The influence of surface roughness on electronic transport in thin films

(1)

Surface Science 269/270 (1992) 772-776

North-Holland

"surface science

The influence of surface roughness on electronic transport in thin films

G. Reiss a n d H. Briickl

lnstttut f~ir Angewandte Phystk, Unu ersttdt Regensburg Unwersitatsstrasse 31, D-8400 Regensburg, Germany Received 9 September 1991, accepted fo~ pubh,.,tlon 10 September 1991

In thin films, the structure of the surfaces considerably mfhences the transport of conduction electrons. For mesoscopic roughnesses m the range of a few nm, this ~s due to the varying film thickness, which gwes rise to a spatially fluctuating conductance Moreover, mxroscop~c roughnesses can contnbute to the scattermg of the electrons and therefore addmonally enhance the thm-film resistivity. For a quanmat~ve understandmg of the transport m these systems, a detatled investigation of the surface roughness combined w~th measurements of the electro,tic properties are necessary. Here, v,e discuss STM imaging of various metal films and the application ef these results to the interpretation of electronic thin-film propemes. Provided reasonable resoluhon of STM m the nm range, a good correspondence of STM results with the electrical behavlour ef growing metal films can be estabhshed Furthermore, a detailed two-d,mens~onal analysis allows for a calculation of the potential on current-carrymg thin films On tl,e other hand, th~ method supphes rehable values for the electromc transport parameters

1. Introduction

The conductwlty of thin films has been fre- quently discussed m the hterature. Whereas scm~-classLcal models used the Boltzmann equa- tion with appropriate boundary conditions [1], more recent contributions [2-4] gave quantma- mechanical models especially for the thickness d e p e n d e , c ~ of the conductivity. In these discussions, special regard must be naturally taken on the surface scattering of the conduction electrons which is the mare reason for the Increase of the resistwtty of thin films. In ref. [4], it was sug- gested, that a possible variation of the electron density with thicknesb could additionally influence the conductwlty.

The mare purpose of these d~scussions is the evaluation of reliable transport parameters from the thickness dependent conductivity or(d) (d:

film thickness). In order to obtain these values, the theoretical curve must be fitted to the experiment by a variation of usually at least three parameters, one of them representing th~ ~urface

profile. This, however, usually was performed in one dimension without knowledge of the real surface. Therefore, a detailed STM analysis of the thin-film sin faces can considerably improve these investigations. Since STM directly images the surfaces, the discussion of the conductivity can be extended to two dimensions. With this treatment the distribution of the electrostatic potential on current carrying thin films can be additionally estimated. The precondition for these evaluations, however, are reliable results of STM imaging.

2. Surface imaging

T h e l m . o h ~ o a f ~'hP t h ; n - f H m ~ n r f a c e q wa¢, t3er-

formed with STM under a m h e n t conditions [5]

The materials have been Pt, Au, N~, Co and Cu.

In order to evaluate possible influences of oxidation, Ni and Cu was partly covered with a 1-2 nm thick Au overlayer. In the case of Ni, STM re- vealed identlca! surface features without remark-

(2)

G Relss, H. Bruckl / The influence o f surface raagnnes~ on electrontc transport ,n thin ftlms 773

able signs of oxidation on the unprotected sam- ples. On pure Cu, rehable imaging was possible only for a few hours after the removal of the films from the UHV chamber. The Cu films protected by Au, however, exhibited the same surface features and allowed stable tunneling for considerably longer time.

As mentioned in the introduction, the resolution of nm features is of special interest for an application of STM results to electronic transport. This point has b e e n already stressed in refs.

[6,7]: based on the STM image and a properly evaluated tip shape, the real surface can be re- constructed except those parts, the tip was not in tunneling contact with during scanning the surface. T h e numerical procedure proposed in ref.

[8] has been recently reformulated in terms of Legendre transforms [9]. In the recalculated image, the parts which had not been reached by the tip remain as "black holes". Since the amount of black holes obviously is directly related to the obtained height resolution, we show in fig. la a model surface consisting of semi-elliptical islands.

If the tip indicated in fig. l a scans this surface, the resulting STM profile corresponds to the line shown in fig. lb. Using the definitions of the height resolution indicated in fig. 1, the dependence of this quantity on the amount ot surface without tunneling contact to the tip can be estimated. The result of this calculation is shown in fig. 2 for different ratios a =

R/H.

As can be seen from this estimate (fig. 2), the obtained height resolution depends very critically on the amount of unresolved surface area. Only a few percent of black-hole surface can cause a considerable underestimate of the roughness. For the further discussion, we therefore used only

1

0 9

0 8

0.6 \ ,

0 0.1 0.2 0 3 0.4 0.5

Reladve amount of black-holes

Fig. 2 The hetght resolution of STM images as a funcuon of the surface area without tunnehng contact to the ttp

images with a black-hole area equal or less than 1% of the whole surface.

If, however, the precondition concerning the resolution is fulfilled, STM clearly can provide reliable surface profiles for the discussion of the electronic transport properties.

