- Email: [email protected]
Ferromagnetic surface states of antiferromagnetic nickel sulphide
Surface Science 57 (1976) 241-250 0 North-Holland Publishing Company
FERROMAGNETIC
SURFACE STATES OF ANTIFERROMAGNETIC
NICKEL SULF’HIDE
S.J. GURMAN and J.B. PENDRY Daresbury Laboratory,
Daresbury,
Warrington WA4 4AD, Cheshire, UK
Received 12 December 19’75 We have calculated the surface state density on the (001) surface (assumed to consist of nickel atoms) of the antiferromagnetic phase of hexagonal nickel sulphide. We find that a surface state only exists for one spin and we style it a ferromagnetic surface state. The density of surface states has been calculated and is compared to the bulk density of states. Some possible effects of the surface state band are considered.
1. Introduction
In an earlier paper [l] on general existence criteria for surface states, we showed, using a two-band model, that in the case of a gap at the edge of the Brillouin zone (i.e. with KZc = 0 or n/2) where the surface barrier potential does not cohtain that element of the crystal potential responsible for the gap, a surface state will always exist for one sign of the potential matrix element responsible for the gap, and never for the other sign. We believe that a closely related situation exists in nickel sulphide, where a gap opens at the Fermi energy in the antiferromagnetic phase. This gap is due entirely to V,, the magnetic interaction between the ion core and the electrons, which has opposite sign for opposite spins. The surface barrier is too weak a potential to have an appreciable spin-orbit interaction and therefore does not contain a spin-dependent component. Hence, according to the results given by Pendry and Gurman [I], a surface state will exist in this gap for one spin but not for the other, and we will have a spin polarized surface state band. In fact, nickel sulphide is not described exactly by the simple model treated by Pendry and Gurman [l] : the bands which split to form the gap are doubly degenerate along I’A (corresponding to the F point of the two-dimensional surface Brillouin zone) and so in general we have four bands close together in the gap region and cannot apply the simple two-band model. However, the predictions of this model are so definite that we expect at least a preponderance of one spin in the surface state band of nickel sulphide. There is some uncertainty concerning the surface of nickel sulphide, as to whether
242
S.J. Gurman, J.B. PendryjFerromgnetic
surface states on antiferromagnetic
NiS
the crystal terminates in a layer of sulphur or nickel atoms. For a (001) surface consisting of a layer of nickel atoms we find that a surface state exists over the entire surface zone for the downspin electrons, electrons in these states being localized within the first four atomic layers. There are no surface states for the up-spin electrons. If the surface layer consists of sulphur atoms, then there are surface states for both spins, neither of which is dominant. We have no available experimental data as to which layer forms the surface, but we suggest below some of the possible effects to which the spin polarized surface state band may give rise.
2. The magnetic transition in nickel sulphide Nickel sulphide is one of several transition metal compounds that undergo a first order metal-insulator transition. At high temperatures it is metallic and above 620K exists in the hexagonal NiAs structure. Below this temperature, a metallic rhombohedral phase is the stable form, but the metallic hexagonal form may be maintained by quenching to room temperature. If such a crystal is further cooled, it goes through a first order phase transition at 263 K [2]. At this transition the conductivity falls by a factor of about 40 and magnetic moments of 1.5~~ appear, these moments being aligned antiferromagnetically down the c-axis. They are located on the nickel atoms. The transition is very sensitive to the composition of the crystal, disappearing completely if there is as little as 5% excess nickel. The crystals themselves are difficult to grow and maintain, and for this reason there is a great deal of variation in the experimental data. There is no observable change in the crystal structure or dimensions at the transition [2]. Several models have been proposed to explain the transition, either attributing it to the effects of the sudden appearance of the magnetic moments or to a change in the electron-electron correlation. These models are discussed by Adler [3]. We follow the calculation of Mattheiss [4] and simply use an ad hoc spin dependent potential in the antiferromagnetic phase: to the usual non-magnetic nickel potential we add a constant I’, within the muffin tin, positive or negative depending on the spin direction. This is a simplified form of the Slater model [S] for exchange effects in itinerant ferromagnets. Thus in the antiferromagnetic phase, the two nickel atoms in the unit cell are no longer equivalent and extra band gaps appear. Mattheiss [4] found that for V, equal to 0.02 hart a band gap opened up at the Fermi energy leading to semiconducting behaviour in the low temperature phase. Above the transition, with V, equal to zero, the calculation showed a metallic behaviour: thus the sudden change rn conductivity can be explained as due to the change in crystal symmetry induced by the antiferromagnetic ordering. The gap at the Fermi energy is caused by the exchange potential Vs which has different signs for up and downspins. (In the following, we define I’, as positive for an up-spin electron: this choice is completely arbitrary.)
S.J. Gurman, J.B. Pendry/Ferromagnetic
Crystal
surface states on antiferromagnetic
NiS
243
structure
Fig. 1. Crystal structure and Brillouin zone of hexagonalNiS. nickel atoms in the AF phase are shown.
