The influence of thickness on the resistivity, the temperature coefficient of resistivity and the thermoelectric power of evaporated palladium films at 77 K and 273 K

Thin Solid Films, 74 (1980) 1-16 © ElsevierSequoia S.A., Lausanne--Printed in the Netherlands

1

T H E I N F L U E N C E OF T H I C K N E S S O N T H E R E S I S T I V I T Y , T H E T E M P E R A T U R E C O E F F I C I E N T OF RESISTIVITY AND THE THERMOELECTRIC POWER OF EVAPORATED PALLADIUM FILMS A T 77 K A N D 273 K G. WEDLERAND G. ALSHORACHI* Institut fftr Physikalische und Theoretische Chemie der Universiti~t Erlangen-Niirnberg, Egerlandstrasse 3, D-8520 Erlangen (F.R.G.)

(ReceivedJanuary 21, 1980; accepted April 29, 1980)

The resistivity, the temperature coefficient of resistivity and the thermoelectric power of palladium films were measured at 77 K and at 273 K in the thickness range 6.7-97 nm. Existing theories were used to describe the thickness dependence of the experimental data and the relationship between these electrical properties.

1. INTRODUCTION Only a few papers dealing with the electronic properties of thin palladium films have been published. Studies have been made of the conductivity 1-4, the photoelectric behaviour 5, the thermoelectric power 4' 6, and the influence of both thickness 2, 4, 6 and annealing conditions L 4 on these properties. There is, however, no work in which the dependences of the various transport properties on thickness have simultaneously been investigated. The aims of this study were as follows: (1) to measure under ultrahigh vacuum (UHV) conditions the resistivity, the temperature coefficient of resistivity and the thermoelectric power as functions of thickness at 273 K and at 77 K; (2) to look for correlations between these electronic properties; (3) to interpret the experimental results by means of the existing theories. The films were annealed under conditions which are adequate from the following points of view: state of order, cleanness and use of the films for adsorption studies. 2. EXPERIMENTALTECHNIQUES The apparatus and the cell used in this work are identical to those described in a previous paper 4. The films were deposited onto a glass substrate 4 at 77 K at a rate of 1 nm min -1. They were annealed at 373 K for 1 h. The procedure of the measurement is described in detail in ref. 7. All the experiments were carried out under a pressure of less than 4 x 10 -a Pa.

* Present address : Universityof Baghdad, ScienceCollege, ChemistryDepartment, Adamiya-Baghdad, lraq.

2

G. WEDLER, G. ALSHORACHI

3. RESULTS AND DISCUSSION

3.1. Structural characterization of the films A prerequisite for the discussion of the electron transport properties of thin metal films is the knowledge of their cleanness and their structural properties. These were investigated and checked for the films under question in various ways. The cleanness of the films was ensured by the use o f " M A R Z " grade palladium (Materials Research, Munich), by the outgassing procedure and by the UHV conditions maintained from the beginning of the evaporation until the end of the experiment. It was verified by the results of adsorption experiments carried out with the same films and with films produced in the same manner. In these experiments calorimetric determinations of the heat of adsorption of hydrogen 8 and measurements of the changes in the resistivity and in the thermoelectric power due to the adsorption of hydrogen 9 were performed. Both studies are very sensitive to impurities. The effect and reproducibility of the annealing procedure has been investigated in a previous paper 4. Information on the medium size of the crystallites of the polycrystalline films and on their orientation can be obtained from X-ray diffraction measurements, as has been shown in detailed investigations with nickel films 1°' 11. Corresponding experiments have been carried out with palladium films produced in the same 12 or in a similar 13 manner as the films studied in this paper. The structural characteristics of nickel and palladium films agree to a great extent. From these experiments it can be concluded that the mean size of the crystallites in the palladium films increases with increasing thickness. Up to about 40 nm the mean crystallite size is identical with the film thickness. More than 90~o of the crystallites exhibit a (111) fibre structure. The crystallites themselves exhibit a rather high degree of order. In order to obtain a hint of the crystallographic orientation in the surface planes of the crystallites, photoelectric determinations of the work function ~ohave been performed14; this was found to be 5.20 +0.005 V. This value agrees very well with the work function reported by Sachtler and coworkers is, 16 for palladium films of thickness 10 nm after an anneal at 373 K. Ert117 has derived work functions for single-crystal faces from UV photoelectron spectroscopy measurements. He obtained q~(111) = 5.8 V and ~o(110) = 5.1 V. A comparison of these values shows that despite the (111) fibre structure the surface does not consist only of(111) planes. It has to be considered, however, that when using the photoelectric method emission from the planes with the lowest work function predominates, so that a rather small percentage of such planes can be responsible for the low value of the work function.

