- Email: [email protected]
Ab initio calculation of the reflectance anisotropy of GaAs(110): the role of nonlocal polarizability and local fields
31 July 2000
Physics Letters A 272 Ž2000. 264–270 www.elsevier.nlrlocaterpla
Ab initio calculation of the reflectance anisotropy of GaAs ž110/ : the role of nonlocal polarizability and local fields P.L. de Boeij a , C.M.J. Wijers b,) a
Theoretical Chemistry, Material Science Centre, UniÕersity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Faculty of Applied Physics, Computational Optics, UniÕersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
b
Received 24 May 2000; accepted 14 June 2000 Communicated by V.M. Agranovich
Abstract We demonstrate that the description of the optical reflectance anisotropy of GaAsŽ110. requires a complete microscopic treatment of both surface and bulk, which is feasible in the discrete cellular method. This method is an extension of standard discrete dipole calculations and accounts for non-locality in the electro-dynamical and quantum-mechanical interactions through the use of both real space local fields and ab-initio nonlocal polarizabilities. The results of our calculations are in excellent agreement with experiment and we show that the anisotropy is surface induced. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Sq; 03.50.De; 42.25.Ja; 71.10.Li; 78.20.Bh
One of the key issues in the microscopic treatment of optical response is the behavior of the short range interactions. Discrete dipole models are very suited instruments to address these interactions theoretically, since they account easily for electro-magnetic nonlocal behavior. The promise to develop discrete dipole models into truly universal microscopic optical problem solvers, has not been accomplished however, since they treat successfully only trivial systems like alkali-halogenides and solid noble gases. The success for this limited class of problems finds its origin in the basic assumption of discrete dipole theory that the system can be made
)
Corresponding author. E-mail address: [email protected] ŽC.M. Wijers..
up from independently polarizable entities. We state that the main reason why discrete dipole models fail for other systems is, that there this independency results in a poor handling of short-range interactions. It means that for those systems no entities can be found which can be polarized, without affecting also the Žnearby. environment. As a result induction, usually treated as a local process, becomes nonlocal and the introduction of nonlocal polarizabilities becomes a necessity. It is not possible to verify this hypothesis for the traditional cases, i.c. bulk simple cubic configurations under static conditions. There short-range dipole-dipole interactions cancel on symmetry grounds. Only cases where symmetry gets broken can reveal the properties of these short range interactions. This can be bulk spatial dispersion w1x or Žanisotropic. surface optical reflectance. Our choice
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 2 7 - 8
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
has been the latter, since it is more prominent and better studied at present. Mochan ´ and Barrera w2x applied successfully a local discrete dipole model to the anisotropic reflectance of the GeŽ110. surface. However, they had to make the ad hoc assignment of a single dipole to a pair of atoms to get reasonable agreement with experiment. In this Letter we show that the use of nonlocal polarizabilities is essential to make discrete dipole models work, particularly for the preferred assignment of one dipole to one atom. Further ‘Žnon.local’ will always be associated in this Letter with polarizability, unless otherwise specified. We present an expression for these nonlocal polarizabilities and show how to calculate them from first principles. We will also show how a discrete dipole calculation has to be modified to incorporate nonlocal polarizabilities. The result is used to calculate the reflectance anisotropy ŽRAS. of the GaAsŽ110. surface, where experimentally a rich spectroscopic structure has been observed w3x. The source of radiation in classical electrodynamics and hence also in discrete dipole models is the induced polarization density PŽr,t .. It can be obtained correctly up to first order using linear-response theory from the constitutive equation,
P Ž r, v . s x Ž r,r X , v . P E Ž r X , v . dr X .
H
Ž 1.
Here EŽr, v . is the self-consistent perturbing electric field and x Žr,r X , v . the nonlocal susceptibility or kernel. It can be obtained w4–6x from density-functional calculations in the local density approximation ŽDFT-LDA.. We use the Kohn–Sham equations to obtain the single-particle wave functions cn and energy eigenvalues e n belonging to the ground-state. For the excited states Žstates above the Fermi energy. we add a simple scissors operator Ža rigid shift of the energy of the conduction bands., such that the calculated band gap matches the experimental value. From these data we obtain the susceptibility as,
265
where we have used the velocity operator matrices j n nX , j n nX Ž r . s
ie"
Ž cn) Ž r . =cn Ž r . y cn Ž r . =cn) Ž r . . . X
2m
X
Ž 3. We assumed adiabatic onset of the perturbation, and retained the conductivity sum rule in Eq. Ž2. by correcting properly for the v s 0 singularity. It turned out to be not necessary to include the scissors adjustment w7x of the velocity operator matrices. The induced polarization P contributes to the perturbing electric field as, E Ž r, v . s E ext Ž r, v . q f Ž r y r X , v . P P Ž r X , v . dr X ,
H
where the transfer kernel f Žr, v . describes the full nonlocal retarded electro-magnetic interactions, f Ž r, v .
mn s
k 2dmn q =m=n
exp Ž ikr . 4pe 0 r
,
x Ž r,r , v . s
j n nX Ž r . j nX n Ž r X .
