dielectric function eels optic solid

Optical Constants of Solids by Electron Spectroscopy

J. DANIELS, C. v. FESTENBERG, H. RAETHER and K. ZEPPENFELD

Contents

1. I n t r o d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2. T h e o r e t i c a l R e m a r k s a n d D e s c r i p t i o n of the M e t h o d . . . . . . . . . . . . . 80

2.1 V o l u m e E x c i t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . . . 80

2.2 F u r t h e r I m p o r t a n t F a c t o r s . . . . . . . . . . . . . . . . . . . . . . 81 2.2.1 Sur face E x c i t a t i o n s . . . . . . . . . . . . . . . . . . . . . . . 81 2.2.2 R a d i a t i o n Losses . . . . . . . . . . . . . . . . . . . . . . . . 83

2.3 E v a l u a t i o n o f the E x p e r i m e n t a l D a t a . . . . . . . . . . . . . . . . . . 86

2.3.1 E x p e r i m e n t a l T e c h n i q u e . . . . . . . . . . . . . . . . . . . . . 86 2.3.2 D e t e r m i n a t i o n of the Loss F u n c t i o n f r o m the O b s e r v e d Q u a n t i t i e s . . 88

2.3.3 C o r r e c t i o n for D o u b l e Losses . . . . . . . . . . . . . . . . . . 90 2.3.4 A c c u r a c y o f the E x p e r i m e n t a l D a t a . . . . . . . . . . . . . . . . 92

2.4 K r a m e r s - K r o n i g Ana lys i s ( K K A ) . . . . . . . . . . . . . . . . . . . 93 2.4.1 G e n e r a l D e s c r i p t i o n . . . . . . . . . . . . . . . . . . . . . . 93

2.4.2 N u m e r i c a l E v a l u a t i o n o f the K K - i n t e g r a l . . . . . . . . . . . . . 95 2.4.3 D e t e r m i n a t i o n o f the O p t i c a l C o n s t a n t s . . . . . . . . . . . . . . 96

3. Resul t s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1 Me ta l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1.1 M o d e l s . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.1.2 E n e r g y Loss F u n c t i o n s of Pd, Pt, Cu , Ag, A u . . . . . . . . . . . . . 100

3.1.3 T h e O p t i c a l C o n s t a n t s of Pd, Pt, Cu, Ag, A u . . . . . . . . . . . . 102

3.1.4 D i s c u s s i o n of the C o n s i s t e n c y o f the Resu l t s . . . . . . . . . . . . 106

3.2 I n s u l a t o r s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.2.1 Sol id X e n o n . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.2.2 K B r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.3 D i a m o n d . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

3.3 I I I /V C o m p o u n d s . . . . . . . . . . . . . . . . . . . . . . . . . . 116

3.4 A n i s o t r o p i c C r y s t a l s . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4.1 D e s c r i p t i o n o f the M e t h o d . . . . . . . . . . . . . . . . . . . . 122 3.4.2 G r a p h i t e . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

3.4.3 M o S 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

4. C o n c l u s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 2

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

78 J. Daniels et al.:

1. Introduction

During the past few years considerable progress has been made in understanding the optical properties of solids by correlating them to their band structure. The calculation of the imaginary part of the dielectric constant ez(e) ) is of special interest since structure in e2(e)) corresponds to critical points in the band transitions. Observation of such peaks allows comparison of experimental data with theoretical results.

The dielectric constant e(e)) is determined from light reflection measurements at well-prepared surfaces over a wide region of wave lengths. From these measurements one can extract the dielectric constant by means of the Kramers-Kronig Analysis (KKA). Also other direct optical methods avoiding the KKA are used [30]. The optical methods, which are in general very accurate, may run into difficulties: the reflectivity is very sensitive to surface conditions especially in the far- ultraviolet region. Therefore existing optical determinations of ~;2(e)) are in some cases not always very precise.

For this and other reasons we decided to study the possibility of determining the optical constants from energy loss measurements of electrons in thin crystal films over a wide range of photon energies (1 to ~ 100 eV). This range covers the interesting region of electronic absorption. The properties of bulk material are determined by measuring the transmission of electron beams through thin crystal films. The method, its physics, and some results are described in the following article. The procedure is described in detail in Chapter 2. It is based on the fact that the energy loss spectrum of electrons having passed a thin crystal foil can be quantitatively described by the dielectric theory [ 12, 39, 56]. The intensity of electrons which have suffered an energy loss A E and a momentum transfer hq in a single collision is given by the energy loss

1 1 function ~ - Im e ( ~ - ( - Im 1/5 = e2/JeJ2), in contrast the optical ab-

sorption depends on e2(e)). This loss function can be used for obtaining the dielectric constant of an isotropic crystal.

The incoming electron transfers besides the momentum hq the energy A E to the crystal electrons; this corresponds in an optical process to the absorbed photon energy he). The field acting on the crystal electrons can be represented by a longitudinal electric field of direction q. The transverse electric field of light moves the crystal electrons as a whole perpendicular to its direction of propagation without changing their density. However the longitudinal field produced by an electron beam induces density changes of the crystal electrons in the q direction. This is the reason why the interaction of electrons with the crystal electrons becomes proportional to ~2/1~[ 2.

Optical Constants of Solids by Electron Spectroscopy 79

The loss function considers only the excitation of the crystal electrons in the volume of the crystal and is rather simple to evaluate. Surface excitations, always present, and the production of Cerenkov photons which are of importance in substances of high dielectric constants in certain frequency regions e.g. in the III/V compounds, complicate the situation. However by measuring the energy loss spectrum not in the direction of the primary beam (~ = 0) but under certain scattering angles

these two effects can be neglected and the above relationship is re- established.

The preparation of thin films does not allow in general the use of thicknesses small enough to guarantee the condition of a single energy loss. Corrections for double inelastic collisions however can be readily applied. In addition, the finite energy width and the finite angular distribution of the primary electron beam have to be considered. The experimental technique has made sufficient progress to verify that the energy loss spectrum is reproducible in its very details, so that the loss function is available over a wide energy range A E with good accuracy. The values ofe 1 and ~2 can be derived by a KKA from this loss function.

The dielectric constant obtained from these electron measurements represents the response to a longitudinal field (elong), whereas the "optical" dielectric constant represents the displacement under the action of a transverse field (et .... ). Detailed experiments have been made to compare the energy loss function ( - Im 1/~)e I with that calculated from the known optical constants ( - Im 1/e)opt. Since one has found in general agreement in the detailed structure of the two loss functions, no experimental evidence exists to date for a difference between elong and 6 t . . . . "

If the crystal is optically anisotropic the dielectric constant becomes a tensor (6i~) so that the loss function becomes different from that given above. In the case of an uniaxial crystal with e(co, O) = 6 • sin 2 0 + e tl cos 20 one obtains for the loss function - 1/q 2. Im 1/e(co, O) where O means the angle between q and the c-axis of the crystal, e II and 6 • are the principal values of the dielectric tensor parallel and perpendicular to the c-axis, respectively. Because of the longitudinal type of excitation by electrons, the q-direction corresponds to the polarization direction of the electric field vector in optical experiments. The angle O can be varied, e.g. by turning the crystal, and the values of 6 tr and 6 • independently obtained. Thus one can measure optical constants in cases where the optical measurements encounter difficulties. For example the application of the energy loss method to layer compounds such as graphite and MoS2 avoids the difficulties inherent to optical experiments: good surfaces can only be obtained by cleaving these crystals parallel to the hexagonal layers.


In Chapter 3 of this article one finds examples of loss functions, el(oJ), e2(co), reflection and absorption spectra of different crystals obtained by the procedures described above. The substances under consideration are noble metals (Ag, Cu, An) and some transition metals (Pd, Pt), insulators (KBr, solid Xe, diamond) and semiconductors such as III/V compounds. While the substances discussed are optically isotropic we present also the spectra of some anisotropic materials such as graphite and M o S 2 . This chapter demonstrates the possibility of using electron energy loss spectroscopy to determine optical constants.

2. Theoretical Remarks and Description of the Method

In this chapter we describe the method to derive the loss function from the energy loss spectrum and the underlying physical assumptions.

2.1 Volume Excitations

A fast electron with the energy Eo and the momentum hk o penetrates a solid film of thickness D and may loose in one inelastic process the energy A E (A E ~ E0) and transfer the momentum hq (Ihq[ ~ Ihkol) to the crystal. The energy loss probability PI(A E, ,9)dA E dO can be described by the dielectric theory [35, 27] :

PI(AE, 9)dAEdY~= Im - ~ - D , 9 2 + 0 2 d A E d O . (1)

Here v means the velocity of the incoming electron scattered by an angle 8 (0 = 1~1) into the infinitesimal solid angle dr2 and OaE = e)/kov = A E/pv, p=hko relativistic momentum of the electron, e=electron charge, h = Planck constant (see also Fig. 1). Neglecting the relativistic correction, pv becomes 2Eo. The momentum hq transferred by the electron is given by:

q2 = q~ + q~ = kg(o~ + 0~) (2)

with [q• = [kol O and [qll] = [kol ,.,a~. (3)

Eq. (1) shows two important features: P1 decreases with O as 0 -2, if 0 > 0a~. Oat is in general a small quantity: assuming AE= 10eV and E o = 50 keV, one has OaF = 0.1 mrad. Further P1 is proportional to the energy loss function - I m 1/e(AE), which describes the individual


character of the crystal given by the quantities e 1 (A E) and e2(A E). The energy loss function - I m 1/~ has a strong maximum due to the excitation of the volume plasmon he)p, if el ~ 0 and E 2 is small. Further peaks will appear at those regions of A E where ~z has maxima which are modified however by dividing e2 by le[2: band transitions and excitons.

Y _

.L

Fig. 1. Inelastic collision of an electron in the foil of thickness D. Eo and E 0 - A E are the energies of the electron before and after the collision, hk o the momentum of the incident electron. The momentum hq is transferred to the crystal

There are a number of effects which disturb the simple relation between P1 and - Im 1/e given by Eq. (1). Their influence is discussed in the following sections.

2.2 Further Important Factors

2.2.1 Surface Excitations

Besides the energy losses in the volume of the solid foil (see Eq. (1)), the electrons suffer additional energy losses at the boundaries of the foils, caused by surface excitations. Restricting the consideration to a foil in vacuum, the surface contribution which has to be added to Eq. (1), can be described by [57]

with

2O d AE d ~ (4)

R = sin2 d~e cos 2 dne + (5) }

6 Springer Tracts Modern Physics 54

82 J. Dan ie l s et al.:

and D dE

d"E-- 2 by"

The factor R contains the coupling of the two boundaries of the foil. This effect can be neglected if

0 w dA E J ~9,tE

In this case R becomes

Iqb .D >> 1. (6)

2

1 R - ~ + 1 (7)

Eq. (7) is valid for thicknesses D > 500 • under the experimental conditions described in Sec. 2.3.

In the case of a free electron gas el = - 1 and ez small represents the surface mode co S = c%/1/~ on the boundary of a rather thick film.

0.1 \ \ X X ~ x

I I I I I I i ' ~ ' ~ 0'010 0.5 mrad 1.0

Fig. 2. Dependence of the intensity of the scattering angle 0 calculated for A E = 6 eV, demonstrat ing that the total loss intensity II ~ approaches the volume intensity I~ ~ very rapidly. Surface- and volume losses are assumed to have the same intensity at g = 0

It exists an important difference in the angular dependence between volume and surface losses. While the volume loss intensity decreases with 9-z , the surface loss intensity decreases with 9-a . Therefore the energy loss probability observed in large scattering angles is only due to volume excitations as is shown in Fig. 2. Here the angular dependence of the total


loss intensity (volume and surface) is plotted for the case that the surface and volume losses have the same intensity in the forward direction (,9 = 0). Both intensities (Eqs. (1), (4), and (7)) are convoluted with the angular resolution of the spectrometer described in Sec. 2.3. Fig. 2 shows that the contribution of surface losses to the total energy loss probability has decreased to 10% for an angle ,9=0.5mrad and to 2% for ,9 = 0.9 mrad. The situation is more favourable in the experiments where the contribution of surface losses to the energy loss probability is generally concentrated around a small energy region only. From these considerations it follows that for angles ,9 > 0.6 mrad the measured energy loss intensity can be determined to a good approximation by Eq. (1).