3.2. Electrical resistivity Figure 1 shows the resistivities measured at 77 K and at 273 K as a function of the thickness d in the range 6.7-97 nm. For thicknesses below 30 nm the resistivity increases markedly with decreasing thickness. Both the shape of the curve and the absolute values are in good agreement with the results obtained earlier 4 when the different annealing conditions are taken into account (see also Table I). For films thicker than 16 nm the dependence of the resistivity on thickness at 77 K and at 273 K can be expressed as

I N F L U E N C E OF THICKNESS ON ELECTRICAL PROPERTIES OF

Pd

FILMS

3

,1, as can be seen from the plot of the film resistivity Pr against the reciprocal of the thickness d (Fig. 2). The constants Po and K have the dimensions of resistivity and length respectively. The full curves drawn in Fig. 1 have been calculated from the linear relationship given by eqn. (1). The resistivities of films thinner than 16 nm do not obey this equation.

~

0

o

20

bulk Pd 273K

77K

Fig. 1. D e p e n d e n c e • f t h e r e s i s t i v i t y p r • n t h e t h i c k n e s s d a t m e a s u r i n g t e m p e r a t u r e s • f 7 7 K ( • ) a n d 2 7 3 K (A). The values for b u l k p a l l a d i u m are m a r k e d on the r i g h t - h a n d side. TABLE I VALUES OF Po FOR PALLADIUM FILMS AS A FUNCTION OF THE ANNEALING TEMPERATURE

Annealing temperature (K)

po(10 - 8 ~ m) at 77 K

Po (10- a D m) at 273 K

80 a 300 a 373 440 a Bulk P d b

25.4 __ 6.5 5.6 1.69

-22.2 14.3 11.5 9.77

a See ref. 4. b See ref. 18.

A relationship of the form of eqn. (1) follows from various theories which have been used to describe the thickness dependence of film resistivities. These are as follows. (1) The Fuchs-Sondheimer approximation for thick films leads to the expression 19

f

/o}

PF = Po 1 + ~ ( 1 - p ) ~

(2)

where Po is the resistivity of the bulk material with the same density of defects as in the film, p is the percentage of electrons specularly reflected at the surface, and lo is the mean free path of the electrons.

4

~

G. WEDLER,

30

G. ALSHORACHI

J

~,--> z-o°

j

1

o._.~o o--"-f~o /

10 J

0

0

i

~

k

i

9

b

lid ?07m -1

Fig. 2. A plot of film resistivityPFagainst the reciprocal 1/d of the thickness for measuring temperatures of 77 K (O) and 273 K (A). (2) Mayadas and Shatzkes 2° have derived the expression P F = PO

(1-t- 2 1 __.R D,o/[

(3)

where R is the percentage of electrons specularly reflected at the grain boundaries, and D is the mean size of the crystallites which may be assumed to be equal to the film thickness d 1~ (3) Wigmann 21 has obtained the relation PF = po{I+(ZAs+Z*As*)~}

(4)

where Z and Z* are the numbers of scattering centres per square centimetre at the surfaces and at the grain boundaries respectively, and A and A* are the mean scattering cross sections at the surface and the grain boundaries respectively. For further discussion it is practical to express the relations given by the three theories as

In all these theories Po has the same meaning. Indeed, it is observed that Po approaches the resistivity of the bulk undisturbed material (see Table I) when the films approach a well-ordered state after an appropriate annealing; this can be checked by the methods discussed in Section 3.1. This indicates that eqns. (1)-(5) are well suited to an extrapolation from film resistivities Pv to the bulk resistivity Po when PF is measured for different thicknesses. F r o m the slopes of the straight lines in Fig. 2 the coefficient K'l o in eqn. (5) can be calculated (Table II). The factor K' has a different meaning depending on the theory used. Since it is not possible to give a reliable value of l0, because the electrons in palladium do not behave as a free electron gas, the factor K' cannot be calculated. Therefore it is not possible to determine p, R and ZA s + Z'As* in order to check the validity of the three theories in the case of palladium.