2
v2
Ýe X
nn
X
n
y e n y " vq
,
Ž 5.
with m , n the Cartesian components, k s vrc and r s
HV P Ž r, v . dr. i
X
Ž 4.
Ž 2. 1
We have used symmetry conserving Voronoi cells.
Ž 6.
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
266
Assuming a uniform field E i Ž v . within each cell Vi , we obtain from Eqs. Ž1., Ž4. the set of equations, p i Ž v . s Ý a i j Ž v . E ext j Ž v . q Ý f jk Ž v . p k Ž v . . j
k
Ž 7. Notice that this induction rule is only different from the usual local one in the occurrence of a second index j and its associated summation. This difference howeÕer is crucial and reflects the essence of this Letter. The nonlocal polarizability a i j Ž v . is defined as,
ai j Ž v . s
X
X
HV dr HV dr x Ž r,r , v . i
j
HV dr j
2 s
v
X
nn
Ž r . H dr j n nX Ž r . Vj
i
2
Ý
e n y e n y " vq
.
Ž 8.
X
X
nn
In this cell concept, the nonlocality of the susceptibility ŽEq. Ž2.. becomes inherited by the polarizability. The presence of a perturbing field in one cell causes neighboring cells to get polarized simultaneously. In general the continuity and differentiability of the wave-functions is at the basis of this nonlocal response and extends well beyond the characteristic inter-atomic spacing. Whenever the atomic wavefunctions do not overlap considerably, this range will be small and the polarizability is effectively local. This holds especially for the classical case, where the eigenfunctions cn are rigorously confined to a single cell. From inspection of Eq. Ž3. it is clear that then for all i / j at least one of the volume integrals in Ž8. has to disappear, leaving a purely local description. This explains in a transparent way the success of local discrete dipole calculations in trivial systems and its failure elsewhere. The transfer tensors f i j Ž v . can be derived from Eq. Ž5., by taking the appropriate cell averages
°f Ž r y r , v . i
fi j
for i / j
j
Ž v . s~ g Ž v . y 1
¢
The majority of the f i j are intercellular Ž i / j . and are determined by the transfer kernel between the cell positions r i and r j . For the intracellular transfer tensor f i i we are faced with the singularity in the transfer kernel. This problem can be overcome by using the approximation of a uniform polarization density, for which the traditional Lorentz field solution holds. This approximation has turned out to be already very good by itself. Since our aim however is to provide accurate surface optical data, we can allow ourselves the freedom to use bulk parameters to refine our intracellular approximation. The refinement is in the term g i Ž v ., which accounts for the deviation from a homogeneous system. We can use the Clausius–Mossotti relation, which is valid for uniform polarizations, to obtain an accurate estimate for the value of the correction g i Ž v .. We use this relation to derive the g i Ž v . from a comparison between experimental and theoretical bulk data. If v < cVy1 r3 with V s Ý i Vi the volume of the primitive cell, we obtain then,
i
3 e 0 < Vi <
ik
3
y 6pe 0
for i s j.
g Ž v . s1y
er Ž v . q 2 er Ž v . y 1
q 3 e 0 Vay1 Ž v .
where e r Ž v . is the experimentally observed relative dielectric function, and a Ž v . the theoretically obtained mean polarizability of the primitive cell given by a Ž v . s Ý i g V , j a i j Ž v .. The second term in the f i i Ž v . tensor restores the energy balance through Lorentz’ radiation loss. The technical advantage of this cellular approximation is that we can solve the system of Eqs. Ž7. for a semi-infinite crystalline system and a given incident field, E ext Žr, v . s E 0 expŽ ik P r., by using the double-cell method w8,9x which can readily be extended to include nonlocal polarizabilities. In this method, the system is built up as a semi-infinite stack of ordered dipole layers, each obeying parallel translational symmetry. Each layer i can be described by a single characteristic dipole p i which is positioned at r i , according to, p Ž r i q s I , v . s exp Ž ik P s I . p i Ž v . .
Ž 9.
Ž 10 .
Ž 11 .
