 
surface states, can give surfaces properties different from the bulk.<cit.> Moreover, since
surfaces lack inversion symmetry, even non-magnetic (i.e. time-reversal invariant) materials can
have surface states with a non-vanishing spin polarization. Hence, surface 
states might be practically useful for instance in spintronics applications, and it is worthwhile to 
characterize them. 

The energy dispersion of surface states with respect to k_∥, the wave-vector parallel to 
the surface, and in some cases also their spin polarization, have been analyzed for many surfaces of different materials, 
by both theoretical and experimental techniques (Density Functional Theory, DFT,<cit.> and photoelectron spectroscopy, PES,<cit.> angular- and 
spin-resolved). For instance, for the heavy metal surfaces, the L-gap surface states are well known.<cit.> They show a characteristic split 
parabolic energy dispersion, that can be interpreted by the Rashba model <cit.> as a relativistic effect due 
to spin-orbit coupling.

Recently, we found theoretically that Os(0001) surface <cit.> should host Rashba split surface states around
Γ̅, below a gap similar to the L-gap of the (111) fcc surfaces, but with an inverted dispersion
as in Ir(111).<cit.> Moreover, energy splittings due to spin-orbit coupling are present in other surface states 
of Os(0001), such as the S_2, S_3', and S_4 states analyzed in Ref. os0001.

Re(0001) is another interesting surface used <cit.> both as a support 
for other metallic layers and as a reactive catalytic surface. Very recently, it has been shown 
that artificially constructed Fe chains on top of Re(0001) surface exhibit a spin spiral state. <cit.> 
Though Re(0001) has been widely studied, surprisingly little information is available about its 
electronic structure. Being similar to the other surfaces discussed above, Re(0001) could have states similar to the Rashba 
split surface states with an inverted dispersion as, e.g., in Ir(111) and Os(0001). Also the other surface states
could be similar, but both the energy dispersion and their spin polarization, are poorly known.

In this work, we study by ab-initio techniques the electronic structure of Re(0001). We characterize 
its main surface states and follow, for the most interesting ones, the average direction of the spin
polarization as a function of k_∥. About Γ̅, we find Rashba-like states with
negative curvature, which cross the Fermi energy, and we characterize their spin texture at the Fermi level. 
We find also several states already familiar from the study of the other surfaces. We analyze the main surface 
states that appear in the electronic band structure: in particular, at variance with Au(111), Pt(111), and Ir(111),
but as in Os(0001), we do not find the S_8 Dirac-like surface states, studied in Ref. dirac.

The paper is organized as follows: in Section <ref>, we describe the methods and the computational
parameters. In Section <ref>, we present the Re(0001) electronic band structure and analyze the main
surface states and resonances. In Section <ref>, we discuss the spin polarization of some selected states and finally,
in Section <ref>, we present our conclusions.



§ METHOD


First-principle calculations were performed by means of DFT <cit.> within the 
Local Density Approximation (LDA) scheme, as implemented in the Quantum ESPRESSO <cit.> and 
.<cit.> packages The Perdew and Zunger's <cit.> parameterization for the exchange and
correlation energy is used. Spin-orbit coupling effects are included by using the Fully Relativistic (FR) PAW  
method,<cit.> with 5d and 6s valence electrons and 5s and 5p semicore states (Pseudopotential
Re.rel-pz-spn-kjpaw_psl.1.0.0.UPF from pslibrary.1.0.0 <cit.>).
Calculations on the bulk system, were performed with an hexagonal close-packed (hcp) structure at the theoretical LDA lattice constants: 
a=5.175 a.u., c=8.338 a.u. (c/a = 1.611), which are respectively 0.8% and 1% smaller than experiment 
(a_exp = 5.217 a.u., c_exp = 8.425 a.u.). <cit.> 
The surface has been simulated by both a 24-layers and a 25-layers slab perpendicular to the [0001] direction 
in order to check the stability of the results with respect to the breaking of the inversion symmetry. 
The slab replicas have been separated by a vacuum space of 44 a.u.. The slab crystal structure has been obtained 
from the bulk, with a further relaxation along the [0001] direction, which has the most relevant effects on the 
first three atomic layers: in particular, the distance between the first two layers decreases by 5.4 % with respect 
to the idealized interlayer distance in the bulk, while the distance between the second and the third layer increases 
by 2.9 %. At a first stage, we performed a calculation with a starting non-zero magnetization, but the 
self-consistent ground state of the slab ended up to be non magnetic. The pseudo wavefunctions are expanded 
in a plane waves basis set with a kinetic energy cut-off of 60 Ry, while the charge density with a cut-off of 400 Ry.
BZ integrations were performed using a shifted uniform Monkhorst-Pack <cit.> k-point mesh of 16 × 
16 × 1 points for the slab and 16 × 16 × 10 points for the bulk. 
The presence of a Fermi surface has been dealt with by the Methfessel-Paxton method <cit.> with a smearing parameter 
σ=0.02 Ry. With these parameters the total energy is converged within 10^-3 Ry and the crystal parameters 
within 10^-3 Å.





