Tuning independently Fermi energy and spin splitting in Rashba systems: Ternary surface alloys on Ag(111)
aa r X i v : . [ c ond - m a t . m t r l - s c i ] M a y Tuning independently Fermi energy and spinsplitting in Rashba systems: Ternary surface alloyson Ag(111)
H Mirhosseini, A Ernst, S Ostanin, and J Henk
Max-Planck-Institut f¨ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle (Saale),GermanyE-mail: [email protected] (H Mirhosseini)
PACS numbers: 71.70.Ej, 73.20.At, 71.15.Mb
Abstract.
By detailed first-principles calculations we show that the Fermi energyand the Rashba splitting in disordered ternary surface alloys Bi x Pb y Sb − x − y /Ag(111)can be independently tuned by choosing the concentrations x and y of Bi and Pb,respectively. The findings are explained by three fundamental mechanisms, namelythe relaxation of the adatoms, the strength of the atomic spin-orbit coupling, andband filling. By mapping the Rashba characteristics, i. e. the splitting k R and theRashba energy E R , and the Fermi energy of the surface states in the complete range ofconcentrations, we find that these quantities depend monotonically on x and y , witha very few exceptions. Our results suggest to investigate experimentally effects whichrely on the Rashba spin-orbit coupling in dependence on spin-orbit splitting and bandfilling. ernary surface alloys
1. Introduction
In the emerging field of spin electronics, proposed device applications often utilize theRashba effect [1] in a two-dimensional electron gas (2DEG). A prominent example is thespin field-effect transistor [2] in which the spin-orbit (SO) interaction in the 2DEG iscontrolled via a gate voltage [3, 4]. Other examples are a high critical superconductingtemperature which shows up in materials with a sizable spin-orbit interaction [5] andthe spin Hall effect [6–9].The Rashba effect relies on breaking the inversion symmetry of the system and,consequently, shows up in semiconductor heterostructures and at surfaces. The breakingof the inversion symmetry results—via the spin-orbit coupling—in a splitting in thedispersion relation of electronic states which are confined to the interface [1]. In asimple model for a two-dimensional electron gas, a potential in z direction confinesthe electrons to the xy plane. The Hamiltonian of the spin-orbit coupling can thus bewritten as ˆ H so = γ R ( σ x ∂ y − σ y ∂ x ) , (1)where the strength of the SO interaction is quantified by the Rashba parameter γ R .Employing a plane-wave ansatz yields the dispersion relation E ± ( ~k k ) = E + ~ k k m ⋆ ± γ R | ~k k | , (2)where m ⋆ is the effective electron mass. The split electronic states are labeled by +and − ; their spins lie within the xy plane, are aligned in opposite directions, and areperpendicular to the wave vector ~k k .In a real system, the Rashba parameter γ R comprises effectively two contributions[10]. The ‘atomic’ contribution is due to the strong potential of the ions (atomicspin-orbit coupling). The ‘confinement’ contribution is due to the structural inversionasymmetry which can be viewed as the gradient of the confinement potential in z direction. The larger this gradient and the atomic spin-orbit parameter, the larger γ R and the splitting k R = | m ⋆ | γ R ~ , (3)which is defined as the shift of the band extremum off the Brillouin zone center ( ~k k = 0).Another quantification of the splitting is the Rashba energy E R = − ~ k m ⋆ = − m ⋆ γ ~ , (4)that is the energy of the band extremum with respect to the energy E for which thebands cross at ~k k = 0.The above dispersion relation suggests to distinguish two energy ranges. Region Iis defined as the energy range between E and the band extrema ( E ∈ [ E − E R , E ]for positive m ⋆ or E ∈ [ E , E + E R ] for negative m ⋆ ) [11]. Region II comprises theother range of band energies ( E > E for positive m ⋆ or E < E for negative m ⋆ ). The ernary surface alloys E while in region II it is constant.In view of designing device applications and investigating fundamental effects, it isdesirable to tune both the strength γ R of the Rashba spin-orbit coupling and the Fermienergy E F of the 2DEG. In a semiconductor heterostructure, this can be achieved by anexternal gate voltage and by doping of the semiconductor host materials. At a surface,these quantities can be affected by adsorption of adatoms [12,13], by surface alloying [11],and by changing the thickness of buffer layers (e. g. in Bi/(Ag) n /Si(111) [14]). Recently,a ferroelectric control has been proposed [15].Surface states in surface alloys show an unmatched Rashba splitting [16], ashas been investigated in detail by scanning tunneling microscopy as well as by spin-and angle-resolved photoelectron spectroscopy. They are convenient systems fortesting fundamental Rashba-based effects. The ordered surface alloys Bi/Ag(111),Pb/Ag(111), and Sb/Ag(111) have been investigated by first-principles calculationsand in experiments [16–18]. These three systems differ with respect to their Rashbacharacteristics k R and E R , and by E . The challenge we are dealing with is how to tunethese properties independently .The basic idea is as follows. Bi/Ag(111) has a large splitting and occupied sp z surface states, while Pb/Ag(111) has a large splitting and unoccupied sp z surface states.In a disordered binary alloy Bi x Pb − x /Ag(111) the Fermi energy can be tuned by theconcentration x , while keeping a large spin splitting. In contrast, Sb/Ag(111) hasoccupied surface states with almost the same binding energy as those in Bi/Ag(111)but a minor splitting. This allows to tune mainly the spin splitting but keeping theFermi energy in Bi x Sb − x /Ag(111). Thus, by an appropriate choice of concentrations x and y in a ternary alloy Bi x Pb y Sb − x − y /Ag(111) we expect to tune the Fermi energyand the splitting independently . In particular, one could access the region I between E and the band maxima which is important for high-temperature superconductivity [5].We report on a first-principles investigation of disordered surface alloysBi x Pb y Sb − x − y /Ag(111), performed along the successful line of our previous works onboth ordered and disordered alloys [11, 16, 18]. Since all ordered and disordered binaryalloys show a √ × √ R ◦ surface reconstruction, we assume this geometry also for theternary alloys. The resulting substitutional disorder is described within the coherentpotential approximation.The paper is organized as follows. Our computational approach is sketched insection 2. The results are discussed in section 3, for binary alloys in section 3.4 and forternary alloys in section 3.5. We give conclusions in section 4.
2. Computational aspects
We rely on our successful multi-code approach, based on the local density approximationto density functional theory. Because this is described in detail elsewhere [15], wedeliberately sketch it in this paper. ernary surface alloys
Ab-initio
Simulation Package (VASP) [19], well-known for providing precise totalenergies and forces. The relaxed structural parameters serve as input for first-principlesmultiple-scattering calculations. Our Korringa-Kohn-Rostoker (KKR) method alreadyproved successful for relativistic electronic-structure computations of Rashba systems[14, 20].The central quantity in multiple-scattering theory is the Green function [21] G ( ~r n , ~r ′ m ; E, ~k ) = X ΛΛ ′ Z n Λ ( ~r n ; E ) τ nm ΛΛ ′ ( E, ~k ) Z m Λ ′ ( ~r ′ m ; E ) ⋆ − δ nm X Λ Z n Λ ( ~r < ; E ) J n Λ ( ~r > ; E ) ⋆ , (5)where Z and J are regular and irregular scattering solutions of sites n and m at energy E and wavevector ~k , respectively. ~r n is taken with respect to the position ~R n of site n ( ~r n = ~r − ~R n ). r < ( r > ) is the lesser (larger) of r n and r ′ n . Λ = ( κ, µ ) comprisesthe relativistic spin-angular-momentum quantum numbers [21]. The scattering-pathoperator τ is obtained in standard KKR from the so-called KKR equation [21], or inlayer-KKR from the Dyson equation for the Green function [22].The local electronic structure is analyzed in terms of the spectral density N n ( E ; ~k ) = − π ℑ Tr G ( ~r n , ~r n ; E, ~k ) . (6)By taking appropriate decompositions of the trace, the spectral density providesinformation on spin polarization and orbital composition of the electronic states.Substitutional ternary alloys Bi x Pb y Sb − x − y /Ag(111) are described within thecoherent potential approximation (KKR-CPA), in which short-range order is neglected.From the agreement of the theoretical data with their experimental counterparts forthe binary alloys Bi x Pb − x /Ag(111) [11], we conclude that short-range order is of minorimportance in these systems. Hence, we applied the KKR-CPA also for the ternaryalloys.The effect of the disorder can be understood as a self-energy [23]. As a consequence,the spectral density of the disordered alloys becomes blurred (or smeared out) ascompared to that of the ordered alloys.