I N F L U E N C E OF THICKNESS ON E L E C T R I C A L PROPERTIES OF

Pd

FILMS

5

T A B L E II VALUES DETERMINED FROM THE SLOPES OF THE STRAIGHT LINES IN FIG. 2

Equation

Quantity

Value 77 K

273 K

(1) (5) (2)

poK (10 Is f~ m 2) K'l o (10 -9 m) (1 - p)lo (10- 9 m)

2.06 31.7 84.5

2.53 17.7 47.2

(3) (4)

{R/(1- R)}/o(10- 9 m)

21.1 31.7

11.8 17.7

(ZAs+Z*As*)l o (10 -9 m)

There is, however, another observation which shows that the FuchsSondheimer theory must not be used to describe the dependence of the resistivity of palladium films on thickness. In deriving this theory it is assumed that disorder in the films is independent of the film thickness. This is not the case since the crystallite size depends on the thickness (see Section 3.1). In contrast, this observation is the basis of Wil3mann's considerations which lead to eqn. (4).

3.3. Temperature coefficient of resistivity The temperature coefficient of resistivity (TCR) ct defined by 1 dp ct = - - -

(6)

pdT

was determined from the resistances measured at 373 K and at 273 K under the assumption that the thermal expansion coefficient can be neglected 21. Figure 3 shows that the TCR ~v of the films is dependent on thickness. For thick films ctr approaches a limiting value: with decreasing thickness ctv also decreases. 40 butk

~g

Pet

1.o

-

o

3.O 1~

u

I 0.8 ~6

-

~o'~--'~"

%j4 ~

o

o

2.0 l 1.0

0.2 0

~o ko Jo io

~o 8o ~o ~o 0o

100

Fig. 3. The T C R ctr as a function of thickness.

Equations (2)-(5) can be written as

PF = Pof(lo/d)

(7)

F r o m logarithmic differentiation it follows that GtF = ~0 "1"

1 df(lo/d ) d/o f(lo/d) d/o d T

(8)

6

G. WEDLER, G. ALSHORACHI

where ~o is the TCR of the bulk material having the same density of imperfections as the film. Taking into account that

1 d/o 1o d T

-

~o

(9)

and that

,lO) eqn. (8) can be transformed into

1 c~F = aOl + K,lo/d

(11)

Usually the thickness dependence of the TCR is not described by eqn. (11) but by the equation22, 23

which is derived in the same manner, but starting with eqn. (2); this expression is the approximation of the exact Fuchs-Sondheimer equation for thick films. In the case that lo/d ~ 1 eqns. (I 1) and (12) become identical. It has been shown 24, however, that eqn. (2) approximates the exact equation fairly well up to lo/d .~ 10. Since l0 is of the order of 10 nm 25 eqn. (12) must not be used in this work because l0 and d are of the same order of magnitude. Equation (11) is well suited to describe the experimental results, as can be seen from Fig. 4 in which 1/~ F is plotted as a function of 1/d. The linear relationship is valid for thicknesses greater than 14.5 nm, which is in good agreement with the observation made concerning the thickness dependence of the resistivity. The full curve drawn in Fig. 3 was calculated from the linear relationship in Fig. 4. It proved to be impossible to describe the thickness dependence of ~F by means of eqn. (12). O.B

O.B D

O.Z D

0.;

o

Fig. 4. A plot of 1/o~F against

1/d to check the validity ofeqn. (11).

I N F L U E N C E OF THICKNESS ON E L E C T R I C A L PROPERTIES OF

Pd

FILMS

7

Since in the derivation of eqn. (11) the quantity K in eqn. (1) was split into a factor K' and the mean free path lo, the validity of eqn. (11) points to the correctness of this splitting; this is also found to apply in eqns. (2)-(4) which have been theoretically derived. This is true irrespective of the model used. In Table III the data determined from the thickness dependence of the TCR are compared with those obtained from the thickness dependence of the resistivity and with the bulk data. The K'l o values differ by 14Yo. This is comparable with the observations made with copper films 26. It is, however, a much better agreement than that found for nickel films 2v. The :to value is 14yo smaller than the value of the bulk undisturbed palladium. An observation generally made is that :to depends strongly on the annealing conditions and approaches the value of the undisturbed bulk material only when rather high annealing temperatures have been used 26' 27. TABLE III VALUES DERIVED FROM THE THICKNESS DEPENDENCE OF THE RESISTIVITY AND THE TCR FOR

T = 273 K

Equation

K'l o (10 -9 m)

% ( 1 0 - a K - 1)

cta (10-3 K - 1)

(5) (ll) (16) Bulk P d

17.7 15.2 ---

-3.24 3,35 3.77 ~

--0.09 --

See ref. 18.

