Optical constants of chromium
by Lawrence Chung-kuen Chor
A thesis submitted to the Graduate Faculty in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE in Physics
Montana State University
© Copyright by Lawrence Chung-kuen Chor (1968)
Abstract:
The frequency-dependent optical constants are derived in the range 0 - 25 eV for Chromium film by
reducing the normal incidence reflectance data with the Kramers-Kronig relation.
The film is prepared in a high vacuum reflectometer having a base pressure of about 5 x 10^-5 torr. The
film is irradiated with a H2 discharge light source between 850 Å and 3100 Å, and with a He source
below 850° dispersed by a vacuum ultraviolet monochromator. Because of the high energy
measurements, the reflectometer is connected directly to the monochromator without a window in
between.
The plasma energy of Chromium, found at 22.8 eV, agrees reasonably well with the published results
obtained from characteristic energy loss experiments. OPTICAL CONSTANTS OF CHROMIUM
by
LAWRENCE C h u NQ=KUEN CHOR
A thesis submitted to the Graduate Faculty in partial
fulfillment of the requirements for the degree
of
MA STE R OF SCIENCE
in
Physics
Approved:
Head, Major Department
MO NT ANA STATE UN IV ERSITY
Bozeman, Montana
March 1968
-iiiACKNOWLEDGMENT
I
Dr.
wish to express my sincere thanks and appreciation to
Gerald J . Lapeyre for his advice and constructive c r i t i ­
cism throughout the course of this study.
I would also like
to express my gratitude to D r s . Arthu r J. M. Johnson and
Georgeanne R. Caughlan for their parts
possible.
I am also thankful for the typing help of M r s .
Lyle Hammer,
agement.
in making this thesis
and to my wife for her understanding and e n c o u r ­
•'ivTABLE OF CONTENTS
CHAPTER
PAGE
List of Tablss
List of Figures
I.
II.
III.
IV.
v
. . . ...................
vi
A b s t r a c t ...................................
vii
■ INTRODUCTION
...................
I
T H E O R Y ......................................
3
■The Definition of Optical Constants
. .
The Technique of M e a s u r e m e n t ...........
The Energy Loss Function Im e ~ . . . .
The Computer Program for the
Evaluation of the Kramers-Kronig
Integral
.....................
3
6
8
• EXPERIMENTAL SET-UP AND APPARATUS
. . .
12
General Experimental Arrangement . . . .
Reflectometer
E v a p o r a t o r ................................
12
13
16
" R E S U L T S ......................
Data R e d u c t i o n ............................
Experimental Results . . . . . . . . . .
Discussion ............................... ■.
V.
9
' S U M M A R Y ............
REFERENCES
. '....................... . . .
17
17
18
24
29
30
-vLIST OF TABLES
Page
Table I
Comparison of the plasma energy values
obtained from the optical measurement
and the energy loss measurements
. .
27
-viLIST OF FIGURES
Figure
.
Page
1
Plan Vi e w of the Reflectometer
2
Reflectance Data of C r .................
19
3
The Spectral Dependence of the Complex
Dielectric Constant . . . ............
20
The Spectral Dependence of the Energy
Loss F u n c t i o n .........................
21
The Spectral Dependence of the
Optical Conductivity
................
22
Comparison of the Computer Output with
the Data of Lenham and.Treherne . . *
23
4.
5
6
. . . .
14
-vii "
ABSTRACT
The frequericy-dependent optical constants are derived in
the range 0 - 25 eV for Chromium film by reducing the normal
incidence reflectance data with the K r a m e r s -iKronig r e l a t i o n .
The film is pre pared in a high va cuum re flectometer having a
base pressure of about 5 x 1 0 ™^ torr.
The fijm is irradiated
with a H 2 discharge light source between 850 A and 3100 A,
and with a He source bel ow 850° dispersed by a va cuum u l t r a ­
violet m o n o c h r o m a t o r . Because of the high energy measurements,
the reflectometer is connected directly to the monochromator
without a wi nd ow in b e t w e e n .
The plasma energy of Chromium, found at 22.8 eV, agrees
reasonably well with the published results obtained from
characteristic energy loss e x p e r i m e n t s .
I.
INTRODUCTION
Within the last d e c a d e , the optical properties o£ s e m i ­
conductors and metals,
particularly the alkali and noble
metal s,1 have been investigated rather thoroughly with r e f l e c ­
tance measurements.