3. Results and discussion
Relaxations have been determined by VASP for the ordered alloys, with √ × √ R ◦ reconstruction and face-centered-cubic (fcc) stacking (VASP cannot treat substitutionaldisorder within the CPA). It turns out that the relaxations of Sb, Bi, and Pb are inaccord with their atomic radii. To be more precise, the outward relaxations are 9 . .
33 ˚A), respectively, withrespect to the positions of the Ag atoms in the topmost layer. Being negligibly small,in-plane displacements of Ag atoms are not considered. ernary surface alloys √ × √ R ◦ surfacereconstruction, we assume this geometry also for the ternary alloys. The relaxationsof the disordered surface alloys were linearly interpolated, in dependence on theconcentrations of the constituting elements Bi, Pb, and Sb. This assumption is withinthe spirit of the CPA; being a mean-field theory, a disordered system is described by aneffective medium. Likewise the relaxation should be taken as a concentration-weightedaverage. We are aware, however, that in real samples, the relaxations of the constitutingindividual atoms could differ, as might be checked by scanning tunneling microscopy. Before presenting details of our calculations, a brief discussion of the general trends andmechanisms is in order. For tuning the Fermi energy and spin splitting independently,the underlying mechanisms should be independent as well.A first mechanism is relaxation. The outward relaxations of Sb, Pb, and Bi arein accord with their atomic radii; the larger the atomic radius, the larger the outwardrelaxation. The relaxation is accompanied by a charge transfer from the atomic sphereto the surrounding: the larger the relaxation, the larger the charge transfer [24]. Thismechanism determines the energy position of the degenerate point E —cf. (2)—and,consequently, the Fermi energy or band-filling of the surface states (2DEG).A second mechanism is the atomic spin-orbit parameter. Bi and Pb are heavyelements with large SO parameter (1 .
25 eV for Bi and 0 .
91 eV for Pb [25]), in contrastto the lighter element Sb (0 . Z Pb = 82, Z Bi = 83). Within a rigid-band model, the surface states in Pb/Ag(111)are shifted to higher energies, as compared to those in Bi/Ag(111). This picture isconfirmed by experiments and first-principles calculations [11]. The ordered surface alloys Bi/Ag(111), Pb/Ag(111), and Sb/Ag(111) have been studiedpreviously in detail [16–18]. They show two sets of surface states; a first set is unoccupiedand consists mainly of p x p y orbitals (for Bi/Cu(111), see [26]). In this paper, we focuson the other set which is either completely or partially occupied and consists of sp z orbitals. The effective mass m ⋆ of both sets is negative, implying a negative dispersion. Sb/Ag(111).
We address briefly the abovementioned relaxation mechanism byconsidering two cases for Sb/Ag(111): (i) an Sb relaxation as calculated by VASP ernary surface alloys . .
05 % as comparedto 0 .
94 %, with respect to the nominal valence charge; cf. [24]). Consequently, the surfacestates are shifted towards higher energies by 0 .
16 eV, as obtained from the degeneracypoint E . Further, the spin splitting k R becomes increased as well (0 .
03 ˚A − as comparedto 0 .
02 ˚A − ). This corroborates that the relaxation mainly affects the crossing point E (or Fermi energy) rather than the spin splitting. x Pb − x /Ag(111). In the disordered binary alloy Bi x Pb − x /Ag(111), which has beenstudied previously [11], the ratio of the Rashba energy E R and the Fermi energy E F canbe chosen within a wide range, in dependence on the Bi concentration x . For both Biand Pb, 0 .
99 % of the atomic charge atom is removed from the muffin-tin sphere, whichis in agreement with the close outward relaxation of Bi and Pb (15 % and 18 %). Asnoted before, Pb has one valence electron less than Bi, which explains the sizable shiftof the surface states to higher energies (band-filling mechanism; cf. the panels on theright-hand side of figure 1). Although the relaxation is of the same order, the splittingis smaller for Pb (topmost panel in figure 1). This can be attributed to the smalleratomic spin-orbit parameter of Pb (0 .