3. The thickness dependence of the resistivity For the evaluetlon of the transport parameters, we use the model of Teganovid et al [2].

Here, the thickness dependence of the conductwlty is given by:

,./,-.

~_~ l=h k~ n

o-= z.., 1 + 6~-d(x) , (1)

llc n = l

where the sum includes the occupied subbands (index n),

d(x)

is the local film thickness, kf is the Fermi wavevector, o'= is the conductivity of

rip I R I

H

^--'~/~- ^-.~.:~ ^{~ _ _ _}

h

Fig 1 (a) Model surface for esttmatmg the height resolution of STM (b) STM profile corresponding to the up/surface combmatxon of (a). The height resolution is defined as the ratio h / H

(3)

774 G Rets~, H Bruckl / The influence of surface roughnes3 ¢.n e;e:tron,c trampott u, thin films

E 10

S

_z-

• ~ 5 e- o o

~ 0

0

OCu ONi

O Au

Pt, C o

I I ....

0 5 10

STM roughnesses (nm)

Ftg 3. Onset thtckness of the ohmtc conductwtty of films growing on glass substrates at r o o m temperature c o m p a r e d

~t lth the correspondm ~ rot~ghness found by STM. The ramus of the dots corresponds to the spread of the experimental

data

infinitely thick material and l=h z is the product o f the intrinsic electronic m e a n free path and the microscopic "oughness of the surface. Rough- nesses eonsiaecably larger than the Fermi wave- length can be included by averaging over a vary- mg film thickness d ( x ) similar to the discussions given in refs. [4,7,10].

The application of STM results can be performed an one dm~enslon simply by using a corresponding one-dimensional rough,Jess distribution The p r e c o n d m o n s and results of th~s t r e a t m e m have been already d,~cussed m rcf [7] for thin NI films. In order to d e m o n s t r a t e the relevance of the m e t h o d for different materials, fig. 3 shows the correlation of roughnesses estimated by two methods:

First, the roughness can be simply obtained via highly resolved STM images. Second, the m e a n roughness should correspond to the thickness of the growing film, where the onset of the ohmic conductivity can be found [7] (see eq. ~1)).

From fig. 3, a suprislngly good agreement between these values of the thin film roughness can be estabhshed for different metals in a range from about 1 to 10 nm.

On the other hand, the one-dimensional model cannot reproduce the conductiwty especially for very small tmcKncsses i7]. Thi~ ~ du~ tu "" ~ '°-' [ . I l k ! e l k 1~

of percolation, ~.e, withm this treatment the con- ductivtty a~proaches zero at a thickness corresponding to the maxtmum corrugation found by

STM. In o-de~ to achmve a better descripuon, we t h u s , xtenuefl o u r discussion to two dime asions.

For this purpose, the film was modeled as a (128 x 128) networks of resistors, each of which represents a small portion of t h e integral film resistance. T h e corresponding local conductances can be o b t a i n e d from eq. (1) in combination with the local thicknesses supplied by t h e STM image.

In order to obtain ~he conductivity, the voltage has to be fixed at two edges o f t h e STM image.

The resulting voltages V, at n o d e i of the network can be o b t a i n e d selfconsistently using the local conductances g,j between node i a n d node j [11]:

4 Vj 4

V~= ~ -2--/ Y'- g,,. (2)

3=1 61.1 J = l

This self-consistent formalism quickly converges towards the distribution {Vj} of t h e potential on the thin-film surface. Using this potential, the current can be evaluated using t h e local conductances defined before. From these values of current and voltage, the integral conductivity can be obtained m the usual way.

h f i g . 4, we show the results of thl~ two-dimensional e s n m a t e together w~th the s~mpler one-dimensional model and the experimental re- suits for the reslsnvity of a i '~ thin film

As can be seen from fig. 4, the one-dimensional fit approaches the experimental lcsults at a thickness of about 7 nm. T h e m a x i m u m depth of

80

Eo

60

:::£

:~ 40 er I1

20

2-d model 1-d model

#

N exp results 7nm

0 0 08 0.12 0 16

1 I d ( nm -1)

Nt

Fig 4 Re',ults o~ biting the theoretical expre',slons for the ltn~krle~-dcpcndcnl thin-him conducnv~ly to the experimen-

tal cur~e

(4)

(; Retss. H Btuc£1 / "fhe tnfhwtwe ol surface touglmes~ on eh'c trontc ttan~pcnt m thin ]tht~s 775 the b u m p s found by S T M on this surface was

a b o u t 6 nm [7]. On the o t h e r hand, the two-dimensional t r e a t m e n t can r e p r o d u c e the experi- m e n t a l curve down to n m c h smaller thicknesses A s a l r e a d y m e n t i o n e d , this is due to the p e r c o l a - tive b e h a w o u r for thicknesses smallec than the m a x i m u m roughnesses, which can be t r e a t e d solely by a two-dimensional description.