The magnetic moments on the
3. Method of calculation Nickel sulphide in its hexagonal form has the NiAs structure, which is shown in fig. 1 together with the corresponding Brillouin zone. The crystal consists of layers perpendicular to the c-axis, each of which consists of one type (Ni or As) only. These layers have a hexagonal structure, the nearest neighbour distance in the layer being a = 6.5064 a.u. These layers are stacked uniformly in the c-direction, the interlayer spacing being c/4 = 2.5125 a.u. The repeat unit is four layers which together form what we term a “sandwich”. The nickel atoms are all vertically above one another and the sulphur atoms lie in the hollows between adjacent nickel atoms, the sulphur layers being rotated by 60” with respect to one another. The two-dimensional surface zone is hexagonal, of the same size and orientation as the face of the bulk zone. In the non-magnetic state above 263 K, all the nickel atoms are equivalent, and the rotation symmetry between the top and bottom halves of the sandwich means that all bands on the upper face ALH of the bulk Brillouin zone are doubly degenerate. The appearance of the anti-ferromagnetically ordered moments removes the equivalence of the nickel atoms and lifts this degeneracy: it is this which gives the new gap at the Fermi level. In the antiferromagnetic state, all spins in a given nickel layer are aligned ferromagnetically [2] , the adjacent layers having opposed spins. There are
244
S. J. Gurman, J.B. PendrylFerrornagnetic
surface states on antiferromagnetic
NiS
no spin moments on the sulphur atoms. A complete description of the symmetry properties of both states is given by Mattheiss [4]. For our purposes, the only effect of the antiferromagnetic ordering we need consider is the lifting of the degeneracy on the Brillouin zone face. We use the crystal potential of Mattheiss [4] in the muffin-tin approximation. This potential is derived from a superposition of atomic charge densities with Slater exchange. In his calculation, Mattheiss also includes non-muffin-tin terms, but we have omitted these: they are of some importance in the detailed bandstructure, but we do not expect them to seriously affect our conclusions. From Mattheiss’ muffin-tin potential, phase shifts were calculated in the usual way and parametrized according to the scheme of Cooper et al. [6] for convenience in performing the energy scan. Three phase shifts were used for both nickel and sulphur. The use of the constant Vs within the muffin tins to represent the spin dependent part of the potential is a major approximation, but is sufficient for our purposes: in a more accurate calculation V, would vary with the electron density. The reasoning leading to the value V, = 0.02 hart is given by Mattheiss [4]. He notes that various estimates ranging up to 0.06 hart can be justified, but use of the value @.02 hart gave a small gap at the Fermi energy which is in approximate agreement with experimental data. Thus both optical data [7] and conductivity data [2] give a gap of 0.005 hart. Since we are more concerned with the general behaviour of the surface states than with the detailed energies of the system, we also have used the value V, = 0.02 hart. We calculate the scattering within the crystal using the layer scattering method of Kambe [8]. In the region of constant potential between ion cores we expand the wavefunction in a series over the reciprocal lattice vectors of the two-dimensional surface net and consider the scattering of each component in this expansion by the atoms in a layer. The method assumes perfect periodicity in the two directions parallel to the layer, and hence k,, , the momentum in these directions, is conserved. We calculate at a fixed input energy and parallel momentum. The repeat unit in the nickel sulphide crystal is a sandwich of four layers, and the method of combining layers to give the scattering matrices for the sandwich is given by Pendry [9] . By applying the Bloch condition to the waves a sandwich apart we can calculate the bandstructure at a given value of k . To check our calculation, we calculated the bandstructure fork,, = 0 (the F point o 1 the surface zone, corresponding to the bulk FA line) over a wide energy range for several different numbers of reciprocal lattice vectorsg,, in the wavefunction expansion. Using nineteen vectors (the first three hexagonal rings) the bandstructure had converged to within 0.003 hart and agreed with the calculation of Mattheiss to within 0.005 hart. The resulting bandstructure is shown in fig. 2. In the surface state calculations up to 20 vectors were used to obtain a similar degree of convergence. To calculate the surface state energies we use the reflection coefficient method described earlier [l] . Here, the allowed states correspond to the poles in the reflection matrix for an electron incident from outside the crystal below the vacuum level. In the bandgap of the semi-infinite crystal at a given value of k,, there will be a small
S.J. Gurman, J. B. PendrylFerromgnetic
surface states on antiferromagnetic
NiS
245
@Hart)
K,ajn
%a
K,a=n
Fig. 2. Bandstructure of AF NiS along the rA line. Calculated using the layer method teen+ Energies are measured relative to the muffin tin zero at -0.522 hart.