2.2.2 Radiation Losses

In some substances, e.g. III/V compounds and diamond, el can reach values of about 20 in the region A E from 3 to 8 eV. Using electrons of 50 keV or v ~ 0.4 c the condition for energy loss by emission of Cerenkov radiation is fulfilled for wavelengths in the above energy region. The

Diamond (lla type) O = 2620A . . . ~

= 0

..... ............. .: ,..e

1'0 2'0 3'0 4~0 ,dE

s'0 eV

Fig. 3. Energy loss intensity as function of the loss energy A E for diamond compared with Kr6ger's formula (dashed, Eq. (8)) and Ritchie's formula (dotted, Eq. (1), (4) and (7)). The optical constants are taken from [51]. Absolute intensities are compared. E o = 60 keV

energy loss spectrum is therefore not described any longer by the energy loss function - Im 1/E. There appears an additional radiation peak as demonstrated in Fig. 3 which compares the experimental curve with the theoretical one obtained from Eqs. (1), (4), and (7). One recognizes that Ritchie's formula fails to describe the experimental result because it neglects the retardation of the fields. Considering retardation, Kr@er [40] obtained the following expression which includes all surface effects and 6*

84 J. Daniels et al. :

the coupling of both surfaces:

Pre1(A E, ,9) dAEd l2 = - ' Im D- e(o2

+

(8) 1 1 i) sin2 2da~-fi402a~o2 + fl2 @- OaE(Pgl ( L + L -

c~ tanh [ ~aEJ ,ga~J dAEd f2

�9 L + + L -

A b b r e v i a t i o n s : e = el + i52 dielectric constant of the foil, q = ql + it/2 dielectric constant of the medium out of

the foil

= (~2 - - gtq2E f l2 )1 /2

~'0 = ( t'~2 - - /~0A2E f l 2 ) 1 / 2

with Im 20 < 0,

# 2 = 1 - e f t 2 ,

#2 = 1 - qfl2 .

L + = 2 o S + 2 q t a n h ( Oa~J '

@2 = 22 + 02 AE~

2 2 ~0o = 2o + 0 ~ , ,p2 =.92 + 02(1 _ (e + , ) fiz).

V

C

D AE d a ~ - 2 h v '

L- = 2oe + 2t/coth {2 ~ } .

We shall limit ourselves to the discussion of the volume term of Eq. (8) for reasons of simplicity:

Vol Prel (AE, g) d A E d ~

= ( e ) 2 ~ I m l D O2+02E[(1--51f12)2+(52f12)2"] 5 " [O2+O~E(l_51f12)]=+(O2AEe2f12) 2 dAEd f2 (9)

The most important difference between Eq. (1) and Eq. (9) is caused by the denominator of Eq. (9) which can lead under the condition that el fi2> �89 (52 small) to a sharp maximum at about O ~ 0.05 mrad in the angular dependence of the intensity in Eq. (9), which does not occur in Eq. (1). This is illustrated in Fig. 4. Because of the finite angular resolution of the spectrometer (marked by the horizontal lines in Fig. 4) this maximum


l ~...retarded

1~ /

n" J '~\X~, unretaraea

C) 012 014 016 018 m rad

Fig. 4. Energy loss probability P (& d E) as function of the scattering angle & Solid curve with retardation (Eq. (9)), dashed curve without retardation (Eq. (1)). The horizontal lines show the angular resolution of the apparatus. ~1 = 12.5; e 2 = 0.13; A E = 2.8 eV; D = 2100 ~; E o = 50 keV

t I1 GaP /-. ,/:

D=2100~ ~ k ' / ~, / ." I ', ./i

t-" "-,/ ', // ....

, "a. -8, =0 rad

/

,../ 0 2 ]+ ; 8eV 0

.4E

l

/-w/~ ! ,

I

/ /

,f 2,_,:/

" '\.." 4 rad

2 4 6 8 eV .4E

Fig. 5. Energy loss intensity I s (dE) for different scattering angles in GaP. Solid line: experimental curve [22]. Kr6ger's formula (dashed) and Ritchie's formula (dotted), both calculated with optical constants taken from [48]. The peak at 3 eV due to retardation (a) disappears at higher scattering angles (b) E o = 56 keV


is observed also in the measurements in the forward direction (0--= 0). This leads to an additional peak (radiation peak) in the energy loss spectrum as is shown in Fig. 5 a. Furthermore one can see in Fig. 4, that the influence of retardation decreases with increasing scattering angle 0 and vanishes for larger 0.

The angular dependence of the relativistic intensity of the surface losses is similar to the nonrelativistic one: The intensity of surface losses can be neglected for ~ > 0.6 mrad too (see Fig. 5 b). It is also interesting to point out that the dependence of the surface loss intensity on the foil thickness D is different in the relativistic and the nonrelativistic case for the energy loss spectrum observed at 0 = 0. More details see [40, 23].

In conclusion one can say that Eq. (1) can be applied to determine the energy loss function - Im 1/e of substances with high dielectric constants e 1 if the measurements are performed at larger scattering angles.

2.3 Evaluation of the Experimental Data

2.3.1 Experimental Technique

The energy loss function is determined experimentally by the ratio of the number of electrons inelastically scattered to the number of electrons which have passed the foil without energy loss ("no-loss beam"). For this purpose one needs an energy loss spectrometer which fulfills the following conditions:

a) an electron beam of good current density and small divergence,

b) an analyser of the energy of the electrons with accurate energy calibration and good energy resolution,

c) a high angular resolution of this analyser to take measurements in different well defined angles,

d) a recording system of the electrons of high sensitivity and of linear response for an intensity ratio of 1 : 105.

For these purposes various types of spectrometers have been con- structed. A review including also recent developments is given in [28].

The type of spectrometer [46], shown in Fig.6, is used for the experiments reported in this paper; it has the following qualities in respect to the points mentioned above:

a) Electrons thermally emitted from a tungsten cathode are acceler- ated to an energy Eo of about 60 keV and are focussed by a magnetic lens. The crystal foil is surrounded by a cooling trap filled with liquid nitrogen, in order to avoid surface contamination of the target. The primary electrons have a Maxwellian energy distribution with a half


width of about 0.4 eV. Monochromatisation of the electrons becomes therefore necessary for the investigation of fine structures in the energy loss spectrum. This is performed by means of a filter lens [32] as monochromator, which is installed instead of the focussing lens.

cathode

m anode

J monochromator I sample

retarding field

, ~ l m l ~ d l J ~ l u [t i plier

spherical condensor

Fig. 6. Scheme of a spectrometer with electrostatic analyser for the study of energy losses of electrons

b) The energy of the electrons is analyzed by deflecting them in an electrostatic field of a spherical condenser. Since the energy resolution of the condenser is proportional to the reciprocal value of the electron energy, it is increased by decelerating the electrons in a retarding field to energies of about 150 eV, before they enter the spherical condensor. The energy loss spectrum is registered by varying the potential of the retarding field: if the analyzer has been adjusted to the no loss beam of the energy E o with a retarding voltage AEo, an electron which has lost the energy A E is counted behind the exit diaphragm when the retarding voltage is reduced by the energy A E. Varying continuously the potential of the retarding field the whole energy spectrum is obtained. The value of the energy loss A E is given by the difference of the two retarding voltages A E o and A E o - A E and can easily be calibrated with an accuracy of 10 .2 eV.


The energy resolution of this type of spectrometer is constant for all energies A E and has values about 0.05 to 0.4eV, with or without monochromatisation, respectively.

c) The angular intensity distribution of the no loss beam, recorded behind the analyzer, is the result of a convolution of the angular ac- ceptance function of the analyzer with the divergence of the no loss beam behind the foil. The resulting angular resolution of the spectrometer, defined by the angular half width of the no loss beam, is about 0.3 mrad. It can be reduced, but at the same time the intensity decreases too. If the loss intensity has to be measured at scattering angles 0 > 0, the upper part of the apparatus is rotated around the foil by an angle 0 or the scattered beam is redeflected into the analyzer electrostatically.

d) The electrons having passed the exit slit of the analyzer are ampli- fied by an open multiplier and counted either by a ratemeter or, for higher intensities, by a d.c. ammeter. Using a multichannel analyzer, one has the advantage that the influence of fluctuations of the intensity of the no-loss beam and statistical errors in the counting rates become smaller.

2.3.2 Determination of the Loss Function from the Observed Quantities

In the energy loss experiment one obtains two quantities which are necessary for the determination of the loss function:

a) the number of electrons having passed the foil without energy loss, b) the number 11 (A E, 9̀) of electrons having lost the energy A E and

being deflected into the angle ,9 by inelastic collisions. It has been pointed out in Sec. 2.3.1 that the no loss beam cannot

be regarded as a delta function with respect to A E and 9̀. Since it can be verified in the experiments that the energy and angular spectrum is independent of each other, the no loss intensity can be written as Io .J(AE ) J(`9). J(AE) represents the energy distribution function, whereas J(`9) is the angular distribution of the no loss beam. J(A E) and J(`9) are normalized to 1 at AE = 0, Io means the maximal intensity at A E = 9̀ = 0. The intensity of the primary beam before crossing the foil is not needed, see [55].

In the angular and energy spectrum of the no loss beam, electrons are comprised which have suffered phonon losses [6, 34]. For the convolution processes treated below this is of no interest, since elastic and inelastic scattering are independent of each other.

The intensity 11 (AE, `9) of the inelastically scattered electrons (here we confine~ ourselves to single scattering processes) is then connected with the energy loss probability PI(AE, ,9) Eq. (1) by convolution of P1

Optical Cons tan t s of Solids by Elect ron Spectroscopy 89

with J(A E) and J('9) in the following way [12]:

II(AE, "9)= I o . ~ ~ J(AE')o p I (A E - AE', "9-,9') J(,9') dAE' d$2' . (10)

Eq. (10) must be integrated over all values of AE' and d~2' =0 'd0 ' dqo' for which J(A E') and J('9') are > 0.

In the experiment J(AE) and J('9) are recorded until they have decreased to about 1% and 0.1%, respectively. The error which results from this approximation is negligibly small.

In the energy loss function - I m l/e, e may be dependent not only on AE, but also on '9 or, since 0 ~ q/ko, on the transferred momentum q. If such dependence can be neglected in the region where J('9) is > 0, a reasonable approximation in general, then the integration in Eq. (10) can be separated into one over dA E' and a second integration over dO': the variation of Ode from OaE-~E' to O~e+AE' is negligible in the region where J(A E) > 0. Thus one has

e2 ( e (AE- )J(AE')dAE' II(AE,,9)= rcZh2v~. D. Io. ~ Im

J(,9') (11)

�9 ~ 0A2E q_ ('9 __ '9,) 2 dg]'.

The first factor can be approximated by I r a ( - 1/~(A E)). ~ J(A E')dA E'. This approximation holds, as long as the variation of - I m 1/5 is small in the energy region of the integral, a condition which is in general fulfilled, even for the investigation of sharp structures if one uses mono- chromatic electrons.

In the second factor the actual scattering angle of the electrons is given by ,9-'9', which depends on the azimut cp. Since d('9) is symmetric around the axis '9 = 0, the integration over ~o can be performed.

Using 1'9 - - '9,]2 = ~2 _]_ 0 ,2 __ 200' cos ~p'

this integral can be written as

J(O') ,9' dO' F(AE) = 27r y (402 ~ 02 + (0,2 + 02 ~ _ 02)2)1/2 . (12)

The function F(A E) is plotted in Fig. 7. The energy loss function thus becomes [-12]:

1 ) II(AE, O) T~2 h2 V 2 Im e(AE) -- I o e 2. D [~ J(AE')dAE'. F(AE)]-I (13)


which shows that - Im 1/~ is proportional to the ratio of l l / I o but the dependence of F(A E) on the energy is also important. As shown in Fig. 7 F(A E) changes rapidly with energy, approximately like 1/A E 2 for ~ = 0, whereas for large scattering angles F(A E) is nearly a constant, as can be seen from Eq. (12).