In Figs. <ref>a-b we show the first two atomic layers of the 24-layers slab and the 25-layers slab, respectively. 
The 24-layers slab has a D_3d point group. In particular, the z axis, normal to the surface, is a 3̅ 
rotoinversion axis, while the axes [100], [110], and [010] in Fig. <ref>a are two-fold rotation axes. 
There are also three mirror planes, (1̅20), (21̅0), and (110) shown in Fig. <ref>a. 
The 25-layers slab has instead, a D_3h point group. The z axis is a 6̅ axis, while the axes
[210], [120], and [1̅10], shown in Fig. <ref>b, are two-fold rotation axes. Moreover, there are 
three mirror planes, whose traces coincide with the C_2 axes [210], [120], and [1̅10].
The electronic band structure was calculated along the path Γ̅-K̅-M̅-Γ̅ 
(that is along the T̅, T̅', and Σ̅ high-symmmetry lines) of the 
Surface Brillouin Zone (SBZ), shown in Figs. <ref>c-d.
The small point group of k of the two slabs is indicated in the band structure in Figs. <ref>a and <ref>b, 
both for the high symmetry points (Γ̅, K̅, and M̅) and for the high symmetry lines 
(T̅, T̅', and Σ̅).
In particular, for the 24-layers slab, at Γ̅, K̅, and M̅ the small group of k is 
D_3d, D_3, and C_2h, respectively. Along the high symmetry lines T̅, T̅', and Σ̅ 
it is C_2, C_2, and C_s, respectively. Along T̅ the rotation axis coincides with the x-axis, while along
T̅' the rotation axis is the [1̅20] axis, shown in Fig. <ref>c. Finally, along Σ̅ the trace 
of the mirror plane of C_s is Σ̅.
On the other hand, for the 25-layers slab, at Γ̅, K̅, and M̅ the small group of k is
D_3h, C_3h, and C_2v, respectively, while along the high symmetry lines T̅, T̅', and Σ̅
it is C_s, C_s, and C_2v, respectively. In particular, along T̅ and T̅' the mirror plane is σ_h.
The two slabs have more symmetry elements than the Re(0001) surface, since they have symmetry operations that
exchange the two surfaces. Removing these elements, the surface point group is C_3v, while the small groups of 
k are C_3v, C_3, and C_s for Γ̅, K̅, and M̅ respectively and C_1, C_1, 
and C_s along T̅, T̅', and Σ̅. Actually, they are the same for both slabs.



§ RESULTS






In this section, we analyze the Re(0001) 24-layers slab FR band structure, shown in Fig. <ref>a. We characterize the 
main surface states, indicated with red dots in Fig. <ref>a, and compare them with Os(0001) and other previously studied surfaces (e.g. Au(111), Pt(111), and
Ir(111)). We use the same names as in Ref. os0001.
Moreover, at the end of the section we discuss the band structure of a 25-layers slab (Fig. <ref> b).





