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The optical properties of manganese thin films
PHYSICA Physica B 205 (1995) 285 290
ELSEVIER
The optical properties of manganese thin films S.S. Fouad*, A.H. Ammar Physics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
Received 16 May 1994; revised 13 July 1994
Abstract The optical constants n and k of thin manganese films of different thicknesses ranging from 20 up to 50 nm were determined in the spectral range 3.5 20 ~tm. The results were compared with the predictions of the Drude free electron theory and anomalous skin effect theory. The previous theories were used to evaluate some micro characteristics of Mn thin films such as the free charge concentration N, the relaxation time ~, the static conductivity as, and the electron velocity at Fermi surface Vv and the value of the effective area of Fermi surface Av. Using the optical constants, the optical conductivity (a = al + i~r2) was also estimated. It was found that the contribution of the effective absorption Aeff equals 69%, i.e. the absorption due to bound electrons may be 31%.
1. Introduction Investigation of the optical properties of thin metallic films is of great importance in studying the properties of solids. However, a limited data on the optical properties of Mn,thin films have been found in the literature [1 4]. In the present work an attempt has been made to determine the optical cofistants of manganese thin films in the spectral range 3.5 20 lam. The obtained optical constants of M n thin films were used in conjunction with both Drude's theory of free carriers and a n o m a l o u s skin effect theory to calculate the following microcharacteristics: (i) the free charge concentration N, (ii) the relaxation time ~, (iii) the static conductivity as, (iv) the electron velocity at Fermi surface Vv, (v) the absorption due to free and b o u n d electrons. The obtained optical constants of thin M n films have led to the determination of some recent theor* Corresponding author.
etical calculations of the interband optical conductivity as well as the lattice dielectric constant.
2. Experimental procedure and results Manganese of purity 99.97% was thermally evaporated at a pressure of 10-5 Torr onto a KBr substrate held at r o o m temperature ( ~ 30°C). In all cases the substrates were masked until the source was at evaporation temperature, and the evaporation rate was made as high as possible (5 nm s - 1) in all individual evaporations to insure the surface smoothness of the films [-5]. Several of metals including the Mn oxidize rapidly in air. To minimize this effect, the sample was measured directly after being prepared. Transmission values did not change for any of the films over at least a 2 h period. Since the measurements were all made within 10 min of the time of formation of all films, any correction would have been well within our error of measurements. Transmittance measurements were carried out at normal incidence for four
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 0 8 - 2
286
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
samples of thicknesses 20, 33, 42 and 5 0 n m using P Y E U N I C A M SP-300 Infrared Spectrophotometer. Our estimated error in t was + 0.2 nm and that in n, k was less than + 2% over most of the spectral range.
.42
12
3B
x
11 x 10
x
~
x
34
x
x
q
3° l
x
19-
2.1. The optical constants of M n thin films
E
For high absorbing thin metallic films, the transmittance can be used to determine the absorption index k using the following formula [6]: 2 ln(l/T2) - ln(1/Tt) k
=
- -
4Tt
t2 - tl
,
(1)
where 2 is the wavelength at which the transmittance T2 and TI were measured for two samples of thicknesses t2 and tl, respectively. The same procedure can be repeated for the whole spectrum under test. The refractive index n for such samples can be determined [7] using T=
16 nl
n2
(n 2 ~-
k 2)
[(r/1 4- n2) 2 4- k 2] [(n2 4- n) 2 4- k 2]
where nl and n2 are the refractive indices of air and substrate, respectively. The obtained results for n and k are illustrated in Fig. 1. The vertical bars in this figure represent the discrepancy in the calculated values of n and k of a set of films in the thickness range 20-50 nm. One can see that this discrepancy is relatively small in comparison with the whole range of wavelength. The error due to any contaminating layer on the surface is considered to be small since all our data were taken immediately after the evaporation. The obtained results indicate that both parameters n and k were independent of the film thickness t. The present results of the variation of n and k with 2, given in Fig. 1, in comparison with the results obtained by Lenham and Treherne [8] are listed in Table 1. It seems that there is a slight difference between our data and those given by Lenham and Treherne [8] for the value of 2nk/2. This difference in our opinion arises from the different experimental
x
x
26
~-
t x
8-
22 x
7.
t
6-
5:
x
.t
t
18
14
× ×
•t
10
•
6 2.5
5
7.5
10
12.5 ~,
15
17.5
20
(pro)
Fig. 1. Relation between n (o) and k ( × ) against 2 for thin Mn films.