1.0

0.5 - - _ x 1..0 O=0.Tmrctd

9=0

0 10 20 30 4.0 eV AE

Fig. 7. Typical shape of F (A E) as given by Eq. (12) for two scattering angles. Calculated for an angular half width of the no loss beam of 0.3 mrad

2.3.3 Correction for Double Losses

So far we have assumed that electrons had suffered one energy loss at most, after having traversed the foil. Already in relatively thin foils multiple inelastic scattering processes must be taken into account; then the observed intensity at higher values of A E exceeds the value given by Eq. (11) and the - I m 1/e calculated from this uncorrected energy loss intensity is too high.

It is therefore necessary to find a procedure, which eliminates the observed energy loss intensities due to multiple scattering. In general it is sufficient to calculate the energy loss intensities resulting from twofold inelastic scattering processes.

We now calculate the intensity for twofold inelastic scattering processes of electrons which loose in two collisons the total amount A E and are scattered into the angle g (see Fig. 8). We start by considering those electrons which have suffered one loss with A El and ~91. This first collision may have happened in the element of the foil dD~ after the electron has passed a length D~ in the foil. The intensity of these electrons coming from dD~ is given by

dD~ Ii(AEi"gl)" D


[/+/7~ -5. / / / ~1

D

/ / v /

~,E 1 1,collision

, .& E 2 2.cott is ion

J / / \ x J / ' f "

Fig. 8. Scheme of two inelastic collisions

While penetrating the rest of the foil, the length of which is D - D 1, the electron can suffer a second inelastic collision with an energy loss A E2 and deflection by an angle 92: A E 1 + A E 2 --AE, and 91 + 92 = 9. The probability for such a process is

D - D 1 Pa (A E2, 92)" D

The total intensity of the twofold energy loss I2 (AE, 9) is obtained by convolution of I1 with P1 and by integration over all foil elements, so that

D A E oo

I2(AE, 9)= ~ ~ ~ I I (AE1,91) 'P l (AE-AE1,9-S l ) o o o ( 1 4 )

( D - D1) dD 1 D2 dAEI dO1

and after integration over dD 1 :

17 ; I I (AE1 ,91 ) .P l (AE-AE1 ,9 -91)dAEld~ 1 (15) 1 2 ( A E , 9 ) = ~ - 0 o

with PI(AE - AE 1, 9 - 91)

e2"O ( ! -I 1 - 7ceh2v~Im e(AE-AE1)] (9- '91) 2+ ~ - ~ l Oz (16)

For the numerical evaluation of I2(A E, ,9) one starts at very low energies A E, e.g. A E = 1 eV, where the intensity of the twofold losses is negligibly small, so that the energy loss function calculated without correction is accurate. Then the intensity of the double losses can be determined for


the energy AE + 1 eV by Eq. (15) and the corrected energy loss function is obtained. This procedure is carried on step by step for all higher energies.

1.5

1.0

E I

0.5

\

r'\ j it\\

~,)i, /i \ \ ' , /// '-~ XX \

,J \ \ -.

/ \ uncorF- - - 500AGok

c o r r . - - 500 ~Gotd 1000 A "

o ~b 2b 3'o ~b 5o eV Fig. 9. Influence of twofold inelastic scattering for gold for different thicknesses. The solid lines obtained after correction agree within the experimental errors and thus represent the energy loss function [14]

The influence of twofold energy losses is illustrated in Fig. 9. The energy loss functions calculated from the uncorrected energy loss intensities of electrons in two foils of different thicknesses disagree at high energies A E. Correction for twofold inelastic scattering however, leads to curves which differ only by less than the experimental error. This result proves that losses of higher order can be neglected under these conditions.

2.3.4 Accuracy of the Experimental Data

The errors in the determination of the absolute value of the energy loss function are due to errors in each factor of Eq. (13). The primary energy E 0 and the value of the energy A E are known with high accuracy,


so that the error resulting from these quantities is negligible. Other factors such as the foil thickness D, the no loss intensity I0, and the distribution functions of the no loss beam J(AE) and J(,9) are known less accurately and are responsible for the main error in Im 1/e. It may be pointed out that these errors are independent on A E. A further inaccuracy comes from the statistical fluctuations of the counting rates representing 11 (A E, ~) which is more important the lower the counting rates. (To a certain extent these functions influence also the structure of the loss function.)

On the whole, one can say that the energy loss function is usually determined correctly within • 115 %. For energies above 50 eV, the error increases to about • 20 % because of the decreasing loss intensities and the correction for twofold losses.

2.4 Kramers-Kronig Analysis (KKA)

2.4.1 General Description

To derive the dielectric constant from the energy loss function, a Kramers-Kronig dispersion relation can be used: the real part of the complex function 1/e = 1/(e 1 + ie2) is related to the whole frequency spectrum of its imaginary part by

l 1 ~ 1 de)' -- J Re g(~o) - 1 = - P Im , (17)

where P indicates the Cauchy principal part of the integral. To evaluate the contribution of negative frequencies to the integral Eq. (17) one uses the relation e ( -a~)= e*(~o) and obtains

Im i / e ( - co) = - Im 1/~(@. (18)

Then, Eq. (17) can be transformed into

Re 1 _ l : 2 p ~ i m 1 co'dco' (19) ~(~o) ~ o ~(~o ' ) ~o ' 2 - co 2 "

For anisotropic crystals however, the loss function depends on the direction of the wave vector q. For small scattering angles ~ of the order of gaE the q-direction is strongly dependent on m (or d E = h@. Therefore the spectra may be different for q(o3) and q ( - @. For the integration in the region of negative frequencies one uses the equivalence q(-co, g) = - q(o~, - ~) which can be derived from the relation tanfl = ~/SaE with ,gaE = m/ko v and fl the angle between k o and q (see Fig. 1). In this case two measurements at 1) and - & on opposite sides of the undeflected


electron beam, have to be analyzed and the corresponding loss functions have to be introduced into the following relation. Neglecting pole contributions one gets [3, 69]

1 1 i { Im [l/s(co', 0)] Im[1/s(co',-O)]}dco ' Re s(co,0) - 1 = --re P c o ' - - co + . . . . co' +co "

(19a)

Some further relations are of importance to perform the KKA: For the limit co-*0 one obtains

Re = 1 - - - 2 P Im 1 do)' rc s(co') co, , (20)

1 1 p ; [ i 1 1 ] d c o ' Res (0 ,0 ) - 1 - m - - + I m rc 0 s(co',0) s (co ' , -0) ~ (20a)

for the isotropic and anisotropic case, respectively. The left hand side of Eqs. (20), (20 a) is known in many case, s or can be estimated theoretically. For metals e.g., one obtains Re 1/e(co-*0)= 0. For insulators, s(co~0) becomes real and can be obtained by extrapolating the refractive index from the visible region to co--* 0:

1 1 Re - - - (21)

s(~--,0) soo

The infrared contribution can be neglected for this purpose [39].

1

(eV) -~

/,~II t

I IO

i J

i ~_~2 T

I 20 eV

I~co'

Fig. 10. Contribution function of the KK-Integral in Eq. (19) (solid lines) and in Eq. (22) (broken lines)


Making use of this additional information, Eq. (20) can be subtracted from Eq. (19) (or Eq. (20a) from Eq. (19a)), and one gets relations for the difference [3]

1 1 2co 2 ~ 1 do)' Re e(co) - Re 5(0) - ~z p jo Im e(co,) CO'(co 'Z - - (.02) ' (22)

1 1 Re - Re - -

e(co, 2) ~(0, 0) (22a)

i t Im [l/e(co', 0)] Im[1/e(co', - 0)] } - co P ~ m~(~;--_o ~ - co,(co,+co) dco'

for the isotropic and the anisotropic case, respectively. 1

The weighting function co,(co,2 _coz) in Eq. (22) decreases more

rapidly for large co' than that in Eq. (19). This is of interest for the calculation. For anisotropic crystals, Eq. (22a) has been recommended [69]. The two weighting functions of Eq. (19) and Eq. (22) are given in Fig. 10 for hen = 10 eV.

2.4.2 Numerical Evaluation of the KK-integral

The integrals of Eqs. (19), (20), (22), and (22a) are computed by dividing the experimental loss function into intervals of different width Ahco'i--- 0.01 up to 2eV in such a way that one can interpolate within A hco'i by a straight line. Then the integrals are calculated analytically. An expression for the two intervals near co can also be found to obtain the principal value of the integral.

In order to perform the integrals in the regions co' ~ 0 and co'---, oo one needs extrapolations: A linear increase of - I m 1/e(co') from zero to finite values in the neighbourhood of co' = 0 is assumed for metals. Since this interval is very narrow in the frequency scale, its contribution to the integral has no much influence. An extrapolation for the region co'~ o9 is possible for materials showing free electron like behaviour at high energies. In this case the loss function decreases as co'- 3 for co' >> cop, coy being the plasma frequency. This shape of the loss function has been observed e.g. for A1 [50] and KBr [39]. The contribution of this term to the integral is usually of the order of a few percent even at high co. If the shape of the loss function at high energies cannot be estimated and a larger contribution is expected, this part of the integral can approximately be replaced by a constant which can be determined by fitting the result at the limit co--, 0, using Eq. (20), (20 a). Such an co-independent contribution disappears in Eq. (22) and (22a) by subtracting the

96 J. Danie ls et al. :

two terms in this formulas. It may be mentioned that a similar procedure has been used in analysing optical reflectance data [61].

The Re 1/e(co) evaluated in this way from the energy loss function by means of Eqs. (19) or (19a) is more sensitive to errors in the - I m 1/e(co) the more the value of the integral

2 ~ Im 1/e(co'). co' P J COt2 - - (/)2 de)'

0

approaches unity. This is the case e.g. for low values of co in metals. The experimental accuracy of about 15 % in the loss function is then not sufficient to get reasonable results of Re 1/e(co). One can overcome this difficulty by using the information resulting from the knowledge of Re l/coo (Eq. (21)) which is obtained from other experiments or theoretical estimates:

Either one normalizes the experimental loss function, so that Re 1/~(0) calculated from Eq. (20) becomes identical with the Re l/coo or one takes Eqs. (22) or (22 a) for the KK analysis. The latter method is preferable if the values of the loss function are correct for lower values of co', but not well known at higher co'.

Another possibility for normalization is obtained by evaluating the f-sum rule (see Eq. (29)). This has been carried out [42] for some materials the spectra of which are similar to those of the free electron gas. Because of the weighting factor co' in Eq. (29) (instead of 1/co' in Eq. (20)), this method requires high accuracy at large co', but is useful, on the other hand, to check the results in this co'-region.

2.4.3 Determination of the Optical Constants

Having now the real and the imaginary part of 1/e(co), one calculates the complex dielectric constant

Re 1/e(co)- i- Im 1/e(co) e(CO) = e 1 (CO) -}- ie2(CO ) -=-- [Re 1/g(CO)] 2 -+- Jim 1/g(CO)] 2 " (23)

From this one gets the optical constants: the refractive index

N = ~/~ = n + i k with

n = V�89 1 + ~]/~+e2), k = 1/1(]/~12 + e22 - el), (24)

the absorption coefficient #, which is usually obtained from optical transmission experiments:

2o) # = - - . k (25)

C

Optical C o n s t a n t s of Solids by Electron Spect roscopy 97

and the complex ampl i tude reflectivity

r = ]r[ e ia = ]//R- e ~ ,

r is the ampl i tude reflection coefficient, R is the intensity reflection coefficient. Fo r n o r m a l incidence one obtains

(n - 1) 2 + k 2 - - 2k R - (n + 1) 2 + k 2 and t ang = n2 + k2 _ 1 (26)

Fo r the in terpre ta t ion of the results the f - sum rule is of interest (see e.g. [53]):

~2 (co ' ) co 'dco '= 2~zeZN ~ 1 �9 - - . n o = - I m ~ - . co'dco', (27) 0 m o

where N is the density of a toms (or molecules) in the material , n o is the n u m b e r of electrons per a t o m (molecule), and m the free electron mass. Regarding Eq. (27) as a function of the upper limit of the integral, one gets the co-dependent function, s [19].

m neff(co ) = j co'. e2(co') dco' = neff (g2) ,

2rc2e2N 0 (28)

2zc2e2N o - - I m e - ~ ; i - d c o ' = n e f f Im (29)

which indicate the effective n u m b e r o fe lec t rons contr ibut ing up to the frequency CO.

