Temperature effects in thin metal films

Thin Solid Films, 148 (1987) 343-353 GENERAL FILM BEHAVIOUR

343

T E M P E R A T U R E E F F E C T S IN T H I N M E T A L F I L M S F. WARKUSZ Institute of Physics, Technical University of Wroclaw, Wybrze~e Wyspiahskiego 27, 50-370 Wroclaw (Poland) (Received July 7, 1986; accepted September 25, 1986)

The electronic thermal conductivity and the temperature coefficient of thermal resistivity for grainy metallic films were calculated in this work. The influence of electron scattering at grain boundaries and at the two external film surfaces was considered. The two surfaces were treated as rough and the angular dependences of the specularity parameters P(cos 0) and Q(cos 0) were taken into account.

1. INTRODUCTION When a temperature difference prevails between two parts of a metallic film, heat transfer occurs as a rule. The heat flows along the highest temperature gradient, the flow rate being proportional to the gradient. In good high purity metallic conductors, lattice vibrations contribute in only a small degree to the heat transfer so that the flow of energy is carried almost entirely by quasi-free electrons. In the case of semi-metals, however, the lattice conduction of heat is comparable with the electronic conduction. The thermal conductivity of thin continuous metallic films is known to depend on the ratio of the film thickness to the electronic mean free path in the bulk material. The observed decrease in conductivity with decreasing specimen size is known as the "external size effect". In a film formed from an isotropic bulk material, the external size effect occurs when a significant fraction of the thermal conduction electrons are scattered in a non-specular manner at the external surfaces1-3. In films formed from non-isotropic bulk materials, size effects have been theoretically predicted and experimentally observed even when the surface scattering is completely specular. This effect has been considered by including scattering of thermal conduction electrons at the grain boundaries as well as the normal background scattering by defects and phonons. This problem is known as the internal size effect3'4. The films have one surface in contact with the substrate material and the other in air or vacuum. While the substrate-film interface has a roughness characterized, to a large extent, by the substrate, the free surface is dominated by the reactivity of the metal with the surrounding gases, often air, particularly oxygen. Thus it is 0040-6090/87/$3.50

© ElsevierSequoia/Printedin The Netherlands

344

F. WARKUSZ

unlikely that it is physically acceptable to describe the two film surfaces in terms of the same specularity parameter. Furthermore, the thermal conductivity depends on the surface roughness. For films where the two external surfaces have independent, and often different, roughnesses it is natural to introduce two r.m.s, surface height deviations. The purpose of this paper is to present a theory of the longitudinal thermal conductivity (Section 2) and the temperature coefficient of thermal resistivity (Section 3) of thin metal films, taking into account both the external and the internal size effects and the geometrical roughness of the external surfaces. 2. THE THERMAL CONDUCTIVITYOF THIN METALLICFILMS The current density Jx and the heat flux JQx for a metal film subject to an electric field E and a temperature gradient OT/Ox in the x direction are given by 5'6

(m)3f

•Ix = --2e ~ Jex = 2 ~-

(fl + + f l - ) v x d 3 v

(fx

+

(1)

+ f l - )(e - eF)Vxd3v

(2)

Here f l is the deviation of the electronic distribution function obtained by solving the Boltzmann transport equation vz

+ ~ = evx

(3)

E4 e=F-~x

where z* is the total relaxation time given by 7

1 r~=

(

a* kF ~ 1 1+

(4)

IkF,IJT

with 2 R* D l-R* Hence the physical scattering parameter ~* is dependent on the grain size D, the bulk electron mean free path 2 and a coefficient R* defined as the grain boundary reflection coefficient, kF is the magnitude of the Fermi wavevector, kFx is the x component of the wavevector and z is the relaxation time for background scattering processes. Considering the Fuchs-Sondheimer boundary condition for two different external surfaces 3'6 (two different surface specularities P and Q), we obtain fl +

-- ez* t3fo sin 0 cos q~( E + -l 8--SF~=.-xT) m 0v

x{1

e

~'-

1-'+'(1-Q'exp(-a/z*vc°sO)exp(-,*vcosO)}i----~ex~~ for vz > 0

(5)

345

TEMPERATURE EFFECTS IN THIN METAL FILMS

and

fl

--

~

-

ez* t?fo sin 0 cos 4~ E + -

-

m

-

av

xJ" 1

1 e - eF t?T) e

7"- -~x

1-Q+Q(1-P)exp(a/z*vcosO)