with nine-
finite number of such poles which give the surface state energies. The system which we consider consists of the surface barrier, which we take in the form of an abrupt step placed half a layer spacing from the end layer, plus several sandwiches. The barrier height was fixed at 0.522 hart, the muffin tin zero of Mattheiss [4]. We take our energy zero at the muffin tin zero. If we are at an energy in the band gap we need include only a finite number of sandwiches since the decay in the Bloch wave amplitudes as we penetrate the crystal means that the deep lying layers do not contribute. We calculate the matrices for 1,2,4,8,. . . sandwiches until we obtain convergence in the trace of the reflection matrix: normally 8 or 16 were required. By the use of the crystal doubling method, originally developed for LEED calculations [9] we can find the reflection matrix very quickly: using fourteen g,, vectors in the wavefunction expansion convergence was usually reached in about 5 set on an IBM 370/165 at a single energy. To find the poles in the reflection matrix, we calculate the determinant of its inverse and plot it by hand across the gap seeking the zeroes. Outside the gap, the reflection matrix does not converge as we add more sandwiches, but has an osciliatory behaviour since the propagating bands penetrate throughout the crystal. We use this property to locate the edges of the gap.
246
S.J. Gurman, J.B. Pe~dry~Ferr~~gnctic
i
surface states on a~tiferro~gnetic
NS
0!3
0:1
Sandwtch
N(E) States/Hart
0 245
0.235
0.225
E(Hart)
0~255
(b) N(E) States/Hart
Sandwtch
Ef 1
240
E(Hart) I
I
I
-3
-2
-1
I 1
I 2
E-E,(eV)
Fig. 3. (a) Surface state energy contours for the irreducible segment of the surface BriIlouin zone for down-spin electrons. (b) Density of surface states, averaged over the first sandwich. The psition of the Fermi energy which gives a half filled band is shown. (c) Density of surface states, superposed on a sketch of the bulk density of states, both averaged over the first sandwich.
S.J. Gurman, J.B. PendryJFerromagnetic
surface states on antiferromagnetic
NiS
241
4. Results Using V, = 0.02 hart and nineteeng,, vectors, we calculated the bandstructure between zero and 0.300 hart, with the results shown in fig. 2. The eigenvalues at r and A agree with those of Mattheiss [4] to within 0.005 hart. We note in fig. 2 all the usual properties of the complex bandstructure, in particular the way all turning points are saddle points in complex-K, space: thus the complex loops can only leave real-K, bands at maxima or minima and link successive real-K, bands of the same symmetry. The gap at the Fermi leve! f.Ef = 0.238 hart) between the 3+ and 3- states at A is caused by VS (we note however that its width is not 2VS as a simple theory would require): in the non-magnetic state these two points are degenerate, as are l+ and 2- at A. The position of the surface state for this value of k,, is indicated. This state is at the extreme upper limit of the surface state band. In the scan over the two-dimensional surface zone, we found that there was a small overlap between the valence and conduction bands: the valence band maximum is 0.238 hart, at A, and the conduction band minimum is 0.236 hart at K. This suggests that VS should be increased slightly to obtain the semiconducting state. The minimum direct gap is 0.005 hart and lies on the line AL, in agreement with the result of Mattheiss. The results of
248
S.J. Gurman, J.B. Pendry/Fcrromagnetic
surface states on antiferromagnetic
NiS
of the order of c/2. Therefore virtually all of the density is localized within the first sandwich, within 5 A of the surface. The surface states are of the same symmetry as the bands which split to form the gap. They are therefore composed of nickel d-functions. At the bottom of the gap, the appropriate d-functions are localized on the surface layer, at the top they are localized on the inner nickel atom. The surface state lies halfway up the gap and is therefore probably equally distributed over both atoms, except for the decay in intensity due to the imaginary part of its Bloch momentum. Each of the nickel atoms in the first sandwich has about one half of an electron in a surface state. With this extra electron density localized near the surface, for the down-spin electrons only, there is a possibility that self-consistency requirements will drastically alter our assumed potential in order to maintain charge neutrality. In fact we find that the surface state electrons are drawn from the bulk bands and there is little or no charging at the surface. In a simple tight-binding model we find that the sum of surface state and bulk electrons in a single band always sums to two irrespective of the surface perturbations. If we take the surface layer as consisting of a layer of sulphur atoms, we find that there is little spin polarization of the surface state bands. Both spins have surface states in this case, both occurring near the edge of the gap and consequently being poorly localized. The average localization length is rather larger than c. The down-spin electrons are more highly localized than the up-spin if the nickel layer nearest the surface has a negative Vs (and vice versa) but the resulting spin polarization is not large. To summarise: we have shown that a spin polarized surface state band exists on the (001) face of hexagonal antiferromagnetic nickel sulphide, if this face is composed of nickel atoms. The surface state band contains about one electron, almost entirely localized in the first sandwhich, within 5 A of the surface, and is about 0.023 hart wide. Since the centre of the surface state band is very close to the bulk Fermi energy, there will be little or no band-bending at the surface.