Table 1 2 (pm)
5 7.5 10 15 17.5
2 nk/2 (pm- 1) Present work
Lenham and Treherne [8]
27.9 36.8 41.2 37.8 43.8
26 30 35 36 37
techniques used for measuring the optical constants, and conditions of preparation; the latter greatly affects the properties of the film.
2.2. Optical constants of M n thin films in conjunction with Drude's theory According to Drude's theory [9], there are two relations correlating the optical constants n and k to the wave number of the incident radiation n. These two relations are (k 2 _ n 2 +
1)-1 = (%)-2 [(~)2 + (~g)2]
(3)
287
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
and (2nk~)- 1 = (9o)- 2 (YR)- ' [(~)2 + (9R)2],
(4)
where Ne 2 (90) 2 - _ _
(5)
7zrn*c 2
(b) 3/.
(o)
32.
28
30,
o
32
iI
24
•
T
x
and
28
(6)
Here, N is the number of conduction electrons per unit volume, e is the electron charge, z is the relaxation time, m* is the effective mass and c is the velocity of light in vacuum. Fig. 2(a) represents a plot of ( k 2 - - n 2 -F 1)- 1 versus (~)2 from which one can get a linear relation of slope 1/~ 2. The plot of (2nk~) 1 versus (9) 2 (Fig. 2(b)) yields a straight line of slope equal to [(~o) -2 (gR)-1]. Accordingly, the obtained values for 90 and ~R are 1.84× l04 c m - 2 and 1.22×104cm -~, respectively. Again, using the values of ~o and 9R in conjunction with Eqs. (5) and (6), the number of conduction electrons per unit volume N and the relaxation time z are found to be 1.8 × 1 0 2 2 cm -3 and 4.3 x 1 0 - 1 6 S, respectively. Knowing the values of N and z the static conductivity a~ in Mn films can be calculated using (7)
a, = S e 2 z / m *
and was found to be as = 4.56 x 1014 (e.s.u.). Also the electron velocity at the Fermi surface VF can be determined using the following equation:
(3(%)2h3c2~1/3 8m----~Ee-~ -}
,
•
,---,_
.12
24
22
20
18
20
3'0
4'o
so
~2 xlO5 (cm -2)
6o
t,
Fig. 2. Relation between: (a) (k 2 - n 2 + 1)- ' and 92 for thin Mn films;(b) (2n kf)- 1 and ~2 for thin Mn films. where m * = m = 9 . 1 × 1 0 - 2 S g , e = 4 . 8 x l 0 -1° e.s.u, and N = 1.8 x 1 0 2 2 cm -3 was previously determined. It was found that COp= 7.57× 10 ~5 s -1 while 2p = 249 nm. The contribution of the effective absorption A~ff can be evaluated using the approximation [10] given by the relation
COpF m* ] 1/2 h e tf = ~ L - - - - ~ ] .
(10)
It was found that Aeef has a value of 0.69 (i.e. 69%). Therefore, absorption due to bound electrons may be 31%. It seems that Drude's theory agrees with the former suggestion, The Fermi energy in case of Mn polycrystalline thin films was found to be equal to 2.51 eV using the relation
h2 Ef = ~ (37z2N) 2/3,
A v = 4 X ( 3 n 2 N ) 2/3
and was equal to 8.26 × 1 0 1 6 e c m - 2 . The plasma frequency (COp= 2/~Vp = 27ZC/,~p) can be easily determined from the relation 4rtNe 2 m* '
~= 2 6
16
(8)
where h is Planck's constant. It was found that VF = 9.36x 107cms -1. The value of the effective area of the Fermi surface AF can be calculated using the relation
(°P)2-
2O t
I VR - 2ncz"
Vv = \
•
(9)
(11)
while the Fermi wave vector was found to be 8.11 x 107 cm using the relation 3rt2"] /a N 1/a. Kr = \--~F/
(12)
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285 290
288
2.3. Optical constants of Mn thin flms in conjunction with complex dielectric constant
0.9. Q.8-
It is well known that the electronic transition in a solid is more directly related to the complex dielectric constant ~ = el + iCE instead of the complex index of refraction fi = n + ik. These are related as ~ = fi2, so that e~ = n 2 - k 2 and e2 = 2nk, respectively. As the optical constants of a metal are mainly governed by the contribution of both bound and free electron regions in that metal, el can thus be written as
07. 06-
i '~
0.124 eV
e~o e°
%eQe e
.7. 0.5b
o
0.4
k "x.x.