When eqn. (5) is directly differentiated, it follows that dp dT

dpo polo dK' ÷ dT d dT

(13)

when as usual 2a polo is taken to be temperature independent. Equation (13) can be transformed into :tV =

dlnK'dT

Po(

b~F 0%

dlnK'/dT]

(14)

A similar expression is obtained when PF is split into a thickness-independent bulk value Po and a thickness-dependent value Pd, i.e. PF = P0+Pd

(15)

and this equation is inserted into eqn. (6): :tF = :td +-P2(Ct0 -- :td)

(16)

PF

From a comparison of eqns. (14) and (16) it follows that :td -

dlnK' dT

(17)

Therefore :td can be interpreted as the contribution of the additional thicknessdependent scattering processes to the TCR.

8

G. WEDLER, G. ALSHORACHI

When eqns. (14) and (16) are valid the plot of ~v against 1/p v should give a straight line. Figure 5 shows that this is indeed the case. s d is obtained from the intercept, whereas So follows from the slope taking into account the known values of Po and s d. s o is in good agreement with the value derived from eqn. (11) (see Table III). The small value of s d shows that according to eqn. (17) K', and therefore the scattering at the surface and at the grain boundaries, is only slightly temperature dependent. It is interesting to note that experiments with nickel films 27 have shown that s d depends strongly on the annealing temperature. The ratio sd/s o for nickel films at 273 K drops from 0.4 to 0.3 when the annealing temperature is increased from 290 K to 333 K. For annealing temperatures above 373 K Sd/~0 seems to be zero. This would mean that K' becomes temperature independent, i.e. that wellannealed films exhibit a thickness-dependent additional resistivity Po that does not depend on temperature. This would be in agreement with Matthiessen's rule. For the palladium films studied in this work Sd/So is 0.027. D D

°~/

%

o

o

o/

0

0

Obl

0'.02

ON

dO4

dO5

tlgF

0.'06

Fig. 5. A plot of~v against 1/p v to checkthe validityof eqns. (14)and (16). It is interesting to note that eqns. (14) and (16) describe the relationship between s v and PF for films with thicknesses between 6.7 and 90 nm. This shows that the validity of eqns. (14) and (16) extends to smaller thicknesses than eqns. (5) and (11). Nearly identical values of c~o are obtained from the application of eqns. (11) and (16) (see Table III) the first of which was derived using a distinct thickness dependence of Pv (eqn. (5)) whereas the second was obtained without such a relationship. This confirms the applicability of both equations to interpolation, to extrapolation or to determination of the behaviour of the bulk material with the same density of imperfections. As in the case of the thickness dependence of the resistivity it is not possible to determine l0 and K' separately. As further information, however, eqn. (17) gives the temperature dependence of K'. 3.4. Thermoelectric power

The thermoelectric power S of a metal consists of two terms which are independent of each other, namely the diffusion thermoelectric power SD and the phonon drag Sg: S = SD+Sg

(18)

INFLUENCE OF THICKNESS ON ELECTRICAL PROPERTIES OF

Pd FILMS

9

At 77 K there is a superposition of both terms. At 273 K, however, the phonons can be scattered by mechanisms other than electron-phonon interactions, so that the phonon drag contribution to the thermoelectric power becomes negligibly small in the case of palladium 28. 3.4.1. Diffusion thermoelectric power at 273 K The most general expression for So in metals is So=

~2k2T/t~ In a(E)/ 31elEvl OlnE ]E=E~

(19)

where k is the Boltzmann constant, T the absolute temperature, E F the Fermi energy, E the energy and a (= 1/p) the conductivity. Figure 6 shows the diffusion thermoelectric power of the palladium films as a function of thickness d. The values obtained are in good agreement with earlier results4 when the change in the annealing conditions is taken into account.