5. Discussion
We have shown that within the model proposed by Mattheiss [4], ferromagnetic surface states exist on nickel sulphide, and thus our earlier speculation [l] is now established on a firmer footing. It is therefore appropriate to discuss what consequences such a state might have. The first question we might ask is whether the magnetisation forms domains, by analogy with ferromagnetism in bulk materials. This cannot happen in our instance because the polarization of the surface state is locked to the antiferromagnetism of the bulk, and in any case the depolarizing fields responsible for bulk domains do not build up to any great strength in a two-dimensional situation. Nevertheless, we can expect the surface to contain domains of up- and down-spin regions but for a different reason. Most surfaces are not perfectly smooth, but have steps in them. If the step contains an odd number of nickel planes, on the far side of the step, roles of up- and
S.J. Gurman, J.B. PendryfFerromagnetic
surface
states on antiferromagnetic
NiS
249
down-spin are interchanged, and the surface state has opposite magnetisation. There will be domains whose boundaries follow edges of steps. Since the surface state lies in the gap, the Fermi level will fall within the surface state band, and we expect the surface to conduct. However, since the surface state band is spin polarized, the transport of charge implies transport of spin-magnetisation. The complete polarization of eiectrons at the Fermi level has a consequence that has no bulk analogue: our two-dimensional domains will not allow current to cross them. To jump across a domain boundary the electron must reverse its spin, and spin-reversing processes are very weak, especially in light elements where spin-orbit coupling is small. This contrasts with the situation in a bulk ferromagnet where spin reversal takes place easily across a domain boundary, the difference being that in the bulk case domain boundaries are much wider and contain powerful magnetic fieds which can reverse the sign of the spin easily as an electron traverses a boundary. A macroscopic sample of nickel sulphide without special preparation of the surface can be expected to contain a mosaic of domains on the surface, the boundaries of which will prevent conduction. On the other hand, if we suppose that the surface has only one domain - obtained either by careful surface preparation or selection of a small sample - conduction will take place and the surface state will act as a spin-sieve. Suppose we make up a composite surface in the form of a plate, the middle portion consisting of a silicon (111) surface (which has unpolarized surface states) and the outer portions made of nickel sulphide (001) surfaces oppositely polarized. On passing a current along the plate, spin up electrons (say) will be transported into the silicon and spin down electrons out of the silicon, thus magnetising the silicon surface, Of course, spin-flip processes, though small, are finite and equilibrium would be established at some magnetisation (probably small) proportional to the current. Next consider a monovalent atom falling on the surface. The level in which the single electron sits (assumed s-like) will split and the level with spin parallel to the surface state polarization will probably be the lower because of exchange forces. If this is the case the parallel spin level will hybridize with the surface state and fill with polarized electrons. On desorption the atoms would emerge in a polarized state, which could be used to detect the effect. The influence of magnetic forces on ad-atom binding has recently been considered by Paulson and Schrieffer [13] . Apart from these more esoteric consequences, the surface state could also be detected by the use of photoemission at high photon energies (ca. 40 eV) where the mean free path for the photoelectron is only about 10 A, which may show up the surface state band from a comparison of data from clean and contaminated surfaces. Such a technique has been used to observe the surface states of tungsten (Feuerbacher and Fitton [l 1] ) and silicon (Rowe and Ibach [ 121). Angular resolved photoemission, which picks out one value of k,, only, offers a better prospect, since we can choose a point where the surface state is below the Fermi level but well clear of the valence --. band: the point k,, = (0.35,0.20) on the line I% 1s about the best. Here the surface state is at 0.237 hart and the top of the valence band is at 0.225 hart. We might also
25’0
S.J. Gut-man, JIB. PendryjFerromagnetic
surface states on antiferromagnetic
NiS
look for the surface state band by measuring the spin polarization of the emitted electrons: if the surface state band exists, this should peak near the Fermi level.
Adtnowledgement We are indebted sulphide.
to L.F. Mattheiss for supplying us with his potentials
for nickel
References [1] J.B. Pendry and S.J. Gurman, Surface Sci. 49 (1975) 87. [2] J.T. Sparks and T. Komoto, J. Appl. Phys. 40 (1968) 752. [ 31 D. Adler, Rev. Mod. Phys. 40 (1968) 714. [4] L.F. Mattheiss, Phys. Rev. BlO (1974) 995. [5] J.C. SIater, Phys. Rev. 82 (1951) 538. [6] B.R. Cooper, E. Kreiger and B. Segall, Phjrs. Rev. B4 (1971) 1734. [7] A.S. Barker and J.P. RameIka, Phys. Rev. BlO (1974) 987. [8] K. Kambe, Z. Naturforsch. 22a (1967) 322. [9] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). [lo] S. Hufner and G.K. Wertheim, Phys. Letters 44A (1973) 133. [ll] B. Feuerbacher and B. Fitton, Phys. Rev. Letters 29 (1972) 786. [12] J.E. Rowe and H. Ibach, Phys. Rev. Letters 32 (1974) 421. [ 131 R.H. Paulson and J.R. Schrieffer, Surface Sci. 48 (1975) 329.