3. Results

3.1 M e t a l s

3.1.1 Models

If the electrons in a solid can be t reated as a Drude gas, the optical behav iour can be described by the complex dielectric constant e with

2 1 ~1 = 1 - - cop

o)2 (30a) 1 + 1/r 2 '

2 1 1 COp e2 - o)2 - (30b) coz 1 + l/co2z 2

Springer Tracts Modern Physics 54


H e r e (l)p means the plasma frequency ( g e 2 ] 1/2

COp = - - ( 3 1 )

\ me0 /

with g = electron density, So = (4n. 9 .10 l l)-I Coul/V cm is the dielectric constant of the vacuum and z the relaxation time which describes the damping of the electron motion.

In Fig. 11 the characteristic frequency dependence of e 1 and e2 is reproduced for a model substance with one free electron per atom�9 The parameters are hCOp = 16 eV, ~ = 3.1023 c m - 3 and z = 1.64.10 - i 6 s e c .

20-

g]O

0

10 6

--4. E

1. o

0.8

0.6 R

O.L

0.2

0.8

neff O.L

I

, 1

2O

lOx

0.8

0.6

0.4.

0.2

0

-10 6

2 Im{-{-

1.8-

I

0.8 neff(s2 ~ O.L ~ ~.~neff(I,m(-~))

0 5 10 15 20 25eV 0 5 10 15 20 eV z5 AE AE

Fig. 11. Characteristic features of a free electron gas (left) with hoop = 16 eV and r = 1.64 �9 10 - i 6 sec. O n the right an oscillator with he% = 10 eV a n d zo = 6 . 6 . 1 0 -16 sec is super- posed to the free electron behaviour


The loss function - I m l/e, which is zero for co = 0 and for co--, 0% reaches a maximum at co = COp or e~ = 0 (volume plasmon) in the case that copZ ~> 1 or e2(COp) ~ 0. Its energy half width is then given by h/z. For higher e 2 the position of the plasma peak is displaced to

((hcop) 2 --h2/4T2)l /2 .

The value Re 1/e is zero for co = 0 and for co = cop, see Fig. 11. The plasma frequency becomes apparent in the spectral dependence

of the reflectivity R which drops from 1 to zero at co = cop if copZ = oo. For finite copz the decrease of R in the region of cop is rather smooth (see Fig. 11).

This free electron model is a rough approximation, since in general the electrons are bound with certain eigenfrequencies to the crystal. If we assume that there is one oscillator with the resonance frequency coo, relaxation time To, and the density of electrons participating at the oscillation go, its contribution to the dielectric constant can be written

e2 no e b = - - - (32)

m~o co2 _ co02 + i ---~ "C 0

This must be added to the contribution of the free electrons (see Eq. (30)) to give the total dielectric constant in such a way that the total electron density remains constant. Fig. 11 demonstrates the spectral dependence of el and/32, as well as that of - Im 1/e and Re 1 ) for the special case hcoo = 10 eV, ~i o = 9 .10 22 cm -3 and Zo = 6.6.10 -16. If the value of go is high enough, as is the case in Fig. 11, the contribution to el near coo is sufficient to produce positive values of el. In this frequency region one obtains an increasing el, crossing the zero line (el = 0 at 7 eV) and a decreasing e2 being small, a behaviour like in the free electron gas near cop. One can speak in this case of a low lying plasma-like oscillation. As examples one can mention Ag, graphite, and MoS2. If the contribution of the oscillator is smaller, so that el remains negative for co < cop, the low energy peak is rather small, examples are Au and Cu. The plasma peak near cop is displaced to energies higher than cop, since the contribution of e b shifts the frequency where ~1 = 0 to higher values.

The oscillator is seen in the reflectivity as a strong decrease in this frequency region.

For both models nef f with its frequency dependence is also shown in Fig. 11. For free electrons nr increases continuously till the value of 1, n~ff(- Im l/e) however remains small for co < COp, since the value of - I m 1/e is small in this energy region, and grows quickly near the 7*

100 J. Danie l s e t al. :

plasma frequency. In the case of an additional oscillator neff(e2) goes up steeply where eb2 becomes finite; neff(-Im l/s) grows nearly like in the free electron model, since the low lying peak of - Im 1/s has only small values.

In energy loss experiments on a free electron gas, the energy loss function and the optical constants could be determined in a simple way from the energy loss spectrum: the position A E of the plasma loss and its energy half width provide the essential quantities cop and z. Whenever oscillators are superimposed to the free electron behaviour, a case which occurs in real crystals, the energy loss function has to be determined according to the procedure described in Sec. 2.3.2. It is possible to describe the dielectric function of a real crystal in the classical form as a sum of oscillator contributions characterized by cooi, zoi, ~0i. One can obtain [54] these parameters by varying them until the - I m 1/s calculated with them agrees with the energy loss function derived from the experiment. It may be of interest for numerical calculations to have at hand a formula of the frequency dependence of s.

3.1.2 Energy Loss Functions of Pd, Pt, Cu, Ag, Au

The transition metals Pd and Pt and the noble metals Cu, Ag and Au were investigated. Polycrystalline, self supporting films were prepared. Their thicknesses, from 400 to 900 ~ were determined by the multiple- beam interference technique.

The energy loss functions, evaluated from the absolute energy loss intensities by means of the formulas in Sec. 2.3, are shown in Fig. 12. They extend from about 2 to 90 eV, except for Ag, for which the energy loss function has been determined only above 4 eV. The well-known plasma loss [19] at 3.78 eV [29, 13] is not represented in Fig. 12. Cor- rections for twofold energy losses have been applied; their magnitude is shown in Fig. 9.

The energy loss functions of the noble and the transition metals have a structure with a number of broad maxima; a high plasma excitation peak, as in many other substances, is absent.

For energies below 10 eV, there are distinct differences in the shape of the energy loss functions of the noble and the transition metals: In the transition metals the - Im 1/s increases slowly with energy A E and reaches a weak maximum at about 7 eV which is due to a strongly damped plasma resonance (see 3.1.3). In the noble metals, however, the loss function is characterized by a sharp onset at energies of about 2 eV in Cu and Au and about 3.7 eV in Ag.

For energies above 10 eV the loss functions generally have the same structure and reach their highest values at about 20 to 33 eV. At even


higher energies the - Im 1/e decreases again, until at about 60 eVa very broad maximum is observed. This is due to transitions from deeper bands, see Sec. 3.1.3.

1,0-

O5

1.0- 1.0.

Pd

' 2o ~ v . v ' 6'o~v ' o ,..!., I 1.o. lO, 0.5 .5

,"-;~--YS

Oo ~o ~o go~v ' o o

cu

20 z,o @ev

' ' 2 o ~ ~ i o ' . ~ ~/o~v ' /"

2o go 6o~v

Fig. 12. Energy loss functions Im ( - l/s) for the transition metals Pd and Pt and for the noble metals Cu, Ag and Au. The full lines show the results of energy loss experiments [14, 15], the other curves were calculated from optical experiments: [-4] B, [8, 11] CHH, [19] EP, [593 R, [70] VA,/-74] YS

These metals already have been investigated to some extend by energy loss measurements. The values of the maxima in the energy loss function, obtained by transmission experiments [12, 14, 15], are compared with those observed in the energy loss spectra in reflection experiments [60, 66] in Table 1. They are found at nearly the same positions. The absolute values of the energy loss functions have been derived for the noble metals [12] and agree for Cu and Au within the error of [12] with the values in Fig. 12.

On the other hand, the energy loss function can be calculated from el and s2 measured with light. The optical values ofe 1 and e2 are available


Table 1. Positions of the maxima of the energy loss function in the noble and the transition metals ~ (in eV)

Pd 7.4 18 25.2 33.3 60 [14] 6.8 16 25.5 31.9 [60] 7.4 25 35 [66]

Pt 6.8 17 27 35 60 [15]

Cu 2.2 4.5 7 12 20.5 27.6 [15] 4.2 7.6 11.4 20 27.6 [12]

Ag (3.78) 7.6 17.3 25.2 32.5 [14 l 8 18 25.5 [12] 7.3 17.2 25 33.5 [60]

Au 2.7 6.4 16.8 25.4 33 60 [15] 2.95 6.3 16.7 25.6 33.5 [12~

6.3 16 25.8 32.6 [60]

The results of Ref. [60] and Ref. [66] are obtained from reflection measurements with slow electrons which integrate over all scattering angles.

for a limited range of energy [4, 8, 1 l, 19, 59, 70, 74]. On the whole, there is agreement between the values calculated from optical measurements and the results of the energy loss experiments (see Fig. 12) regarding both the absolute value and the structure of the energy loss function. Major discrepancies exist in Ag for the values of Ref. [19-1 which were found to be too low compared with the other results for energies above 4 eV. This may be due to surface contamination of the specimens, to which the results of the optical reflection experiments are very sensitive in the higher photon energy region, while the electron loss experiment in transmission is not influenced by the surface conditions. The 50 eV-maximum in Au, observed in Ref. [8] and the lack of the splitting of the maximum in Ag at 2 5 - 33 eV in the results of [59] are attributed to errors in the optical experiment, since in the electron loss measurements - Im 1/e is a directly measured quantity, and its structure is well reproducible. The reason for the discrepancies in the intensity of the 7 eV loss in Pd is not clear.

3.1.3 T h e Op t i ca l C o n s t a n t s o f Pd, Pt, Cu, Ag, Au

The optical constants have been calculated from the energy loss function as indicated in Sec. 2.4 (for extrapolation in Ag to low energies see [14]).

The values of~ 2, shown in Fig. 13 in the energy region from 2 to 40 eV, are in good agreement with those obtained by optical measurements.

In the noble metals ~2 is increasing very steeply above 2 eV, in Ag 4 eV. In Ag and Au, this leads to a high maximum at 5.2 and 3.3 eV, respectively, whereas in Cu there are two maxima at 2.7 and 5 eV. Then


~2 3.

I

0

Pd 4

3 t ' 3

2- 2 "\ " ~ ,Z,

1: ]

C

! '~ Cu

10 20 30eV

10 20 3OeV 20 30eV

5- i ~ Pt [ Au

S

0 [ 0 10 20 30 eV 0 10 20 30 eV

Fig. 13. ~2 for the transition metals Pd and Pt and the noble metals Cu, Ag and Au in an energy range from 2 to 40 eV. The full lines are the results from energy loss measurements [14, 15], the other lines refer to optical experiments (see Fig. 12)

~2 decreases wi thout p ronounced structure, but remains relative high: e2 ~ 0.5 at 40 eV. In Pd and Pt e 2 is decreasing without any structure till energies of 10 eV; at higher energies e2 is similar to the noble metals.

This behaviour can roughly be unders tood, as follows (see Ref. [18]): In the transit ion metals the conduct ion bands mix with low lying d-bands. Therefore in terband transit ion become possible, starting at very

104 J. Daniels et aI.:

low energies: The structure of e 2 is determined by these transitions together with free-electron effects. In the noble metals, however, the conduction bands are well separated from a flat d-band lying 2 - 4 eV lower. For energies smaller than this energy gap, e2 is determined only by free

E~

~,0-

0,5 -

-0,5 -

0,S-

-0,5.

Pt

E 4

0,5.

- 0 5'

~ ;0 1~ 20 2'5 io 3'5 ~V

0,5-

- 0 , 5 -

0 I0 I'5 eV

i i . . . . . 10 15 eV

Fig. 14. el for Cu in an energy range from 2 to 40 eV and for the o ther meta ls up to 18 eV. Fo r h igher energies the s t ruc ture of el is s imi lar for the nob le and t rans i t ion meta ls [ 14, 15]


electron effects (Drude behaviour). For energies corresponding to the energy gap, the first interband transition leads to a steep increase in ez, well separated from the free electron structure.

For energies above 10 eV, the same behaviour of ez in the noble and transition metals indicates that the energy bands are quite similar because of the same electronic configuration. The fact that s2 remains high till 40 eV shows, that the transitions are spread over a wide range of energy.