Here s I is any vector from the commensurate surface lattice. For the first few layers all interactions
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
are described using these characteristic dipoles. Together they build the surface cell Ždenoted by r is .. In the bulk, where the layers are regularly spaced with period d, we can make use of a finite number of modes from the following normal mode expansion, originally derived in w10x: p Ž r ib q s, v . M
s
Ý
nm Ž v . exp Ž ik m Ž v . P s . u i m Ž v . ,
Ž 12.
ms1
Here r ib locates the characteristic dipoles in the first bulk unit cell and s is now any vector from the semi-infinite bulk lattice. Each normal mode m is characterized by the propagation vector k mŽ v . s k I qqmŽ v .n ˆ where nˆ is the inward surface normal, the mode strength nmŽ v . and the polarization vectors u i mŽ v ., one for each characteristic bulk dipole. For notational convenience the v-dependence will be dropped in the sequel. We will only have to consider weakly attenuating modes which have 0 F I qm 4 < 2prd. The electric field for such a stack of dipoles is obtained by summing the transfer tensors of Eq. Ž9.. These sums can be evaluated efficiently using Ewald transformations, the result of which can be expressed as an evanescent wave expansion w8,9x. The only components in this expansion which do not attenuate strongly, propagate in the direction of the incident field k. Deep inside the bulk the self-consistency of Eq. Ž7. has to hold for each component separately, which leads for the normal modes of Eq. Ž12. to the following bulk secular equation Žimplied summation over repeated indices.,
d i j 1 y A bi k Ž h . Fkbj Ž h . P u j s 0 A bi j Ž h . s Ý exp Ž ih P s . a ib, jqs .
Ž 13 .
s
This equation yields the bulk normal modes k m , u ˆ i m 4. The tensors Fibj Žh. comprise all contributions of transfer tensors between the bulk sites r ib and r jb to the field component that propagates with velocity h. The A bi j Žh. is the phase corrected sum of nonlocal bulk polarizabilities a i,b jqs . A similar condition for the field component k reveals that deep in the bulk the induced and incident
267
field have to cancel exactly - the familiar Ewald– Oseen extinction theorem. This yields two equations, one for each polarization direction t. Additional conditions can be generated similarly for the other evanescent field components E n . Together with the self-consistency condition for the surface dipoles at r is this yields a manageable set of interaction equations,
d i j 1 y A si k Fksjs
yA si k Fksbm
yt P Fnbjs
yt P Fnbmb
P
A si j E ext pj j s ext nm t PEn
Ž 14 . The solution determines the strengths of the surface dipoles and normal modes, p i Ž v ., nnŽ v .4 . The local fields have to be evaluated Žthrough the various tensors F. for all characteristic dipoles in the surface, but also for a finite range of the neighboring normal mode region. This range corresponds to what extent the polarizability matrix elements A si j s Ýs I expŽ ik P s I. a isjqs I , affect this region. With the polarization distribution now fully determined, we obtain the reflected field in the yn ˆ direction w8,9x. The GaAs bulk and surface polarizabilities have been obtained from single particle energies and wave functions using accurate periodic DFT-LDA calculations w11,12x. In the bulk calculation we used a ˚ The surface properties lattice constant a s 5.613 A. have been obtained from a single asymmetric slab containing 16 atomic layers. The surface reconstruction has been taken from w13x and the backside has been made bulk terminated by adding 4 frozen core atomic layers ŽGa and As. and 2H termination layers w15x. Two additional frozen-ion layers were added to model the bulk Madelung potential. We used the scissors operator to fix the bulkŽ-like. band-gap to 1.52 eV. The cut-off range for the nonlocal polariz˚ resulting in eight shells of ability was set to 8 A, neighboring atoms. The value of g Ž v . in Eq. Ž10. was found to be nearly constant and real, g Ž v . s 0.02, confirming that it builds only a minor correction to the Lorentz field. ŽSee Fig. 1.. The first step in our calculations has been to obtain a value for the bulk dielectric constant. We used Clausius Mossotti and a to obtain the bulk dielectric constant. Compared e.g. to other calcula-
268
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
Fig. 1. Calculated and experimental dielectric constant of GaAs. Parameters: EGa p s1.52 eV, f s 0.98.
tions w14x similar agreement with experiment is found, but the spectra are somewhat more smeared out. This has to be expected, since our nonlocal calculation explicitly accounts for the Žsmall. photon wave vector. The E1 and E2-type transitions are well resolved, have the right magnitude and are at the correct energetic location after application of the scissors operator. The normal mode wave number qm agrees in all cases within 0.1 % with the Fresnel value. In summary this indicates that the nonlocal character of the polarizability hardly influences the average bulk optical properties, as expected. In doing the normal mode calculations it turned out that the usual Fresnel resembling modes were insufficient. Until 2.6 eV we found for each Fresnel mode exactly one extra normal mode which did not decay rapidly enough. We have examined the behavior of the extra Žnon-Fresnel. normal modes in some detail. They start at very low frequencies already, but are too much damped to play a role, but at 1.1. eV they become perfectly transparent and have to be taken into account explicitly. Meanwhile the real part of the qm moves from a value of prd towards zero. One extra normal mode happens to be purely transverse Žu i m P k m s 0., the other has also a longitudinal component Žu i m P k m / 0.. All extra normal modes start to become increasingly absorbing beyond 1.5 eV Žcoinciding with the bulk band gap., and their influence can be discarded after 2.6 eV. In Fig. 2 we show the individual dipole strengths at " v s 1.9 eV for the external field at normal
incidence and polarized parallel to the Ž110. direction. The dipole strength varies clearly going from Ga to As sites, as expected. Matching between surface and normal mode region Žarrow in Fig. 2. is perfect. Neglect of the extra modes destroys this matching. Dipole strengths for the same type of atom show a modulation also for the normal mode region. This is the influence of the extra normal modes. The dipole strength for the surface dipoles is substantially smaller than for the bulk. This is a general trend and has been observed also for other surfaces w8,9x. Accurate reflectance anisotropy spectra have been measured and studied recently w3,19x. We have calculated the reflectance anisotropy for GaAs Ž110., using the previously outlined discrete cellular method. Due to the limited slab thickness, we used bulk values for a i j Ž v . throughout the system except for i, j both top-layer atoms. The convention we have used to define the anisotropic difference is
D RrR s 2 Ž R w110x y R w001x . r Ž R w110x q R w001x . .