Little information is available,
on the optical behavior of the transition metals
spectral range above 5 eV.
however,
in the
Cr is selected for this study
because it is typical of this group.
The object of the study
.of the optical properties of these metals
is to gain an
understanding o f ^ the electronic structure in the solid form.
The optical properties of a solid in the ultraviolet
region are determined principally by the interaction of
electromagnetic radiation with the electrons
Classically,
in the solid.
the interactions are described in terms of the
optical constants defined by Maxwell's
equations.
Since
the definitions of the constants are not unique, many different
constants are found in the literature.
To characterize a
response of a solid to radiation in a limited frequency range,
two frequency-dependent constants are needed,
phase,
one for the in-
(energy a b s o r b i n g ) , .and one for the out-of-phase,
energy a b s o r b i n g ) , response of the electron.
constants can be writt en as a complex number.
(non­
The pair of
If one optical
-2constant is k n o w n , h o w e v e r , for all frequencies,
any pair of
constants can be calculated with the Kramers-Kronig d i s p e r ­
sion relations.
This paper is concerned with the determination of the
optical constants of Cr obtained from normal incidence
reflectance data.
Data are obtained in the spectral range
from 4 eV to 25 eV.
The data of Lenham and Treherne are
used to extend the measurements below 4 eV..2
The real and
imaginary parts of both the index of refraction and the
dielectric constants,
the optical conductivity,
and the
energy loss function are deduced from the reflectance data
by the application of the Kramers-Kronig relation.
The
computer pro gra m for the evaluation of the Kramers-Kronig
integral is taken from the work of E. L. K r e i g e r , D. J.
Olechna,
and D . S . S t o r y . 3
In Section II, different definitions of the optical
constants wi th their interrelations are outlined and physical
interpretations given.
The computer programming techniques
are also outlined in Section II.
The equipment designed and
constructed for this experiment is described in Section III.
In Section IV, the experimental results are prese nt ed and
discussed wi th emphasis on the pl asma energy.
II.
A »
THEORY
Definition of Optical Constants
In a m e d i u m , Maxwell's
electromagnetic wave equations
are
eIy 92S , 4mop 9@
- .2 + C 2 9t
c2■ 91
CD
g2S
F r at2
(2 )
and
B 1P
V iH
in which
4mon 9?I
91
C2
is the real part of the dielectric constant,
is the conductivity of the medium,
a
and p is the permeability.
The constants are frequency dependent and characterize the
me diu m through which the wave is propagating.
magnetic materials,
For n o n ­
such as C r ,■ p = I .
The solutions of E q s . (I) and
S = Sqx
exp[iu)(t~z)]
S =
ex p[i w(t -^ z) ]
(2) can be w r i tt en as
(3)
and
,
(4)
where N is the complex index of refraction and z is taken as
the direction of propagation.
The complex index of refraction N is defined by
4
N
T-.
n - ilc
(5)
where n is the usual index of refraction and k is the
extinction c o e f f i c i e n t .5
The following relations between the electromagnetic and
optical constants are obtained by substitution of the plane
wave solution
(3) into the wave equation
of the definition
(I) and making use
(5):
n2 - k2
(6)
2 n k
(7)
and
4TTO
CO
Another useful wa y to describe the effects of the
electromagnetic wave on the solid is to define a complex
dielectric constant
8 (co) -=
With this definition,
•
the wa v e equation
(2) may be written
as
V 2B =
4
'
c% St^ .
On substituting the plane wave solutions,
■relation is obtained:
the following
Thus
= -2nk
(9)
Combining E q s . C7) and
(d), we h a v e :
coe2
a =
(10 )
4 TT
The extinction coefficient,
k , is a measure of the
decrease in the amplitude of the electromagnetic wave.
It
■is simply related to the absorption coefficient wh i c h is a
measure of the electromagnetic energy, absorbed per unit
distance and is defined by:
a =
I
s
dill
where s is the Poynting vector.
and
(4) into
(H)
dz
By substituting E q s . (3)
(11), one can express a in terms of k:
47rk
A
(12)
which is the rate of decay as the wave travels through the
medium.
-
Of particular interest is the optical constant whi ch is
most naturally related to the number of photons absorbed per
unit time.