1

P-Q ~x~-2a/--~v ~os ~

x f

a-z

e P[xz*v~os 0 ) } for v~ < 0

(6)

Equations (1) and (2) are integrated using the formula (7)

k &= J . . . . Hence

-~o ~"~(~)~d~ = ~"~(~)+ ~{°In-1)~"-~(~) +2.~F"-'\ ~ : .... +~F Le-7-) .... ~

(8)

The average current density .Ix and heat flux density JQx are given by the equations Y ~ = a1 .f£ J~ dz

J~x

(9)

1 I" jex d z a Jo

(10)

where a is the film thickness. Solving eqns. (9) and (10), having inserted expressions (1), (2), (5) and (6), we obtain the complete transport equation: ]~ = afAfE + trrSf(-VT ) ]Qx =

o'fSf r E +

o'fL 0 T(

-- V T)

(11) (12)

where af is the electrical conductivity of the thin metallic film, Sf is its thermoelectric power, Af is the temperature effect coefficient, L o is the Lorentz coefficient and VTis the temperature gradient. The first two are given as follows: af = aoF(K,P,Q, ct*)

(13)

Sf = SoB(K, P, Q, ~*)

(14)

where a o and So are the electrical conductivity and the thermoelectric power of the bulk metal respectively and K is the ratio of the film thickness a to the electron mean free path 2 in the bulk. These are given by 8~e2m2~-l)F 3 ao =

3h 3

(15) lr2k2T So -

2eev

F. WARKUSZ

346

F(K,p,Q, ot,)= G(c(,)_3_~ f~/2dq~ f~ - c°sEc~.

3-

1--exp(-KH/t)

a t - - - - ~ ( t -- t )1 -- ~ - ~ / t )

x {2 - (P + Q) + (P - 2PQ + Q) exp(- KH/t)}

(16)

where

f2do.c°s2cbsin30 3 f ~ f~dc~ G(a*) = ~ H =l-}e*+3e*2-3e*aln /-/= 1 -~

(') 1+~-~

(17)

(1 - t~) 1/~ c o s ¢

c(* is given by eqn. (4), t being equal to cos 0, where 0 is the angle which the electron mean free path makes with the normal to the film. Af and B are given by (~k r) 2

Af=l+--

8eF

x { 1 - 3150-5Kd~//dK+K2d2O/dK2+7g*d~b/dct*+ct*2d2~k/dct*2}F(K, P, Q, ~*) • (18)

1 ~b- Kd~/dK + a*dff/da* 3 F(K,P,Q,o~*)

B=I where V, = ~ 3g x

/2dr~

f,cos2¢.

a t - - - ~ ( t - - ta)l

(19)

1-exp(-KH/t) PQexp(-2KH/t)

{2 - (P + Q) + (P - 2PQ + Q) exp( - KH/t)}

(20)

It is convenient to write eqns. (11) and (12) in matrix form:

°f(s '

(21)

Actually we do not perform experiments to obtain E = 0. It is easier to place the sample in an open circuit so that there is an electric field across the sample:

E = ~VT

(22)

Inserting eqn. (22) into eqn. (12) we obtain .Tex= t r r T ( L o - ~ ) ( - -

VT )

(23)

which leads to the equation for the thermal conductivity of a thin metal film in the following form: xf =

of( o

r Af,/

(24)

TEMPERATURE EFFECTS IN THIN METAL FILMS

347

It is easy to derive the W i e d e m a n n - F r a n z law from eqn. (24):

K,f (Lo_Sf2~r =

(25)

The second term in the parentheses on the right-hand side of eqn. (25) is very small and can be neglected in the case of sufficiently thick films. Then eqn. (25) takes the form

r.f = aoLoTF(K,P,Q, ot*)

(26)

where ao is given by eqn. (15) and Lo = ~2k2/3e2. In the case of thermal conduction each electron carries its thermal energy of k T and is subjected to the thermoelectric power of k V T. The heat flux per unit temperature gradient is proportional to k 2 T. In the case of electrical conduction each electron carries its electric charge e and is subject to a force eE. The ratio of the thermal conductivity to the electrical conductivity has to be of the order of k2T/e 2 (x/o oc kT/e 2, cf. eqn. (25). The factor ~2/3 comes from the fact that we deal only with electrons on the Fermi surface, which are subject to Fermi-Dirac statistics. F(K, P, Q, 0(*) is a function of electron scattering in a thin film (eqn. (16)). Equation (26) can be rewritten in the form

xf = KoF(K, P, Q, ~t*)