O3 02 01 0
el = eL
e2NI
- to m* (1
"+- ( D 2 1~2
,1
O1
(13)
"
For metals at near-infrared frequencies, one finds ~o>>l/z, therefore el = eL --//22 where / / = e2N/ 47t2eom*c 2. Fig. 3 illustrates el =f(22), where the point of intercept of the straight line on the y-axis yields the required value of eL (the lattice dielectric constant); accordingly, eL = 27. Knowing the slope of the linear part of the straight line, the concentration of the free charge carriers N can
800
700
600 ®
500
400 ®
300
o
02
03
04
0.5
h v (eV)
Fig. 4. Optical conductivity in 1014s 1 for M n thin films as a function of photon energy in eV.
be calculated; the resulting value of N was = 3.86 x 1022 c m - 3 . In the intraband region of the spectrum, the optical conductivity (0. = 0.1 + i0.2) is more convenient than the dielectric constant. They are related to the components of the complex dielectric constants by 0.1 = e2(o/47z and - - 0 " 2 = ( e I - - 1 ) ~ o / 4 n . The optical conductivities 0.1 and - 0.2 calculated from the optical constants are plotted as a function of hv (eV) in Fig. 4. It is seen in Fig. 4 that 0.1 and -0.2 show peaks at 0.124 eV. These peaks may be attributed to spin-orbit splitting in view of the assumption given by Craven [11] in his study of the optical conductivity for tin thin films, as well as for Wang and Callaway 1-12] in their study of 0.1 for ferromagnetic nickel. Since we did not find any recent measurements on manganese except those previously referred to, we cannot compare our results with others.
200
2.4. Optical constants of Mn thin films in conjunction with the anomalous skin effect
100
o o
,oo
2~0 ~,;
(Fro 2 )
300 ,,
Fig. 3. Plot of the real part of the dielectric constant - e l versus (wavelength) 2.
The optical constants n and k for Mn thin films in conjunction with the theory of the anomalous skin effect are considered to determine the number of conduction electrons per unit volume N and the relaxation time t. According to Dingle's dispersion
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
relations, the second approximation may be written [13, 14] as
Table 2
k 2 _//2
- 22
= A - B2 z,
Dingle [13] Drude [9]
(14)
nk 2--x = G - D2 z,
(15)
289
N (cm-3)
z (s)
3.46 × 1021 1.8 × 1022
5.2 x 10-16 4.3 x 10 16
where
If the static conductivity a and the intercept A are known, it is also possible to determine the relaxation time from the relation
A=
z
t7 ~C2~ '
(16) G = c--3 47t22----~
The dependence of (k 2 - n 2 ) / 2 2 and (nk)/2 3 o n 22 should be linear, with intercepts yielding A and G, respectively. Figs. 5(a) and (b) represent simultaneously the two relations (k 2 - n2)/22 =f(22) and (nk)/23 = 9(22). The observed intercepts of A and G are equal to 3.1 x 10s cm -2 and 5.5 × 101° cm -3, respectively. The magnitude of A determined in this way may be used in conjunction with the expression for the static conductivity a = N e 2 r / m * to calculate the number of conduction electrons per unit volume N, provided the effective electron mass m* is known; the relation of N is given as rtc2m* A
N =
e2
(17)
5 l
0
100
200
300 h
4;0
1 0
(~um) z
Fig. 5. Relation between: (a) ((k 2 - n2)/2 2) cm -2 and 2 2 (p-m)2; (b) (n k/2) cm -a and 22 (gm) 2.
ncZA
(18)
Using these two values A and G in conjunction with Eqs. (17) and (18) the number of conjunction electron N and the relaxation time z can be evaluated. The present results of N and r obtained by using Dingle's theory in comparison with that obtained by using Drude's theory are listed in Table 2.
