Optical and Electrical Properties of Au and Ag in Relation to Free Electron Theory

Optical and Electrical Properties of Au and Ag

in Relation to Free Electron Theory

T he results of new measurements of the refractive in d ex m and the absorption coefficient %; of A u and A g are discussed with respect to free electron theory. This perm its for the calculation of the concentration A of free electrons and D .C . conductivity <r„, which are com pared with values obtained from the electrical conductivity an d H a ll effect measurements taken fo r the same films.

2. Results and discussion

1 . Introduction

Outside the range of interband transition due to hound electrons, the optical constants % and A at wave length A (frequency m) are related to the electronic parameters of the metal in the following equations, based on the free electron theory of metals [1, 2, 3]. 47t — e = — %2 = — l _ j---m3 47T<7„M' = - l + --- ^ - = - 1 + --- ( i ) 7te-= (7 2TT " m3 <7„ M U , w'3 47t3p2 A3, (2)

where, e and u are the frequency-dependent dielectric constants and electrical conducti­ v ity of the metal, respectively, A is the number of free electrons par unit volume, is the effective mass of the electron, m' is the self frequency of the electron defined as the recipro­ cal of the relaxation time T, is the D.C. conductivity, and r is the velocity of light in vacuum. * **

* Assist. P ro f., Physios D ep t., U niversity College for W o m e n , A in Shams U n iversity , Heliopolis, Cairo.

* * Lecturer, Physios D e p t., College of Education, A in Shams U niversity, H eliopolis, Cairo.

The results of the optical constants for Au and A g have been reported in the previous paper. According to equation (1), the relation between (%3 — %3) and. A3 is linear. This is verified in case of Au and Ag, as shown in Fig. 1 and Fig. 2,

---

---*-Fig. 1. Relation betw een (%;3 — a nd A3 f or A u

respectively, with data of previous authors for comparison [1, 2, 4]. jV (optical) has been calcu­ lated from the slope of the straight line, consi­ dering /a* = w = 9.1 x 10 gin. (theore­ tical) has been also calculated. (A „ = db/A, where d is the density of the metal, A its atomic weight and A A\ugadro's number) considering one free electron per atom. A (electrical) was deduced from the present Hall effect

measure-ments for both An and A g films, which were used before in the optical measurements. The data thus obtained are listed in Table 1. As it is visible A (optical) is in fair agreement with A (electrical). The effective number of free electrons per atom A / A „ (optical) being also given. Hence, the optical effective mass w*/w (opt.) may be calculated. The resulting values are comparable with that obtained by CoHAK [5], B E A G L E H O L E [6] for Au (1.10 ±0.08) and that reported by GiVExs [3] for A g (0.98), respec­ tively.

T a b l e 1

A n A g

A (optical I 4.79 x lO^^elec./c.c. ' 5.2 x 10^ elcc./c.c. A „ (theoretical) 5.89 x 10-^elcc./c.c. 5.9 x 10-2 elcc./c.c. A (electrical) ; 4.96 x 10^elec./c.c. i 5.2 x 10^ elcc./c.c.

A / A „ (opt.) 0.81 ¡0.88

?M.*/m,

= A ^ / A (opt.) ¡1.23 ¡1.13

Fig. 3 represents the dependence of the conductivity <7 = Mdao/27i: on the wavelength A for Au, giving a threshold of interband transi­ tion at 0.6 gin, corresponding to an energy A = 2.07 eV, due to the excitation of d electrons

--- ---*-Fig. 3. The dependence of <r on 1 for A u

to the conduction band [7, 8]. A t wavelength region longer than the absorption edge, the condutivity <y increases with increasing A, as expected from eq. (2).

Fig. 1 represents the dependence of 2ab/7. = 2c/e on 7.2 for Ag, showing similar behaviour

Fig. 4. V ariation of 2 wl'/A w ith A2 io r A g

as Au, and indicating a peak at A = 0.95 gm, which is possibly associated with interband transition of electrons either from the Ferini- surface to the next higher empty hand or from a lower lying filled band to the Fermi-surface [1, 9, 10, 11]. Beyond 1.5 ^.m, the curve shows a continuous increase of u with increasing /. as the theory (eq. (2)) expects.

According to eqs. (1) and (2), the Argand diagram for Au in Fig. 5 represents (/r^ — t?À+l) against <7 = -nA'm/2^ showing two straight lines

Values thus obtained are listed in Table 2 with the values of a. deduced from the present electrical measurements on the same films used before in the optical measurements.

Table 2 T (opt.) "o (<'pt.) f7„ (elect.) Au 1 x 10 ** s 1.3 x 10'7 e.s.u. 4.2 x 10*7 e.s.u. Ag 0.6H x 10-4 s 0.9 x 10*7 e.s.u. 1.92 < 10*7 e.s.u.

Fig. 5. The A rg a n d diagram for A u

It is clear that (opt.) < (elect.). This is attributed to the fact that the electrons near the surface have frequent collisions; therefore they have a shorter mean free path and a smaller relaxation time r than the electrons located deeper in the metal, which determine (elect.). Since the light waves penetrate to a very short distance into the metal they interact only with the electrons near the surface, there­ fore a, (opt.) is reduced [15, 16].

L e s propriétés o p tiqu es et électriqu es de A u et de A g rap p o rtées à la théorie des e lectro n st

lib res

On a exam ine, p ar rapp ort à la théorie des électrons libres, les résultats des mesures effectuées pou r A u et A g et concernant l'indice de réfraction de la lumière a et le coefficient d'absorption A;, t'eci perm et de cal­ culer la concentration des électrons libres A ainsi que la conductivité spécifique en courant continu <r„. Les résultats ont été com parés avec les valeurs de A et de obtenues des mesures de la conductibilité électri­ que et de l'effect fiai! qu'on avait effectuées sur les mêmes couches.

of different slopes (slope = Ix/ o' = lx ? ) cor­ responding to two values of relaxation time r = 0.99

x

The relaxation time r of the free electron in Ag, calculated from the slope of Fig. 2 and the slope of the linear part of Fig. 4, gives T = 0.68

x

Using the values of V (opt) and r, the D.C. conductivity <?(, is calculated ((?„ = Оптические и электрические свойства А и и A g с точки зрения теории свободных электронов Результаты измерений коэффициента преломления све­ та и коэффициента поглощения Аг, произведенных для Аи и Ag, обсуждены с точки зрения теории свободных элек­ тронов. Благодаря этому становится возможным расчет концентрации свободных электронов А и удельной про­ водимости для постоянного тока и„. Результаты сопостав­ лены со значениями А и ио, полученными путем изме­ рений электропроводности и эффекта Холла, проведен­ ных на тех же пленках. ОГТИ'А APPLÏCATA У Д

19

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Received, JTarcA 24, 7974