FERROMAGNETIC
SURFACE STATES OF ANTIFERROMAGNETIC
NICKEL SULF’HIDE
S.J. GURMAN and J.B. PENDRY Daresbury Laboratory,
Daresbury,
Warrington WA4 4AD, Cheshire, UK
Received 12 December 19’75 We have calculated the surface state density on the (001) surface (assumed to consist of nickel atoms) of the antiferromagnetic phase of hexagonal nickel sulphide. We find that a surface state only exists for one spin and we style it a ferromagnetic surface state. The density of surface states has been calculated and is compared to the bulk density of states. Some possible effects of the surface state band are considered.
1. Introduction
In an earlier paper [l] on general existence criteria for surface states, we showed, using a two-band model, that in the case of a gap at the edge of the Brillouin zone (i.e. with KZc = 0 or n/2) where the surface barrier potential does not cohtain that element of the crystal potential responsible for the gap, a surface state will always exist for one sign of the potential matrix element responsible for the gap, and never for the other sign. We believe that a closely related situation exists in nickel sulphide, where a gap opens at the Fermi energy in the antiferromagnetic phase. This gap is due entirely to V,, the magnetic interaction between the ion core and the electrons, which has opposite sign for opposite spins. The surface barrier is too weak a potential to have an appreciable spin-orbit interaction and therefore does not contain a spin-dependent component. Hence, according to the results given by Pendry and Gurman [I], a surface state will exist in this gap for one spin but not for the other, and we will have a spin polarized surface state band. In fact, nickel sulphide is not described exactly by the simple model treated by Pendry and Gurman [l] : the bands which split to form the gap are doubly degenerate along I’A (corresponding to the F point of the two-dimensional surface Brillouin zone) and so in general we have four bands close together in the gap region and cannot apply the simple two-band model. However, the predictions of this model are so definite that we expect at least a preponderance of one spin in the surface state band of nickel sulphide. There is some uncertainty concerning the surface of nickel sulphide, as to whether
242
S.J. Gurman, J.B. PendryjFerromgnetic
surface states on antiferromagnetic
NiS
the crystal terminates in a layer of sulphur or nickel atoms. For a (001) surface consisting of a layer of nickel atoms we find that a surface state exists over the entire surface zone for the downspin electrons, electrons in these states being localized within the first four atomic layers. There are no surface states for the up-spin electrons. If the surface layer consists of sulphur atoms, then there are surface states for both spins, neither of which is dominant. We have no available experimental data as to which layer forms the surface, but we suggest below some of the possible effects to which the spin polarized surface state band may give rise.
2. The magnetic transition in nickel sulphide Nickel sulphide is one of several transition metal compounds that undergo a first order metal-insulator transition. At high temperatures it is metallic and above 620K exists in the hexagonal NiAs structure. Below this temperature, a metallic rhombohedral phase is the stable form, but the metallic hexagonal form may be maintained by quenching to room temperature. If such a crystal is further cooled, it goes through a first order phase transition at 263 K [2]. At this transition the conductivity falls by a factor of about 40 and magnetic moments of 1.5~~ appear, these moments being aligned antiferromagnetically down the c-axis. They are located on the nickel atoms. The transition is very sensitive to the composition of the crystal, disappearing completely if there is as little as 5% excess nickel. The crystals themselves are difficult to grow and maintain, and for this reason there is a great deal of variation in the experimental data. There is no observable change in the crystal structure or dimensions at the transition [2]. Several models have been proposed to explain the transition, either attributing it to the effects of the sudden appearance of the magnetic moments or to a change in the electron-electron correlation. These models are discussed by Adler [3]. We follow the calculation of Mattheiss [4] and simply use an ad hoc spin dependent potential in the antiferromagnetic phase: to the usual non-magnetic nickel potential we add a constant I’, within the muffin tin, positive or negative depending on the spin direction. This is a simplified form of the Slater model [S] for exchange effects in itinerant ferromagnets. Thus in the antiferromagnetic phase, the two nickel atoms in the unit cell are no longer equivalent and extra band gaps appear. Mattheiss [4] found that for V, equal to 0.02 hart a band gap opened up at the Fermi energy leading to semiconducting behaviour in the low temperature phase. Above the transition, with V, equal to zero, the calculation showed a metallic behaviour: thus the sudden change rn conductivity can be explained as due to the change in crystal symmetry induced by the antiferromagnetic ordering. The gap at the Fermi energy is caused by the exchange potential Vs which has different signs for up and downspins. (In the following, we define I’, as positive for an up-spin electron: this choice is completely arbitrary.)
S.J. Gurman, J.B. Pendry/Ferromagnetic
Crystal
surface states on antiferromagnetic
NiS
243
structure
Fig. 1. Crystal structure and Brillouin zone of hexagonalNiS. nickel atoms in the AF phase are shown.