 


We start our analysis from the Γ̅ point, where we find two gaps in the PBS. Taking the Fermi energy as a
reference, the first is located 4 eV above it and the second approximately from -7 eV to -3 eV. The first gap, 
higher in energy, is similar to the L-gap of the fcc surfaces and is found in Os(0001) as well. It extends partly along
the T̅ and Σ̅ lines. The second gap, deeper in energy, extends up to half of the T̅ line and
along the whole Σ̅ line. 
Similarly to Os(0001) and Ir(111), but at variance with Au(111) and Pt(111), no surface states are found in the L-gap.
Below the L-gap, near the Fermi energy, we found two couples of states (L in Fig. <ref>a) that transform as the 
Γ_4^- and Γ_4^+ representations of the D_3d group. Their energy dispersion around Γ̅ is parabolic with negative 
curvature, as for the Rashba split states in Os(0001) and Ir(111). In these surfaces we could fit their dispersion with 
the equation:


    E_± = ħ^2/2 m^* k_∥^2 ±γ_SO k_∥,


where k_∥ is the modulus of the wave-vector parallel to the surface, m^* is the effective electron mass and 
γ_SO is the spin-orbit coupling parameter. However, here their dispersions do not cross at Γ̅, as shown 
in Fig. <ref> a, and even neglecting this splitting it is not possible to fit them with Eq. (<ref>). 
Nevertheless, at the Fermi energy the two states show a splitting along k_∥, due to spin-orbit coupling: indeed,
a comparison with the Scalar Relativistic (SR) band structure (Fig. <ref> b), shows that this splitting emerges only in 
the FR picture. Moreover, the spin texture of the L states at the Fermi energy is well predicted by the Rashba model 
(see Section <ref> for more details), so they behave as Rashba states.
In Fig. <ref>a we show the contour plots and the planar average of the sum of the charge densities at Γ̅ 
for the L_a states, the couple higher in energy. The contour plots suggest that it has mainly s character hybridized
with some d states. The planar average is maximum around the surface and shows a very slow decay towards the center of the
slab, indicating that the L states are resonances. The gap at Γ̅ could have several causes: 
among them, the evident hybridization with bulk states, possibly together with the finite size of the slab. Yet, a 
calculation with a 40-layers slab shows that the gap at Γ̅ between L_a and L_b is the same as for the 
24-layers slab, thus finite-size effects do not play a relevant role in this case.

At lower energies at Γ̅, there are two couples of states in a PBS gap, similar to the previously studied S_2 
states of the other metal surfaces. At Γ̅, they have symmetry Γ_4^+ and Γ_4^-. Their energy dispersion
has a positive curvature and can be fitted with (<ref>), with: 
γ_SO= (0.200 ± 0.005) × 10^-9 eV cm and m^*/m = (0.661 ± 0.003), with identical values, within the error bar,
along T̅ and Σ̅. In particular, γ_SO is 30 % lower than in Os(0001), while the effective mass is 
approximately 10 % lower. The charge density contours and planar average of the S_2a states, those higher in energy, 
are shown in Fig. <ref>b. The states are surface states mainly localized on the first two atomic layers.

Finally, at Γ̅ there are two couples of empty localized surface states called S_13, which have been characterized in
Os(0001) surface. At Γ̅ they transform as the Γ_4^+ and Γ_4^- representations of the D_3d group. Similarly to
Os(0001), they are resonances and they have mainly d character, with main contributions from the first two atomic layers (Fig. 
<ref>c).

The states L, S_2, and S_13 extend partially also along the T̅ line, where they all transform as the Γ_3 ⊕Γ_4
representation of the C_2 group. Along T̅ we find some PBS gaps as well: the widest ones host the S_10, S_4, and 
the previously mentioned S_2 states. The S_10 states are two couples of degenerate states with symmetry Γ_3 ⊕Γ_4. They 
cross the Fermi level around k_∥ = 0.51  Å^-1. As in Pt(111), Ir(111), and Os(0001), they merge with the S'_3 states at K̅. The 
S_4 states are located inside a PBS gap, they cross the K̅ point and extend along the T̅' line as well. They
have symmetry Γ_3 ⊕Γ_4.