Ž 15 .
Theoretical results are obtained and shown in Fig. 3. Realistic results can only be obtained when dipole strengths match perfectly from surface to normal mode region. Curve A shows the theoretical nonlocal RAS for the reconstructed surface. Curve B equals A but with the RAS of a GaAs bulk terminated system subtracted. Although both curves are in close agreement with the experimental observations, the agreement is best for curve B. Difference between A and B is mainly due to the anisotropy of the bulk. In view of the experimental evidence for bulk spatial dispersion, also for GaAs along the Ž110. direction
Fig. 2. Modulus
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
Fig. 3. Reflectance Anisotropy Spectra of GaAsŽ110.. Experiment: w3x, A: nonlocal, B: Aybackground, C: as B, but reduced to local.
w16x, and its theoretical justification w17x, the influence of a bulk contribution to the optical anisotropy cannot be ignored. For this reason we have added curve B. For the same reason we strongly recommend that the experimental calibration procedures are carefully reexamined as to this point w18x. All calculated spectra have the right sign and order of magnitude. Also the spectral structure is remarkably well reproduced apart from a rigid energy shift of 0.3 eV. This holds in particular for curve B, where only the small shoulder at 3.0 eV is missing. Therefore we conclude that the RAS signal finds its basic origin in the surface atoms, meaning that the spectral structure vanishes mostly if these atoms are left out. However one has to be aware, that within a Žtwofold!. nonlocal calculation, no clear distinction can be made anymore between bulk and surface. The shift indicates that the scissors operation may have to be different for surface and bulk states as noted by others w19x. Finally curve C shows the result when the nonlocal polarizabilities are made effectively local, according to the substitution a i j Ž v . s d i j Ý k a i k Ž v .. Result C shows clearly that surface optical anisotropy depends strongly on the nonlocal character of the polarizabilities, opposite to the behavior of the bulk optical properties. The discrete cellular method, as we prefer to call the method given in this Letter contains very few
269
adjustable parameters, all of them Žscissors energy shift, g i-term. related to bulk properties. The method is intrinsically connected to a high quality ab initio electronic structure method ŽADF-BAND. and enhances the results of these methods for new fields of application. This holds particularly for the description of the optical properties of strongly inhomogeneous systems, such as surfaces, and there particularly for the correct determination of the intensity of the response, rather than an accurate determination of the position of the energy levels. As such it builds a major improvement beyond what traditional discrete dipole models that can take only effects of electro-magnetic nonlocality into account, are able to achieve. The addition of the influence of quantum mechanical nonlocality through nonlocal polarizabilities has been shown to be vital for the case of semiconductor surfaces. We suggest that a merger of this method and methods to correct for the proper location of energy levels, such as GW or Bethe– Salpeter, might result in truly outstanding results for the optical properties of inhomogeneous systems.
Acknowledgements We wish to thank G. te Velde and E.J. Baerends for letting us use their ADF-BAND w11,12x code.
References w1x J. Pastrnak, K. Vedam, Phys. Rev. B 3 Ž1971. 2567. w2x W.L. Mochan, ´ R.G. Barrera, Phys. Rev. Lett. 55 Ž1985. 1192. w3x N. Esser, R. Hunger, J. Rumberg, W. Richter, R. Del Sole, A. Shkrebtii, Surf. Sci. 307–309 Ž1994. 1045. w4x Z.H. Levine, D.C. Allen, Phys. Rev. B 44 Ž1991. 12781. w5x Z.H. Levine, D.C. Allen, Phys. Rev. B 48 Ž1993. 14768, ŽE.. w6x Z.H. Levine, Phys. Rev. B 49 Ž1994. 4532. w7x R. Del Sole, R. Girlanda, Phys. Rev. B 48 Ž1993. 11789. w8x C.M.J. Wijers, G.P.M. Poppe, Phys. Rev. B 46 Ž1992. 7605. w9x G.P.M. Poppe, C.M.J. Wijers, A. van Silfhout, Phys. Rev. B 44 Ž1991. 7917. w10x O. Litzman, P. Rosza, Surf. Sci. 66 Ž1977. 542. ´ w11x G. te Velde, E.J. Baerends, Phys. Rev. B 44 Ž1991. 7888. w12x E.J. Baerends, J. Comp. Phys. 99 Ž1992. 84. w13x C. Eckl, J. Fritsch, P. Pavone, U. Schroder, Surf. Sci. 394 ¨ Ž1997. 47.
