Mound surface roughness effects on the thermal capacitance of thin filmsG. Palasantzas and J. Th. M. De Hosson Citation: Journal of Applied Physics 89, 6130 (2001); doi: 10.1063/1.1368390 View online: http://dx.doi.org/10.1063/1.1368390 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/89/11?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Thermal convection in a rotating porous layer using a thermal nonequilibrium model Phys. Fluids 19, 054102 (2007); 10.1063/1.2723155 Design of a film surface roughness-minimizing molecular beam epitaxy process by reduced-order modeling ofepitaxial growth J. Appl. Phys. 95, 483 (2004); 10.1063/1.1632554 Evolution of normal stress and surface roughness in buckled thin films J. Appl. Phys. 93, 893 (2003); 10.1063/1.1528299 Self-affine and mound roughness effects on the double-layer charge capacitance J. Appl. Phys. 92, 7175 (2002); 10.1063/1.1519952 Numerical and experimental study on the thermal damage of thin Cr films induced by excimer laser irradiation J. Appl. Phys. 86, 4282 (1999); 10.1063/1.371358

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Mound surface roughness effects on the thermal capacitance of thin filmsG. Palasantzasa) and J. Th. M. De HossonDepartment of Applied Physics, Materials Science Centre and Netherlands Institute for Metals Research,University of Groningen, 9747 AG Groningen, The Netherlands

~Received 28 November 2000; accepted for publication 7 March 2001!

We investigate the influence of roughness at a nanometer scale on the thermal properties of thinfilms. It is shown that the roughness causes an increase of the thermal capacitance. For mound roughsurfaces the increase of the thermal capacitance depends strongly on the relative magnitude of theaverage mound separationl and the system correlation lengthz. Indeed, a rather complex behaviordevelops forz.l, while for z,l a smooth decrease of the capacitance as a function of the averagemound separationl takes place. Finally, the roughness strongly affects the thermal capacitance asa function of the film thickness as long asz,l, while a precise determination of the actual effectrequires a more-detailed knowledge of the thickness dependence of the involved roughnessparameters during film growth. ©2001 American Institute of Physics.@DOI: 10.1063/1.1368390#

I. INTRODUCTION

Many device geometries require the growth of high-quality films. However, kinetic effects may induce a substan-tial roughness depending on the material, the substrate, andthe deposition conditions. Deviations of surfaces/interfacesfrom flatness, as well as the presence of material defects~e.g., dislocations, impurities, etc.! may alter the operationcharacteristics of microelectronic devices.1,2 For example,the presence of a rough metal/insulator interface was shownto influence the field breakdown mechanism,3 as well as thecapacitance and leakage currents in thin-film capacitors.4

Random rough surfaces have been shown to affect the imagepotential of a charge situated in the vicinity of the vacuum/dielectric interface.5 Such roughness effects could have astrong influence on inversion layers at the semiconductor/oxide interface, because they cause a shift of electroniclevels5 and, consequently, alter the device operation. In ad-dition, surface/interface roughness has been shown to influ-ence strongly the electrical conductivity of semiconductingand metallic thin films.6

In contrast to electric transport properties, only a smallamount of progress has been achieved for roughness effectson thermal transport properties in thin films. Indeed, thetopic of thermal management issues in electronic and opticaldevices has been a topic of intense research in recentyears.7–10 For example, the lifetime of metallic interconnectsin integrated circuits depends strongly on the operating tem-perature because the resistive heating increases their tem-perature. In photothermal analyses,11 Au films are depositedon dielectric substrates for heat adsorption because theirproperties are necessary for deducing the thermal propertiesof the dielectric film. It appears that the thermal conductivityof thin films ~thicknessh0,1000 nm! is less than its bulkvalue due to surface and grain-boundary scatteringmechanisms.11,12 Although much work has been concen-

trated on the effect of film thickness on thermal conductivity,only recently the effect of surface roughness and film micro-structure was investigated in order to understand the thermalconductivity results.7 It was found that roughness develop-ment yielded a lower thermal conductivity, with the film tex-ture also playing a significant role.7