91 eV for Pb and 1 .
25 eV for Bi [25]). Bi x Sb − x /Ag(111). Recently, the surface states of the disordered binary alloysBi x Sb − x /Ag(111) were mapped out by angle-resolved photoelectron spectroscopy. Themomentum offset k R evolves continuously with increasing Bi concentration x . Thesplitting decreases sizably for x < .
50 [27].In theory, the outward relaxation of Bi is larger than for Sb (15 % and 9 . .
94 %) issmaller than that of Bi (0 .
99 %). Since Bi and Sb are iso-electronic, with valence-shellconfiguration 5 p and 6 p , E remains almost unaffected by x , as can be seen in thebottom row of figure 1. The spin splitting for Sb is much less than for Bi, in agreementwith the atomic spin-orbit parameter (0 . .
25 eV). In accord with experimentalresults, the Rashba splitting k R evolves with Bi concentration x .To elucidate further the effect of the relaxation, we calculated the splittingof Bi . Sb . /Ag(111) for two relaxations. The interpolated relaxation forBi . Sb . /Ag(111) is 12 . x, y, z ) = (0 . , . , .
0) in figure 1), for theartificial relaxed system the outward relaxation is taken as 19 % (not shown here). Thecharge transfer for the two systems is very close, and the difference in the splitting isnegligibly small. Hence, the splitting is negligibly sensitive to the relaxation, as wasalready established for Sb/Ag(111). Pb y Sb − y /Ag(111). To complete the picture of the binary alloys we turn toPb y Sb − y /Ag(111), for which experimental results are not available. The trends which ernary surface alloys Figure 1.
Surface states of disordered ternary alloys Bi x Pb y Sb − x − y /Ag(111) along¯Γ– ¯K of the two-dimensional Brillouin zone. The spectral density at a heavy-elementsite Bi x Pb y Sb − x − y is depicted as linear gray scale, with dark gray corresponding tohigh spectral weight; cf. (6). ernary surface alloys E shifts down from E F + 0 . E F − . y = 0 .
3. As for Bi x Sb − x /Ag(111), the spin splitting increases with y . x Pb y Sb − x − y /Ag(111) Having established the ingredients which are necessary for independently tuningthe Fermi energy and the spin splitting in the surface alloys—by investigating thedisordered binary surface alloys—we now mix them to disordered ternary alloysBi x Pb y Sb − x − y /Ag(111). By choosing appropriate concentrations x and y , thedegeneracy point E and the Rashba splitting are tuned. Note that the splitting k R and the Rashba energy E R are not fully independent; both can be expressed (in a free-electron model) in terms of the effective electron mass and the Rashba parameter [cf.(3) and (4)].In figure 1 the surface-state dispersions of ternary alloys Bi x Pb y Sb − x − y /Ag(111)are shown. The concentrations x and y have been varied in steps of 0 .
2. A commonfeature of the spectral density of the binary and ternary alloys is a finite lifetime of thespectral density, which is the consequence of the substitutional disorder.The Rashba characteristic of the ternary alloys follow the general trends of thebinary alloys which have been discussed before. In the ternary alloys with larger outwardrelaxation (i. e. the Bi- and Pb-rich compounds), the degenerate point E shifts towardhigher energies (main mechanism: relaxation). The larger the concentration of heavyelements Bi and Pb as compared to the Sb concentration, the larger the splitting k R (main mechanism: atomic spin-orbit parameter). The degenerate point E shifts upwardwith increasing Pb concentration (main mechanism: band filling).The shift k R of the surface states in reciprocal space versus concentrations x and y is shown in figure 2 (top). As expected, the smallest splitting (dark blue) shows up forSb/Ag(111) ( z = 1 − x − y = 1), while the largest (dark red) corresponds to Bi/Ag(111)( x = 1). For Pb/Ag(111), k R is of intermediate order (green/yellow). Surprisingly, thesplitting is not monotonic, as one might have expected in a rigid-band picture. Forexample, k R shows a local minimum at ( x, y, z ) ≈ (0 . , . , . k R , the Rashba energy E R depends monotonously in a large range ofconcentrations (bottom in figure 2). Sizable Rashba energies are found mainly for Bi-rich alloys, say for x > .