Using the correspondence principle,
S p i c e r ,7 by
- 6equating the classical and quantum mechanical expressions for
the energy density associated w i t h an electromagnetic field,
has shown that
1_ 1_ dN]
(14)
4Tr n ' dt
where N
is the density of photons.
Therefore, a is the
pr obability per unit time of a p ho to n being absorbed by the
solid.
Since Eg Is simply related to
0
, it also describes
the absorption behavior of a solid.
B.
The Technique of Measurement
The measurement technique used in this study obtains
the optical constants from the normal incidence reflectance
data, R ( w ) .
The reflectance is the ratio of the reflected
to incident power and is equal to the square of the ratio of
reflected to incident amplitude,
r.
The optical constants
are related to reflectance by the Fresnel equation for normal
i n c id en ce ,8
r - r1/2 e16 = g ; g
; i
(15)
if the interface is between vacuum and solid..
tance is known for all frequencies,
If the r e f l e c ­
the phase angle 6 can be
calculated wi th the Kramers-Kronig r e l a t i o n : 9
-7-
2a).
£nR (co)
dw
2 2
o (Ji)
(16)
Wg
where P stands for principal v a l u e s .
range of laboratory data is limited,
Since the spectral
an extrapolation procedure
is required to evaluate the integral in (16) .
The e xt ra po la ­
tion procedure is discussed in Section II, D.
For computer
integration,
it is convenient to write
(16)
in the following
way . i o
I
2tt
6<u0)
Using equations
(15),
(17)
,
(6), and
(9), the complex index of
refraction and complex dielectric constant are obtained in
terms of R and 6:
(18)
1+ R-2 R1/2Cos6
- 2R1/ 2sin9
. '
(19)
1+R-2R1/2C o s 6 " ’■
(I-R)2 - IR s i n 2G
(2 0 )
(1+R - 2R1/2C o s 6)2
4R1 / 2 (1-R)sin9
(1+R
(21)
2R1/2C o s 0)2
Thus measurements of the spectral reflectance over a large
energy range are used to calculate the optical constants.
-8C•
The Energy Loss' Function Ime
It is known that when a beam of electrons of fixed energy
is incident upon a thin film of m a t e r i a l , some of the t r a n s ­
mitted electrons will lose energy by a discrete a m o u n t .
The
energy loss spectrum is characteristic of the target material
The determination of the characteristic energy losses is
usually achieved by measurements of scattering through thinsolids..12
The same information can also be obtained through
the knowledge of the optical constants.
This, is due to the
fact that the energy loss probability function P(E),
the
probability that an electron will loose an amount of energy
between E and E + dE in interacting with the medium,
can be
related to the complex dielectric constant in the following
w a y :1 3 ■
P(E)
oc Im e
.
(24)
Thus the plasma energy given by the peak position in the
energy loss measurements
should correspond to the plasma
energy obtained op t i c a l l y . 14
-c2 / (.ei + e P >
Since
orc^er for Ijn £
Im e ^ is equal to
to have a maximum,
necessary that both c^ and e^ be small.
In fact,
it is
the general
condition for the existence of plasma frequency is that
c = O. 15
For the simple free electron model, w h i c h approximates
-9 metals rather well,
the plasma energy is given by
AE = h U p = h ( - Pe-^l1/ 2 .
where
is the plasma f r e q u e n c y , p the electron density,
the electronic change,
D.
(25)
e
and m the electronic mass.
The Computer Program for the E v a l u a t i o n "of the K r a m e r s Kronig Integral
The computer program used in this study to evaluate the
phase angle 0 with the Kramers-Kronig integral.is
the work of K r e i g e r , O l e c h n a , and S t o r y .3
6, the pr og ram computes n, k,
taken from
After calculating
, Im e'
and a.
The
rationale for the computer prog ra m is outlined in t h e .following
discussion.
'
in
The difficulty found in the evaluation of the integral
(17)
is due to the fact- that the experimental curve for
the reflectance cannot be written in terms of elementary
functions.
Even though it is possible to fit a polynomial to
the experimental reflectance curve., we may still have problems
in integration.
A n alternative wa y is to do the integral
graphically, whic h is very time-consuming and inaccurate.
The best procedure seems to be a numerical integration with a
digital computer.