0

-2

\ °\ oo°',x,~o

o

~

'

-8 -~'0 0

ko

io

~o

~

-

-

o oo.~__~_o bulk Pd

~o

_a_ ~bo ngn

Fig. 6. The diffusion thermoelectric power SoF as a function of thickness d at 273 K. The value for bulk undisturbed palladium is marked on the right-hand side.

In order to describe the thickness dependence of the thermoelectric power SDF of the film eqn. (5) is inserted into eqn. (19). It follows 29 that SO

n2k2T I 1

d ~-~/dln/o

dlnK'~

when the energy dependence of both the mean free path lo and K' are considered. So ° is the thermoelectric power of the bulk metal having the same density of imperfections as the film. Usually the relationship •2kZT [d In 1o \

K, lo

(21)

SDF = S°° + ~ l d - T n E n e/~=~~ d is applied; this has been derived by Justi et al. 3° from the exact Fuchs-Sondheimer theory in the case of thick films (d/l o >> 1). As already pointed out in Section 3.2, this approximation is not justified for the thicknesses used in this paper. In Fig. 7 So v is plotted as a function of (1 + d/K'lo)- x using the K'l o value obtained with eqn. (5) (see Table II). The thickness dependence of Sov is well described by eqn. (20) for d in the range 10-97 nm. The application ofeqn. (21) to the

10

G. WEDLER, G. ALSHORACHI

experimental results failed for thicknesses smaller than 25 nm. SD° has been determined from the intercept in Fig. 7 to be 9.18 ~tV K - 1, in good agreement with the value reported in the literature for bulk palladium 31. The full curve drawn in Fig. 6 was calculated with the aid of the linear relationship found in Fig. 7. o

-I /

-3

o J

°I

-5 -7 /

-9 /

-11 0,0

0.1

42

0.4

03

0.5

0.6

{I+ ~1o ) -7

Fig. 7. A plot of So ~ against (1 + d/K'lo)- 1 to check the validity of eqn. (20).

Ifpo and Pd in eqn. (15) represent two different scattering mechanisms, SDF can be written as + pdSD d SD F = poSD ° = So d @ ~ ( S o 0 - S D d) PO -~- Pd

(22)

[F

where SDd is the contribution of the surface and grain boundary scattering to the diffusion thermoelectric power. This relationship is known as the Nordheim-Gorter rule 32. Its validity has been verified for both impurity and imperfection scattering 33'34. Several authors have also applied this rule to bulk and surface scattering26, 29, 35. Figure 8 shows the appropriate plot of So F against 1/p F. It gives a value for SD~ of + 2.3 ~tV K - 1 and a value for SD° of -- 8.9 ~tV K - 1 when the Po value of Table I is inserted. Equation (22) proves to be applicable to films with thicknesses between 19 nm and 97 nm.

2

-2

~

o

° %~----e -5 -8

0

0.01

0.02

003

0.01,,

0.05

Fig. 8. A plot OfSD F against 1/'PF to check the validity ofeqn. (22).

0.06

INFLUENCE OF THICKNESS ON ELECTRICAL PROPERTIES OF

PdFILMS

11

Another relationship which follows immediately when eqn. (11) is inserted in eqn. (20) is

SDO_~_(1~tv~rt2k2Tldlnlo dlnK'~

SD F =

It is similar to an expression derived by Leonard and Lin 36 and by Thompson 37, who did not, however, take into account the energy dependence of K'. Figure 9 shows the plot of SDv against 1 - ~v/~0. Values determined with thicknesses between 9 nm and 97 nm are well described by eqn. (23). SD° is found to be - 8.85 IxV K - a

o/

-2 ,o

J o°S

o

-6

-8

'

~2

'

~.~

'

b8

~.6

¢t0

Fig. 9. A plot ofSD r against (1 -- ctr/%) t o c h e c k t h e v a l i d i t y o f e q n . (23).