The magnetic moments on the
3. Method of calculation Nickel sulphide in its hexagonal form has the NiAs structure, which is shown in fig. 1 together with the corresponding Brillouin zone. The crystal consists of layers perpendicular to the c-axis, each of which consists of one type (Ni or As) only. These layers have a hexagonal structure, the nearest neighbour distance in the layer being a = 6.5064 a.u. These layers are stacked uniformly in the c-direction, the interlayer spacing being c/4 = 2.5125 a.u. The repeat unit is four layers which together form what we term a “sandwich”. The nickel atoms are all vertically above one another and the sulphur atoms lie in the hollows between adjacent nickel atoms, the sulphur layers being rotated by 60” with respect to one another. The two-dimensional surface zone is hexagonal, of the same size and orientation as the face of the bulk zone. In the non-magnetic state above 263 K, all the nickel atoms are equivalent, and the rotation symmetry between the top and bottom halves of the sandwich means that all bands on the upper face ALH of the bulk Brillouin zone are doubly degenerate. The appearance of the anti-ferromagnetically ordered moments removes the equivalence of the nickel atoms and lifts this degeneracy: it is this which gives the new gap at the Fermi level. In the antiferromagnetic state, all spins in a given nickel layer are aligned ferromagnetically [2] , the adjacent layers having opposed spins. There are
244
S. J. Gurman, J.B. PendrylFerrornagnetic
surface states on antiferromagnetic
NiS
no spin moments on the sulphur atoms. A complete description of the symmetry properties of both states is given by Mattheiss [4]. For our purposes, the only effect of the antiferromagnetic ordering we need consider is the lifting of the degeneracy on the Brillouin zone face. We use the crystal potential of Mattheiss [4] in the muffin-tin approximation. This potential is derived from a superposition of atomic charge densities with Slater exchange. In his calculation, Mattheiss also includes non-muffin-tin terms, but we have omitted these: they are of some importance in the detailed bandstructure, but we do not expect them to seriously affect our conclusions. From Mattheiss’ muffin-tin potential, phase shifts were calculated in the usual way and parametrized according to the scheme of Cooper et al. [6] for convenience in performing the energy scan. Three phase shifts were used for both nickel and sulphur. The use of the constant Vs within the muffin tins to represent the spin dependent part of the potential is a major approximation, but is sufficient for our purposes: in a more accurate calculation V, would vary with the electron density. The reasoning leading to the value V, = 0.02 hart is given by Mattheiss [4]. He notes that various estimates ranging up to 0.06 hart can be justified, but use of the value @.02 hart gave a small gap at the Fermi energy which is in approximate agreement with experimental data. Thus both optical data [7] and conductivity data [2] give a gap of 0.005 hart. Since we are more concerned with the general behaviour of the surface states than with the detailed energies of the system, we also have used the value V, = 0.02 hart. We calculate the scattering within the crystal using the layer scattering method of Kambe [8]. In the region of constant potential between ion cores we expand the wavefunction in a series over the reciprocal lattice vectors of the two-dimensional surface net and consider the scattering of each component in this expansion by the atoms in a layer. The method assumes perfect periodicity in the two directions parallel to the layer, and hence k,, , the momentum in these directions, is conserved. We calculate at a fixed input energy and parallel momentum. The repeat unit in the nickel sulphide crystal is a sandwich of four layers, and the method of combining layers to give the scattering matrices for the sandwich is given by Pendry [9] . By applying the Bloch condition to the waves a sandwich apart we can calculate the bandstructure at a given value of k . To check our calculation, we calculated the bandstructure fork,, = 0 (the F point o 1 the surface zone, corresponding to the bulk FA line) over a wide energy range for several different numbers of reciprocal lattice vectorsg,, in the wavefunction expansion. Using nineteen vectors (the first three hexagonal rings) the bandstructure had converged to within 0.003 hart and agreed with the calculation of Mattheiss to within 0.005 hart. The resulting bandstructure is shown in fig. 2. In the surface state calculations up to 20 vectors were used to obtain a similar degree of convergence. To calculate the surface state energies we use the reflection coefficient method described earlier [l] . Here, the allowed states correspond to the poles in the reflection matrix for an electron incident from outside the crystal below the vacuum level. In the bandgap of the semi-infinite crystal at a given value of k,, there will be a small
S.J. Gurman, J. B. PendrylFerromgnetic
surface states on antiferromagnetic
NiS
245
@Hart)
K,ajn
%a
K,a=n
Fig. 2. Bandstructure of AF NiS along the rA line. Calculated using the layer method teen+ Energies are measured relative to the muffin tin zero at -0.522 hart.