The el-curves, reproduced in Fig. 14 allow interpretation of the low energetic structure in the energy loss function. In Pd and Pt the monotonic increase of e~ makes el = 0 at about 6 - 7 eV. This leads to a plasma resonance, strongly damped because of the high value of e2 (e2 is about 2 -3 ) at this energy. In the noble metals an oscillator contribution, due to the interband transitions mentioned above, is superimposed to the free electron part of el in the low energy region. This leads to two maxima in the loss function of Ag which can be interpreted as plasma-like losses (Sl = 0 is fulfilled at 3.78 and 6.5 eV). In Cu and Au, the oscillator contribution to e~ lies at lower energies and thus cannot compensate the negative value of the free electron part to fulfill the condition Sl = 0. No sharp plasma-like peak is therefore observed at low energies. Only one peak

105~ 10

Ia

5

0

lOSe 10-

V

/ m- I ~ ......... "~"'J"-~. Ag

lot/ A "%,

5 xx'c-S

m-1

4pl/2 3/2

10-

Pt

~o 60 8o eV

5pS/2 5pl/2 5~/2

0 4.0 6o 80eV 0 - - , ~0 60 80 eV

Fig. 15. Absorption coefficient y for Pd, Ag, Pt and Au. The full lines are obtained from energy loss experiments [14, 15], the broken lines from synchroton radiation experiments [65]. The energy levels of the atoms are also indicated


due to st = 0 at 5 eV in Au and 7.8 eV in Cu is possible, but it is strongly damped because of the high values of 52.

In addition to the values of ax and 52 the absorption coefficient/~ can be deduced from the energy loss experiments. It is shown for energies above 40 eV in Fig, 15 for Pd, Pt, Ag and Au. At energies of about 50 eV, an increase in # indicates that transitions from deeper bands come into play. Comparing the onset of absorption with the energy levels of atoms [5], the transitions can be attributed to those of the p-electrons in the inner subshell (4p in Pd and Ag, 5p in Pt and Au). In this energy region # can also be determined directly by means of synchroton radiation [65]. The agreement with the energy loss results is rather good.

3.1.4 Discussion of the Consistency of the Results

To demonstrate the consistency of the results we have applied the following comparison: The values of el and 52 are determined from energy loss spectra at higher scattering angles O which are produced by volume excitations. With these optical constants we calculate the absolute intensity of the spectra at g = 0, which include also surface effects. The surface effects depend on the optical constants in a way different from the volume excitations, see Eqs. (4) and (7). The comparison of the observed intensities with the calculated ones is a good control of the consistency of the results. Good agreement is found, see the example of Ag in Fig. 16. This fact indicates further that the optical constants are

Ag

o Ib 1's ov

Fig. 16. Comparison of the measured energy loss intensities at ,9 = 0 (full line) with those calculated from the optical constants obtained from energy loss measurements at ~ about 0.6 mrad (dashed line) [14]. The agreement is good also in the region where surface losses are of great influence. The dotted line represents the volume contribution only


not dependent on the transferred momentum hq in these cases. The same is valid for Pd.

The sum rules for nef f as well as the value Re l/e(0) have to fulfill the conditions given by Eqs. (20), (28), and (29): For metals Re l/e(0) calculated with the loss function should be zero if the energy loss function is determined correctly. Errors in - I m 1/s lead to a value of Re l/e(0) different from zero, and the amount of this difference is a test for the

neff

lO

8 / / / /

d

2 ~ / / / / / / / i

,/ 0 i i i I I

neff -" /

~ J / / Ag

. ' / / 8 ."1 I

4 ..4 /

//~//i / I

i

0 210 40 60 80 lC)0 eV

Fig. 17. The values of nef f for Pd and Ag obtained from g2 (full lines) and from Im ( - l/e) (broken lines) [14]. Results for Ag from optical experiments are shown too (from e2 . . . . [59], from # . . . . [65])


error in the loss function. In the experiments values of Re 1/5(0) were found between 0 and 0.2, indicating that the energy loss function was determined with an accuracy of at least 20 %, a figure which is the same as that given in Sec. 2.3.4. The evaluation of the sum rules for nef f calculated from e2 and - Im 1/5 does not show any kind of saturation in the number of electrons participating at the transitions, as can be seen for Ag and Pd in Fig. 17. From this plot one can only deduce that the transitions do not come from well separated bands. The consistency between nef f calculated from - Im 1/5 and nef f from e2 is demonstrated by the fact that for high energies both quantities reach nearly the same value. Agreement with nef f calculated from optical experiments 1-59, 65] is found, as expected from the agreement in the e2-values.

3.2 Insulators

3.2.1 Solid Xenon

Thin polycrystalline films of Xe (ca. 300- 2000 • thick) were obtained by condensation on thin carbon substrates at 5 ~ K. The lattice is fcc with a = 6.13 A at 5 ~ K. The loss spectrum in the low energy region is characterized by sharp exciton peaks which start at about 8 eV [-37]. The band edge lies at 9.28 eV. At higher energies (,,~ 15.2 eV) a broad maximum is found which corresponds to a volume plasma oscillation of the 6p electrons of the valence band, followed by humps due to transitions from deeper levels. The energy values of these peaks are listed in Table 2: in the first line the energy loss data of the solid, in the second line the peaks in the optical absorption #, in the third line the maxima of the loss spectrum in the gaseous state and in the fourth line the difference of the atomic energy levels. These last figures allow to control the energy scale of the loss apparatus. The small difference of the values of line 1 and 2 are due to the different position of the maxima of # and - Im 1/~, see Eq. (25).

Table 2. Comparison of solid and gaseous Xe (energies in eV)

/'13/2 r 3 / 2 fill2 L3/2 L1/2 z25

crystal energy loss [39] 8.53 9.10 9.8 10.55 11.3 15.2 # [1] 8.36 9.07 9.53 10.30 11.14

gas energy loss [39] 8.42 9.57 10.39 11.17 11.60 12.49 energy level 8.43 9.57 10.40 11.17 11.60 12.47

(6 s) (6 s') (5 d) (6 d) (5 a') (6 d')


From these measurements the loss function - I m 1/e was deduced. Its mean value is reproduced in Fig. 18; above approximately 23 eV it decreases as (1/A E) a. The determination of its absolute value had certain difficulties since the thickness of the solid Xe film could not be obtained. The same was true in the optical measurements referred to below. In this case the thickness determination can be replaced by using e (o )~0)

�9 . . . . . , , r , ' ,

-Ira I / r

1.0

O.8

0,6

0.4

0.2

i i i J = i i ~ ~ , i i i i , i i

B 9 10 11 12 12 13 14 15 16 17 18 19 20 (eV)

Fig . 18. E n e r g y l o s s f u n c t i o n o f s o l i d X e a t c a . 5 ~ K [ 3 9 ]

�9 , ,,,,, , . . . . . , . . . . ~ ,,, ,

6y 4

0 . . . . . . . " /

- 2

E I - -

~2 ....

\ /)]/~ ',, p~

" \ F -/-"

\

\

I f , , ~ t i i L i i J = i = i i , i i ,

7 9 10 11 12 12 14 16 1B (eV)

F i g . 1 9 . R e a l a n d i m a g i n a r y p a r t o f t h e d i e l e c t r i c c o n s t a n t ~ f o r s o l i d X e [ 3 9 ]

2 . 0

1 .6

1 .2

0 . 8

0 . 4

0 . 0

- 0 . 4


see Eq. (21), however, the value of ~(~ ~ 0) is not known from experiments and it must be deduced from theoretical considerations [39].

This loss function enables us to deduce the values of ~ (e)) and % (r reproduced in Fig. 19 [38, 39]; there exist no optical values to compare with. In the low-energy region one recognizes the excitons; in the region of 15 eV a~ increases from small values whereas e2 decreases, a behaviour typical for the appearance of a plasma oscillation in the loss spectrum.

For a comparison with optical data there exist measurements of/~D [1], # absorption coefficient, D film thickness, dotted line of Fig. 20. The value of # deduced from the loss function (full line) can be compared with the optical values #D, assuming a suitable thickness D. The agreement is good except that the loss data do not yield the fine structure perhaps by lack of energy resolution.

I " \ ^ I / �9

, 9 ,0 ,, ,2~2 ,'3 ,~ ;5 ; ,'7 ~8 ,; i .~

(10S/cm

16

12

8

4

Fig. 20. Absorption coefficient # from energy loss data (full) [39] compared with optical absorption (dashed) [1]

In the case of Xe the uncertainty in the knowledge of e(e)--,0) was reduced by the fact that the absolute value of - I m 1/s influences the position of the absorption peak in Fig. 20. The most prominent peak lies at 8.36 eV in the optical measurements, whereas the values of the loss function Fig. 18, normalised with e(co~0)= 2.16, give the maximum at 8.27 eV as indicated in Fig. 20. This value permits also an optimum fit of the two absorption curves at higher energies [39].

3.2.2 KBr

The polycrystalline KBr films deposited on thin carbon substrates had thicknesses between 290 and 2700 ~, as measured by interferometric methods. The loss spectrum is rich in details and can be divided into the


exciton region (~ 5 eV - ~ 10 eV), the plasma loss region (~ 13 eV) and the high energy region with band transitions and high energetic excitons (> ~ 16 eV) [39]. In the low energy region one recognizes the F3/2, F1/2 excitons (a,b in Fig. 21), the band edges F~/sZ~FI, F~/52--+FI, and the X

5 I0 15 2O (eV)

Fig. 21. Energy loss spectra of electrons in KBr at different temperatures [39]

exciton (c) which is split up like the F exciton by spin orbit-coupling (0.24 eV at 80 ~ K). Since the fine structure is strongly dependent on the temperature of the crystal, the spectra are taken at low temperatures. For details on temperature dependence of the loss position, of the half width of the excitons etc. see [39]. The loss at 13.5 eV, see d in Fig. 21, corresponds to a collective oscillation of the six p electrons of the valence band. It is further remarkable that one finds high energetic F and X excitons g (20.03 eV) and h (21.45 eV) (transition from deeper levels than


the valence band into the exciton states) as they were observed in the loss spectrum of KBr and of other alkali halides already by [-12]. The splitting of the Fexciton g (19.89 eV and 20.12 eV) has a value of 0.23 eV at 80 ~ K.

i

-Im l& !

2.0

0.8

OA

0 0

i 1 i ~ i . . . . x , "~ i i " ~ ' 1 , '"

f

/ /

!" & !

~ J

/ ' x i I 'x I , / ,, /

( / "\. ;

, i

l i . 1

iv

212 ' 6 8 10 12 l& 16 '18 20 (eV)

i.0

0.8

0.6

0.4

0.2

Fig. 22. Energy loss function of KBr, deduced from electron experiments (full line) [39] and loss function calculated from optical values (broken line [623 scale right side)

The loss function deduced from the spectra at 270 ~ is reproduced in Fig. 22. The following points are of interest:

a) Values of el and ez from optical reflection measurements [49] are available; the structure of the loss function calculated from these values is in good agreement with the electron energy loss function as Table 3 demonstrates for the position of the maxima. At higher energies discrepancies between the two structures are observed which are probably due to the optical measurements.

Table 3. Comparison of the position of the maxima in the loss function of electrons (line 1 and 2) with those in the loss function calculated from optical data (line 3 and 4) in KBr, 20 ~ C (energies in eV)

1 Ref. [12] 6.70 7.25 8.90 13.5 16.2 18.55 20.00 21.50 2 Ref. [39] 6.70 7.28 8.95 13.52 16.3 18.56 20.03 21.45 3 Ref. [49] 6.70 7.30 8.90 13.55 4 Ref. [62] 6.63 7.18 8.74 13.64


I~ O0~Icm)

12

l.O

0,8

0.6

i 0.4

II 02

0

i

I " ' %

J

ETO_//

i . . . . . , : ,'

I / \ ;

i I i '

i I \ I

............. ' -"- i . . . . " T , " ......... k./ - ' ' < ' . . /

8 10 12 14 I 18 (eV)

Fig. 23. Comparison of the absorption coefficients of KBr obtained from electron experiments (full) [-37, 39] with optical measurements (dashed by [62] R and dotted by [17] ETD and [16] MD)

28

R(%I

24

20

~ ~E.~. ................. ,:. ........ ,., ~'~: ......................... '~,. ......./~\.,?\,~ -..

r , , , , , , , , , t i i , , , I ,

6 ,El 10 12 14 16 18 20 22 (eV)

Fig. 24. Reflectivity of KBr calculated from electron experiments (full) [37, 39] and optical data (dotted by [49] PE and dashed by [72] WNNM)

8 Springer Tracts Modern Physics 54


Concerning the absolute values of - Im 1/~ the following can be said: The two curves in Fig. 22 coincide well below ,-~ 15 eV (they are fitted in the energy region of ~ 8 eV). If one normalizes the loss function with ~ = 2.33, see Sec. 2.4.2, one obtains an agreement between the two curves in Fig. 22 in between 10 % for A E < 15 eV.