Furthermore, during epitaxial growth of thin films thegrowth front can be rough in the sense that multilayer stepstructures are formed.13,14 In this case, the existence of anasymmetric step-edge diffusion barrier~Schwoebel barrier!inhibits the down-hill diffusion of incoming atoms, leadingeffectively to the creation of multilayer step structures in theform of mounds~unstable growth!.13,14 So far, a theoreticalinvestigation of the mound surface roughness, and more gen-eral thin-film growth mechanisms, on thermal properties re-mains widely unexplored, and will be the topic of the presentwork.

II. THERMAL FIELD AND CAPACITY CALCULATIONFOR A FILM WITH A SINGLE ROUGH BOUNDARY

A. Thermal field calculation

Consider a film with its rough surface at temperatureT2

and its smooth one~film/substrate interface! at temperatureT1 , as is shown in Fig. 1. In order to calculate the tempera-ture distribution under steady heat flow conditions (]T/]t50), one needs to solve the Laplace equation1 for filmthickness varying betweenz50 andz5 f (x,y):

¹2T~x,y,z!50, ~1!

and boundary conditions

T~x,y,z50!5T1∧T@x,y,z5 f ~x,y!#5T2 , ~2!

where T(x,y,z) is the temperature distribution, andz5 f (x,y) is the rough film surface. Assumingf (x,y)5h0

1lh(x,y), we can apply a perturbation theory to solve Eq.~1! by expanding Eq.~2! andT(x,y,z) in Taylor series of theform

a!Author to whom correspondence should be addressed; electronic mail:g.palasantzas@phys.rug.nl

JOURNAL OF APPLIED PHYSICS VOLUME 89, NUMBER 11 1 JUNE 2001

61300021-8979/2001/89(11)/6130/5/$18.00 © 2001 American Institute of Physics

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T~x,y,h0!1Tz~x,y,h0!lh~x,y!

1 12!Tzz~x,y,h0!l2h2~x,y!1¯5T2 , ~3!

T~x,y,z!5T~0!~x,y,z!1lT~1!~x,y,z!1l2T~2!~x,y,z!1¯ .~4!

For the zero-order termT(0)(x,y,z), the boundary conditionsT(0)(x,y,z50)5T1 andT(0)(x,y,z5h0)5T2 yield the solu-tion T(0)(x,y,z)5(z/h0)(T22T1). For first- and second-order termsT(n)(x,y,z)(n>1) we have, respectively,

T~1!~x,y,z50!50∧T~1!~x,y,z5h0!

52h~x,y!Tz~0!~x,y,h0!, ~5!

T~2!(x,y,z50)50∧T~2!(x,y,z5h0)

52h~x,y!Tz~1!(x,y,h0)2 1

2h2~x,y!Tzz

~0!

3~x,y,h0!. ~6!

Further, the Fourier transform technique is employed tosolve the Laplace equations¹2T(n)(x,y,z)50 (n>1),which are obtained by substituting Eq.~4! into Eq. ~1!. Set-ting l51 in Eq. ~4! and using Eqs.~A3!–~A4! from theAppendix, we obtain

T~x,y,z!'~T22T1!

h0z

2~T22T1!

h0E dk

sinh~kz!

sinh~kh0!h~k!e2 ikr

1~T22T1!

h0E dkE dk8

3cosh~k8h0!sinh~kz!

sinh~k8h0!sinh~kh0!

3k8h~k8!h~k2k8!e2 ikr. ~7!

B. Thermal capacitance calculation

The specific thermal capacitance~for constant volume!‘‘ c’’ is given by the energy balance equation

Vc~T22T1!5K thctE ~2“T•n!dS, ~8!

with n5(¹h2 z)/@11(¹h)2#1/2 the unit vector normal tothe surface, K thc the thermal conductivity, dS5A11(¹h)2d2r the differential area,t the time that theheat flux is passing through the film, andV the film volume.Note that2¹T represents the thermal field that drives heatflow, andK thc(2¹T•n) the heat density on the rough filmsurface.