with nine-
finite number of such poles which give the surface state energies. The system which we consider consists of the surface barrier, which we take in the form of an abrupt step placed half a layer spacing from the end layer, plus several sandwiches. The barrier height was fixed at 0.522 hart, the muffin tin zero of Mattheiss [4]. We take our energy zero at the muffin tin zero. If we are at an energy in the band gap we need include only a finite number of sandwiches since the decay in the Bloch wave amplitudes as we penetrate the crystal means that the deep lying layers do not contribute. We calculate the matrices for 1,2,4,8,. . . sandwiches until we obtain convergence in the trace of the reflection matrix: normally 8 or 16 were required. By the use of the crystal doubling method, originally developed for LEED calculations [9] we can find the reflection matrix very quickly: using fourteen g,, vectors in the wavefunction expansion convergence was usually reached in about 5 set on an IBM 370/165 at a single energy. To find the poles in the reflection matrix, we calculate the determinant of its inverse and plot it by hand across the gap seeking the zeroes. Outside the gap, the reflection matrix does not converge as we add more sandwiches, but has an osciliatory behaviour since the propagating bands penetrate throughout the crystal. We use this property to locate the edges of the gap.
246
S.J. Gurman, J.B. Pe~dry~Ferr~~gnctic
i
surface states on a~tiferro~gnetic
NS
0!3
0:1
Sandwtch
N(E) States/Hart
0 245
0.235
0.225
E(Hart)
0~255
(b) N(E) States/Hart
Sandwtch
Ef 1
240
E(Hart) I
I
I
-3
-2
-1
I 1
I 2
E-E,(eV)
Fig. 3. (a) Surface state energy contours for the irreducible segment of the surface BriIlouin zone for down-spin electrons. (b) Density of surface states, averaged over the first sandwich. The psition of the Fermi energy which gives a half filled band is shown. (c) Density of surface states, superposed on a sketch of the bulk density of states, both averaged over the first sandwich.
S.J. Gurman, J.B. PendryJFerromagnetic
surface states on antiferromagnetic
NiS
241
4. Results Using V, = 0.02 hart and nineteeng,, vectors, we calculated the bandstructure between zero and 0.300 hart, with the results shown in fig. 2. The eigenvalues at r and A agree with those of Mattheiss [4] to within 0.005 hart. We note in fig. 2 all the usual properties of the complex bandstructure, in particular the way all turning points are saddle points in complex-K, space: thus the complex loops can only leave real-K, bands at maxima or minima and link successive real-K, bands of the same symmetry. The gap at the Fermi leve! f.Ef = 0.238 hart) between the 3+ and 3- states at A is caused by VS (we note however that its width is not 2VS as a simple theory would require): in the non-magnetic state these two points are degenerate, as are l+ and 2- at A. The position of the surface state for this value of k,, is indicated. This state is at the extreme upper limit of the surface state band. In the scan over the two-dimensional surface zone, we found that there was a small overlap between the valence and conduction bands: the valence band maximum is 0.238 hart, at A, and the conduction band minimum is 0.236 hart at K. This suggests that VS should be increased slightly to obtain the semiconducting state. The minimum direct gap is 0.005 hart and lies on the line AL, in agreement with the result of Mattheiss. The results of
248
S.J. Gurman, J.B. Pendry/Fcrromagnetic
surface states on antiferromagnetic
NiS
of the order of c/2. Therefore virtually all of the density is localized within the first sandwich, within 5 A of the surface. The surface states are of the same symmetry as the bands which split to form the gap. They are therefore composed of nickel d-functions. At the bottom of the gap, the appropriate d-functions are localized on the surface layer, at the top they are localized on the inner nickel atom. The surface state lies halfway up the gap and is therefore probably equally distributed over both atoms, except for the decay in intensity due to the imaginary part of its Bloch momentum. Each of the nickel atoms in the first sandwich has about one half of an electron in a surface state. With this extra electron density localized near the surface, for the down-spin electrons only, there is a possibility that self-consistency requirements will drastically alter our assumed potential in order to maintain charge neutrality. In fact we find that the surface state electrons are drawn from the bulk bands and there is little or no charging at the surface. In a simple tight-binding model we find that the sum of surface state and bulk electrons in a single band always sums to two irrespective of the surface perturbations. If we take the surface layer as consisting of a layer of sulphur atoms, we find that there is little spin polarization of the surface state bands. Both spins have surface states in this case, both occurring near the edge of the gap and consequently being poorly localized. The average localization length is rather larger than c. The down-spin electrons are more highly localized than the up-spin if the nickel layer nearest the surface has a negative Vs (and vice versa) but the resulting spin polarization is not large. To summarise: we have shown that a spin polarized surface state band exists on the (001) face of hexagonal antiferromagnetic nickel sulphide, if this face is composed of nickel atoms. The surface state band contains about one electron, almost entirely localized in the first sandwhich, within 5 A of the surface, and is about 0.023 hart wide. Since the centre of the surface state band is very close to the bulk Fermi energy, there will be little or no band-bending at the surface.