If one uses the experimental thicknesses of the films one obtains values of - Im l/e which are too high by a frequency independent factor 2,3 compared with the (- Im 1/~)opt. The source of these discrepancies could not yet be found; it may be due to the structure of the polycrystalline films produced by vacuum deposition.

b) A comparison of (q, e2) derived from the loss function can be made with the (el, ~2) values obtained from light reflection measurements using the KKA. The agreement is very, good. More interesting is to compare the loss data with original optical results as the absorption (#) and the reflectivity (R). Fig. 23 shows the spectral dependence of the absorption coefficient #. Between 6.5 and ca. 15 eV the value of ~ [62] (R) is in good agreement with the values from loss data. Between 6.5 and 11.5 eV we found the same good agreement with the spectral dependence of #D observed by [17] (ETD). (The absolute values of # cannot be compared since the thickness is not measured in [17].) Adapting the curve (ETD) to the #-curve from loss data (full line), both structures fit rather well in this energy region. Above 15 eV the optical data of Ref. [16] (MD) agree much better with the loss data than those of Ref. [62] (R).

By means of the (el, e2) values derived from the energy loss function the reflection coefficient R can be calculated and compared with the optical data of Ref. [49] (PE) as well as of Ref. [72] (WNNM) Fig. 24; it shows a g o o d fit till ~ 15 eV with Ref. [49] (PE) and an excellent fit with Ref. [72] (WNNM) for energies above 15 eV.

3.2.3 Diamond

The experiments were performed with thin monocrystals of ca. 2000 A thickness, prepared from the bulk material by grinding with a steel ball followed by cathodic sputtering with argon ions. The energy loss spectrum taken at a scattering angle O = 0, shows radiation loss intensity at energies A E < 12 eV. Its structure agrees with that calculated with the relativistic formula (Eq. (8)) using the optical constants [51] as shown in Fig. 3.

The energy loss function Im 1/~ given in Fig. 25 was determined from measurements at 0=0 .8 mrad where surface and radiation effects vanish. It is in agreement with that calculated from optical data [51, 58] except for deviations near the plasmon peak. Similar deviations exist in

Optical Constants of Solids by Electron Spectroscopy 20[ [~ Diamond

15

[

10- i ' ~ / ~ I

\\

-s- II // \ \ i, y , \ -,0- I

T

E I

115

-1- ~ 1 r 1 -0 0 10 20 30 40 eV

Fig. 25. Dielectric constant and energy loss function for diamond [77]. The broken lines are optical data of [51]

reflection measurements with electrons [73]. This plasmon peak is found at 34 eV [77]; its half width is about 14 eV, much broader than in the semiconductors Ge and Si having the same structure and in the III/V compounds. Assuming 4 electrons per atom one calculates hop = 31 eV. The shift to a higher value is due to the interband transition at 23 eV, which can be seen in the loss function. This transition is not yet identified in the band scheme but the transmission loss experiment con- firmes that it is intrinsic to diamond.

In the lower energy region the loss function of diamond is small because of the screening due to the high s-values, so that the relative error is increased, and the KK-analysis does not give accurate values for s in this region; however, the general shape is reproduced as compared with optical data (see Fig. 25). Several attempts have been made to coordinate these results with the band structure of diamond [30, 52]. 8*


3,3 III/V Compounds

In this section results will be reported for the III/V compounds GaP, GaAs, GaSb, InAs, InSb and on Ge. The technique of flash evaporation was used for preparing the foils. For details see [23]. Only the GaP samples have been prepared from single crystal material by polishing and Argon ion sputtering similar to the preparation of diamond foils (see Sec. 3.2.3).

The determination of the energy loss function - I m 1/e had to be made at scattering angles above 0 =0.6mrad, since the energy loss spectrum is significantly modified at energies below 10 eV due to radiation losses (Cerenkov radiation) as in the case of diamond.

In Fig. 26 the energy loss functions - Im 1/e are plotted for different III/V compounds, the structures show the following general features [48] :

1. For energies A E below 10 eV the structure of - Im 1/e is determined by interband transitions. The absolute values of - Im 1/e are small due to the large values of ~1-

2. The highest maximum at about 15 eV is ascribed to the excitation of volume plasma oscillations.

3. A doublet structure at about 22 eV is due to transitions from the d-band.

In Fig. 26 the energy loss functions - Im 1/e calculated from optical reflectivity measurements are plotted for comparison. There is good agreement in the positions A E of the maxima due to band transitions as can be seen in Table 4. The doublet structure ascribed to the d-band transitions at about 22 eV is not observed in the optical energy loss function. It is interesting to point out that this splitting corresponds to a splitting seen in the density of conduction states calculated for InAs, GaSb, and InSb [33 a].

For energies below 10 eV differences in the absolute values of the loss functions are larger than the experimental errors of the energy loss measurements. The accuracy of the optical values is not known. The loss function calculated from the optical constants of InSb measured by Ref. [24] are in better agreement with the energy loss measurements.

Above 10 eV there are important discrepancies between the energy loss and the optical measurements: In the region of the volume plasma loss the absolute value of the optical - Im 1/e is considerably smaller than that derived from energy loss measurements. These discrepancies are probably due to surface contaminations of the specimen in the optical reflectivity measurements. This is demonstrated for Ge: Fig. 27 compares - I m 1/~ obtained on single crystals from loss measurements [78] and from optical refiectivity measurements [-48, 63]. The samples in the optical experiments were exposed to air for some time. In the same


s I

,l a. GaAs I

/ I 10 /, I /// - .

/ /

1(1 2'0 eV AE

~1~2 E I

3

1// b. GaSb

\

10 210eV AE

3

2

T W

1

~ c. lnSb

':i !'I, 2 !l o

o' 1'0 2'o.v ,dE

Fig. 26. Energy loss function Im( - l/e) obtained with electrons (full [23]) and from optical measurements (broken [48], dotted [24])


Table 4. Position of the maxima in the loss function in eV

Region of interband transitions d-band region

G aP 4.4 (4.2) 6.4 (6.4) 9.1 (8.8) 21.3 23.3 (22.9) GaAs 4.0 (3.6) 6.3 (6.2) - (9.4) 21.0 23.2 (21.8) GaSb 3.0 ( - ) 5.2 ( - ) - ( - ) 20.8 22.5 ( - ) lnAs 3.6 (3.3) 5.8 (5.8) - (9) 19.6 21.6 (19.5) InSb 3.2 (2.9) 5.1 (5.6) - ( - ) 19.1 21.0 (21.1)

(5.0)

The values in the brackets are due to optical measurements by [48].

Fig. 27 two optical curves [47] which were obtained on Ge-films vaporized in a vacuum of 10- 6 torr are reproduced. They were measured in the same vacuum at different intervals after evaporation without exposing the films to air. These curves show that the absolute value of the optical - Im 1/e in the region around 15 eV is closer to the energy loss result the shorter the time of exposure to the vacuum.

E

f . ~ Z R

//

AE

~-1-2min MT j3Omin MT

*~ j S

~--........___.__ I I

20 25 eV

Fig. 27. Energy loss function Im ( - l/e) of Ge in the plasma region. Energy loss measurements: [78] ZR. Optical measurements: [47] MT, 163] S and [48] PE

The value AE of the maximum in - Im 1/e is the same for the optical [47] and the electron measurements [78] obtained on the single crystal. (Since the volume plasma loss of amorphous Ge lies ~ 0.5 eV lower than in crystalline Ge [78], one can verify that the foils used for the optical measurements [47] were not amorphous.) The application of the sum


rule (Eq. (29)) to the curve obtained by energy loss measurements in Fig. 27 gives an effective electron number r/ef t ~ 4 demonstrating that the four valence electrons in Ge contribute to the volume plasma excitations. The optical measurements Ref. [48, 63] give only neef( - Iml /e)~2.

The optical constants el and e 2 were determined from the energy loss function - Im 1/e as described in Sec. 2.4. To control the absolute value of - I m 1/e the value Re (l/e(0)) obtained from the loss experiments is compared with the optical dielectric constant e~o, taken from the review

1 - 1/zo~ did not deviate much from unity, Ref. [33]. The quotient 1 - Re l/e(0)

it lies between 0.89 and 1.06. The el and ez curves obtained in this way are compared in Fig. 28

with the optical measurements Ref. [48]. The first two maxima in the e2-curves are ascribed either to band transitions at different symmetry points of the Brillouin zone (L, F, X and S) or to extended critical regions. For details see [30, 52, 9]. The weak maximum at 5 - 6 eV is caused by the transition L 3, ~ L 3 .

Another transition Z 3 ~ L 1 is possible for zinc sulfide structures. It appears in GaP, at 9 eV. In the other substances it is not observed because it is probably too weak. In the diamond structure this transition is forbidden.

For the III/V compounds considered here el becomes zero between 12 and 17 eV and e2 is small too. This means that in this energy region the valence electrons are able to carry out volume plasma oscillations leading to the large maximum in - Im 1/e. The agreement between the values of the volume plasma losses and those determined from the free electron model (lowest line in Table 5) is probably due to the fact that the influence of the transitions lying energetically lower is compensated by those lying energetically higher (d-band transitions) than the volume plasma energy. The agreement with the plasmon energy positions deduced from the optical loss function is not so good due probably to surface contaminations as mentioned above.

In the energy region between 1 9 - 24 eV there are two maxima in e2 obtained by the energy loss measurements. See also Fig. 26. They are interpreted as transitions of the d-electrons of Ga or In into the conduction band.

Table 5. Position AE of the volume plasma losses (in eV)

GaP GaAs GaSb InAs InSb

Energy loss [23] 16.5 15.7 14.3 13.8 12.8 Optical measurements [48] 16.9 14.7 - 13.0 12.0 Calculated 16.57 15.70 13.88 13.85 12.79

t_.

Q

D"

ul

ii'i ..

....

....

....

....

..

....

....

....

...

-3~-

_2

__

-~

7

Optical Constants of Solids by Etectron Spectroscopy 121

s

-2O

-10

.[

i'i d. InAs

i _ _ ; I0 "I'I I'5 20 eV

AE

2O

g2 10

j/ 0

t:1 /'~ e. InSb

i.J i

~, k x x 5

5 9 10 15 20 eV zlE

Fig. 28. e 1 and e 2 as function of the energy A E. Full curve obtained by energy loss measurements [23], dotted by optical reflectivity measurements [48]

GaP D = 1900,~

- - I I I

0 2.5 5.0 eV 75 AE

Fig. 29. Experimental energy loss spectrum taken at 0 = 0 (full) compared with that calculated by Eq. (8) using the optical constants determined from electron experiments at 3 4= 0. Absolute intensities are compared [22]


The consistency of the optical constants el and ~2 Obtained by the energy loss measurements can be checked in the following way. el and e2 were determined by measurements at scattering angles ~>0 .6 mrad where the retardation can be neglected. With these values of q and e2 the energy loss spectrum at the scattering angle 0 = 0 was calculated by Eq. 8, see Fig. 29. It turned out, that the retardation effects which occur only in the forward direction can be described with respect to structure and absolute value by the optical constants derived from the measurements at 0 > 0.6 mrad. This proves further that the dielectric constant derived from energy loss is independent of the transferred momentum hq within this range.

Recently, optical constants were determined from energy loss experiments [25] on the IV/VI compound SnTe. General agreement was found with optical measurements.

3.4 Anisotropic Crystals

3.4.1 Description of the Method

For anisotropic crystals the excitation probability for electronic transitions depends on the direction of the electric field in the crystal. The dielectric constant has therefore to be written as a tensor such that D i = ~ e~jEj. The subscripts i, j indicate vector components in an ortho-

J gonal coordinate system.