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P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
w14x S.N. Rashkeev, W.R.L. Lambrecht, B. Segall, Phys. Rev. B 57 Ž1998. 3905. w15x A.F. Wright, C.Y. Fong, L.P. Batra, Surf. Sci. 244 Ž1991. 51. w16x P.Y. Yu, M. Cardona, Solid State Commun 9 Ž1971. 1421. w17x V.M. Agranovich, V.L. Ginzburg, Spatial Dispersion in
Crystal Optics and the Theory of Excitons, Wiley, New York, 1966. w18x N. Esser, private communication. w19x O. Pulci, G. Onida, R. Del Sole, A.L. Reining, Phys. Rev. Lett. 81 Ž1998. 5374.
Physics Letters A 272 Ž2000. 264–270 www.elsevier.nlrlocaterpla
Ab initio calculation of the reflectance anisotropy of GaAs ž110/ : the role of nonlocal polarizability and local fields P.L. de Boeij a , C.M.J. Wijers b,) a
Theoretical Chemistry, Material Science Centre, UniÕersity of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands Faculty of Applied Physics, Computational Optics, UniÕersity of Twente, P.O. Box 217, 7500 AE Enschede, The Netherlands
b
Received 24 May 2000; accepted 14 June 2000 Communicated by V.M. Agranovich
Abstract We demonstrate that the description of the optical reflectance anisotropy of GaAsŽ110. requires a complete microscopic treatment of both surface and bulk, which is feasible in the discrete cellular method. This method is an extension of standard discrete dipole calculations and accounts for non-locality in the electro-dynamical and quantum-mechanical interactions through the use of both real space local fields and ab-initio nonlocal polarizabilities. The results of our calculations are in excellent agreement with experiment and we show that the anisotropy is surface induced. q 2000 Elsevier Science B.V. All rights reserved. PACS: 03.65.Sq; 03.50.De; 42.25.Ja; 71.10.Li; 78.20.Bh
One of the key issues in the microscopic treatment of optical response is the behavior of the short range interactions. Discrete dipole models are very suited instruments to address these interactions theoretically, since they account easily for electro-magnetic nonlocal behavior. The promise to develop discrete dipole models into truly universal microscopic optical problem solvers, has not been accomplished however, since they treat successfully only trivial systems like alkali-halogenides and solid noble gases. The success for this limited class of problems finds its origin in the basic assumption of discrete dipole theory that the system can be made
)
Corresponding author. E-mail address: [email protected] ŽC.M. Wijers..
up from independently polarizable entities. We state that the main reason why discrete dipole models fail for other systems is, that there this independency results in a poor handling of short-range interactions. It means that for those systems no entities can be found which can be polarized, without affecting also the Žnearby. environment. As a result induction, usually treated as a local process, becomes nonlocal and the introduction of nonlocal polarizabilities becomes a necessity. It is not possible to verify this hypothesis for the traditional cases, i.c. bulk simple cubic configurations under static conditions. There short-range dipole-dipole interactions cancel on symmetry grounds. Only cases where symmetry gets broken can reveal the properties of these short range interactions. This can be bulk spatial dispersion w1x or Žanisotropic. surface optical reflectance. Our choice
0375-9601r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 Ž 0 0 . 0 0 4 2 7 - 8
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
has been the latter, since it is more prominent and better studied at present. Mochan ´ and Barrera w2x applied successfully a local discrete dipole model to the anisotropic reflectance of the GeŽ110. surface. However, they had to make the ad hoc assignment of a single dipole to a pair of atoms to get reasonable agreement with experiment. In this Letter we show that the use of nonlocal polarizabilities is essential to make discrete dipole models work, particularly for the preferred assignment of one dipole to one atom. Further ‘Žnon.local’ will always be associated in this Letter with polarizability, unless otherwise specified. We present an expression for these nonlocal polarizabilities and show how to calculate them from first principles. We will also show how a discrete dipole calculation has to be modified to incorporate nonlocal polarizabilities. The result is used to calculate the reflectance anisotropy ŽRAS. of the GaAsŽ110. surface, where experimentally a rich spectroscopic structure has been observed w3x. The source of radiation in classical electrodynamics and hence also in discrete dipole models is the induced polarization density PŽr,t .. It can be obtained correctly up to first order using linear-response theory from the constitutive equation,
P Ž r, v . s x Ž r,r X , v . P E Ž r X , v . dr X .
H
Ž 1.
