Theoretical study of ultraviolet photoemission spectra of noble metals

157

Journal of Elecfron Spectroscopyand RelatedPhenomena, 68 (1994) 157-166 036%2048/94/$07.00 @ 1994 - Elsevier Science B.V. All rightsreserved

Theoretical

E.E. Krasovskii,

study of ultraviolet

A.N. Yaresko,

photoemission

spectra

of noble metals

V.N. Antonov

Institute of Metal Physics, Academy of Sciences 36 Vernadsky blvd, Kiev 252142, Uktaine

of Ukraine,

The formalism of a new version of the LAPW method is presented. The new method (ELAPW) is shown to yield accurate eigenenergies and wavefunctions in the energy interval up to 4Ry above the Fermi level. A semi-relativistic vetsion of ELAPW is used to calculate the band structure of Cu and Ag. The band sttucture of Au is calculated by the fully relativistic LMTO. The ab inifio band structure is transformed so as to generate the measured optical spectra. The model band structure is used to calculate the ultraviolet photoemission spectra of noble metals.

1. INTRODUCTION The computational methods of band theory have been widely used to explain the spectral ptoperties of crystals in terms of transitions between the one-electron states. In early studies the KKR and APW methods were used [1,2]. These methods have the advantage that they provide an arbitrarily high accuracy of one-electron eigenenergies and wavefunctions. However, these methods ate inconvenient to use because they are time-consuming. Besides, both of the methods employ energy-dependent basis functions, which causes considerable in calculation of the momentum difficulties matrix elements (MME). That is why the so-called linear methods (LMTO and LAPW) are more popular (see Ref.3 for LMTO calculations, Refs.4,5 for relativistic LMTO (RLMTO), and Refs.6,7 for LAPW). Linear methods employ fixed basis i.e., basis functions do not depend upon the ( energy of electron state to be calculated), which makes them suitable for the calculation of MME. The shortcoming of linear methods is that they yield accurate eigenenergies only in a finite energy region about 1Ry in width [8]. Moreover, even for the states whose energies

SSDIO368-2048(94)02113-E

ate close to the energy parameters of method the accuracy of well-converged wavefunctions is questionable [9]. In the present wotk we use the new version of LAPW. The new method employs an extended basis set, which markedly improves its accuracy. For noble metals it yields eigenenergies accurate to within SmRy in the energy range from the bottom of valence band to 4Ry above the Fermi level. In Sect.2 we describe the method and discuss its accuracy. We have used this method to perform the semi-telativistic calculations for Cu and Ag. As the spinorbit coupling is most important for Alc, the RLMTO have been used in this case. In the present work we use the local density approximation (LDA) with exchange-correlation potential proposed by Hedin and Lundqvist [lo]. In Sect. 3 we demonstrate that the ab initio calculations based on this form of one-electron potential are incapable of reproducing the optical properties of noble metals. The sources of errors are: (i) the application of ground-state one-electron theory to excited states is questionable by itself (i.e., the excitation of many-body system is treated as the transition of an electron frbm the state in valence band to unoccupied state, both of which ate calculated using the same potential), and

158

(ii) the LDA overestimates the energies of localized d-states owing to the self-interaction being not excluded [Ill. We consider (ii) to be most important. In this study we switch from the a6 initio band structure to a model one, which is adjusted so as to generate the measured optical spectra. In Sect.3 we describe the fitting procedure and,demonstrate that the model calculations are in excellent agreement with experiment for all the three metals. In Sect.4 we use the model band structures to calculate the photoeIectron energy distribution curves, The theoretical UP spectra are compared with experiment. 2. THE EXTENDED

LAPW

METHOD

In the LAPW method originally proposed by Andersen [8] the wavefunction for a given Bloch vector k is represented as a sum of en; augmented plane waves ergy-independent (APW’s)

ytiPW(t-) where

= &(G,)y/(k n

+ G,,r),

G, are the reciprocal

The C,+(G,,) are obtained

lattice

variationally.

tive at this energy,

4“1,

are used to construct

the radial derivative

with

a proper

~l(D,4 =#VI+ e%h

s(&J) @l, 4

u;(&S)

= j,@,J), r) = &Qr(&

is the radial

r),

= j;(Q),

i$ E &%,/ &

logarithmic

derivative

(4)

Thus the LAPW basis set is uniquely mined by a set of energy parameters,{&}.

deter-

To improve the representation of the partial waves of a given angular momentum I, we introduce a set of radial functions which vanish in the interstitial region (zl;(r); i=1,2,...M,} and use them to construct the additional basis set. The additional basis function is written as G&J = z&-)%#). (5) For the trial function to retain the continuity in value and slope at the sphere boundary, we requite that zii(r) have both zero value and zero slope at the sphere: zi;(s) = cl.