5. Discussion
We have shown that within the model proposed by Mattheiss [4], ferromagnetic surface states exist on nickel sulphide, and thus our earlier speculation [l] is now established on a firmer footing. It is therefore appropriate to discuss what consequences such a state might have. The first question we might ask is whether the magnetisation forms domains, by analogy with ferromagnetism in bulk materials. This cannot happen in our instance because the polarization of the surface state is locked to the antiferromagnetism of the bulk, and in any case the depolarizing fields responsible for bulk domains do not build up to any great strength in a two-dimensional situation. Nevertheless, we can expect the surface to contain domains of up- and down-spin regions but for a different reason. Most surfaces are not perfectly smooth, but have steps in them. If the step contains an odd number of nickel planes, on the far side of the step, roles of up- and
S.J. Gurman, J.B. PendryfFerromagnetic
surface
states on antiferromagnetic
NiS
249
down-spin are interchanged, and the surface state has opposite magnetisation. There will be domains whose boundaries follow edges of steps. Since the surface state lies in the gap, the Fermi level will fall within the surface state band, and we expect the surface to conduct. However, since the surface state band is spin polarized, the transport of charge implies transport of spin-magnetisation. The complete polarization of eiectrons at the Fermi level has a consequence that has no bulk analogue: our two-dimensional domains will not allow current to cross them. To jump across a domain boundary the electron must reverse its spin, and spin-reversing processes are very weak, especially in light elements where spin-orbit coupling is small. This contrasts with the situation in a bulk ferromagnet where spin reversal takes place easily across a domain boundary, the difference being that in the bulk case domain boundaries are much wider and contain powerful magnetic fieds which can reverse the sign of the spin easily as an electron traverses a boundary. A macroscopic sample of nickel sulphide without special preparation of the surface can be expected to contain a mosaic of domains on the surface, the boundaries of which will prevent conduction. On the other hand, if we suppose that the surface has only one domain - obtained either by careful surface preparation or selection of a small sample - conduction will take place and the surface state will act as a spin-sieve. Suppose we make up a composite surface in the form of a plate, the middle portion consisting of a silicon (111) surface (which has unpolarized surface states) and the outer portions made of nickel sulphide (001) surfaces oppositely polarized. On passing a current along the plate, spin up electrons (say) will be transported into the silicon and spin down electrons out of the silicon, thus magnetising the silicon surface, Of course, spin-flip processes, though small, are finite and equilibrium would be established at some magnetisation (probably small) proportional to the current. Next consider a monovalent atom falling on the surface. The level in which the single electron sits (assumed s-like) will split and the level with spin parallel to the surface state polarization will probably be the lower because of exchange forces. If this is the case the parallel spin level will hybridize with the surface state and fill with polarized electrons. On desorption the atoms would emerge in a polarized state, which could be used to detect the effect. The influence of magnetic forces on ad-atom binding has recently been considered by Paulson and Schrieffer [13] . Apart from these more esoteric consequences, the surface state could also be detected by the use of photoemission at high photon energies (ca. 40 eV) where the mean free path for the photoelectron is only about 10 A, which may show up the surface state band from a comparison of data from clean and contaminated surfaces. Such a technique has been used to observe the surface states of tungsten (Feuerbacher and Fitton [l 1] ) and silicon (Rowe and Ibach [ 121). Angular resolved photoemission, which picks out one value of k,, only, offers a better prospect, since we can choose a point where the surface state is below the Fermi level but well clear of the valence --. band: the point k,, = (0.35,0.20) on the line I% 1s about the best. Here the surface state is at 0.237 hart and the top of the valence band is at 0.225 hart. We might also
25’0
S.J. Gut-man, JIB. PendryjFerromagnetic
surface states on antiferromagnetic
NiS
look for the surface state band by measuring the spin polarization of the emitted electrons: if the surface state band exists, this should peak near the Fermi level.
Adtnowledgement We are indebted sulphide.
to L.F. Mattheiss for supplying us with his potentials
for nickel
References [1] J.B. Pendry and S.J. Gurman, Surface Sci. 49 (1975) 87. [2] J.T. Sparks and T. Komoto, J. Appl. Phys. 40 (1968) 752. [ 31 D. Adler, Rev. Mod. Phys. 40 (1968) 714. [4] L.F. Mattheiss, Phys. Rev. BlO (1974) 995. [5] J.C. SIater, Phys. Rev. 82 (1951) 538. [6] B.R. Cooper, E. Kreiger and B. Segall, Phjrs. Rev. B4 (1971) 1734. [7] A.S. Barker and J.P. RameIka, Phys. Rev. BlO (1974) 987. [8] K. Kambe, Z. Naturforsch. 22a (1967) 322. [9] J.B. Pendry, Low Energy Electron Diffraction (Academic Press, London, 1974). [lo] S. Hufner and G.K. Wertheim, Phys. Letters 44A (1973) 133. [ll] B. Feuerbacher and B. Fitton, Phys. Rev. Letters 29 (1972) 786. [12] J.E. Rowe and H. Ibach, Phys. Rev. Letters 32 (1974) 421. [ 131 R.H. Paulson and J.R. Schrieffer, Surface Sci. 48 (1975) 329.