5. This implies that for accessing region I, Bi-rich surfacealloys are inevitable. For smaller x (blue areas in the bottom panel of figure 2), theenergy range of region I could be too small to be employed in experiments.The energy E of the degeneracy point depends almost linearly on the heavyelements’ concentrations x and y (figure 3). For equal Bi and Sb concentrations ( x = z )it is nearly constant; upon adding Pb, E shifts up. For systems with about 40 % of Pbconcentration, E is very close to the Fermi level E F , so that the latter lies in regionI [24]. ernary surface alloys x, y, z ) ≈ (0 . , . , . E and the Fermi energy E F coincide. Keeping the Sb concentration constant and changing the Pb concentrationof about 10 % is accompanied by transition between region I and region II, while k R and E R are almost constant. It is also possible to tune E R and k R while keeping theposition of degenerate point constant. The changes of k R and E R are not independentbut k R depends more sensitive on the concentrations than E R .
4. Conclusions
Disordered ternary surface alloys Bi x Pb y Sb − x − y /Ag(111) allow to fabricate a two-dimensional electron gas with specific Rashba spin-orbit splitting and Fermi energy Bi 10.2 y z Bi 10.2 y z Figure 2.
Spin splitting in disordered ternary alloys Bi x Pb y Sb − x − y /Ag(111). Top:The surface-state displacement k R (in reciprocal space) is depicted as color scale as afunction of Bi concentration x , Pb concentration y , and Sb concentration z = 1 − x − y .The color bar on the right is in units of ˚A − . Bottom: Same as in the top but for theRashba energy E R . The color bar is in eV. ernary surface alloys Bi 10.2 y z Figure 3.
Surface-state energy in disordered ternary alloys Bi x Pb y Sb − x − y /Ag(111).The degeneracy energy E of the surface state, with respect to the Fermi level E F , isdepicted as color scale as a function of Bi concentration x , Pb concentration y , and Sbconcentration z = 1 − x − y . The color bar on the right is in eV. At negative energies,the surface states are fully occupied (blue area). which can be investigated by surface-scientific methods (scanning tunneling probes andespecially photoelectron spectroscopy). In particular, the important transition fromenergy region I (that is, the Fermi energy E F lies above the degeneracy point E ) toregion II ( E F below E ) can be studied for different strengths of the Rashba spin-orbitcoupling. Thus, the present study may stimulate further experiments on Rashba systemsand their unique properties.
5. Acknowledgment
We gratefully acknowledge very fruitful discussions with Christian Ast, Hugo J Dil,Isabella Gierz, and Fabian Meier.
References [1] Yu. A. Bychkov and ´E. I. Rashba. Properties of a 2D electron gas with lifted degeneracy.
Sov.Phys. JETP Lett. , 39:78, 1984. Translated from Ref. Bychkov84c.[2] S. Datta and B. Das. Electronic analogue of the electronic modulator.
Appl. Phys. Lett. , 56(7):665,1990.[3] J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki. Gate control of spin-orbit interaction in aninverted In . Ga . As/In . Al . As heterostructure.
Phys. Rev. Lett. , 78:1335, 1997.[4] Takaaki Koga, Junsaku Nitta, Hideaki Takayanagi, and Supriyo Datta. Spin-filter device based onthe Rashba effect using a nonmagnetic resonant tunneling diode.
Phys. Rev. Lett. , 88:126601,2002.[5] E. Cappelluti, C. Grimaldi, and F. Marsiglio. Topological change of the Fermi surface in low-density Rashba gases: Application to superconductivity.
Phys. Rev. Lett. , 98:167002, 2007.[6] J. E. Hirsch. Spin hall effect.
Phys. Rev. Lett. , 83(9):1834, 1999.[7] Y. Kato, R. C. Myers, A. C. Gossard, and D. D. Awschalom. Coherent spin manipulation withoutmagnetic fields in strained semiconductors.
Nature , 427:50, 2004. ernary surface alloys [8] D. S. Saraga and D. Loss. Fermi liquid parameters in two dimensions with spin-orbit interaction. Phys. Rev. B , 72:195319, 2005.[9] S. O. Valenzuela and M. Tinkham. Direct electronic measurement of the spin hall effect.