3. Summary A technique of measuring transmission from evaporated films to determine accurate values for optical constants has been used in the spectral range 3.5-20 gm. Our precautions used during the preparation of the Mn films is in good agreement with those previously quoted in the literature. The optical constants of Mn thin films determined by our thin film measurements were compared with Drude free electron theory and anomalous skin effect theory. Along with these two theories, the microcharacteristics, i.e. N, z, % Vv, AF, c a n be determined. We derived the optical constants, the optical conductivity and the lattice dielectric constant.
References [-1] [2] I-3] 1-4] 1-5]
M.A. Angadi, J. Mater. Sci. 20 (1985) 761. G.E. Sabine, Phys. Rev. 55 (1939) 1064. F. Bueche, J. Opt. Soc. Amer. 38 (1948) 806. R. Atkinson, Thin Solid Films 37 (1976) 195. L. Holland, Vacuum Deposition of Thin Films (Chapman and Hall, London, 1966) p. 178, 208 and 244. [6] P.B. Johnson and R.W. Christy, Phys. Rev. B 9 (1974) 5056.
290
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285 290
[7] O.S. Heavens, in: Physics of Thin Films, Vol. 2 (Academic Press, New York, 1964) p. 193. [8"] A.P. Lenham and D.M. Treherne, J. Opt. Soc. Amer. 56 (1966) 1137. I-9] P. Drude, Appl. Sci. Res. B 2 (1953) 169. [10"] V.L. Jinzlurg and G.P. Molulevich, Usp. Khi. Fiz. Nauk, 55 (1955) 469.
[11] J.E. Craven, Phys. Rev. 182 (1969) 693. [12] C. Wang and J. Callaway, Bull. Amer. Phys. Soc. 18 (1973) 397 and to be published; J. Callaway and C. Wang, Phys. Rev. B 7 (1973) 1096. 1-13] R.B. Dingle, Appl. Sci. Res. B 3 (1953) 69. [14] S.S. Fouad, M.H. E1-Fazary, A.A. E1-Shazly, F. Sharaf and K.M. Nassr, J. Mater. Sci. 26 (1991) 5843.
ELSEVIER
The optical properties of manganese thin films S.S. Fouad*, A.H. Ammar Physics Department, Faculty of Education, Ain Shams University, Cairo, Egypt
Received 16 May 1994; revised 13 July 1994
Abstract The optical constants n and k of thin manganese films of different thicknesses ranging from 20 up to 50 nm were determined in the spectral range 3.5 20 ~tm. The results were compared with the predictions of the Drude free electron theory and anomalous skin effect theory. The previous theories were used to evaluate some micro characteristics of Mn thin films such as the free charge concentration N, the relaxation time ~, the static conductivity as, and the electron velocity at Fermi surface Vv and the value of the effective area of Fermi surface Av. Using the optical constants, the optical conductivity (a = al + i~r2) was also estimated. It was found that the contribution of the effective absorption Aeff equals 69%, i.e. the absorption due to bound electrons may be 31%.
1. Introduction Investigation of the optical properties of thin metallic films is of great importance in studying the properties of solids. However, a limited data on the optical properties of Mn,thin films have been found in the literature [1 4]. In the present work an attempt has been made to determine the optical cofistants of manganese thin films in the spectral range 3.5 20 lam. The obtained optical constants of M n thin films were used in conjunction with both Drude's theory of free carriers and a n o m a l o u s skin effect theory to calculate the following microcharacteristics: (i) the free charge concentration N, (ii) the relaxation time ~, (iii) the static conductivity as, (iv) the electron velocity at Fermi surface Vv, (v) the absorption due to free and b o u n d electrons. The obtained optical constants of thin M n films have led to the determination of some recent theor* Corresponding author.
etical calculations of the interband optical conductivity as well as the lattice dielectric constant.