The energy loss function for volume excitations is then given in the nonrelativistic case by [35]

- Im 1 /~ qieijqj (33)

instead of - Im 1/e.

The loss spectra are thus dependent on the direction of the transferred wave vector q whose components q~ can be determined from the momentum scheme of the scattering process (see Fig. 1).

With respect to the incident beam direction k o, the direction of q is described by an azimuthal angle q~ and an angle fl determined by tan fl = qi/qll = O/OdE. From this one gets that q is perpendicular to ko for large scattering angles 0>> OdE, and that the q-direction becomes dependent on e) or A E for small 0 ~ Ode; the consequences for the KK- analysis have already been discussed (see Sect. 2.4.1).

Considering the finite angular resolution, the actual energy loss spectrum is given as explained (see Sec. 2.3.2) by the convolution integral Eq. (10). This integral is calculated in analogy to Eq. (11) by splitting up the loss function into ( - 1/q 2) Im 1/glieijglj (q is unit vector in the q-direc-

Optical Constants of Solids by Electron SpectrOscopy 123

tion) and by neglecting the convolution in the second factor. In this way however, only the variation of 1/q 2 is taken into account, but not the variation of the q-direction within the angular resolution function. This leads to an average over the q-directions in the loss function. This is of importance for the measurements at small scattering angles where the direction of q depends strongly on 0 [75]. To make the uncertainty in the determination of the q-direction small, the angular resolution half width must fulfill the condition d01[ 2 ~ ] / / ~ 4-~q2 .

For uniaxial crystals such as the layer structures graphite and molybdenite, the tensor e has only two different diagonal elements e• and e II (co), perpendicular and parallel to the c-axis, respectively. The direction of q is then described by an angle O with respect to the c-axis, and the sum in the loss function Eq. (32) can be written as

e(o, O) = e• sin 2 0 + e It (o9) cos 2 0 . (34)

The angle O is for spherical geometry:

cos O = [cos/~ cosec - sin/~ sine cos ~0f, (35)

where c~ is the angle between c-axis and ko,/3 and ~o are defined above, see Fig. 30.

~ c-axis

8

q

~o=0

Fig. 30. Representation of the angle O between c-axis and q-direction. The angles ,9 and/3 lie in the dashed plane which is rotated about (p with k o as axis


As an example for the determinat ion of the dielectric constants of uniaxial crystals we describe the method used in the case of graphite [-77]:

A thin cleaved foil (800 ~) is used: the c-axis coincides with the normal to the foil, ~ being thus the angle of incidence. Three spectra have been taken at fixed ~ = 61 ~ and at ~ = 0.8 mrad, but at different azimuth (p, see Fig. 31. This a r rangement allows the direct compar ison of the intensities for different O, since the effective foil thickness and the other experimental parameters remain constant.

I~t= 8 "10 "4 md ~= 61 ~

~ :90 ~

50eV 40 30 20 10 0

(9, 5,5o 5,0 ~ 4,5 ~ 4,0 ~ 3,5 ~ 3bo

~,,~P '~" % ~ - 0

50 eV 40 30 20 10 0

, / • ~0=180 ~

50 eV 40 30 20 10 0

Fig. 31. Energy loss spectra at different azimuth q~, from which the dielectric tensor is derived. The broken line is the calculated intensity for double losses [77]


The twofold intensity (broken line) is calculated as described in Sec. 2.3.3; this procedure is only an approximation for the case of an anisotropic crystal, since the twofold scattering process is assumed to belong to the same 0 as the corresponding single process. It is estimated that the broken line is correct within 30 %, but no detailed calculation has been made.

The spectrum at (p---90 ~ corresponds to q_Lc-axis. This holds nearly independently of hco (or A E) [a variation of ho) (via the angle/3) has less influence on O the greater e (see Eq. (35))].

For this azimuth the loss function does not differ for (+ 0) and ( - ~9), so that el(co) is obtained from this single spectrum by the KK-analysis of Eq. (22).

- I rn ~'*- (Lo',--~) ~'(w'+w)

eV 25 20 15 10 5

(.dl = ~t)

W'(W'-(~)

V

Fig. 32. The integrand of the KK-integral for hco = 10 eV, using the two lower spectra of Fig. 31

To obtain ell(co) the spectra at cp= 180 ~ and ~0=0 ~ are used which correspond to ~9 and -~9, being consistent with Eq. (35). The spectrum at ~o = 180 ~ corresponds to a q-direction close to the c-axis or small O, as indicated in Fig. 31 ; O varies with A E as indicated, since ~ is a function of A E, see above. In order to obtain ~(co, O) one needs in addition for the KK- analysis - I m 1/e(co, - 0) which is deduced from the spectrum taken at (p = 0 ~ see Fig. 31. Both spectra become equal with smaller values of hco. To illustrate this procedure, the integrand of Eq. (22a) is plotted in Fig. 32 for hto = 10 eV: the difference of the two dashed areas gives Re 1/e(hco


= 10 eV, O = 23~ Finally one calculates

~ll(co ) = e(CO, (9) -- ea(co) sin 2 0 COS z O

(36)

separately for the real and imaginary part. An analysis of spectra taken at different c~ but fixed (p has been made

[3], neglecting that the structure becomes different for 0 and - 0 with increasing energy. The data of the dielectric constant are however nearly the same as obtained by the procedure described above, since the weighting function in Eq. (22a) decreases rapidly with higher energies A E, see also Fig. 32.

For biaxial crystals analogous considerations are made. Energy loss spectra were taken at monocrystals of anthracene [40a, 70a]. The components eu of this substance have been deduced [70a].

3.4.2 Graphite

The dielectric constants ~a and e II are given in Fig. 33. For E• the results are in close agreement with those obtained in Ref. [67, 7] from optical reflectance data. The interpretation of 52 in the band scheme [2] may roughly be summarized as follows: The peaks in e~ are attributed to interband transitions at the point Q, where a logarithmic singularity in the joint density of states is expected. Transitions either between 7t-bands or between or-bands are allowed for E • c in the two-dimensional approximation, i.e. neglecting the interaction between the hexagonal layers which build up the crystal. The maxima at 4.5 and about 14.5 eV correspond to the separation of the r~-bands (Qzo~Q2u) and of the a-bands (mainly + -, + Qau Q20), respectively.

For the polarization direction Ell c-axis, only transitions r c~ r or o-~ ~ are allowed by the selection rules. The sharp peak arising at 11 eV can be explained by transitions Q[g~ Q~, [3], but other interpretations are also possible (see Ref. [31] and [68]). In the lower energy region however, ~ is expected to be zero according to the theory [2]: no absorption is found below 5 eV in optical experiments [20, 31]. This is not confirmed in Fig. 33, and this discrepancy is not yet explained.

Using the data of Fig. 33, the loss function can be computed for any q-direction. The result (Fig. 34) agrees with direct observations for each O, and the position of the peaks as function of O is the same as in former experiments [75]. For qZc, the 7 eV peak in the loss function can be interpreted as a plasma oscillation of the 7z-electrons [36] which is calculated to be at 12.6 eV in the free electron model, but is shifted due


12

10

8

6

4

2

0

-2

-4

0

' i Graphite J opt. E z c \ ..',:

a I i _ _ ] / i i

10 20 30 40 eV --r---- I ~ - i - - i - - I

8

4

2 ,

0

-2

EIIc 4

I I I - - t I _ _ . ~ I

- 40 10 20 30 40 eV

Fig. 33. Real and imaginary part of the dielectric constant in graphite derived from energy losses for E• i.e. the electric field vector being in the layer plane, and for El[ c [77]. Optical data [67] are given for e~ by the dotted line

to the pol~irizability of the a-electrons [67]. A further plasmon at 27 eV in- volves re- and a-electrons. Its position and half width have been confirmed recently by energy loss experiments [45] and optical measurements with synchroton radiation [64]. In former energy loss measurements this peak was observed at lower energy [71, 43], presumably due to a mixing of different q-directions. These measurements were made with poor angular resolution at 0 = 0, and it was assumed that the directions q_l_ c contribute essentially to the loss function. The 0 -z decrease of the loss intensity however makes the small scattering angles ~9 also important. This gives a contribution of ell to the loss function which leads to a displacement of this loss peak to lower energy.


4 I ~ - - ~ ,

Graphite j ~

"7 /

@ = 90 ~ / / '~ \

oL7 L e = 0 ~

i (qlic) ~, /

o i ~ , ~ , , ~ _ _

0 10 20 30 40 eV

Fig. 34. The energy loss function for different q-directions as calculated from the energy loss data in Fig. 33 [77]. The dashed line represents - I m 1/s • from optical reflectivity measurements [67]

Further information is obtained by evaluating the f-sum rule (Fig. 35). It has been shown 1-67] that nef f computed from ~2 ~ shows a plateau near nef f = 1 electron per atom at hco = 9 eV and saturates at ne f f=4near 30 eV. The same values are also calculated from energy loss data [68]. The present data however, yield lower values of nef f (solid line in Fig. 35). New evaluation [77] of the optical data [67], however, yields the same values of nef f (/~) as in Fig. 35. The dashed line gives nef f obtained from the present e~-curve. The main feature is that the interband transitions for E il c lie in the most part at higher energy than for E 3_ c, and no much contribution to nr arises from the energy region below 10 eV.

The reflectivity computed from the present energy loss data is shown in Fig. 36. The result for E l c is compared with that measured directly in optical experiments [67, 31], and is shown to be consistent up to about


3 Graphite

f / f I"

0 10 20 3O eV 4O "hm

Fig. 35. The effective number of electrons contributing to s~ (full) and to s~ (broken) as evaluated from the electron experiments [77]

A

0 ' 10 ' 2; ' 3'0 ' 40eV

Fig. 36. The normal incidence reflectivity for E J_ c and Ell c obtained from energy loss data (full) [77], and values for E J_ c from optical measurements (triangles [67] and circles [31]). For Ell c, some values are calculated from n II- and kll-values (crosses [31])

12 eV. A deviation appears near 15 eV and is visible in the e~-curves too. In this energy region, the accuracy of the energy loss data is reduced because of uncertainties in separating the twofold rc-plasmon loss the intensity of which is of the same order as the loss function itself at this energy. On the other hand, a s t rong q-dependence of e~ in the 15 eV region at larger scattering angle has been found [76]; an indication of such effect in these energy loss data may exist. Fo r he) > 26 eV the dotted curve is the extrapolated one used for the K K A of the optical data. 9 Springer Tracts Modern Physics 54

130 d. Daniels et al.:

For EILc, experimental difficulties arise in measuring the normal incidence reflectivity with light because of the lack of surfaces parallel to the c-axis. An analysis of the non normal reflectivity has been carried out for hm < 5 eV [31], giving n- and k-values for Ell c. Some values of R II calculated from these data are indicated by crosses in Fig. 36. In the higher energy region (12 to 30 eV), preliminary results seem to confirm the present energy loss data [64].

3.4.3 MoS 2

Measurements have been made on Molybdenite in a similar way as described for graphite. The loss functions obtained for q• and for q II c are given in Fig. 37.

i i i i

-Ira ~'e•

2 - Im~.

2

/ I I i _ _ ~ _ _ o

x3 1

/ t I I I L i 0

0 5 10 15 20 25 30 ev �9 "h~0

Fig. 37. The energy loss function of MoS z for q L c-axis ( - Im 1/e • and q II c ( - Im 1/d I) [77]

A plasmon peak can be seen at about 23 eV, which is nearly independent of the direction of the electric field. Assuming that the 18 valence electrons of the molecule are free, the energy of this plasmon becomes 21.5 eV, which is in good agreement with the observed value. The interband transitions have no much influence as in the case of other semiconductors (see Sec. 3.3). A strong anisotropy appears however at lower energy: a sharp peak arises at 8.75 eV only for E J_ c, which has been interpreted

O p t i c a l C o n s t a n t s o f Sol ids by E l e c t r o n S p e c t r o s c o p y 131

30 i r I i ~ i

l Mo S 2 20 Eic

o

-1C t p p I f I 0 5 10 15 20 25 ) ~ ( n 3 0 eV

20 t l t I I I t I

Jt fr r 1 EI IE~ i t I

/l',i ~ E,c

'I 2 x i 0

0 i - - t 1 - E - I

-101 a ~ 110 I I [ I

5 15 20 25 ~o} 3 0 e V

Fig. 38. T h e d i e l ec t r i c c o n s t a n t s o f M o S z d e r i v e d f r o m the e n e r g y loss e x p e r i m e n t [ 7 7 ]

as a n-electron plasmon involving 6~-electrons [45]. For E lJ e, this peak vanishes and a different structure is observed, see the lower curve in Fig. 37, as well as an additional maximum near 15 eV.