Nature ,442:176, 2006.[10] L. Petersen and P. Hedeg˚ard. A simple tight-binding model of spin-orbit splitting of sp -derivedsurface states. Surf. Sci. , 459:49, 2000.[11] C. R. Ast, D. Pacil´e and L. Moreschini, M. C. Falub, M. Papagno, K. Kern, M. Grioni, J. Henk,A. Ernst, S. Ostanin, and P. Bruno. Spin-orbit split two-dimensional electron gas with tunableRashba and Fermi energy.
Phys. Rev. B , 77:081407(R), 2008.[12] F. Forster, S. H¨ufner, and F. Reinert. Rare gases on noble-metal surfaces: An angle-resolvedphotoemission study with high energy resolution.
J. Phys. Chem. B , 108:14692, 2004.[13] L. Moreschini, A. Bendounan, C. R. Ast, F. Reinert, M. Falub, and M. Grioni. Effect of rare-gasadsorption on the spin-orbit split bands of a surface alloy: Xe on Ag(111)-( √ × √ ◦ -Bi. Phys. Rev. B , 77:115407, 2008.[14] Emmanouil Frantzeskakis, St´ephane Pons, Hossein Mirhosseini, J¨urgen Henk, Christian R. Ast,and Marco Grioni. Tunable spin gaps in a quantum-confined geometry.
Phys. Rev. Lett. ,101(19):196805, Nov 2008.[15] H. Mirhosseini, I. V. Maznichenko, S. Abdelouahed, S. Ostanin, A. Ernst, I. Mertig, and J. Henk.Toward a ferroelectric control of rashba spin-orbit coupling: Bi on batio
Phys. Rev. B , 81(7):073406, Feb 2010.[16] Chr. R. Ast, J. Henk, A. Ernst, L. Moreschini, M. C. Falub, D. Pacil´e, P. Bruno, K. Kern, andM. Grioni. Giant spin splitting through surface alloying.
Phys. Rev. Lett. , 98:186807, 2007.[17] G. Bihlmayer, S. Bl¨ugel, and E. V. Chulkov. Enhanced Rashba spin-orbit splitting in Bi/Ag(111)and Pb/Ag(111) surface alloys.
Phys. Rev. B , 75:195414, 2007.[18] L. Moreschini, A. Bendounan, I. Gierz, C. A. Ast, H. Mirhosseini, H. H¨ochst, K. Kern, J. Henk,A. Ernst, S. Ostanin, F. Reinert, and M. Grioni. Assessing the atomic contribution to therashba spin-orbit splitting in surface alloys: Sb/ag(111).
Phys. Rev. B , 79:075424, 2009.[19] G. Kresse and J. Furthm¨uller. Efficient iterative schemes for ab initio total-energy calculationsusing a plane-wave basis set.
Phys. Rev. B , 54:11 169, 1996.[20] J. Henk, A. Ernst, and P. Bruno. Spin polarization of the L -gap surface states on Au(111): Afirst-principles investigation. Surf. Sci. , 566–568:482, 2004.[21] J. Zabloudil, R. Hammerling, L. Szunyogh, and P. Weinberger, editors.
Electron Scattering inSolid Matter . Springer, Berlin, 2005.[22] J. Henk. Theory of low-energy diffraction and photoelectron spectroscopy from ultra-thin films.In H. S. Nalwa, editor,
Handbook of Thin Film Materials , volume 2, chapter 10, page 479.Academic Press, San Diego, 2001.[23] P. Weinberger.
Electron Scattering Theory of Ordered and Disordered Matter . Clarendon Press,Oxford, 1990.[24] L. Moreschini, A. Bendounan, H. Bentmann, M. Assig, K. Kern, F. Reinert J. Henk, C. R. Ast,and M. Grioni. Influence of the substrate on the spin-orbit splitting in surface alloys on (111)noble-metal surfaces.
Phys. Rev. B , 80:035438, 2009.[25] K. Wittel and R. Manne. Accurate calculation of ground-state energies in an analytic Lanczosexpansion.
Theor. chim. Acta. , 33:347, 1974.[26] H. Mirhosseini, J. Henk, A. Ernst, S. Ostanin, C.-T. Chiang, P. Yu, A. Winkelmann, andJ. Kirschner. Unconventional spin topology in surface alloys with rashba-type spin splitting.