At K̅ we find four main gaps in the PBS: the highest in energy is located above 3.5 eV, the second one crosses the Fermi 
level and does not host any surface state, the third one contains the S_4 states, while the fourth one extends down to -3.5 eV. 
The main surface states at K̅ are the S'_3 and S_4 states. S'_3 are made up of three couples of empty states, 
that are named S'_3a, S'_3b, and S'_3c in decreasing order of energy. S'_3a transforms as the Γ_5 ⊕Γ_6 representation of the D_3 group, while S'_3b and S'_3c have symmetry Γ_4. As in Os(0001), they are not 
in a PBS gap, they are localized in the first two atomic layers and project mainly on d_3z^2-r^2 states, as can be seen from
the charge density contour lines shown in Fig. <ref>d.

The S_4 states are located in the PBS gap found at -2 eV < E < -0.3 eV. S_4a, S_4b, and S_4d transform as the 
representation Γ_4, while S_4c transforms as Γ_5 ⊕Γ_6. In Fig. <ref> we show their charge density,
which is peaked on the top atomic layer, with a small contribution on the third atomic layer for S_4a, S_4b, and S_4d.
They have very similar features in Os(0001), though S_4b and S_4c exchange their character, 
as can be argued from the symmetry and the charge density plots. 

The PBS gaps and surface states described at K̅ extend also along the T̅' line. Both S'_3 and S_4 states, along
T̅' have symmetry Γ_3 ⊕Γ_4 of the C_2 group. Moreover, S_4b and S_4c anticross near K̅ 
(k≈ 1.53  Å^-1), similarly to Os(0001). Along T̅', near M̅, we find another PBS gap, located 
around -6 eV, that hosts the S_7 states. It extends up to M̅ and along the whole Σ̅ line, as well as the S_7 
states, that connect to the S_2 states at M̅.

At M̅, besides the previously mentioned PBS gap and S_7 states, we find the S_12 states. They are a couple of degenerate
states with symmetry Γ_3^+ ⊕Γ_4^+ and project on many d states (Fig. <ref> e). The S_7 states, instead, 
are made up of two couples of states, that belong to the representations Γ_3^+ ⊕Γ_4^+ and Γ_3^- ⊕Γ_4^- 
of the group C_2h, respectively. They have a strong contribution to the charge density (Fig. <ref>f) coming from d_x^2-y^2 and
d_xy orbitals localized in the first atomic layer. 

The band structure of the 25-layers slab (Fig. <ref> b) is overall very similar to the one of the 24-layers slab and the surface states
are located at the same energies in both slabs. Nevertheless there are minor differences, due to the different symmetries of the two
slabs. In particular, since the 25-layers slab lacks inversion symmetry (its point group is D_3h), only the k - -k Kramers
degeneracy remains, and a spin splitting may appear, along some lines. This is the case of the lines T̅ and T̅', in which states 
of different symmetry (in our case, even and odd with respect to the mirror plane σ_h) are split. The spin splitting is different for 
different states: it can be very small as, e.g., ≈ 10^-6 eV for the S_4 states, or larger as ≈ 0.03 eV for the S'_3 states, 
and it decreases increasing the slab thickness.<cit.> At variance with the states along T̅ and T̅', the states along 
Σ̅ are doubly degenerate because the C_2v double group has only one two-dimensional irreducible representation, Γ_5.