Here EŽr, v . is the self-consistent perturbing electric field and x Žr,r X , v . the nonlocal susceptibility or kernel. It can be obtained w4–6x from density-functional calculations in the local density approximation ŽDFT-LDA.. We use the Kohn–Sham equations to obtain the single-particle wave functions cn and energy eigenvalues e n belonging to the ground-state. For the excited states Žstates above the Fermi energy. we add a simple scissors operator Ža rigid shift of the energy of the conduction bands., such that the calculated band gap matches the experimental value. From these data we obtain the susceptibility as,
265
where we have used the velocity operator matrices j n nX , j n nX Ž r . s
ie"
Ž cn) Ž r . =cn Ž r . y cn Ž r . =cn) Ž r . . . X
2m
X
Ž 3. We assumed adiabatic onset of the perturbation, and retained the conductivity sum rule in Eq. Ž2. by correcting properly for the v s 0 singularity. It turned out to be not necessary to include the scissors adjustment w7x of the velocity operator matrices. The induced polarization P contributes to the perturbing electric field as, E Ž r, v . s E ext Ž r, v . q f Ž r y r X , v . P P Ž r X , v . dr X ,
H
where the transfer kernel f Žr, v . describes the full nonlocal retarded electro-magnetic interactions, f Ž r, v .
mn s
k 2dmn q =m=n
exp Ž ikr . 4pe 0 r
,
x Ž r,r , v . s
j n nX Ž r . j nX n Ž r X .
2
v2
Ýe X
nn
X
n
y e n y " vq
,
Ž 5.
with m , n the Cartesian components, k s vrc and r s
HV P Ž r, v . dr. i
X
Ž 4.
Ž 2. 1
We have used symmetry conserving Voronoi cells.
Ž 6.
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
266
Assuming a uniform field E i Ž v . within each cell Vi , we obtain from Eqs. Ž1., Ž4. the set of equations, p i Ž v . s Ý a i j Ž v . E ext j Ž v . q Ý f jk Ž v . p k Ž v . . j
k
Ž 7. Notice that this induction rule is only different from the usual local one in the occurrence of a second index j and its associated summation. This difference howeÕer is crucial and reflects the essence of this Letter. The nonlocal polarizability a i j Ž v . is defined as,
ai j Ž v . s
X
X
HV dr HV dr x Ž r,r , v . i
j
HV dr j
2 s
v
X
nn
Ž r . H dr j n nX Ž r . Vj
i
2
Ý
e n y e n y " vq
.
Ž 8.
X
X
nn
In this cell concept, the nonlocality of the susceptibility ŽEq. Ž2.. becomes inherited by the polarizability. The presence of a perturbing field in one cell causes neighboring cells to get polarized simultaneously. In general the continuity and differentiability of the wave-functions is at the basis of this nonlocal response and extends well beyond the characteristic inter-atomic spacing. Whenever the atomic wavefunctions do not overlap considerably, this range will be small and the polarizability is effectively local. This holds especially for the classical case, where the eigenfunctions cn are rigorously confined to a single cell. From inspection of Eq. Ž3. it is clear that then for all i / j at least one of the volume integrals in Ž8. has to disappear, leaving a purely local description. This explains in a transparent way the success of local discrete dipole calculations in trivial systems and its failure elsewhere. The transfer tensors f i j Ž v . can be derived from Eq. Ž5., by taking the appropriate cell averages
°f Ž r y r , v . i
fi j
for i / j
j
Ž v . s~ g Ž v . y 1
¢
The majority of the f i j are intercellular Ž i / j . and are determined by the transfer kernel between the cell positions r i and r j . For the intracellular transfer tensor f i i we are faced with the singularity in the transfer kernel. This problem can be overcome by using the approximation of a uniform polarization density, for which the traditional Lorentz field solution holds. This approximation has turned out to be already very good by itself. Since our aim however is to provide accurate surface optical data, we can allow ourselves the freedom to use bulk parameters to refine our intracellular approximation. The refinement is in the term g i Ž v ., which accounts for the deviation from a homogeneous system. We can use the Clausius–Mossotti relation, which is valid for uniform polarizations, to obtain an accurate estimate for the value of the correction g i Ž v .. We use this relation to derive the g i Ž v . from a comparison between experimental and theoretical bulk data. If v < cVy1 r3 with V s Ý i Vi the volume of the primitive cell, we obtain then,
i
3 e 0 < Vi <
ik
3
y 6pe 0
for i s j.
g Ž v . s1y
er Ž v . q 2 er Ž v . y 1
q 3 e 0 Vay1 Ž v .
where e r Ž v . is the experimentally observed relative dielectric function, and a Ž v . the theoretically obtained mean polarizability of the primitive cell given by a Ž v . s Ý i g V , j a i j Ž v .. The second term in the f i i Ž v . tensor restores the energy balance through Lorentz’ radiation loss. The technical advantage of this cellular approximation is that we can solve the system of Eqs. Ž7. for a semi-infinite crystalline system and a given incident field, E ext Žr, v . s E 0 expŽ ik P r., by using the double-cell method w8,9x which can readily be extended to include nonlocal polarizabilities. In this method, the system is built up as a semi-infinite stack of ordered dipole layers, each obeying parallel translational symmetry. Each layer i can be described by a single characteristic dipole p i which is positioned at r i , according to, p Ž r i q s I , v . s exp Ž ik P s I . p i Ž v . .