(27)

where Xo = aoLoTis the thermal conductivity of the bulk. Softer s and Sambles and Elsom 9 have suggested that a rough metal surface could be characterized by an r.m.s, height deviation h and a mean lateral correlation length. In this work, as is often the case, the lateral correlation length is taken to be zero, i.e. the surface is uncorrelated. Taking the surface roughness as r = h/2,, where 2, is the electron wavelength, Softer was able to show that the specularity parameter could be expressed as P = exp{ -(4r~r) 2 cos20}

(28)

For films where the two surfaces have independent, and often different, roughnesses it is natural to introduce two r.m.s, surface height deviations, the corresponding roughnesses r 1 and r2, and specularities P and Q. We can see from eqn. (27) that the dependences of the thermal conductivity on the external size effect (the film thickness) and on the internal size effect (the grain structure) have an analogous form to those of electrical conductivity. The d ~ c e of xr/Xo (cf eqn. (27)) on K ( = a/2) for the fixed parameters of rl = r 2 = r and for the fixed parameter of electron scattering at grain boundaries (0t*) is shown in Fig. 1. We can see that xf approaches x o for K ~ oo and for 0t* ~ 0. With increasing r the curves in Fig. 1 fail since according to eqn. (28) the specularity parameters P and Q for the external film surfaces approach zero for r = 10. The dependence of xr on the parameter r is particularly remarkable for small K. For r = 0 we have total mirror-like reflection of electrons at the external film surfaces and P = Q = 1. It is interesting to study how the thermal conductivity changes with the film thickness for grainy films, for which the following calculations are useful. We can

348

p

F. W A R K U S Z

f /

0.8///"J/~ ~ w

-

-

-

S

~

.

.

.

.

.

.

.

.

.

.

.

.

.

-

oe-V3

-

oe*-Z/5

oe%2

0

2

~

6

8 ~ 0 K~

Fig. 1. •f/••vs.••mthicknessKf•rvari•usva•ues•fthegrainb•undaryparameterandtw•va•ues•fthe r o u g h n e s s r = h/2c: , r = 1 0 ; - - - , r = 0.1.

obtain the following expression from eqn. (27): 1 d~:f 1 dxo 1 dF(K,P,Q,~*) 1 ~F(K,P,Q,~*)dot* xf d K - x o d K ~ F(K,P,Q,~*) ~K + F(K,P,Q,~*) ~* dK (29) The expression (1/x o)dxo/dK equals zero since Ko is the thermal conductivity of the bulk. The third right-hand term in eqn. (29) is responsible for the internal size effect, i.e. scattering at grain boundaries. However, when we deal with a film with grains of a constant diameter which is independent of the film thickness and when the electron scattering at grain boundaries is constant (R* is constant), then d~*/dK = 0 and the last term in eqn. (29) vanishes. Typically in real films the average grain diameter D and the coefficient R* vary with the growth of the film t°'11. Experiment shows t° that D oc a for not very thick films and then for constant R* we obtain dot* 1 dK- oc -- K~

(30)

and for sufficiently thick films da*/dK ---,0. In this case the last term in eqn. (29) has a negligible contribution to the product (1/xf)dxf/dK and thus 1 dtcf 1 ~F(K,P,Q,a*) t¢r d K F(K,P,Q, ot*) dK

(31)

349

TEMPERATURE EFFECTS IN THIN METAL FILMS

3.

TEMPERATURE COEFFICIENT OF THERMAL RESISTIVITY FOR THIN METALLIC FILMS

The solution of the equation Jex = x f ( - VT)

(32)

with respect to VTis as follows: VT=

--

(33)

rfJQx

where rf is the thermal resistivity and for isotropic film is given by 1

(34)

rf = - Kf

Considering eqn. (24) we obtain Pf

rf = (Lo _ S f 2 / A f ) T

(35)

where pf = po/F(K, P, Q, ct*) is the thin film resisitivity. The resisitivity Po of the bulk is proportional to Tat high temperatures (practically above the Debye temperature) and this is why the thermal resistivity r 0 of the bulk approaches a constant at high temperatures in accordance with the Wiedemann-Franz equation. The temperature coefficient of thermal resistivity for a thin film is defined as 1 drf ~(f -