2. Experimental procedure and results Manganese of purity 99.97% was thermally evaporated at a pressure of 10-5 Torr onto a KBr substrate held at r o o m temperature ( ~ 30°C). In all cases the substrates were masked until the source was at evaporation temperature, and the evaporation rate was made as high as possible (5 nm s - 1) in all individual evaporations to insure the surface smoothness of the films [-5]. Several of metals including the Mn oxidize rapidly in air. To minimize this effect, the sample was measured directly after being prepared. Transmission values did not change for any of the films over at least a 2 h period. Since the measurements were all made within 10 min of the time of formation of all films, any correction would have been well within our error of measurements. Transmittance measurements were carried out at normal incidence for four
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 9 2 1 - 4 5 2 6 ( 9 4 ) 0 0 9 0 8 - 2
286
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
samples of thicknesses 20, 33, 42 and 5 0 n m using P Y E U N I C A M SP-300 Infrared Spectrophotometer. Our estimated error in t was + 0.2 nm and that in n, k was less than + 2% over most of the spectral range.
.42
12
3B
x
11 x 10
x
~
x
34
x
x
q
3° l
x
19-
2.1. The optical constants of M n thin films
E
For high absorbing thin metallic films, the transmittance can be used to determine the absorption index k using the following formula [6]: 2 ln(l/T2) - ln(1/Tt) k
=
- -
4Tt
t2 - tl
,
(1)
where 2 is the wavelength at which the transmittance T2 and TI were measured for two samples of thicknesses t2 and tl, respectively. The same procedure can be repeated for the whole spectrum under test. The refractive index n for such samples can be determined [7] using T=
16 nl
n2
(n 2 ~-
k 2)
[(r/1 4- n2) 2 4- k 2] [(n2 4- n) 2 4- k 2]
where nl and n2 are the refractive indices of air and substrate, respectively. The obtained results for n and k are illustrated in Fig. 1. The vertical bars in this figure represent the discrepancy in the calculated values of n and k of a set of films in the thickness range 20-50 nm. One can see that this discrepancy is relatively small in comparison with the whole range of wavelength. The error due to any contaminating layer on the surface is considered to be small since all our data were taken immediately after the evaporation. The obtained results indicate that both parameters n and k were independent of the film thickness t. The present results of the variation of n and k with 2, given in Fig. 1, in comparison with the results obtained by Lenham and Treherne [8] are listed in Table 1. It seems that there is a slight difference between our data and those given by Lenham and Treherne [8] for the value of 2nk/2. This difference in our opinion arises from the different experimental
x
x
26
~-
t x
8-
22 x
7.
t
6-
5:
x
.t
t
18
14
× ×
•t
10
•
6 2.5
5
7.5
10
12.5 ~,
15
17.5
20
(pro)
Fig. 1. Relation between n (o) and k ( × ) against 2 for thin Mn films.
Table 1 2 (pm)
5 7.5 10 15 17.5
2 nk/2 (pm- 1) Present work
Lenham and Treherne [8]
27.9 36.8 41.2 37.8 43.8
26 30 35 36 37
techniques used for measuring the optical constants, and conditions of preparation; the latter greatly affects the properties of the film.
2.2. Optical constants of M n thin films in conjunction with Drude's theory According to Drude's theory [9], there are two relations correlating the optical constants n and k to the wave number of the incident radiation n. These two relations are (k 2 _ n 2 +
1)-1 = (%)-2 [(~)2 + (~g)2]
(3)
287
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
and (2nk~)- 1 = (9o)- 2 (YR)- ' [(~)2 + (9R)2],
(4)
where Ne 2 (90) 2 - _ _
(5)
7zrn*c 2
(b) 3/.
(o)
32.