§ SPIN POLARIZATION: RESULTS AND DISCUSSION


In this section we discuss the spin polarization of some of the surface states found above. 
The spin polarization can be obtained integrating the planar average of the magnetization density over half slab:

    m_α=∫_0^L/2 m_α (z)   dz   ,


where the zero of z is taken at the center of the slab and L is its length along z, including vacuum. m_α (z) in Eq. <ref> is
the planar average of the magnetization density m_kn^α (r) associated to the Bloch state 
⟨r | Ψ_k n σ|$⟩ and is defined as:


    m_α(z) = ∫_A m_kn^α (x,y,z)   dx   dy   ,

whereAis the yellow shaded region shown in Fig. <ref>a, and 

    m_kn^α (r)= μ_B ∑_n ∑_σ_1,σ_2⟨Ψ_k n σ_1 | r|σ⟩^σ_1 σ_2_α⟨r | Ψ_k n σ_2|,⟩

whereμ_Bis the Bohr magneton andσ_αare the Pauli matrices. The sum overσ_1andσ_2is over the spin,
while the sum overnis over degenerate states (see Ref. os0001 for more details).



We start our discussion from theLstates. In particular, we consider their contribution to the Fermi surface and their spin texture at 
the Fermi energy. The results are shown in Fig. <ref>. The Fermi surface of the slab is shown in Fig. <ref>a, while the contour 
levels of theLstates, shown in Fig. <ref>b (compared with the SBZ) are magnified in Fig. <ref>c: they have been obtained with a 
cubic interpolation of the energies of the states computed in a14 ×14square mesh ofkpoints centered inΓ̅. 
The spin polarization, computed via Eq. (<ref>), is represented by arrows whose length is proportional to the component of the 
spin parallel to the surface. The arrows are colored according to the magnitude of the spin polarization (Eq. <ref>) perpendicular 
to the surface, as indicated by the color map in the Figure. TheL_bstates have a circular Fermi surface, whereas the shape of theL_astates is more influenced by the underlying lattice. The component of the spin polarization parallel to the surface is perpendicular to the 
wavevector for both states, and it rotates clockwise and counter-clockwise for the two states, respectively. This is in agreement with the 
prediction of the Rashba model,<cit.> so theLstates appear as Rashba split states at the Fermi level, although it has not been 
possible to fit their energy dispersion with Eq. (<ref>). In particular, given the dependence of the Rashba spin texture on the sign of 
both the effective mass and the spin-orbit coupling parameter,<cit.> our results are consistent with a Rashba model withγ_SO > 0. Due to the presence of the underlying atomic layers, the spin polarization shows a non vanishing component perpendicular 
to the surface. As shown in Fig. <ref>d, this component oscillates around zero along the contour levels, with a period of2 π/3as a 
consequence of the symmetry of the lattice, with opposite phase forL_aandL_b. Similar effects can be 
simulated also in the Rashba model by introducing hexagonal warping effects.<cit.>

Along theT̅andT̅'high symmetry lines the spin polarization can rotate in a plane perpendicular to the line, as 
explained in Refs. os0001, symmetry_rashba. In this work we consider the rotation of the spin polarization of the
statesS'_3,S_10(Fig. <ref>), andS_4(Fig. <ref>).



TheS'_3states (Figs. <ref>a-c) have been studied along the wholeT̅'line: atK̅the states have only a non-zeroz(perpendicular to the surface) component of the spin polarization, due to symmetry constraints, while atM̅their spin 
polarization vanishes becauseM̅is a time-reversal invariant point. The spin polarization ofS'_3ais mainly perpendicular 
to the surface: thezcomponent decreases along theT̅'line, in a similar fashion as in Os(0001). TheS'_3bstates show 
a more pronounced rotation: thezcomponent changes sign along the high symmetry line, and the component perpendicular toT̅'spans a wide range of values, at variance with Os(0001). Finally, theS'_3cstates have a rotating spin alongT̅', which 
always points towards the center of the slab: its behaviour is similar to what shown in Os(0001).

TheS_10states show a smooth evolution of the spin polarization in the region0.55   Å^-1 < k_∥ < 1   Å^-1, as
shown in Figs. <ref>d,e: in particular,S_10aandS_10bhave opposite spin. Aroundk_∥ ≈0.55 Å^-1andk_∥ ≈1 Å^-1the spin polarization rotates more rapidly, because the two states anti-cross. Overall, their 
behaviour is similar to that shown by Os(0001).