The integral
in
(17) can be approximated
by a trapasoidal sum as the following:
-IO-
0 Cw O^
^ kN ^x N-I
J-
" xN-1
where the slope,
XN ~ C I I comes from the term
I _3 J Z
d SnR(w)/dm, and
^ N - ’
is related to the term
&n |(o) + a) ) / (or CD )| da) in' (17) . 3
The input- of the computer prog ra m is N pairs of E and R
picked off at an interval of 0.2 eV from the Jln |R | vs. E
curve.
The output is computed at 0.2 eV intervals.
The first
value of E, or E ^ , must not be zero, but must be close to
zero so that the slope
is zero.
evaluated for every desired putput.
is printed,
The integral in
(17)
is
After the first output
the progr am computes the next required output
until all the desired outputs are obtained.
If N pairs of
the energy and reflectance values are the input to the computer
and results are wanted for M values of the energy,
the approxi^
mate computer time can be estimated as N x M/60 minutes.
In
this study the computer time for a complete calculation is
about one hour.
For evaluation of the integral
in
(17), it is
necessary that the reflectance be known over the entire energy
range,
i.e.
from zero to infinity.
In this work,
the experi"
mental values of reflectance are known up to only 25 e V , thus
-11an extrapolation for the high-energy- values
is necessary.
The
reflectance is usually extrapolated linearly in a plot of
&nR^/^ vs 5-nE.16
Without an adequate extrapolation,
the
values of the optical constants are inaccurate and may even
appear to be physically unreasonable.
The extrapolation is
adjusted in such a wa y that the results of the computation
agree with the independently determined low-energy values of
the optical c o n s t a n t s .
In this w o r k , the reflectance is
linearly extrapolated to 0.19% at 100 eV.
<?*
III.
A.
EXPERIMENTAL SET-UP AND APPARATUS
General Experimental Arrangement
The experimental determination of the optical constants
from the reflectance data requires
cleanest possible surface.
that the sample have the
The technique used in this study
for obtaining a clean surface is to form the sample by e v a p o r a ­
tion in vacuum.
This experiment
is concerned primarily with
the optical constants at high photo n energies where no material
exists which can be used for a w in do w to separate the e x p e r i ­
mental vac uum chamber from the m o n o c h r o m a t o r .
windowless
construction,
Because of this
the best va cuum in the r e f lectometer
is limited by the vacuum in the monochromator wh i c h is about
5 x 10 ^ torr.
Therefore the sample is conceivably subject
to some surface contamination.
To minimize the contamination,
the experiment is perfor med as soon as the film is m a d e .
The light source used is a one-meter Mc P h e r s o n model 225
vacuum ultraviolet monochromator which is equipped with a
Hinteregger type DC discharge lamp.
850 - 3100 A, hydrogen is used;
used.
In the spectral range
and for 450 - 900 A, helium is
A quartz filter is used to eliminate the second order
O
effect in the hydrogen spectrum above 1600 A.
The experiment
O
is terminated at 3100 A
(4 e V ) , where the quartz filter begins
to transmit higher order reflections from the grating in the
monochromator.
-13The detector used inside the r e f lectometer is a sodiumsalicylate-coated photomultiplier tube.
The current output
of the detector is mea sur ed with a Gary model 401 vibrating
reed electrometer.
B . . Reflectometer
The r e f l e c t o m e t e r , which is connected to the m o n o c h r o ­
mator,
is shown in Fig.
I.
It is evaculated through the slit
assembly with the monochromator v ac uu m system.
The best
pressure in the monochromator is about 5 x I O ^
torr.
After
the light source is turned on, the vacuum in the r e f lectometer decreases
to 10
-4
torr.
The reflectometer is designed in such a w a y that the same
photomultiplier can be positioned to read either the reflected
or the incident light intensity.
This
is achieved by s u p p o r t ­
ing the photomultiplier on a rotary feedthrough moun te d on the
top flange of the r e f l e c t o m e t e r .
It can be rotated to the
incidence beam position or to the reflected beam position.
The substrate is fixed at the proper angle on the optical
axis.
The evaporator is positioned inside a stainless-steel
shield so the.deposition is directed toward the substrate.
During evaporation the photomultiplier is rotated behind the
evaporator shield.
The electrical feedthrough for the p h o t o ­
multiplier and for the evaporator are brazed into the top
y/\shield
light beam
from monochromator
r
X v^Waporator
filter
-- ^
ii smbstrate
incident /beam position
of photomultiplier
xX
reflected beam position of
photomultiplier
viewport
Fig.