All these examples show that eqns. (20), (22) and (23) are well suited to the extrapolation and interpolation of diffusion thermoelectric powers measured with films of different thicknesses. Table IV summarizes the values obtained. The values of SD° are in good agreement with the thermoelectric power of undisturbed bulk palladium 31. T A B L E IV VALUES DERIVED FROM THE THICKNESS DEPENDENCE OF THE THERMOELECTRIC POWER AT 273 K

SD°

n2k2T[t31nlo t3 In K' I 3[~FIOInE+--OInE ~E=Er

SDd

(~tV K - ~)

(lxV K - ~)

(~tV K - ~)

(20) (22) (23)

-9.18 - 8.9 - 8.85

+ 12.29 -+ 11.97

(+3.11) + 2.3 ( + 3.12)

Bulk Pd a

- 9.0

--

Equation

a See ref. 31.

Sometimes authors 35 • 38 have tried to evaluate (0 In lo/O In E)E~ from equations comparable with eqn. (20) in order to compare this value with that (2.0) of a free electron gas 28. They did not, however, take account of the energy dependence of K'.

12

G. WEDLER, G. ALSHORACHI

Another difficulty arises from the uncertainty concerning the exact value of E F. For the moment it does not seem to be justified to determine d In lo/d In E from

lt2k2Tldlnl o

dlnK'~

The values found for this term with the aid of eqns. (20) and (23) are in good agreement (see Table IV). The quantity SDd can be directly derived only from eqn. (22). There is, however, another possible way of expressing SDa. The term (1 +d/K'lo) -1 in eqn. (20) is identical with Pd/PF, SO that eqn. (20) can be written as SDF____ SD0~ Pd ltek2T[dlnlo

÷

dlnK'l

t 0at

The Nordheim-Gorter rule (eqn. (22)) can be transformed into So F = So ° + Pd(Sod -- SO°) PF

(22a)

Comparison of eqns. (20a) and (22a) shows that So d

S o ~-I--rc2k2T[dlnlo D

dlnK' I + 3 ~ e ~ E F / d l n E + d i n E ]e=e,

(24~

Using this relationship the Sod values in parentheses in the last column of Table IV were calculated from eqns. (20) and (23). They agree fairly well with the value found by means of the Nordheim-Gorter rule.

3.4.2. Thermoelectric power at 77 K As pointed out in the introduction to Section 3.3, the thermoelectric power at 77 K consists of two terms, namely the diffusion thermoelectric power S o and the phonon drag Sg, as expressed by eqn. (18). There is no possibility of separating these terms experimentally. Furthermore there is a strong thickness dependence of the thermoelectric power S F of palladium films at 77 K. Figure 10 shows that S F decreases with increasing thickness for thicknesses below 20 nm. There is a minimum in S F (2.5 pV K-1). As the thickness further increases, SF also increases. Wedler and Chander 4 have observed a curve of similar shape and a minimum at d = 30 nm with palladium films that had been annealed at 300 K. If the annealing temperature exceeded 400 K the minimum vanished, and the curve always had a positive slope. It is interesting to note that the thermoelectric power of the thick films is greater than that (3.7 gV K - 1) of the bulk material. 3.4.2.1. Diffusion thermoelectric power at 77 K. In order to determine the diffusion thermoelectric power at 77 K attempts have been made to extrapolate the diffusion thermoelectric power from regions in which the phonon drag is negligibly small to 77 K, so that Sg can be determined as the difference between the measured value of S and the extrapolated SD. Huebener 3a has applied this procedure to the determination of So and Sg for platinum at low temperatures under the assumption that S o varies linearly with the temperature T. The temperature dependences of the

I N F L U E N C E OF THICKNESS ON E L E C T R I C A L PROPERTIES OF

Pd

13

FILMS

o/< bulk Pd

3.6 3.2

\o

\

o

~ o

2.8 2.~ 2.1]

o

o

o oo

,.~o % o 2b

,~o

go

1~o

do d nm

Fig. 10. The thermoelectricpower Sr of the filmsas a function of thickness d at 77 K. The value for bulk undisturbed palladium is marked on the right-hand side. absolute thermoelectric powers S of platinum (Fig. 2 of ref. 38) and of palladium (Fig. 1 of ref. 39 and Fig. 3 of ref. 4) are very similar to each other, so that such an extrapolation should also be possible in the case of palladium. Values obtained in this way are Sg° = 6.3 ~tV K - 1 and SD° = - 2.6 IIV K - 1. Wohlleben 26 has shown that it is also possible to calculate the diffusion thermoelectric power SD° of the bulk material with the same density of imperfections as the film at 77 K from values measured at 273 K. He followed a similar procedure to that proposed by Sugawara et al.35 Equation (24) can be written as