Assuming statistically stationary surfaces up to the sec-ond order~translation invariant!, the average over possibleroughness configurations can be expressed as^h(k)h(k8)&5@(2p)4/A#^uh(k)u2&d(k1k8), whereA is the average flatmacroscopic surface area. For small surface roughness oru¹hu!1(dS'@11(1/2)(¹h)2

¯#d2r), we obtain fromEqs.~8! and ~7!, and taking into accounth(k)&50, the av-erage thermal capacitance

^c&5^c0&M ~h0!, c05K thct

h02 , ~9!

where

M ~h0!5H 11~2p!4

A F2E0,k,kc

k2^uh~k!u2&d2k

12p

h0E

0,k,kc

cosh~kh0!

sinh~kh0!k^uh~k!u2&d2kG J ,

~10!

with Kc5p/a0 anda0 a lower roughness cutoff of the orderof the atomic spacing, and film volumeV'Ah0 .

III. MOUND ROUGH MORPHOLOGY

Mound rough surfaces have been described in the pastby the interface widthw; the system correlation lengthz,which defines the randomness of the mound distribution onthe surface; and the average mound separationl.14 Such arough morphology can be described by the height–heightcorrelation function C(r )5^h(r )h(0)&5w2e2(r /z)2

J0(2pr /l), where its Fourier transform is ofthe form14

^uh~k!u2&5A

~2p!5

w2z2

2e2~4p21k2l2!~z2/4l2!I 0~pkz2/l!,

~11!

with h(k) the Fourier transform ofh(r ). J0(x) and I 0(x)are, respectively, the Bessel and modified Bessel function ofthe first kind and zero order. Ifz>l, the surface is charac-teristic of that caused by Schwoebel barrier effects.14 Notethat the correlation functionC(r ) of the mound roughnesshas an oscillatory behavior forz>l, which leads to a char-acteristic satellite ring atk52p/l of the power spectrum^uh(k)u2&.14

FIG. 1. Schematic of a film with a single rough boundary~film surface! anda smooth film/substrate interface.

6131J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 G. Palasantzas and J. Th. M. De Hosson

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IV. THERMAL CAPACITANCE DEPENDENCE ONMOUND ROUGHNESS

Our calculations have been performed for small rough-ness (u¹hu,1) or alternatively small rms local surfaceslopesr rms5^u¹hu2&1/2,1, and small rms roughness ampli-tudesw!h0 . Figure 2 shows the dependence of the localsurface slope as a function of the average mound separationl for various system correlation lengthsz. The localsurface slope is given by r rms5$@(2p)4/A#3*0,k,kc

k2^uh(k)u2&d2k%1/2, which is essentially propor-tional to the second term in Eq.~10!. For system correlationlengthsz.l, the local surface slope decays in an oscillatoryfashion asl increases~or the ratiow/l decreases leading tosurface smoothing!. On the other hand forz,l, the localslope decays exponentially as the surface smoothens orw/ldecreases.

Since^uh(k)u2&}w2, the thermal capacitance will have asimple dependence on the roughness amplitudew, while anycomplex dependence will arise as a function of the laterallength scalesz and l through the factorM (h0). Figure 3shows the dependence of the thermal capacitance ratio^c&/c0 as a function of the average mound separationl. Atany rate, c&>c0 , or in other words, the presence of rough-ness increases the thermal capacitance.