I
Plan View of the Reflectometer
-14-
_/
-15flange of t h e ■r e f l e c t o m e t e r .
A viewport is mounted on the
end of a side arm on the reflectometer for observation.
The photomultiplier used is a RCA 1P28 tube.
The window
of the photomultiplier is coated with sodium salicylate by an
atomizer.
The coating is thick enough to make the window
become opaque to the eye.
is 850 v.,
The photomultiplier bias voltage
so that each stage has about 100 v. across
it. • A
series of 87 Kfi resistors are used in a voltage divider to
achieve this dynode bias.
The reflectometer and its components are cleaned and
assembled with conventional vacuum t e c h n o l o g y . 17
The s u b ­
strate is made of copper and requires special handling.
It
is polished w it h aluminum oxide to obtain a smooth surface.
It is then cleaned by dipping into concentrated nitric acid
and rinsed with deionized water.
The measuring procedures start immediately after e v a p o r a ­
tion of the Cr film is completed.
The photomultiplier is
rotated into the path of normal incidence and the intensity
of the He light source is me asured from 450 - 900 K.
Then the
tube is rotated into the path of the reflected b e a m from the
sample- and its spectral intensity is likewise measured.
gas then replaces
The
He gas for longer wa ve length measurements
The spectral dependences of the lamp and the detector cancel
because the dependences are the same In both the measurements
-16o£ incident and reflected spectral intensities.
C.
Evaporator
The Cr film is formed by condensation from the vapor
phase.
The evaporator is made by electroplating C r , 18 onto
a 0.009" diameter tungsten wire.
The tungsten wire
is cleaned
by electropolishing in NaOH s o l u t i o n , and then place d in the
chromic acid solution for electroplating.
The tungsten wire
is connected as the cathode and a pl atinum screen basket is
used for the a n o d e .
A plate current of 6 0 0 :ma is used and
the solution is kept at a temperature of about 53°C by a
controlled heater.
The temperature and the current must be
carefully controlled for good results.
After electroplating
for three to four h o u r s , a mass of about 100 mg of Cr is
plated on the tungsten wire.
The evaporator is then ready for
use after it is rinsed with deionized water and degreased in
acetone. •
-
•
■
The procedure for forming the film is to outgas the
evaporator first by slowly increasing the c u r r e n t .
Then the
evaporator is evaporated for about ten seconds to obtain a
smooth and bright Cr film on the substrate.
about 4 amperes
evaporation.
A current of
is required to give a sufficient rate of
IV.
A.
RESULTS
Data, Reduction
The reflectance at a certain energy is Obtained by
measuring the relative
intensities of reflected and incident
light at that e n e r g y ;
The ratio of these intensities is the
reflectance.
Since the spectral dependence of the lamp is a
strong f u n ct io n 1of the
wavelength, matching the wavelengths
of the two spectral curves
is very important.
The w a v e ­
lengths of the spectra are correlated by matching peaks and
valleys.
In the data reduction the scattered light in the output
of the monochromator must be taken into consideration.
the Hg lamp is used,
When
the scattered light is negligible in
comparison with the monochromatic l i g h t .
For the data taken
at short w a v e l e n g t h s , where He is used in the lamp,
tered light must be accounted f o r .
He terminates at about 500
A,
the s c a t ­
The light spectrum of
thus it is justifiable to assume
that all the signals below 400 A are due to the scattered
light.
It is observed that the scattered light level increases
linearly with decreasing wa ve le ng th below 400
tered light correction,
to long w a v e l e n g t h s .
above
700
A.
At 640
The sc a t ­
t h e n , is made by linear extrapolation
This extrapolation gives
A
A.
zero correction
(19.4 e V ) , the correction is 11.6%
of the recorder level for the incident i n t e n s i t y , and 30%
■
-J.8 ^
for the reflected i n t e n s i t y .
After the data are obtained in the above manner,
the
computer is used to analyze the data to obtain the optical
constants.
B.
Experimental Results
The reflectance data obtained in this study from 4 - 2 5
eV are shown in Fig.
2.
The reflectance is me as ur ed with two
samples and the results are essentially the same.