SDd SDO = SDo[Sodo_l)= ~SD 3-T~FId~nE ~2k2T [dlnl o + dd-~RE In K'~]E=EF Since eqn. (19) is valid 28 for both SD° and

(25)

Sod, it follows after replacing cr by p that

SDd (d In pd/d In E)~=e~ SD° = (d In po/d In E)~=~

(26)

This ratio is independent of temperature and can therefore be calculated from the values determined at 273 K. The combination of eqns. (25) and (26) gives

SoO =

n2k2T /dlnl o

OInK'~

+ d_r _n

~[SDal

-1

~-1

(27)

To a first approximation the term in parentheses in the numerator can be regarded as temperature independent, since the temperature dependence of K' is extremely small (see Section 3.2). Thus it is possible to calculate So ° for each temperature with the aid of the values in Table IV (eqn. (22)) which were measured at 273 K. It follows that SD° = - 2 . 7 2 IxV K - * , which is in good agreement with the value ( - 2 . 6 ~tV K - 1) obtained for the bulk material by linear extrapolation. The thickness dependence of S t [ at 77 K is given by eqn. (20). 3.4.2.2. Phonon drag at 77 K. Equation (18) can be applied both to the thermoelectric power S F of the film s F = SD F - - Sg F

(28)

14

G. WEDLER, G. ALSHORACHI

and to the thermoelectric power S O of the bulk material with the same density of imperfections as the film S O = SD° - Sg°

(29)

Subtracting eqn. (29) from eqn. (28) gives S F - - S D F --[- S O 0 ~--- S O -[- S g F -

(30)

Sg 0

S F is determined experimentally; SOv and So ° can be determined as shown in Section 3.4.2.1 ; S g F - Sg0 is given by 35' 38

SgV-Sg° = - S g ° 1 +

(31)

when the size effect theory of the electrons (see Section 3.1) is applied to the phonons, where 2o is the mean free path of the phonons. Combination ofeqns. (29)-(31) gives S F - SDF = Sg° - Sg° 1 + ~

(32)

which can be written as 1

l

K'2 o

1

(33)

SF_SD F - ~ o - ~ SO d In order to check the validity ofeqn. (33) (S F - SDF)- 1 is plotted in Fig. 11 against 1/d. For films thicker than 20 nm the linear relationship predicted by eqn. (33) is observed. F r o m the slope and the intercept Sg° and K'2 o were determined; their values are 7.22 gV K - 1 and 28.0 nm respectively. The value of Sg° is somewhat higher than that (6.3 gV K - 1) found for the bulk material in Section 3.4.2.

Ofo J

O'35 ~_, to

~ >

0-36

v

0-25

o o o~

/o0.~ -£o°°

0"2~

0-15

j~

0-10 005 0

i

i

I

i

I

2

3

/.

t/a

Fig. 11. A plot of I/(S F -- SD F) against 1/d to check the validity ofeqn. (33).

Huebener 4° has calculated 20 for platinum using the Fuchs-Sondheimer approximation for thick films, i.e. K ' = 3. He obtained 2o(Pt) = 48 nm at T = 59 K. F r o m the K'2o value determined in this paper it follows under the same assumptions that 2 o ( P d ) 7 7 K ---= 74 nm.

INFLUENCE OF THICKNESS ON ELECTRICAL PROPERTIES OF Pd FILMS

15

All the values determined from the measurements at 77 K are compiled in Table V. TABLE V THERMOELECTRIC DATA DETERMINED AT

77 K

So ° (bulk Pd) (laV K - 1)

SD° (eqn. (27)) (I.tV K - 1)

S~° (bulk Pd) (~V K - t)

Sg° (eqn. (33)) (ttV K - 1)

K'2 o (eqn. (33)) (nm)

2.6

2.72

6.3

7.22

28

ACKNOWLEDGMENTS

The authors thank the Deutsche Forschungsgemeinschaft and the Verband der Chemischen Industrie for financial support. One of the authors (G.A.) thanks the Deutscher Akademischer Austauschdienst for a scholarship. We thank Professor Wil3man and his coworkers for performing the X-ray diffraction measurements on palladium films. REFERENCES 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

