Computations of relativistic LEED intensities and spin-polarisations for tungsten (001)

Physica Status Solidi (b)

1973

Motivated by the importance of spin polarization and further relativistic effects in low energy electron scattering from large Z atoms (cf. review article (I)), we have developed a formalism based on Dirac' s equation for calculating intensities and spin polarizations in LEED from crystal surfaces (2,3). Applying this formalism to W(OOl), we find sizeable spin polarization values. In the fol lowing we first state the particular model assumptions underlying our computations and then present and discuss some numerical results.Following (3) we use,a model crystal consisting of our relativistic KKRZtype pseudopotential and an energy-dependent imaginary potential. The atomic phase shifts, which determine the pseudopotential coefficients, are obtained, as in (4), from Mattheiss' APW potential (5). We include five phase shifts in our computation. The effect of inelastic processes i s ideally taken into account by an imaginary self-energy part, which i s calculated by a many-body treatment of the electron gas in W. In the absence of such a calculation we content ourselves initially with an adjustable imaginary quantity, the choice of the energy dependence of which i s guided by experimental loss profiles (6). Since the threshold energy for a particular loss mode i s obtained a s the loss energy minus the work function ( 4 . 6 5 eV for W(OO1) (7)), the large loss around 10 e V (cf. Fig. 1 of (6)) suggests an increase in the imaginary potential V. in the vicinity of E = 5 . 3 5 eV (energy of the incident electron). In the present computation we choose V. = -1 . 5 eV for E 4 . 5 eV, V. = -2 e V for E P 4 . 5 eV. The real inner 'potential i s taken into account by shifting the intensity profiles calculated with zero inner potential to match the measured profiles. Thermal lattice motion is neglected at present.

supporting

confidence: 44%

Relativistic Low Energy Electron Diffraction from W(001)

Physica Status Solidi (b)

1973

“…Since the surface-sensitivity of this peak and therefore the degree of localization of the corresponding loss mechanism is not yet firmly established, we assume, in the vicinity of Eth[lO], a noticeable rise in vi or v, or in both. A further rise in ui is plausible in the vicinity of Eth [22], i.e. the threshold for the loss peak near 22 eV, which is ascribed t o the excitation of bulk plasmons.…”

Section: Assumptions Specific To the Present Calculations For Tungstenmentioning

confidence: 97%

“…e.g. [g]), we follow [22] in adopting an APW potential due to Mattheiss [2315) and thence calculating relativistic and nonrelativistic phase shiftsthe latter being used in the nonrelativistic analogue of our formalismby means of the Bunyan-Schonfelder [24] and the Numerov method (cf. e.g.…”

Section: Assumptions Specific To the Present Calculations For Tungstenmentioning

confidence: 99%

Relativistic Theory of Low Energy Electron Diffraction: Application to the (001) and (110) Surfaces of Tungsten

Physica Status Solidi (b)

1974

A relativistic theory for calculating low energy electron diffraction intensities and spin polarization properties is applied to the (001) and (110) surfaces of tungsten for several angles of incidence. Intensity-energy profiles are obtained in generally good agreement with experimental data. Relativistic effects on intensities are found t o be noticeable, and appreciable spin polarization features are predicted. I n particular, large degrees of spin polarization in conjunction with sizeable intensities are found in the specular beam for large angles of incidence on W(OO1).

“…17 termined from neutron-scattering experiments. 2 TmSb crystallizes in the cubic NaCl structure and the Tm 3+ J=6 CEF splitting gives a I\ singlet ground state followed by a T 4 triplet at 25 K and a T 5 triplet at 56 K, the other excited levels lying higher than 110 K. These higher levels are neglected in our considerations throughout this work. 4 In this Letter we would like to show the first experimental evidence of a CEF effect on the thermal expansion.…”

Section: R Federmentioning

confidence: 99%

Spin Polarization in Low-Energy Electron Diffraction from W(001)

1976

Phys. Rev. Lett.

A relativistic low-energy electron-diffraction theory has been modified to allow computations for energies up to 200 eV. Its application to W(001) yields intensity profiles in reasonable correspondence with experiment, and pronounced polarization profiles which show encouraging agreement with novel and as yet limited experimental data. A high sensitivity of the polarization profiles to displacement of the topmost layer suggests that spin polarization measurements could be valuable for structure determination of surfaces involving heavy atoms.Theoretical studies of spin-polarization effects in low-energy electron diffraction (LEED) from the (001) surface of tungsten 1 * 2 were recently followed by an experiment, 3 in which significant degrees of spin polarization were found for electron energies from 45 to 190 eV at various small angles of incidence. Comparison of these novel data with the theoretical predictions 1 ' 2 is not possible, however, since the latter are confined to energies below about 40 eV. It is the aim of this Letter to report the first theoretical results at higher energies, to compare them with the recent experiment, 3 and to draw conclusions regarding the value of a polarization analysis in surface-structure determination.The theory underlying the present calculations follows to a large extent a relativistic LEED theory described earlier. 4 In particular, the crystal is assumed to consist of a finite number of monatomic layers, for which the Dirac-equation boundary problem is solved to yield the fourspinor amplitudes and thence the intensities and the spin polarization vectors of the reflected beams as functions of the energy, the polar and azimuthal angles of incidence, and the polarization vector of the (normalized) primary beam. A modification was, however, made in the treatment of the monolayer part of the problem. The Korringa-Kohn-Rostoker-Ziman (KKRZ) 5 type relativistic pseudopotential used in Ref. 4 depends on radii R t -associated with the spin-up and spin-down atomic phase shifts 5^-which have to be optimized individually in a numerical application of the method. Since calculations up to 200 eV require phase shifts up to about l-l (cf. also Van Hove and Tong), 6 I decided to avoid the resulting radii parameter problem by developing and applying the Dirac-equation-based analog 7 of a KKR-type Schrodinger-equation-based method, 8,9 which has been used with great success for LEED intensity calculations. 10 The ef-fect of thermal lattice vibrations is taken into account by replacing the actual (real) spin-up and spin-down atomic phase shifts 6j* by effective (complex) phase shifts fy*, which are obtained from the fy* by averaging the corresponding scattering amplitudes over a Debye spectrum. 11 In the computational application to W(001) the following specific model assumptions have been made. Effective phase shifts oV, obtained from Mattheiss's muffin-tin potential 12 and with use of the bulk Debye temperature of 380°K, are included up to 1 = 1. The number of surface reciprocal lattice ...