Ž 9.
Ž 10 .
Ž 11 .
Here s I is any vector from the commensurate surface lattice. For the first few layers all interactions
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
are described using these characteristic dipoles. Together they build the surface cell Ždenoted by r is .. In the bulk, where the layers are regularly spaced with period d, we can make use of a finite number of modes from the following normal mode expansion, originally derived in w10x: p Ž r ib q s, v . M
s
Ý
nm Ž v . exp Ž ik m Ž v . P s . u i m Ž v . ,
Ž 12.
ms1
Here r ib locates the characteristic dipoles in the first bulk unit cell and s is now any vector from the semi-infinite bulk lattice. Each normal mode m is characterized by the propagation vector k mŽ v . s k I qqmŽ v .n ˆ where nˆ is the inward surface normal, the mode strength nmŽ v . and the polarization vectors u i mŽ v ., one for each characteristic bulk dipole. For notational convenience the v-dependence will be dropped in the sequel. We will only have to consider weakly attenuating modes which have 0 F I qm 4 < 2prd. The electric field for such a stack of dipoles is obtained by summing the transfer tensors of Eq. Ž9.. These sums can be evaluated efficiently using Ewald transformations, the result of which can be expressed as an evanescent wave expansion w8,9x. The only components in this expansion which do not attenuate strongly, propagate in the direction of the incident field k. Deep inside the bulk the self-consistency of Eq. Ž7. has to hold for each component separately, which leads for the normal modes of Eq. Ž12. to the following bulk secular equation Žimplied summation over repeated indices.,
d i j 1 y A bi k Ž h . Fkbj Ž h . P u j s 0 A bi j Ž h . s Ý exp Ž ih P s . a ib, jqs .
Ž 13 .
s
This equation yields the bulk normal modes k m , u ˆ i m 4. The tensors Fibj Žh. comprise all contributions of transfer tensors between the bulk sites r ib and r jb to the field component that propagates with velocity h. The A bi j Žh. is the phase corrected sum of nonlocal bulk polarizabilities a i,b jqs . A similar condition for the field component k reveals that deep in the bulk the induced and incident
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field have to cancel exactly - the familiar Ewald– Oseen extinction theorem. This yields two equations, one for each polarization direction t. Additional conditions can be generated similarly for the other evanescent field components E n . Together with the self-consistency condition for the surface dipoles at r is this yields a manageable set of interaction equations,
d i j 1 y A si k Fksjs
yA si k Fksbm
yt P Fnbjs
yt P Fnbmb
P
A si j E ext pj j s ext nm t PEn
Ž 14 . The solution determines the strengths of the surface dipoles and normal modes, p i Ž v ., nnŽ v .4 . The local fields have to be evaluated Žthrough the various tensors F. for all characteristic dipoles in the surface, but also for a finite range of the neighboring normal mode region. This range corresponds to what extent the polarizability matrix elements A si j s Ýs I expŽ ik P s I. a isjqs I , affect this region. With the polarization distribution now fully determined, we obtain the reflected field in the yn ˆ direction w8,9x. The GaAs bulk and surface polarizabilities have been obtained from single particle energies and wave functions using accurate periodic DFT-LDA calculations w11,12x. In the bulk calculation we used a ˚ The surface properties lattice constant a s 5.613 A. have been obtained from a single asymmetric slab containing 16 atomic layers. The surface reconstruction has been taken from w13x and the backside has been made bulk terminated by adding 4 frozen core atomic layers ŽGa and As. and 2H termination layers w15x. Two additional frozen-ion layers were added to model the bulk Madelung potential. We used the scissors operator to fix the bulkŽ-like. band-gap to 1.52 eV. The cut-off range for the nonlocal polariz˚ resulting in eight shells of ability was set to 8 A, neighboring atoms. The value of g Ž v . in Eq. Ž10. was found to be nearly constant and real, g Ž v . s 0.02, confirming that it builds only a minor correction to the Lorentz field. ŽSee Fig. 1.. The first step in our calculations has been to obtain a value for the bulk dielectric constant. We used Clausius Mossotti and a to obtain the bulk dielectric constant. Compared e.g. to other calcula-
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P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
Fig. 1. Calculated and experimental dielectric constant of GaAs. Parameters: EGa p s1.52 eV, f s 0.98.