Z/j(S) = 0,

(6) To obtain a function of this shape one may use a linear combination of an arbitrary function hi(r) and radial functions defined by Eq.(4).

vector-s.

(7)

h. is the

Here Dj is the logarithmic derivative of hi(r) defined as in Eq.(3). The natural choice of additional basis set is to set M,=2 and to define frl=#Gcrl, fiz = Jp,; solutions

satisfy the boundary

logarithmic

(1)

band number. The energy-independent APW is continuous and differentiable at the muffin-tin sphere radius. Inside the sphere the APW is written as

K,=k+G,. The radial functions conditions at the sphere

function

(24 matching

the $.,

for energy

$,

and

$d

are

and its energy

radial deriva-

tive respectively. Using the extended basis set (which now contains 2{21+1) more functions) we obtain the trial function of extended LAPW method (ELAPW)

(3) (3a)

(8)

of the

Bessel function

aim and b,, are new variational Consider

Q = S$#[,_s, (S is the sphere radius) (3b) I The solution of the radial Schrodinger equation for energy Eyl, q&, and its energy deriva-

two

sets

of

coefficients.

energy

parameters,

in which for all but one anguG,J and {%}, lar momenta the parameters are the same for

159

both

(E,,I = EL/ for

sets

M,,

Evlo f E&).

Using the two sets of energies we can construct two LAPW basis sets (see Eqs.(l-4)). Let N be the number of basis functions in each set. If we pool the two sets to form one set with 2N functions, and exclude all the linear dependent functions, we obtain exactly the same basis set as in Eq.(8) (i.e., only N+2(21,+1) functions are independent}. The two basis sets described above may be used to perform the so-called panelling; i.e., for example, with 1,=2, we can choose the Ev2 to be at the center

of Jd-band

and E,,

to be

than

In the region up to 3Ry above is less then 1mRy (see Table lb). In spite of the large basis set, the usual LAPW yields precise eigenvalues only in the interval up to 0.5Ry above E, (see Table la). The easiest way to test the accuracy of

Table la. Comparison of the eigenenergies (in mRy) and velocities (in a-u.) calculated by LAPW and ELAPW. The APW energies are not listed because they do not differ from ELAPW ones.

at the center of 4d-band. This results in two independent calculations, each involving N APW’s. It should be noted that the results of the panelling are not equivalent to those of ELAPW method, as the independent calculations cannot provide the orthogonality of the wavefunctions in different panels. This leads to the eigenenergies in the upper panel being underestimated. Moreover, we will show that with additional parameters Epr several Ryd-

h

Evr, the

bergs

above

the

LAPW

parameters,

extension of the basis affects substantially the shape of the wavefunctions in the “lower panel”. To examine the accuracy of ELAPW method we use CU metal as an example. We calculate eigenenergies for 25 bands at k-point (2x/a) (0.75, 0.75, 0.) using three methods: Slater’s APW, LAPW, and ELAPW. The number of APW’s is the same for all the

0.2mRy.

EF the difference

Eh

EL

v, grad

v,

MME v,

ELAPW

MME

LAPW

ELAPW

LAPW

2 3 4 5 6 7 8

276 300 398 446 477 925 1018 1166

276 300 398 446 477 927 1023

-0.0811 -0.0374 0.1092 0.0525 0.0310 -0.3574 0.6239

1177

9

1711

1726

-0.1083 -0.1082 -1.8483 -0.3165 -0.3164 -0.3138

1

-0.0811 -0.0374 0.1092 0.0525 0.0306 -0.3572 0.6237

-0.0812 -0.0459 0.1092 0.0525 0.0302 -0.3566 0.6096

Table lb. Accuracy of the energies and diagonal MME calculated by ELAPW. The LAPW values are not listed because they are highly inaccurate.