The r e f l e c ­
tance data between 0.1 and 4.2 eV are obtained from the work
of Lenham and T r e h e r n e . 2
The optical constants
e, Im e \
and c determined from the reflectance data are shown in Figs.
3, 4, and 5, respectively.
As discussed in Section II D,
the reflectance curve is extrapolated above 25 eV in such a
way that the computed values of n and k agree wi t h the work of
Lenham and T r e h e r n e .
The comparison is shown in Fig.
general agreement is good.
6.
The
The minor inconsistencies are
explained by the fact that at low photon energies, both n and
k are very sensitive to the reflectance.
For example,
a
difference of less than 1.0% in reflectance can change n and
k by a factor of three or f o u r . 19
Because of the.strong line structure in the He spectrum
between 20 and 24 eV, the determination of the reflectance at
these energies is very difficult.
Althou gh the structures
in
-19-
Fig.
2
Reflectance Data of Cr
150
-50
-IOOL
4
O
8
12
16
hr
Fig.
3
20
24
28
(e V )
The Spectral Dependence of the Complex Dielectric Constant
I m ('4)
-
21
-
Fig.
4
The Spectral Dependence of,the Energy Loss Function
8
-
22
-
4
O
Fig.
4
8
12
16
20
24
hZ/ ( G V )
The Spectral Dependence of the Optical Conductivity
23
LENHAM a
TREHERNE
— COMPUTED
Fig.
6
Comparison of the Computer Output
with the Data of Lenham and Treherne
-24the reflectance at 4.4 eV and 13.2 eV are r e p r o d u c i b l e ,■ it is.
not c e r t a i n , on the basis of this study,
There.is
little doubt,
that they are real.
h o w e v e r , about the main features of
the reflectance spectra.
To resolve this p r o b l e m , a more
detailed study is required.
A suggested improvement would be
O
the use of a different type of lamp below 1000 A, for e x a m p l e ,
a pulse lamp.
C.
Discussion
a, and a , w h i c h are simply
The optical constants
related through E q s . (10) , (11), and
(14), all describe the
absorption behavior in the metal. ■ Thus it is sufficient to
discuss
only one of them.
In this discussion,
o is c h o s e n .
To examine the plasma oscillations at higher energies, h o w ­
ever,
the energy loss function Im e ^ is the important
constant to be investigated.
Since an optical measurement of
the plasma energy is the principal motivation
for this study,
this point is discussed in some detail.
The spectral dependence of the optical conductivity a
is- shown in F i g . 5.
Its features are discussed in terms of
the different physical phenomena w h i c h contribute to the
energy loss in different energy regions.
types of energy loss mechanisms:
interband excitation,
There are three
free-electron excitation,
and plasma excitation.
-25The free-electron effects dominate at the low photon
energies,
from zero to about I eV,
and.cause o to be large.
According to the Crude free-electron theory of metals,
a
should fall off linearly as a function of wav el en gt h in the
free-electron region.
However,
Lenham has shown that cr falls
off as I 0 *74 for Cr., where X is- the wavelength:. ^ 0
The
threshold for interband transitions is very close to zero
photon energy since the Fermi level is in the d-band.
However,
the interband contribution is small in the free-electron
region.
This fact enables one to distinguish rather u n a m b i g u ­
ously the contributions from free-electrons and interband
transitions.
Interband transitions dominate in the range
between one and 8 eV.
The electrons excited in.interband
transition are frequently called bound electrons.
For Cr,
the interband absorption is' due mainly to transitions between
the occupied d-bands belo w the Fermi energy and the empty
d-bands above the Fermi energy. ' From the ph otoemission study,
it is found that the interband transitions in Cr are p r e ­
dominately nondirect,
that is, the conservation of the Bloch
state wavevectors is not an important selection r u l e . 21
Finally,
the region where plasma oscillations occur is c h a r ­
acterized by large values of Im c ^ and small values of
and E 2 .
This region occurs at the high photon energies.
separation of the plasma and interband transition regions
The
is
-
not well d e f i n e d .
-
G e n e r a l l y ,■the onset of the plasma o s c i l l a ­
tion is where a starts
eV for Cr.
26
to rise s h a r p l y , wh i c h .is at about 13
The'method of determination of the plasma energy
is discussed in the next several paragraphs.
The location of the plasma energies
peaks of the energy loss function Im e
is obtained from the
.