T h e bulk resistivities p~ obtained from the fitting calculations are 19.5 and 21 ~ [ l cm for the o n e - and two-dimensional case, respectively. F o r the p r o d u c t of the m e a n free path with t h e microscopic surface roughness, we o b t a i n e d 1.25 nm ~ ( 1 D ) and 1.4 nm 3 (2D). Since the t w o - d i m e n - sional t r e a t m e n t is m o r e realistic, the m e a n free path therefore s e e m s to be somewhat larger than the values o b t a i n e d f r o m the o n e - d i m e n s i o n a l discussion. Nevertheless, already the simple one- dimensional model reveals rather correct values for this f u n d a m e n t a l t r a n s p o r t p a r a m e t e r .

T h e two-dimensional model, however, can sup- ply m o l e information: A s discussed b e f o r e , the distribution of the electrostatic potentmi on the current-car~,mg thin film must be c a l c u l a t e d for fitting the theory to the experiment. O n the o t h e r hand, th~s d~stnbution itself is of c o n s i d e r a b l e

interest In fig. 5 we show the original S T M t o p o g r a p h y of a 10 nm thick Ni film (hg 5a) and the c o r r e s p o n d i n g calctdated distribution of the local electrostatic field d e f i n e d as the m a g n i t u d e of the gradient of the local potential. A g o o d c o r r e s p o n d e n c e of the topographical features with the local field can be established: at locations o f small film thickness, the local field can be en- h a n c e d by a factor two c o m p a r e d with the m e a n field.

Fig. 5 t h e r e f o r e shows, that a rather inhomo- g e n e o u s distribution of the potential on rough thin films can be e x p e c t e d d u e to the spatmlly varyipg conductances. O n the other hand, the d r o p of the potential i~self is not as m h o m o g e - ncous as found by p o t e n t i o m e t r i c STM m e a s u r e - m e n t s on polycrystal|ine thin films [12,13], Thus, it s e e m s to be necessary to include gram b o u n d - ary s c a t t e r m g in the discussion of the experimen- tally e v a l u a t e d potential [14].

4. C o n c l u s i o n s

In this contribut~ol~, we p~escntcd a t~o-dimensional lrcatmcnt ot the t h l c k n e s s - d e p e n d e m

~ F '~

Fig 5 (a) Grey-scale image o1 Ihe topography of a 10 nm thick Ni film (180 nm x 180 nm) The scale l~ Ill nm ram, d,l,k L,, ,,'.~,k (b) The magmtud¢ ol the local Paid calculated tor the surface topography ,,hogan m (a) \~mt¢ corr¢,,pond,, to a field t~lcc a~ large

as d,u k

(5)

7"6 G Relss, H Bruckl / The mfluence of surface roughne~ on electrontc transport m thin fibre

thin-film r.~s:stlviw. The surface profiles necessary for this purpose can be sdpplied by highly resolved STM images. The STM roughnesses of the surfaces have been shown to correspond very well with the thickness of the onset of ohmic conductivity, i.e., just with the stage of coales- cence of the growing film.

A two-dimensional fit of the experimental data of the thickness-dependent thin-film resistivity can be performed using a selfconsistent resistor network model and the topography supplied by STM. Although tt,e realistic two-dimensional model supplies slightly different values, the results for the transport parameters agree rather well with former one-dimensional treatments [7,10]. In contrast with this, however, the extended discussion can include the stage of percolation, i.e., using this formalism, network struc- tures can be treated. Moreover, this formalism naturally supplies the distribution of the potential on a current-carrying thin film. This turns out to be rather inhomogeneous as soon as the roughness becomes comparable to the f:lm thickness.

The potential obtained, however, is still smoother than observed by potentlometnc STM measure-

ment. Therefore it seems to be necessary to include addiuonally grain boundary scattering in order to explain these results.

References

[1] K Fuchs, Proc Cambridge Phd. Soc. 34 (1938) 100.

[2] Z Te~anovl~, M V Jan6 and S. Maekawa, Phys. Rev Lett 57 ¢1986) 2760

[3] D. Caleck, and G FIshman, Surf Scl 229 (1990) ll0.

[4] N Trivedl and N W Ashcroft, Phys. Rev B 38 0988)

! 2298.

[5] J Vancea, G. Relss, F. Schneider, K Bauer and H Hoffmann, Surf. So 218 (1989) 108.

[6] G. Reiss, H Brtickl, J Vancea, R. Lecheler and E.

Hastreiter, J, Appl Phys. 70 (1991) 523

[7] G. Relss, E. Hastreller, H. Bruckl and J Vancea, Phys.

Rev. B 43 (1991) 5176

[8] G. Relss, F, Schnetder, J. Vai~cea and H. Hoffmann, Appl. Phys Lett 57 (1990) 867

[9] D. Keller, Surf Scl. 253 (1991) 353

[10] U. Jacob, J Vancea and H. Hoffmanl~, Phys. Rev B 41 (1990) 11852

[ll] S. Kirkpatnck, Rev. Mod Phys 45 (1973) 574

[12] J P Pelz and R H Koch, Rev. So In~trum. 611 (1989) 301

[13] J P Pelz and R H . Koch, Phys Rev B 41 (1990) 1212 [14] C S Chu and R S Sorhello, Phvs Rcv B 42 (1990) 4928.