APW

ELAPW

grad

ELAPW

10

2019

2018

-0.1027

-0.1027

11

2715

2715

-1.0317

-1.0313

< 14.6

12

3282

3282

0.5132

0.5141

are included. The number of basis functions in APW and LAPW calculations is 296. Apart from these, the ELAPW basis includes 32 additional functions for 1 5 3.The energy parameters E,, used, in both LAPW and ELAPW cal-

13

3364

3364

14

3557

3556

0.6772 0.6641

15

3675

3676

0.5368

0.5352

16

4220

4222

0.4988

0.4936

methods.

All the APW’s for which SlQl

culations, (0.407Ry) rameters

are taken at the center of 3d-band for all 1. The additional energy paof are EP,=7.2Ry, ELAPW EPl=7.6Ry, EP,=5.4Ry, and Ep,=ll.ORy. In the energy interval from the bottom of valence band (-0.094Ry) to about 1Ry above the Fermi energy (E,=0.606Ry), the difference between, the APW and ELAPW values is less

0.6771 0.6664

17

4281

4283

0.5861

0.5861

18

4657

4659

-0.3250

-0.3189

19 20 21

4801 4813 5051

4803 4823 5053

-0.3728 -0.9515

22

5068

5080

-0.7040 -0.1719

-0.4781 -0.9819 -0.7246 -0.1724

23

5134

5141

-0.6514

-0.6460

24 25

5157 5418

5170 5443

-0.6688 -0.7902

-0.6596 -0.7642

6

-8

-6

-4

-2

0

E-EF (eV) Figure 1. Density of states (DOS) curves for Cu, Ag, and Au. ab irtitio DOS VW_-_ model DOS on which the further calculations are based

15

IO

5

20

25

Photon energy (eV) Figure 2. Imaginary

pat-t of DF, E& )

spectrum spectrum based on the model DOS experiment [13]

~6 it~itio

____0000000

161

wavefunctions is to compare the electron velocity, calculated as a diagonal momentum matrix element

with the gradient

value

calculated

14ca jrgrad= -h

dk’

as

the

energy

(10)

obtained by numerical differentiating. The results, presented in Table la and lb, show that the ELAPW wavefunctions are much more accurate than LAPW ones, and are reliable up to 4Ry above EF. 3. OPTICAL

PROPERTIES

The band structure of noble metals was calculated self-consistently using the ELAPW method fot Cu and Ag and RLMTO method for Au. The ab initio density-of-states curves (DOS’s) are presented in Fig.1 (full lines). The ab initio spectra of the imaginary part of the dielectric function, E*(O), obtained in the framework of self-consistent-field approach [12] are shown in Fig.2 (full curves). The results of Sect.2 suggest that the energies and wavefunctions calculated by the ELAPW method are accurate enough to yield optical spectra of desired accuracy over a wide range of photon energies. This enables us to attribute the disagreement between theory and experiment to the inadequacy of the oneelectron approximation. The comparison with experiment of Ref.13 (open circles in Fig.2) indicates that for all noble metals the calculated optical absorption edge occurs a few tenths of an eV lower than in experiment. In the theoretical curves some maxima are also shifted toward lower enetgies. All the structures poorly reproduced by theory are due to transitions from the localized states in the d-band. In our calculations of the optical properties of transition metals and their compounds [6,7] we did not observe any discrepancies of this sort. This leads us to the assumption that it is the form of the ground-

state one-electron potential which is responsible for the discrepancies, as it yields wrong energies of the localized d-states. Although this downward displacement of the peaks in the curves does not prevent the interpretation of the optical spectra, for noble metals the one-electron approach will lead to photoemission spectra which could hardly be used to interpret the experimental spectra. In the present work our study of the photoemission properties is based on the model band structure which is derived from the ab initio one using the following fitting procedure. The purpose is to shift d-states downwards by AE from their calculated positions (AE is the difference between the theoretical and measured location of absorption edge) and at the same time to leave the delocalized states unchanged. To make the distortion continuous we used the following transformation of the energies in the valence band E’=E-A&f(E)

f(E) = 1,

forEb


, for Et < E c EF

(11)

, for Er < E < Eb Hete EF is the Fermi

energy;

E, is the bottom

of valence band (point r); E, is just above the top of d-band (point X); Eb is fitting parameter used to separate the energy region in which the energies are to be shifted. The unoccupied states are left unchanged. The parameters are

Table 2. The parameters (energies in eV, JZ~=O). J%

of the fitting

function.