The plasma resonance
can be distinguished from the interband transition by the fact
that both
in Im E ^ .2 2
and E^ are small in the vicinity of the maximum
In Fig.
4, it is observed that the maximum peak
of Im £ ^ occurs at 22.8 eV.
The narrowness of the peak may
be due to instrumental problems,
as discussed in Section IV B .
This peak is generally accepted as due to the so-called
classical "volume" plasmons whi ch can be explained in terms
of the simple free-electron m o d e l .^ 3
The me asurement of the
plasma energy of Cr has been made by several
investigators
by means of the characteristic energy loss experiments.
results are given in Table I.
It is seen from Table
Their
I that
the agreement between the present optical measurement and the
energy loss measurements
is good.
The electronic configuration of Cr is
5 I
■
(Ar)3d 4s which
has the characteristic unfilled d - shell of the transition
metals as well as an unfilled s-shell.
Thus the Fermi energy
level is located simultaneously w i t h i n both the d- and s - b a n d s ,
a fact that greatly complicates the interpretation of the
-27Table. I .
Comparison of the plasma energy values
obtained from the optical measurement and
the energy loss measurements.
Date
Investigator
1954
Marton and L e d e r 2lt
21.8 eV
1954
W a t a n a b e 25
26.0 eV
1958
Fert and P r a d a l 26
28.0 eV
Robins and S w a n 27
24.3 eV
Present Work
22.8 eV
1960
■
1968
plasma oscillations
in Cr.
Plasma energy
Pines has suggested a two-plasma
model of coupled d and s electrons
in which the low energy
plasma is associated with an in-phase motion of s vs.
electrons;
•
d
the high energy plasma is associated w it h an out-
o f -phase m o t i o n . 28
In the energy loss function,, an additional
peak is observed at about 10 eV.
The two peaks
in the loss
function observed at 10 eV and 22.8 eV are consistent with the
two-plasma model.
The value of the plasma energy of Cr,
calculated by assuming all the 3d and 4s electrons
is 26.1 eV.
to be free,
The difference between the experimental and the
calculated free-electron-model values
is possibly due to the
effects of core polarization and the interband transitions
associated with the d-band e l e c t r o n s . 29
-28Ritchie has shown that there is a plasma loss at energy
(1//2) hu)p
Fig.
due to
the "surface" p l a s m o n s . 3 0
H o w e v e r , in
4, no peak is observed at around 16 eV, but there is
some weak structure at 13.4 eV which is about. I / /3 times
22.8 eV.
It is uncertain whether this
instrumental difficulties.
is real or just due to
If it is real,
it can be explained
by R i t c h i e ’s interpretation that "the resonance frequency of
plasma contained in a small sphere is less than the value
appropriate to an infinite plasma by a factor of 1//3 and
that this shift.is due to the depolarizing effect of surface
charge on the s p h e r e . " 31
Because of the fact that thin
metallic films.may have a strongly granular s t r u c t u r e .3 2
one would expect the surface loss in an actual thin film to
lie closer to.the value
cal grain,
(1//3) hw^,
than to the value
appropriate to a s p he ri ­
(I/ /7)' hw^.
The relative
weakness of the structure at 13.4 eV may be due to the o x i d a ­
tion of the surface whi ch reduced the intensity of the surface
loss . 3 3
V.
SUMMARY
In this s t u d y , a refloctomete'r is designed and c o n ­
structed.
It is designed in such a way that the same p h o t o ­
multiplier detector can be positioned to read either the
reflected or the -incident light intensity.
This design
eliminates the possibility of error due to. the different
spectral dependences of detectors when two or more detectors
are used.
i
The optical constants are calculated in the photon energy
range 0 - 25 eV for thin Cr film by reducing the normal
incidence reflectance d a t a .
The plasma energy is found at
22.8 eV where the energy loss function Im e
peak.
-I
has its maximum
The result of this optical experiment compares f a v o r ­
ably with the published r e s u l t s .of the characteristic energy
loss experiments.
In addition,
the free-electron effect and
interband transitions have been discussed.
REFERENCES
1.
D . B e a g l e h o l e , P r o c . P h y s . S o c . 87, 461' (1966),
cited r e f e r e n c e s ,
and
2.