(36a)

rf d T

or

1 dtcf Zf--

xf dT

(36b)

Taking account ofeqn. (35) we obtain 1 dpf

)~f

-

-

pf d T

1

2(1/Sr)dSf/dT-

(1/Af)dAf/dT

po(LoAf/Sf 2 - 1)

T d

(37)

where pf = pO/~(K,P,Q,~*). The third term in eqn. (37) is very small and can be neglected. The temperature coefficient of film resistivity occurring in eqn. (37) has the form s, xi 1 dpo 1 dpf pf dT Po d r

1

F(K,P,Q,~*)

dF(K,P,Q, ot*) dT

(38)

Thus 1 d& _ l d p o

pf dT

Po dT

(

ld_~ fl-2

K OF(K,P,Q,a*) x. F(K,P,,Q, ct*) OK

ct* OF(K,P, Q, ot*) F(K,P,Q,a*)

(39)

where fl is the linear expansion coefficient for a solid. We assume that the scattering

350

F. WARKUSZ

parameters P and Q are independent of temperature, dP dQ = 0 d--T = d T

i.e.

dr1 dr2 d-T = d--T = 0

where r I and r 2 are the roughness parameters. The value of (1/2)d2/d T can be calculated from eqn. (15) if we assume that the rigid band model of metals is valid and that the number of conduction electrons is independent of temperature: 1 d2 = 2. d T

1 dpo Po d T

(40)

It should be noted that fl ~ (1/po)dpo/dT, i.e. less than the temperature coefficient of resistivity for the bulk material. Taking account of eqns. (39) and (40), eqn. (36) can be rewritten in the form

I ~T ( ;(f=

1

8F(K,P,Q,~t*)~ 1 K tgF(K,P,Q,~t*) or* F(K,P,Q, ot*) t~K ~ F(K,P,Q, ct*) (41)

which for the bulk metal transforms into the expression 1 dr o 1 dpo ro d T = Po d T

1 T

(42)

The size effects so important in thin films are described by the term K tgF ~* t3F -~ F t3K F d,t*

1

(43)

F(K,P,Q, ct* = dpo(1 K t~F)

For non-grainy film F =

1 drf 1 Zf = r-f O-T = p-o d--T

F ~

0), so that eqn. (41) will transform into 1 T

(44)

From the fact that the change in thermal conductivity due to variation in the film thickness is described by eqn. (31), equation (44) can be written for non-grainy films in the form

_ Zf 4.

1 dpo Po d T

( 1 + - -Kd,, rf d K ]

1

(45)

T

DISCUSSION AND CONCLUSIONS

In our considerations a theoretical analysis of the effects of temperature on the thermal conductivity and on some transport parameters in thin grainy films with rough external surfaces was presented. A general expression for the thermal conductivity in these films (eqn. (24)) was derived. It was noted that in the approximate expression (eqn. (26)) the change in thermal conductivity due to the dependence on grain boundaries and surface effects is analogous with the change in electric conductivity if only electrons are responsible for the heat transfer. This will not hold in the case of semimetals since the lattice thermal conductivity in these is

T E M P E R A T U R E EFFECTS IN T H I N M E T A L FILMS

35 1

comparable with the electronic thermal conductivity. As can be seen from the plot in Fig. 1, with increasing electron scattering at grain boundaries (increase in ~t*) the thermal conductivity decreases, especially when the roughness of the external film surface also increases. The influence of the surface roughness on the conductivity, however, is lower than that of grain boundaries (see Fig. 1). The influence of the surface roughness is observable, in general, in very thin films. In order to show how the thermal conductivity changes with the film thickness it is convenient to use eqn. (29). This equation can be simplified when the change in scattering parameter ~t* with the film thickness is known. In general, with increasing film thickness, a* decreases since the grain diameter increases (see eqn. (4)). The thermoelectric power Sf calculated for grainy films has been presented in many works 5"11. The effects of the external and internal size effects are particularly marked for very thin films, i.e. for K < 1. For other films these effects are minimal and can be neglected in expressions for the thermal conductivity. The effects of surface roughness on the thermoelectric power have been calculated by Sambles and Preist 12. In Section 3 the temperature coefficient of thermal resistivity Xfwas calculated for grainy films (eqn. (41)). For non-grainy films the Mayadas-Shatzkes function transforms into the Fuchs-Sondheimer-Lucas function 13 and eqn. (41) can be rewritten in the form of eqn. (44). A plot of the temperature coefficient of thermal resistivity ~f for aluminium film vs. the film thickness K for different temperatures and scattering parameters ~* is shown in Fig. 2, on the assumption that the temperature coefficient of electrical