Upon increasingl ~or decreasingw/l!, i.e., surfacesmoothening at larger wavelengths,^c& approaches the ther-mal capacitancec0 characteristic for a film with a flat sur-

face. However, for large system correlation lengths such thatz.l, significant increments ofc& develop, approaching fi-nally c0 for large mound separationsl~.z!. Such an effect isspecial for mound roughness, while for any other type ofroughness, such as Gaussian or self-affine fractal roughness,a smooth approach to the thermal capacitance for films withflat boundaries will eventually occur~similarly to the case ofl@z!. Such an effect reflects the oscillatory behavior of theheight–height correlation function for mound roughness thattakes place forz.l.14 Finally, the thermal capacitanceshows a complex behavior as a function of the system cor-relation lengthz ~Fig. 4!, i.e., with a smooth decrement to-wards it value for films with flat boundaries as long asl issmall ~,z!. For intermediate values of the average moundseparation a complex behavior develops, as is displayed inFig. 4.

Note that if in Eq.~10! the integration is extended toinfinity (Kcz,Kcl@1), we obtain a simpler thermal capacityexpression

^c&5c0F11w2S 1

z2 1p2

l2 D1

~2p!w2z2e2~p2z2/4l2!

2h0E

0

`

k2cosh~kh0!

sinh~kh0!

3e2k2z2/4I 0~pkz2/l!dkG , ~12!

with the third term in the parenthesis of Eq.~12! becominginsignificant with increasing film thicknessh0 .

Finally, we will investigate roughness effects on thethermal capacitance as a function of the film thickness. Fig-ure 5 shows the dependence of^c&/c0 on film thicknessh0 .The ratio^c&/c0 decreases drastically for small system cor-relation lengths~z,l!, which can be understood from Eq.~12! due to its Gaussian dependence on the ratioz/l. A moreprecise determination of the film thickness effect needs amore precise knowledge of the dependence of the roughnessparameters (w,z,l) on film thickness. For example, the av-erage mound separation has been shown to evolve with filmthickness~or growth time for fixed deposition rate! as l}h0

b ~0.16<b<0.26!,13 and the rms roughness amplitude as

FIG. 2. rms local surface sloper rms5^u¹hu2&1/2 as a function of the averagemound separationl for various system correlation lengthsz, w51 nm anda050.3 nm.

FIG. 3. Thermal capacitance ratioc&/c0 as a function of the averagemound separationl for various system correlation lengthsz, w51 nm, h0

530 nm, anda050.3 nm.

FIG. 4. Thermal capacitance ratio^c&/c0 as a function of the system corre-lation length z for various average mound separationsl, w51 nm, h0

530 nm, anda050.3 nm.

6132 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 G. Palasantzas and J. Th. M. De Hosson

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w}h0b8 ~b8,1!.14 As an example, we consider in Fig. 6 the

thickness dependence of the ratio^c&/c0 for growth withroughness parameters evolving with a film thicknessh0 asw51.0(h/10)0.24(nm), l510(h/10)b (nm), and z520(h/10)b (nm). As Fig. 6 indicates, different growth ex-ponentsb have a strong contribution on the thermal capaci-tance ratio c&/c0 as a function of the film thicknessh0 .

For the actual dependence of the average thermal capaci-tance^c& on thickness, one has also to consider the thicknessdependence of the thermal conductivityK the. Indeed, forsufficiently thick films this is given byK thc5Kbulk /@11(3L/8h0)1(7G/5)#,7,12 with Kbulk the bulk thermal con-ductivity, L the electron mean-free path@i.e., L;41 nm forAu ~Ref. 7!#, andG5(L/h0)@R/(12R)# with R the electronreflection coefficient due to scattering at grain boundaries forthe more general case of polycrystalline films. This equationfor K the is valid if G,10 andL/h0.0.1,7 and incorporatesboth surface and grain-boundary electron scattering in a qua-siclassical sense. Its derivation is based on the electrical con-ductivity via the use of the Wiedemann–Franz law~assum-ing that electrons serve both as electrical and thermal carrierswith phonons having a negligible effect on the thermal con-ductivity of metals7!. Such a law was discovered initially forbulk materials, and has been proven to be valid also for thin

films.10 In contrast, for ultrathin films (h0,10 nm) one hasto consider quantum size effect contributions to the thermalconductivity.6