Finally, theS_4states (Fig. <ref>) show a spin texture alongT̅andT̅'very similar to Os(0001). In particular, 
the smoothest behaviour is shown byS_4d, for which the spin always points towards the slab.S_4bandS_4chave a rapidly 
varying spin, even in a very narrow range ofk_∥as shown in Fig. <ref> b-c, due to their mixing and anticrossing aroundK̅: a comparison with Os(0001) shows that their features are exchanged, as pointed out by their symmetry (see Table <ref>).

Similar calculations have been performed for the 25-layers slab as well. The results are very similar to those discussed 
above, in particular for theS_4andS_10states, which have the same energy dispersion in the two systems. Instead, 
the spin polarization of theS'_3states shows a somehow different behavior, characterized by more rapid variations,
which might be due to the mixing of the states caused by their non-negligible spin splitting. However, as pointed out by 
remark 41_layers, the spin splitting decreases, though slowly, with increasing slab thickness, so we expect a better 
agreement using a thicker slab.
  


§ CONCLUSIONS

We discussed the electronic structure of Re(0001) surface. We analyzed its main surface states and resonances, focusing on the contours
and planar average of their charge density.
AtΓ̅we found a gap similar to the L-gap of the (111) fcc surfaces. Like in the recently studied Os(0001) and at variance
with other well known metal surfaces (e.g. Au(111)), this gap does not contain any surface state. Two states that cross 
the Fermi level, with the same nature as the L-gap surface state of Au(111), have been found: their spin texture at the 
Fermi energy is similar to the one predicted by the Rashba model, though the energy dispersion crossing predicted atΓ̅has not been found. Rashba split states are actually of interest because they can have 
relevant applications in spintronics <cit.> and recently their engineering 
has been discussed, for instance, in ferroelectric oxides.<cit.> 

We foundS_2,S'_3,S_4,S_6,S_7,S_10,S_11,S_12, andS_13states as in other surfaces. In particular,S_2are Rashba split
states whose dispersion has been fitted with parametersγ_SO= (0.200 ±0.005) ×10^-9eV cm andm^*/m = (0.661 ±0.003).
TheS_8Dirac-like states instead have not been found.

AlongT̅andT̅'the spin polarization can rotate in a plane perpendicular to the high symmetry line: for theS'_3,S_4, andS_10states we followed this rotation as a function ofk_∥. Some of them, as theS'_3states, show a smooth behavior, while others 
(e.g.S_4andS_10) have a more rapidly varying spin polarization, due to the anti-crossing and mixing of the states. 

Compared to the recently studied Os(0001) surface, Re(0001) shows similar surface states and resonances, although they are higher in 
energy with respect to the Fermi level because of the lower number of electrons per atom. The main differences are found in theLstates, which
are more hybridized with the bulk, as shown by the planar average of the charge density. Minor differences can be observed in the spin
textures of theS'_3andS_4states: in particular, theS_4bandS_4cstates are exchanged with respect to those of Os(0001),
as pointed out also by their symmetries.

Our work has been developed using the DFT-LDA scheme. The Kohn-Sham eigenvalues are different, in principle, from the quasi-particle energies, 
hence it might be necessary to compute many-body corrections for a more detailed comparison with experimental data. However, since these kinds of 
calculations are more computationally demanding, they are usually performed only when LDA is not enough to explain the experimental results. On 
the other surfaces, the main features of the bands, such as the presence or absence of L-gap states, are well predicted by DFT-LDA, while the 
exact energy positions of the surface states might have small shifts. To the best of our knowledge, 
there are no experimental data to compare with our results. We hope that our theoretical calculations could motivate 
ARPES measurements on this surface and, in case of discrepancies, other theoretical calculations.



§ ACKNOWLEDGMENTS


Computational facilities have been provided by SISSA through its Linux Cluster and ITCS and by CINECA through the SISSA-CINECA 2018 Agreement.

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