The dielectric tensor components ~• and e H are given in Fig. 38. For the KK-analysis, the value e~(~0~0)= 14.9 has been used obtained by extrapolating the refractive index n • [41]. The result is then consistent with these data up to 3.5 eV. For Ejt c, the value ~l(~o~0)= 9 is estimated giving best consistency in the analysis Ref. [77]. Since the result for e rl(h(~ < 3 eV) depends on this value, the curves are given as dashed line in this energy region.

A typical plasmon behaviour can be seen in the e-curves near the position of the strong peaks of Im 1/~, namely a decreasing and small e~ with detailed structure, and an increasing e~, zero near the plasmon 9*


energy. The situation of the rc-plasmon can be compared with that in graphite: The calculated energy is hop = 12.4 eV for 6 free electrons, but it is shifted to 8.75 eV by the interband transitions at higher energies.

The transitions between the a-states of the valence and the conduction band may be responsible for the broad maximum in e~ near 11 eV, while the 7z--+ ~ transitions cause the structure at lower energy for E Zc. This is supported by evaluating the sum rule using ~ (Fig. 39, dashed) which shows a step near 9 eV. At this energy the oscillator strengths of the ~z-electrons are mainly exhausted. Such a separation is not possible for Ell c. The neff-curve obtained from the loss function is not different for both directions, because it depends essentially on the high energy part of the loss function where no anisotropy is found.

2 0 i i i

neff MoS2

10 neff(~) ..

"""/'" " ~.~n eff(E ~

0 Y' '-------~' ' , ' 0 10 20 ~'h~ 30eV

Fig. 39. Effective number of electrons contributing to 82 and to the loss function. Different curves for EZc and E II c are obtained in the low energy region for ~2 [77]

In the energy band scheme as proposed in [10], the valence band is divided in a narrow one near the Fermi energy and a broader one about 2 eV below. The peaks in ~ at 2.5 and 4.2 eV could be related to transitions from these bands into the nearest conduction band. In optical experiments on the other hand, some sharp peaks near the absorption edge are identified as excitons [26, 21]. This structure is not resolved in the present data. The structure in e~ at 15 eV seems to be related to transitions between the bands which are built up from the 3s and 3p orbitals of the sulphur atom [10]. For a more detailed discussion however, a complete band calculation for the whole Brillouin-zone and the knowledge of the selection rules would be required.

4. Conclusion

The preceding summary demonstrates that the inelastic interaction of fast electrons with a solid, leading to a variety of excitations of the crystal is now well understood. The theoretical description, including


retardation, fits quantitatively the observed data. The complex dielectric constant, and its spectral dependence derived from these energy losses, is in good agreement with the dielectric constant derived e.g. from reflectance measurements with light. The described experimental technique thus opens the possibility to measure optical constants over a large energy region in a reliable way. It allows us to determine these data under circumstances where the optical methods have difficulties e.g. in energy regions near the plasma frequency or in anisotropic substances.

The agreement of the dielectric constants derived from energy loss and from optical experiments underlines that the longitudinal and the transverse dielectric constant are equal. A further point is the statement that the dielectric constants ofAg, Pd, GaAs etc. derived from loss spectra in scattering angles ~ ~ 1 mrad, are the same as those for 0 = 0. In other words: they are independent of the momentum h q transferred from the incoming electron to the crystal electrons till values of q ~ (1/10)qBr ( h q s r momentum corresponding to the whole Brillouin-zone; energy of the electrons 50 keV). This result, obtained only for these substances, may not be valid in general. This is of interest since in graphite a q-dependence of ~ has been observed at large 0(q ,-~ qar)" Apparently the small q-values applied in the experiments described above have no influence on the loss spectrum probably on account of the band scheme of these substances or, at least, the effect is so small that it lies within the experimental error.

Further progress will be reached by increasing the experimental accuracy in determining the loss function so that details in its structure become still more reliable. At the same time we hope that the optical measurements will become more accurate so that one may draw information from possible discrepancies between optical and electron energy loss data.

We would like to thank Prof. M. Cardona for a critical reading of the manuscript.

References

1. Baldini, G.: Phys. Rev. 128, 1562 (1962). 2. Bassani, F., Pastori Parravicini, G.: Nuovo Cimento B 50, 95 (1967). 3. - - TosattL E.: Phys. Letters A 27, 446 (1968). 4, Bea#lehole, D.: Proc. Phys. Soc. 85, 1007 (1965). 5. Blochin, M. A.: Physik der R6ntgenstrahlen. Berlin: VEB Verlag Technik 1957. 6. Boersch, H., Geiger, J., Stickel, W.: Phys. Rev. Letters 17, 379 (1966). 7. Carter, J. G., Huebner, R. H., Harnm, R. N., Birkhof f R. D. : Phys. Rev. 111,782 (1968). 8. Canfield, L. R., Hass, G., Hunter, ~q R.: J. Phys. 25, 124(1964). 9. Cardona, M.: In: Semiconductors and Semimetals. NewYork-London: Academic

Press 1966. Vol. 3, p. 125.


10. Conell, G. A. N., Veq'lson, J. A., Yoffe, A. D.." J. Phys. Chem. Solids 30, 287 (1969). 11. Cooper, B. R., Ehrenreich, H., Philipp, H. R. : Phys. Rev. 138, A 494 (1963). 12. Creuzbur 9, M.: Z. Physik 196, 433 (1966). 13. Daniels, J.: Z. Naturforsch. 21 a, 662 (1966); Z. Physik 203, 235 (1967). 14. - - Z. Physik 227, 234 (1969). 15. - - Dissertation Universit~it Hamburg (1969). 16. Duckett, S. W., Metzger, P. H.: Phys. Rev. 137, A 953 (1965). 17. Eby, J. E., Teegarden, K. J., Dutton, D. B.: Phys. Rev. 116, 1099 (1959). 18. Ehrenreich, H.: In: Optical Properties and Electronic Structure of Metals and Alloys.

Amsterdam: North Holl. Publ. Co. 1966. p. 109. 19. - - Philipp, H. R.: Phys. Rev. 128, 1622 (1962). 20. Ergun, S.: Nature 213, 135 (1967). 21. Evans, B. L., Young, P. A.: Proc. Roy. Soc. (Lond.) A 284, 402 (1965). 22. Festenberg, C. v.: Z. Physik 214, 464 (1968). 23. - - Z. Physik 227, 453 (1969). 24. Finkenrath, H., K6hler, H..' Z. Angew. Physik 19, 404 (1965). 25. - - Sch~fer, G.-H.: Phys. Stat. Sol. 34, K 95 (1969). 26. Frindt, R. F., Yoffe, A. D. : Proc. Roy. Soc. (Lond.) A 273, 69 (1963). 27. Fr6hlich, H., Pelzer, H.: Proc. Phys. Soc. (Lond.) A 68, 525 (1955). 28. Geiger, J. : Elektronen und Festk6rper. Braunschweig: Fr. Vieweg u. Sohn 1968. 29. - - Wittmaack, K.: Z. Physik 195, 44 (1966). 30. Greenaway, D. L., Harbeke, G.: Optical Properties and Band Structures of Semi-

conductors. London: Pergamon Press 1968. 31. - - - - Bassani, F., Tosatti, E.: Phys. Rev. 178, 1340 (1969). 32. Hartl, W. A. M.: Z. Physik 191, 487 (1966). 33. Hass, M.: In: Semiconductors and Semimetals. New York-London: Academic Press

1967. Vol. 3, p. 3. 33a. Higginbotham, C. W., Pollak, F. H., Cardona, M.: Proc. Int. Conf. Physics of Semi-

conductors, Moscow, Vol. I, p. 67 (1968). 34. Horstmann, M.: Fortschr. Physik 16, 75 (1968). 35. Hubbard, J.: Proc. Phys, Soc. (Lond.) A 68, 411 and 976 (1955). 36. Iehikawa, Y. H.: Phys, Rev. 109, 653 (1958). 37. KeiI, P.: Z. Naturforsch. 21a, 503 (1966). 38. - - Z. Naturforsch. 23a, 336 (1968). 39. - - Z. Physik 214, 251 (1968). 40. Krfger, E.: Z. Physik 216, 115 (1968). 40a. Kunstreich, S., Otto, A.: Optics Commun. 1, 45 (1969).

41. Landolt-B8rnstein: Band II, 8. Teil./2-159 (Springer-Verlag, 1962). 42. La Villa, R., Mendlowitz, H.: Appl. Opt. 4, 955 (1965); 6, 61 (1967). 43. Leder, L., Suddeth, J. A.: J. Appl. Phys. 31, 1422 (1960). 44. Liang, W. Y. : Phys. Letters A 24, 573 (1967). 45. - - Cundy, S. L.: Phi l Mag. 19, 1031 (1969). 46. Lohff; J.: Z. Physik 171,442 (1963). 47. Marton, L., Toots, J. : Phys. Rev. 160, 602 (1967). 48. Philipp, H. R., Ehrenreich, H.: Phys. Rev. 129, 1550 (1963). 49. - - - - Phys. Rev. 131, 2016 (1963). 50. - - - - J. Appl. Phys. 35, 1416 (1964). 51. - - Taft, E, A.: Phys. Rev. 136, A 1445 (1964). 52. Phillips, J. C.: Solid State Phys. 18, 55 (1966). 53. Pines, D.: Elementary Excitations in Solids. New York-Amsterdam: W. A. Benjamin

Inc. 1963. 54. PowelI, C. J.: J. Opt. Soc. Am. 59, 738 (1969).


55. Raether, H.: Springer Tracts Mod. Phys. 38, 85 (1965). 56. - - Helv. Phys. Acta 41, 1112, t968. 57. Ritchie, R. H. : Phys. Rev. 106, 874 (1957). 58. Roberts, R. A.. Technical Report Univ. of California (Jan. 1967). 59. Robin, S.: In: Optical Properties and Electronic Structure of Metals and Alloys.

Amsterdam: North Holl. PuN. Co. 1966. p. 202. 60. Robins, J. L.." Proc. Phys. Soc. (Lond.) 78, 1177 (1961). 6l. Roessler, D. M.." Brit. J. Appl. Phys. t l , 1313 (1966). 62. - - Private communication. 63. Sasaki, T.: J. Phys. Soc. (Japan) 18, 701 (1963). 64. Skibowski, M.: Private communication. 65. Sonnta 9, B.: Dissertation Hamburg 1969. 66. Staib, Ph., Ulmer, K.: Z. Physik 219, 381 (1969). 67. Taft, E. A., Philipp, H. R.: Phys. Rev. 138, A 197 (1965). 68. Tosatti, E., Bassani, F.: Private communication. 69. - - Nuovo Cimento B 63, 458 (1969). 70. Vehse, R. E., Arakawa, E. T.: ORNL-Report ORNL-TM-2240. 70a. Venghaus, H.: To be published. 71. Watanabe, H.. J. Electr. Microscopy. 4, 24 (1956). 72. Watanabe, M., Nakarnura, Y., Nakai, I~, Murata, T.: J. Phys. Soc. (Japan) 24, 428 (1968). 73. Whetten, N. R.: Appl. Phys. Letters 8, 135 (1966). 74. Yu, A. Y.-C., Spicer, E. W.: Phys. Rev. 171, 834 (1968). 75. Zeppenfeld, K.: Phys. Letters A 25, 335 (1967); Z. Physik 211,391 (1968). 76. - - Optics Commun. 1, 119 (1969). 77. - - Dissertation Universit~it Hamburg 1969. 78. - - Raether, H.. Z. Physik 193, 47l (1966).

Dr. J. Daniels Dr. C. yon Festenberg Prof. Dr. H. Raether Dr. K. Zeppenfeld Institut fiir Angewandte Physik der UniversitM D-2000 Hamburg 36, Jungiusstral3e II

dielectric function eels optic solid

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