V. CONCLUSIONS

In summary, we investigated the effect of roughness onthe thermal capacitance of thin films with one smooth bound-ary and the other rather rough at a nanometer-length scale.Qualitatively, similar results would be expected for filmswith double rough boundaries, as well as for other geom-etries. It is found that the roughness may cause a strongincrease of the thermal capacitance. For mound rough sur-faces such an increase strongly depends on the relative mag-nitude of the average mound separationl and the systemcorrelation lengthz. Indeed, complex behavior develops forz.l, while a smooth decrease of the thermal capacitance asa function ofl occurs forz,l ~Gaussian-type behavior ofboundary roughness!. Finally, as a function of film thicknessthe thermal capacitance decreases with roughness, contribut-ing significantly to small film thickness as long asz,l. Atany rate, a precise determination of the actual effect willrequire a detailed knowledge of the thickness dependence ofthe roughness parameters during film growth.

ACKNOWLEDGMENTS

The authors would like to acknowledge support from the‘‘Nederlandse Organisatie voor Wetenschappelijk Onder-zoek~NWO!,’’ as well as useful discussions with Y.-P. Zhao~Rensselaer Polytechnic Institute.!

APPENDIX

Here, we outline the Fourier analysis procedure.4 Indeed,we have to solve the Laplace equations

¹2T~n!~x,y,z!5Tzz~n!~x,y,z!1¹r

2T~n!~x,y,z!50

~n51,2!, ~A1!

with r5(x,y) representing the position vector in thex–yplane. Performing a Fourier transform in thex–y planeaccording to the equationsF(n)(k,z)5@1/(2p)2#* dr

3T(n)(r,z)eik•r, T(n)(r,z)5* dkF(n)(k,z)e2 ik•r, theLaplace Eq.~A1! takes the formFzz

(n)(k,z)2k2F(n)(k,z)50, which has the general solution

F~n!~k,z!52A~n!~k!sinh~kz!, ~A2!

under the constraint thatF(n)(k,z)uz5050 for z50 @sinceT(n)(r,z50)50#. The coefficientsA(n)(k) can be deter-mined by the boundary condition atz5h0 @Eqs.~5! and~6!#.

For the first-order perturbation, we obtain in Fourierspace F(1)(k,z)52(T22T1 /h0)@sinh(kz)/sinh(kh0)#h(k),and in real space

T~1!~r,z!52~T22T1!

h0E dk

sinh~kz!

sinh~kh0!h~k!e2 ik•r.

~A3!

Similarly, for the second-order perturbation we obtainF(2)(k,z)5@(T22T1)/h0#* dk8@cosh(k8h0)sinh(kz)/sinh(k8h0)

FIG. 5. Thermal capacitance ratio^c&/c0 as a function of the film thicknessfor various system correlation lengthsz. The inset shows a similar plot forvarious average mound separationsl. w51 nm, anda050.3 nm for bothplots.

FIG. 6. Thermal capacitance ratio^c&/c0 as a function of the film thicknessfor the film with the roughness parameters evolving with thickness asw51.0(h/10)0.24(nm), l510(h/10)b (nm), and z520(h/10)b (nm). Thecurves correspond to different growth exponentsb.

6133J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 G. Palasantzas and J. Th. M. De Hosson

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3sinh(kh0)]k8h(k8)h(k2k8) by using the property of convo-lution for the Fourier transform. Thus, in real space we have

T~2!~r,z!5~T22T1!

h0E dkE dk8

3cosh~k8h0!sinh~kz!

sinh~k8h0!sinh~kh0!k8h~k8!h~k2k8!

3e2 ik•r. ~A4!

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6134 J. Appl. Phys., Vol. 89, No. 11, 1 June 2001 G. Palasantzas and J. Th. M. De Hosson

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