%

E,

,AE

CU

-9.53

-1.54

-1.50

0.45

A8

-7.90

-6.15

-2.75

1.20

Au

-10.19

-5.50

-2.00

0.50

162

0 x10'14

(l/set)

cu

cu 0.4

30

11

tr

20

2

0.2

1

3

11 --v

I

4

5

10

CD10-14( l/SK)

,

60 I

I*’

I

I-

I-

43

I\ I

t

\ \

0.4

\

!;s-:@ I

40

I

‘.J

I

\

\

I

\

I

\

_

I

-

N

I

.-

I

I

20

I

I

0.2

I

/\

‘\

\

/’

23

45

80 60 2.0

20

30

10

40

Photon energy (eV) Figure 3. Conductivity

(0)

and reflectivity

(R)

20

2.5

30

3.0

40

Photon energy (eV) ( _I__

experiment

[13] , -theory)

163

listed in Table 2. The DOS’s recalculated from the model band structure are shown in Fig.1 (dashed curves). The corresponding ~~(0) curves are shown in Fig.2 (dashed curves). the are conductivity and reflectivity spectra shown in Fig.3. The reflectivity curves are in better agreement with experiment because it is reflectivity that is actually measured in Ref. 13, by the experimental E? curve being obtained the Garners-Kronig analysis. 4. PHOTOEMISSION

SPECTRA

The photoemission energy distribution curves (EDC’s) for Cu are compared with experiment in Figs.4 and 5. Although the effects of electron transport and escape factor are neglected in the present calculations, our curves are in better agreement with experiment than those of Janak et al [I]. The intensity of the peak at -2eV caused by the transitions from the top of d-band is underestimated in their spectra, which may be due to inaccu-

rate treatment of electron transport and escape. It should also be noted that our treatment

of the band

structure

of Cu is different

CU

Ao = 5.6eV

-4

-3

-2

-1

Initial energy

0

(ev)

Figure 4. Comparison of the experimental (Berglund and Spicet [14]) and theoretical EDC’s for photon energies 4.7eV and 5.6eV.

-8

-6

4

-2

0

Initial energy (eV) Figure S. Comparison of the experimental and theoretical EDC’s for photon energies 10.2eV (experiment by Krolikowski and Spicer [15]) and 21.2eV (a: Shen eb al 1161; b: Evans [17]). from that of Janak et al in that they attach much importance to self-energy corrections (which leads to incteasing the energies of excited states, and are neglected in out study). At the same time (contrary to our approach) it is assumed in Ref.1 that using the X, potential with a-0.77 the d-states can be treated in the same way as s-p-states. However, there is one more uncertainty in comparing our tesults with those of Ref.1 in that there were made no investigations as to the accuracy of MME calculated by KKR method. To illustrate the effect of MME on the photoemission distribution we have calculated the UP spectra for Cu at kF40eV both with MME included and with constant MME. From Fig.6 we notice that the constant MME approximation overestimates the intensity of the peak at -3.2eV so that the curve resembles the DOS curve. Fig.6 thus indicates that the crystal momentum conservation rule by itself may not lead to strong deviations of UPS from the DOS curve, and that the effect of MME on the transitions from the d-band is significant.

164

----

UPS

-

TJPS

- --

II I

-10

The EDC’s for Au are presented in Fig.9. They are in a good agreement with experiment [19] and describe the change in relative intensities of peaks with photon energy. Of all the noble metals the case of Ag is most difficult. This is because the LDA band structure is not as good a starting point for Ag as it is for Cu and Au. The errors in energies ate too large (l.PeV compared to -0.5eV in Cu or Ag) to be corrected adequately by a simple fitting procedure.

*

DOS

II

-8”

.,/:

e

1

-6

jl I

;

,’

-4

‘...

I

I:

I

1

-2

0

Initial energy (eV) Figure 6. Effect of momentum matrix on EDC for copper for bv=40eV.