R. Chander, Ph.D. Thesis, Delhi University, India, 1970. P. Rudolf, Thesis, University ofErlangen-Niirnberg, F.R.G., 1976. P. Rudolf and P. Wi~mann, Verh, Dtsch. Phys. Ges., 11 (1976) 1013. G. Wedler and R. Chander, Thin Solid Films, 65 (1980) 53. P . E . C . Franken, R. Bouwman, B. E. Nieuwenhuys and W. M. H. Sachtler, Thin Solid Films, 20 (1970) 243. C. Reale, J. Less-Common Met., 12 (1969) 167. G. Alshorachi, Thesis, University ofErlangen-Niirnberg, F.R.G., 1979. C. Pluntke, Thesis, University ofErlangen-Niirnberg, F.R,G., 1979. G. Wedler and G. Alshorachi, to be published. G. Wedler and P. Wil~mann, Z. Naturforsch., Teil A, 23 (1968) 1537. G. Wedler and P. Wil3rnann, Z. Naturforsch., Teil A, 23 (1968) 1544. P. Wil3mann and W. Fischer, personal communication, 1976. P. Wi~mann and H. Zitzmann, personal communication, 1980. M. Ritter, Thesis, University ofErlangen-Niirnberg, F.R.G., 1976. P . E . C . Franken, R. Bouwman, B. E. Nieuwenhuys and W. M. H. Sachtler, Thin Solid Films, 20 (1974) 243. B.E. Nieuwenhuys, R. Bouwman and W. M. H. Sachtler, Thin Solid Films, 21 (1974) 51. G. Ertl, personal communication, 1980. Landolt-Bbrnstein, Zahlenwerte und Funktionen, Vol. 11/6, Springer, Berlin, 1959. E.H. Sondheimer, Adv. Phys., 1 (1952) 1. A.F. Mayadas and H. Shatzkes, Phys. Rev. B, 1 (1970) 1382. P. Wil~mann, in G. H6hler (ed.), Surface PhysiC's, Vol. 77 of Springer Tracts in Modern Physics, Springer, Berlin, 1975. K.L. Chopra, Thin Film Phenomena, McGraw-Hill, New York, 1976, p. 366. H. Mayer, Physik Diinner Schichten, Section II, Wissensehaftliche Verlagsgesellsehaft, Stuttgart, 1955, p. 228. G. Wedler and M. Fouad, Z. Phys. Chem. (Frankfurt am Main), 40 (1963) 1. N.F. Mott and H. Jones, The Theory of the Properties of Metals and Alloys, Dover, New York, 1958. W, Wohlleben, Thesis, University ofErlangen-Niirnberg, F.R.G., 1977. H. Reichenberger, G. Wedler, H. Wenzel, P. Wil3mann and C. W61fing, Ber. Bunsenges. Phys. Chem., 75 (1971) 1033.

16

G. WEDLER, G. ALSHORACHI

28 29 30 31 32 33 34 35 36 37 38 39 40

R.D. Barnard, Thermoelectricity in Metals and Alloys, Taylor and Francis, London, 1972. G. Wedler, H. Reichenberger and H. Wenzel, Z. Naturforsch., Teil A, 26 (1971 ) 1444. E. Justi, M. Kohler and G. Lautz, Z. Naturforsch., Teil A, 6 (1951) 544. N.E. Cusack and P. W. Kendall, Proc. Phys. Soc., London, 72 (1958) 889. L. Nordheim and C. J. Gorter, Physica, 2 (1935) 383. D . K . C . MacDonald, Thermoelectricity, Wiley, New York, 1962. A.V. Gold, D. K. C. MacDonald, W. B. Pearson and I. M. Templeton, Philos. Mag., 5 (1960) 765. H. Sugawara, T. Nagano, K. Uozumi and A. Kinbara, Thin Solid Films, 14 (1972) 349. W.F. Leonard and S. F. Lin, J. Appl. Phys., 41 (1970) 1868. J.B. Thompson, Thin Solid Films, 18 (1973) 77. R.P. Huebener, Phys. Rev. A, 140 (1965) 1834. D. Greig, T. K. Brunck and P. A. Schroeder, Philos. Mag., 25 (1972) 1009. R.P. Huebener, Phys. Lett., 15 (1965) 105.