T-6OOIK]-

/ .-%-f-

15

,

soo

~

300

B

K

d J

-15

Fig. 2. Xf v s . film t h i c k n e s s K f o r v a r i o u s t e m p e r a t u r e s T, a r o u g h n e s s r = 10, a n d t w o values boundary parameter: , ~* = 0; - - -, ~* = 1/15.

of the grain

352

F. WARKUSZ

resistivity is constant over the temperature range under consideration. In the calculations presented in this figure the temperature coefficient of electrical resistivity has been assumed.to be 39 x 10 -4 K - 1 for aluminium. The temperature coefficient of thermal resistivity increases with the film thickness, and for very thin films it can assume negative values. From an analysis of the solution ofeqn. (41) it is found that the temperature coefficient of thermal resistivity is lower for grainy films, i.e. when electron scattering at grain boundaries increases, then Xf decreases. This is shown in Fig. 3 (for the bulk metal Xo is expressed by eqn. (42)). In Fig. 4 the dependence of Zf on the surface roughness r = h/2e is shown for a film thickness of K = 0.5 and for fixed parameters of electron scattering at grain boundaries, ~* = 0 and ~* = 1/15. The temperature coefficient of thermal resistivity decreases with increasing surface roughness.

3¢te'J kt aO"[K~

35

20 30

Xf

\\

aS x

5

\\

I0 0

"x x

x-.0<5 " ' .

1

15

~.T-600 [K]

5'

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

-5 ~-

. . . . . . . . . . . . . . . . . . . . . .

0 40

400 . . . . . . . . . . . . . . . . . . . . . . . . .

_c. -'r5

-16 -20

300 ~

. . . . . . . . . . . . . . . . . . . . . . . .

-I c. -25

-2C

I

I

2

4

,

I

~

I

,

10 h/A, Fig. 3. Xf vs. the grain boundary parameter ct* for a roughness r = 10, v a r i o u s f i l m t h i c k n e s s e s K and various temperatures T : - - . - - , K = 0.1; - - - , K = 1 ; - - - , K = 10; c u r v e s a, T = 3 0 0 K ; c u r v e s b, T=

400 K;curves

c, T =

500 K;curves

d, T =

6

8

600K.

Fig. 4. gf for thin films ( K = 0.5) shown as a function of the roughness r = and two values of~*: - - - , ct* = 0; . . . . , ct* = 1/15.

h/2 c

for various temperatures

ACKNOWLEDGMENTS

This work was supported from the CPBP Research Programme. The author is grateful to Professor C. Wesolowska for helpful suggestions.

TEMPERATURE EFFECTS IN THIN METAL FILMS

353

REFERENCES

F.J. Blatt, Physics of Electronic Conduction in Solids, McGraw-Hill, New York, 1968. K. Fuchs, Proc. Cambridge Philos. Soc., 34 (1938) 100. E.H. Sondheimer, Adv. Phys., 1 (1952) 1. F. Warkusz, Prog. Surf Sci., 10 (3) (1980) 287-382. I.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge, 1964. F. Warkusz, Thin Solid Films, 122 (1984) 105. A.F. Mayadas and M. Shatzkes, Phys. Rev. B, 1 (1970) 1382. S.B. Soffer, J. Appl. Phys.,38(1967) 1710. J.R. Sambles and K. C. Elsom, J. Phys. D, 15 (1982) 1459. P. Wissmann, The Electrical Resistivity o f Pure and Gas Covered Metal Films, Vol. 77, Springer, Berlin, 1975. 11 C.R. Tellier and A. J. Tosser, Size Effects in Thin Films, Elsevier, Amsterdam, 1982. 12 J.R. Sambles and T. Preist, J. Phys. F, 14(1984) 1693. 13 M.S.P. Lucas, J. Appl. Phys., 36 (1965) 1632. 1 2 3 4 5 6 7 8 9 10