28
30,
o
32
iI
24
•
T
x
and
28
(6)
Here, N is the number of conduction electrons per unit volume, e is the electron charge, z is the relaxation time, m* is the effective mass and c is the velocity of light in vacuum. Fig. 2(a) represents a plot of ( k 2 - - n 2 -F 1)- 1 versus (~)2 from which one can get a linear relation of slope 1/~ 2. The plot of (2nk~) 1 versus (9) 2 (Fig. 2(b)) yields a straight line of slope equal to [(~o) -2 (gR)-1]. Accordingly, the obtained values for 90 and ~R are 1.84× l04 c m - 2 and 1.22×104cm -~, respectively. Again, using the values of ~o and 9R in conjunction with Eqs. (5) and (6), the number of conduction electrons per unit volume N and the relaxation time z are found to be 1.8 × 1 0 2 2 cm -3 and 4.3 x 1 0 - 1 6 S, respectively. Knowing the values of N and z the static conductivity a~ in Mn films can be calculated using (7)
a, = S e 2 z / m *
and was found to be as = 4.56 x 1014 (e.s.u.). Also the electron velocity at the Fermi surface VF can be determined using the following equation:
(3(%)2h3c2~1/3 8m----~Ee-~ -}
,
•
,---,_
.12
24
22
20
18
20
3'0
4'o
so
~2 xlO5 (cm -2)
6o
t,
Fig. 2. Relation between: (a) (k 2 - n 2 + 1)- ' and 92 for thin Mn films;(b) (2n kf)- 1 and ~2 for thin Mn films. where m * = m = 9 . 1 × 1 0 - 2 S g , e = 4 . 8 x l 0 -1° e.s.u, and N = 1.8 x 1 0 2 2 cm -3 was previously determined. It was found that COp= 7.57× 10 ~5 s -1 while 2p = 249 nm. The contribution of the effective absorption A~ff can be evaluated using the approximation [10] given by the relation
COpF m* ] 1/2 h e tf = ~ L - - - - ~ ] .
(10)
It was found that Aeef has a value of 0.69 (i.e. 69%). Therefore, absorption due to bound electrons may be 31%. It seems that Drude's theory agrees with the former suggestion, The Fermi energy in case of Mn polycrystalline thin films was found to be equal to 2.51 eV using the relation
h2 Ef = ~ (37z2N) 2/3,
A v = 4 X ( 3 n 2 N ) 2/3
and was equal to 8.26 × 1 0 1 6 e c m - 2 . The plasma frequency (COp= 2/~Vp = 27ZC/,~p) can be easily determined from the relation 4rtNe 2 m* '
~= 2 6
16
(8)
where h is Planck's constant. It was found that VF = 9.36x 107cms -1. The value of the effective area of the Fermi surface AF can be calculated using the relation
(°P)2-
2O t
I VR - 2ncz"
Vv = \
•
(9)
(11)
while the Fermi wave vector was found to be 8.11 x 107 cm using the relation 3rt2"] /a N 1/a. Kr = \--~F/
(12)
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285 290
288
2.3. Optical constants of Mn thin flms in conjunction with complex dielectric constant
0.9. Q.8-
It is well known that the electronic transition in a solid is more directly related to the complex dielectric constant ~ = el + iCE instead of the complex index of refraction fi = n + ik. These are related as ~ = fi2, so that e~ = n 2 - k 2 and e2 = 2nk, respectively. As the optical constants of a metal are mainly governed by the contribution of both bound and free electron regions in that metal, el can thus be written as
07. 06-
i '~
0.124 eV
e~o e°
%eQe e
.7. 0.5b
o
0.4
k "x.x.
O3 02 01 0
el = eL
e2NI
- to m* (1
"+- ( D 2 1~2
,1
O1
(13)
"
For metals at near-infrared frequencies, one finds ~o>>l/z, therefore el = eL --//22 where / / = e2N/ 47t2eom*c 2. Fig. 3 illustrates el =f(22), where the point of intercept of the straight line on the y-axis yields the required value of eL (the lattice dielectric constant); accordingly, eL = 27. Knowing the slope of the linear part of the straight line, the concentration of the free charge carriers N can
800
700
600 ®
500
400 ®
300
o
02
03
04
0.5
h v (eV)
Fig. 4. Optical conductivity in 1014s 1 for M n thin films as a function of photon energy in eV.