tions w14x similar agreement with experiment is found, but the spectra are somewhat more smeared out. This has to be expected, since our nonlocal calculation explicitly accounts for the Žsmall. photon wave vector. The E1 and E2-type transitions are well resolved, have the right magnitude and are at the correct energetic location after application of the scissors operator. The normal mode wave number qm agrees in all cases within 0.1 % with the Fresnel value. In summary this indicates that the nonlocal character of the polarizability hardly influences the average bulk optical properties, as expected. In doing the normal mode calculations it turned out that the usual Fresnel resembling modes were insufficient. Until 2.6 eV we found for each Fresnel mode exactly one extra normal mode which did not decay rapidly enough. We have examined the behavior of the extra Žnon-Fresnel. normal modes in some detail. They start at very low frequencies already, but are too much damped to play a role, but at 1.1. eV they become perfectly transparent and have to be taken into account explicitly. Meanwhile the real part of the qm moves from a value of prd towards zero. One extra normal mode happens to be purely transverse Žu i m P k m s 0., the other has also a longitudinal component Žu i m P k m / 0.. All extra normal modes start to become increasingly absorbing beyond 1.5 eV Žcoinciding with the bulk band gap., and their influence can be discarded after 2.6 eV. In Fig. 2 we show the individual dipole strengths at " v s 1.9 eV for the external field at normal
incidence and polarized parallel to the Ž110. direction. The dipole strength varies clearly going from Ga to As sites, as expected. Matching between surface and normal mode region Žarrow in Fig. 2. is perfect. Neglect of the extra modes destroys this matching. Dipole strengths for the same type of atom show a modulation also for the normal mode region. This is the influence of the extra normal modes. The dipole strength for the surface dipoles is substantially smaller than for the bulk. This is a general trend and has been observed also for other surfaces w8,9x. Accurate reflectance anisotropy spectra have been measured and studied recently w3,19x. We have calculated the reflectance anisotropy for GaAs Ž110., using the previously outlined discrete cellular method. Due to the limited slab thickness, we used bulk values for a i j Ž v . throughout the system except for i, j both top-layer atoms. The convention we have used to define the anisotropic difference is
D RrR s 2 Ž R w110x y R w001x . r Ž R w110x q R w001x . .
Ž 15 .
Theoretical results are obtained and shown in Fig. 3. Realistic results can only be obtained when dipole strengths match perfectly from surface to normal mode region. Curve A shows the theoretical nonlocal RAS for the reconstructed surface. Curve B equals A but with the RAS of a GaAs bulk terminated system subtracted. Although both curves are in close agreement with the experimental observations, the agreement is best for curve B. Difference between A and B is mainly due to the anisotropy of the bulk. In view of the experimental evidence for bulk spatial dispersion, also for GaAs along the Ž110. direction
Fig. 2. Modulus
P.L. de Boeij, C.M.J. Wijersr Physics Letters A 272 (2000) 264–270
Fig. 3. Reflectance Anisotropy Spectra of GaAsŽ110.. Experiment: w3x, A: nonlocal, B: Aybackground, C: as B, but reduced to local.
w16x, and its theoretical justification w17x, the influence of a bulk contribution to the optical anisotropy cannot be ignored. For this reason we have added curve B. For the same reason we strongly recommend that the experimental calibration procedures are carefully reexamined as to this point w18x. All calculated spectra have the right sign and order of magnitude. Also the spectral structure is remarkably well reproduced apart from a rigid energy shift of 0.3 eV. This holds in particular for curve B, where only the small shoulder at 3.0 eV is missing. Therefore we conclude that the RAS signal finds its basic origin in the surface atoms, meaning that the spectral structure vanishes mostly if these atoms are left out. However one has to be aware, that within a Žtwofold!. nonlocal calculation, no clear distinction can be made anymore between bulk and surface. The shift indicates that the scissors operation may have to be different for surface and bulk states as noted by others w19x. Finally curve C shows the result when the nonlocal polarizabilities are made effectively local, according to the substitution a i j Ž v . s d i j Ý k a i k Ž v .. Result C shows clearly that surface optical anisotropy depends strongly on the nonlocal character of the polarizabilities, opposite to the behavior of the bulk optical properties. The discrete cellular method, as we prefer to call the method given in this Letter contains very few
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adjustable parameters, all of them Žscissors energy shift, g i-term. related to bulk properties. The method is intrinsically connected to a high quality ab initio electronic structure method ŽADF-BAND. and enhances the results of these methods for new fields of application. This holds particularly for the description of the optical properties of strongly inhomogeneous systems, such as surfaces, and there particularly for the correct determination of the intensity of the response, rather than an accurate determination of the position of the energy levels. As such it builds a major improvement beyond what traditional discrete dipole models that can take only effects of electro-magnetic nonlocality into account, are able to achieve. The addition of the influence of quantum mechanical nonlocality through nonlocal polarizabilities has been shown to be vital for the case of semiconductor surfaces. We suggest that a merger of this method and methods to correct for the proper location of energy levels, such as GW or Bethe– Salpeter, might result in truly outstanding results for the optical properties of inhomogeneous systems.
Acknowledgements We wish to thank G. te Velde and E.J. Baerends for letting us use their ADF-BAND w11,12x code.
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