A, P, Lenhdm and D . M. T r i h s r n e , in Optieal Properties
and Electronic Structure of Metals' and Alloys, edited,
by F. A b e l e s , (North-Holland Publishing Co., Amsterdam,
1966), p p . 1 9 6 t 201.
3.
E . L. K r e i g e r , D. J. Olechna, and D . S . Story, Computer
Det ermination of Optical Constants from Reflectance
D a t a , (General Electric Research Laboratory R e p o r t ) .
4.
Richard Becker, Electromagnetic Fields and Interactions,
Vol. I, (Blaisdell Publishing Co., New YorkT, 1964) ,
p. 228.
5.
N is sometimes defined as
N = n(l - k ) , where k is
called the attenuation coefficient.
6.
W. E . Spicer, Optical Properties of S o l i d s , (Lecture
notes, Stanford University, 19627".
7.
Ibid.
8.
J. C . Slater and N . H. Frank, E l e c t r o m a g n e t i s m , (McGrawHill Book Co., Inc., New York^ 1947), p .. 120.
9.
C . Kittel, Elementary Statistical P h y s i c s , (John Wiley
§ Sons, Inc., Ne w York, 1958) , p . 210.
10.
S . Wang, Solid State E l e c t r o n i c s , (McGraw-Hill Book, Co.,
Inc'. , Ne w York, 1966) , p . 6 74.
11.
E . I . Fisher, I . Fujita, and G . L . W e i s s l e r , Optical
Constants of Silver and Barium in the V a c u u m Ultraviolet Spectral Region, (Technical Report prepared
for NASA, No. US C - V a c U V - 108, May, 1966).
12.
L . Marton,
13.
T. E . LaVilla and J. M e n d l o w i t z , A p p l .
(1965).
Rev. Mod.
P h y s . 2_8, 172
(1956).
Opt.
4, 955
-
31
-
14.
H . Ehrenreich and H. R . Philipp,
• (1963).
15.
D. Pines, Elementary Excitations in S o T i d s , (W.'A.
Benjamin, Inc., New York,. 1964), p . 208.
16.
H. Ehrenreich and H. R. Philipp,
(1962).
17.
F. R o s e b u r y , Handbook of Electron Tube and V ac uu m
' T e c h n i q u e s , (Addison-Wesley Publishing Co., Inc.,
Reading, Massachusetts, 1965)., p p . 3-18.
18.
Nickel-Chr omi um P l a t i n g , Symposium, Institute of Metal
F i n i s h i n g , (Robert Draper, Ltd., London, 1961).
19.
Fisher,
20.
A. P . L'enham, J. Opt.
21.
G. J. Lapeyre and K. A. Kress,
to be published. ■
22.
H. Ehrenreich and H. R. Philipp,
(1962).
23.
Pines,
24.
j. L . Robins and J. B. Swan,
■ 7_6, 857
(1960).
25.
L . Ma rt on and L . B . L e d e r , P h y s . Rev.
26.
H. Watanabe,
27.
C . Fert and F . Pradal,
252 (1958). ■
28.
Pines,
29.
Robins,
30.
R; H. Ritchie,
31.
Ibid.
et.
P h y s . Rev.
129,
128,
1550
1622
a l . , op_. c i t .
ojd . c i t . , p.
Soc. Am.
5(7, 473
(1967).
P h y s . Rev.,
(1968),
P h y s . Rev.
128,
1622
186..
P r o c . P h y s . Soc.
F . Phys.. Soc. Japan,
op_. c i t . , p.
et.
P h y s . Rev.
C . R . Acad.
94,
9_, 920
Sci.
(Paris),
al., o]3. c i t .
106,
874
(1954).
(1954).
217.
P h y s . Rev.
203
(London),
(1957).
246,
-32- .
32.
L . B . L e d e r , H. M e n d l o w i t z , and L. Marton,
1 0 1 , 1460 (1956) .
33«
Pines,
op.
cit.* p,
186,
Phys. Rev.
MONTANA STATE UNIVERSITY LIBRARIES
CO
111I III millin I l IIIII
7 62 100 1332E 5
NITS
Choc, L. C.
C455
Optical constants
cop.?
L-f chromium
N A M g ANO A Q p R g g r
' —r,/'.I 1 <
iWtt-
/Cr ■
-r ^
r ^'? c
3 - I/ - Lct
#
,4 ;
I7
J 1- ?-
gel r
/'
/