be calculated; the resulting value of N was = 3.86 x 1022 c m - 3 . In the intraband region of the spectrum, the optical conductivity (0. = 0.1 + i0.2) is more convenient than the dielectric constant. They are related to the components of the complex dielectric constants by 0.1 = e2(o/47z and - - 0 " 2 = ( e I - - 1 ) ~ o / 4 n . The optical conductivities 0.1 and - 0.2 calculated from the optical constants are plotted as a function of hv (eV) in Fig. 4. It is seen in Fig. 4 that 0.1 and -0.2 show peaks at 0.124 eV. These peaks may be attributed to spin-orbit splitting in view of the assumption given by Craven [11] in his study of the optical conductivity for tin thin films, as well as for Wang and Callaway 1-12] in their study of 0.1 for ferromagnetic nickel. Since we did not find any recent measurements on manganese except those previously referred to, we cannot compare our results with others.
200
2.4. Optical constants of Mn thin films in conjunction with the anomalous skin effect
100
o o
,oo
2~0 ~,;
(Fro 2 )
300 ,,
Fig. 3. Plot of the real part of the dielectric constant - e l versus (wavelength) 2.
The optical constants n and k for Mn thin films in conjunction with the theory of the anomalous skin effect are considered to determine the number of conduction electrons per unit volume N and the relaxation time t. According to Dingle's dispersion
S.S. Fouad, A.H. Ammar / Physica B 205 (1995) 285-290
relations, the second approximation may be written [13, 14] as
Table 2
k 2 _//2
- 22
= A - B2 z,
Dingle [13] Drude [9]
(14)
nk 2--x = G - D2 z,
(15)
289
N (cm-3)
z (s)
3.46 × 1021 1.8 × 1022
5.2 x 10-16 4.3 x 10 16
where
If the static conductivity a and the intercept A are known, it is also possible to determine the relaxation time from the relation
A=
z
t7 ~C2~ '
(16) G = c--3 47t22----~
The dependence of (k 2 - n 2 ) / 2 2 and (nk)/2 3 o n 22 should be linear, with intercepts yielding A and G, respectively. Figs. 5(a) and (b) represent simultaneously the two relations (k 2 - n2)/22 =f(22) and (nk)/23 = 9(22). The observed intercepts of A and G are equal to 3.1 x 10s cm -2 and 5.5 × 101° cm -3, respectively. The magnitude of A determined in this way may be used in conjunction with the expression for the static conductivity a = N e 2 r / m * to calculate the number of conduction electrons per unit volume N, provided the effective electron mass m* is known; the relation of N is given as rtc2m* A
N =
e2
(17)
5 l
0
100
200
300 h
4;0
1 0
(~um) z
Fig. 5. Relation between: (a) ((k 2 - n2)/2 2) cm -2 and 2 2 (p-m)2; (b) (n k/2) cm -a and 22 (gm) 2.
ncZA
(18)
Using these two values A and G in conjunction with Eqs. (17) and (18) the number of conjunction electron N and the relaxation time z can be evaluated. The present results of N and r obtained by using Dingle's theory in comparison with that obtained by using Drude's theory are listed in Table 2.
3. Summary A technique of measuring transmission from evaporated films to determine accurate values for optical constants has been used in the spectral range 3.5-20 gm. Our precautions used during the preparation of the Mn films is in good agreement with those previously quoted in the literature. The optical constants of Mn thin films determined by our thin film measurements were compared with Drude free electron theory and anomalous skin effect theory. Along with these two theories, the microcharacteristics, i.e. N, z, % Vv, AF, c a n be determined. We derived the optical constants, the optical conductivity and the lattice dielectric constant.
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[7] O.S. Heavens, in: Physics of Thin Films, Vol. 2 (Academic Press, New York, 1964) p. 193. [8"] A.P. Lenham and D.M. Treherne, J. Opt. Soc. Amer. 56 (1966) 1137. I-9] P. Drude, Appl. Sci. Res. B 2 (1953) 169. [10"] V.L. Jinzlurg and G.P. Molulevich, Usp. Khi. Fiz. Nauk, 55 (1955) 469.
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