We wish to thank 0. Slobodyan programming assistance. We are indebted Prof. Yu.N. Kucherenko and Dr. A. Perlov useful discussions.

for to for

REFERENCES elements

The UP spectra for Ag are presented in Figs.7 and 8. For ail photon energies theory predicts a maximum at about -6eV, a wide gap between -6 and -SeV, and a set of peaks from 5eV to -3.8eV. The overall agreement with measured spectra [18] is satisfactory, but the discrepancies are considerable. For bvlZOeV the lowermost structure in measured spectra is shifted 0.5eV leftwards, whereas for bvX?leV its position is the same as in theory. It seems likely that in the present model the intensity of transitions from the bottom of d-band (a weak maximum at -7eV) is underestimated, and that in the experiment the superposition of the two structures (at -7eV and at -6.2eV) is observed. In the experiment the gap at -6eV is narrower, and the peaks appear from -5.5 to -3.8eV. At the energies hv>24 there is a disagreement in the intensity of the uppermost peak (at -4eV); the structure is not manifested in the measured curves at all. It is hardly possible to ascribe the discrepancy to the inadequacy of the model band structure at the top of d-band, as the states involved are responsible for the formation of the optical spectrum near the absorption edge, which is in accord with experiment. It is possible that in this case the effects of transport and escape are significant.

1. J.F. Janak af aZ, Phys.Rev. B, ll(1975) 1522. 2. J. Yamashita et al, J.Phys.Soc.Jpn., 42(1977) 1906. 3. E.G. Maksimov ef al, J.Phys.F, 18(1988) 883. et al, Phys. Metals, 4. V.V. Nemoshkaienko lO(1990) 191. 5. V.N. Antonov et al, Fiz.Nizk.Temp. to be published (1993). et al, Phys.Metals, 8(1990) 6. E.E. Krasovskii 882. Proc. of 7. E.E. Krasovskii and V.N. Antonov, the CP90 Europhys Conf. on Computational Physics, Tenner (ed.) p.400. Singapore; World Scientific (1991). 8. O.K. Andersen, Phys.Rev. B, lZ(1975) 3060. 9. E.E. Krasovskii, V.V. Nemoshkalenko, V.N. Antonov, Z.Phys. B, 91(1993) 463. 10.L. Hedin and B-1. Lundqvist, J.Phys.C, 4(1971) 2064. 11.A. Williams and U. Barth in Theory of the inhomogenius electron gas, S. Lundqvist and N.H. March (eds.), Plenum press (1983). 12.B. Ehrenreich and M.A. Cohen, Phys.Rev, 115 (1959) 786. 13. J.H. Weaver et al, Optical properties of 18-2 metals. Physics Data No (Fachinformation Zentrum. Energie-Physik. Mathematic Katlsruhe 1981). 14. C.N. Berglund and W.E. Spicer, Phys.Rev. 136(1964) A1044.

\ c 0

PA

16.5

\

/

ho =25eV

/

--

-

-\

/

=24eV

Ii0

,w

-]il.

.

1 Ao =23eV

t ---

h

/

/

flk

Aa =3OeV

h--j

f’/

\ t i

I

A\/

1 \

Ro =21eV

AU =28eV

-

hw

=ZOeV

AA

tic0 =27eV

A

fio =1111

-6

-4

Initial energy

-2

0

(eV)

Figure 7. Photoemitted electron energy distribution curves for Ag for photon energies from 15 to 25 eV . theory _c--- - experiment [18].

-6

-4

Initial energy

-2

0

(eV)

Figure 8. Photoemitted electron energy distribution curves for Ag for !::I; energies from 26 to 40 eV. ____I_ experiment [MS].

15. W.F. Krolikowski Rev. 185(1969)

and

W.E. Spicet,

Phys.

882.

16.W. Shen et al, J.Phys., 4(1992) 6587. 17,s. Evans, J.Chem.Soc. Faraday Trans II, 71(1975) 1044. 18.C. Norris and G.P. Williams, J.Phys.F, 6(1976) 1167. 19.D.E. Eastman and W.D. Grobman, Phys.Rev. Lett., 28(1972) 1327.

-8

-6

-4

Initial energy

-2

0

(eV)

Figure 9. Comparison of the experimental Eastman and Grobman [19] and theoretical EDC’s for Au for photon energies between 15 and 40eV.