This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 129.8.242.67

This content was downloaded on 03/06/2014 at 05:33

Please note that terms and conditions apply.

Experimental investigation of change in sheet resistance and Debye temperatures in metallic

thin films due to low-energy ion beam irradiation

View the table of contents for this issue, or go to the journal homepage for more

2013 J. Phys. D: Appl. Phys. 46 435304

(http://iopscience.iop.org/0022-3727/46/43/435304)

Home Search Collections Journals About Contact us My IOPscience

IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 46 (2013) 435304 (10pp) doi:10.1088/0022-3727/46/43/435304

Experimental investigation of change insheet resistance and Debye temperaturesin metallic thin films due to low-energyion beam irradiationAbhishek Chowdhury and Sudeep Bhattacharjee

Department of Physics, Indian Institute of Physics, Kanpur 208016, India

Received 24 May 2013, in final form 19 August 2013Published 7 October 2013Online at stacks.iop.org/JPhysD/46/435304

AbstractWe present a systematic experimental investigation of low-energy (0–1 kV) ion irradiationinduced changes in sheet resistivity and Debye temperatures in metallic nano-films of Ag, Cuand Al of thickness d/λo ∼ 2–5, where d is the film thickness and λo is the bulk mean freepath, as a function of ion beam induced defects and impurities in a controlled manner. Ions ofboth atomic (Ne, Ar and Kr) and molecular (H2, N2) gases are employed in the investigationand the number of defects and impurities in the nano-film can be varied in a controlled mannerby varying the ionic mass number (1–84) and beam fluence (8.7 × 1015–1.4 × 1016 ions cm−2).Low-temperature measurements are carried out for pristine and irradiated films to obtain theresidual sheet resistance (RRS). An empirical formula relating the variation of RRS with beamfluence and ionic mass number is proposed for the first time. The change in RRS is due to thelarge diffusion of the impurities inside the nano-films as confirmed from energy dispersivex-ray spectroscopy. The Debye temperature (�D) is determined from Bloch–Gruneisen fittingof the temperature variation of sheet resistance data and it is found that �D decreases withincrease in both fluence and ionic mass number arising primarily from the change in bulkmodulus of the nano-film.

(Some figures may appear in colour only in the online journal)

1. Introduction

In the last few years, low-energy ion beams (LEIBs) haveattracted attention due to their interesting nature of interactionswith matter where primarily the subsurface atomic layerscan be tailored for a variety of research applications [1–9].LEIB is an important tool for non-destructive study of surfacephysics. In recent years LEIBs have been used for surfacemodification [1–4], patterning using ripples [5–8] formed onthe surface, and generation of internal subsurface excitations[9] etc. Hot carrier excitations in gold thin film bombardedby low-energy alkali and noble gas ions [9] have beeninvestigated. Since LEIBs cannot penetrate deep inside thesubstrate, their interaction is mainly limited to surface andsubsurface layers. Therefore, the surface properties e.g.morphology [5–8], electronic [5–8, 10, 11], optical [10, 12–15] and thermal properties of materials can be tailored by

using LEIBs. Ion bombardment of clean single-crystal metalsurfaces is an effective method of increasing the number ofactive sites at a surface through the creation of interstitialsand vacancies [16, 17], which can increase the reactivity of asurface for catalytic applications.

The damage in solids due to ion irradiation (e.g.,impurities and implanted atomic species interacting withvacancies, self-interstitials, and dislocations) have been treatedin the past for quite some time mainly for high-energy (∼MeVrange) ion irradiation e.g. hydrogen and helium ion irradiationof metals for their applications in nuclear technology. Thereare a few works on the resistance changes in metals with theintroduction of metallic impurity e.g. dilute metals [18, 19],where some fraction of other metals are mixed or by implantingmedium energy (>10 keV) metal ions directly in pure samples.

However, the interaction of LEIBs (0–1 keV) with matterhas gained attention in recent years and is a subject of active

0022-3727/13/435304+10$33.00 1 © 2013 IOP Publishing Ltd Printed in the UK & the USA

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

research [1–9]. The primary point defects in metals are defectsin the lattice which consists of, vacant lattice sites (vacancies),interstitial atoms of the host metal, substitution and interstitialimpurity atoms. Since, even small concentrations of pointdefects can have an important influence on many physicalproperties and processes in metals, this investigation has beencarried out where one can control the defect generation andinvestigate the change in residual sheet resistance.

This article presents investigations of LEIB (0–1 keV)irradiation on metallic nano-films where the change inelectrical sheet resistance in the temperature range 300–15 Kand change in Debye temperatures are determined for threedifferent target metallic nano-films of Ag, Cu and Al,bombarded with five different ionic species viz. H2, N2,Ne, Ar and Kr and for fluences in the range ∼8.7 × 1015–1.4 × 1016 ion cm−2. RRS increases both with ionic massnumber and fluence. It is found that the variation of RRS withmass no. follows a parabolic nature, whereas with fluence,RRS shows an exponential behaviour. Empirical relationshipshave been proposed to describe the behaviour. The ratio ofroom temperature to residual resistance (RRR ratio) decreaseswith increasing fluence (maximum 78.9% for copper) as wellas increasing mass number (maximum 83.8% for copper).The Debye temperature (�D) of the irradiated film shows asignificant decrease (10–20%) from that of the pristine film.It is considered that the large change in RRS is because of theimplanted ions diffusing much deeper (than predicted by SRIM(Stopping and Range of Ions in Matter) simulations) inside thefilm as confirmed from cross-sectional energy dispersive x-ray(EDX) experiments.

2. Theoretical background

In metals, resistivity arises due to collisions of the conductionelectrons with vibrating lattice atoms (phonons), impuritiesand defects. Each scattering process has its characteristicrelaxation time and a single relaxation time can be defined interms of the individual relaxation times using the Matthiessen’srule [20], given by

1

τaverage= 1

τlattice+

1

τGB+

1

τdefects+

1

τimpurity. . . , (1)

where τlattice, τGB, τdefects and τimpurity are the relaxation timesdue to phonon induced scattering, scattering induced by grainboundaries, defects and impurities respectively. If equation (1)is multiplied by me/ne2ρ0 we obtain,

ρaverage

ρ0= ρlattice

ρ0+

ρGB

ρ0+

ρdefects

ρ0+

ρimpurity

ρ0. . . , (2)

where ρ0 and ρaverage are the resistivity of the bulk metal andaverage resistivity of the nano-film and ρlattice, ρGB, ρdefects

and ρimpurity are the individual resistivity contribution of thenano-film due to effects of lattice, grain boundary, defects andimpurities, respectively. The mass, charge and number densityof electrons are given by me, e and n, respectively. In a pristinethin polycrystalline film where the thickness is of the order ofthe mean free path of the bulk metal, surface scattering and thegrain boundary scattering are the dominant processes.

For single-crystal films, the theory of electron transport isgiven by Fuchs [21] where the ratio of thin-film resistivity tothe bulk resistivity is given by the formula,

ρlattice

ρ0=

[1 −

(3

2k

)(1 − p)

∫ ∞

d

(1

t3− 1

t5

)

× 1 − e−kt

1 − pe−ktdt

]−1

, (3)

where k = d/λ0, p is the coefficient of specular reflection.p = 1 indicates all the scattering is specular and the resistivityis equal to that of the bulk material, whereas p = 0 meanscompletely diffused scattering. Equation (3) represents thefirst term on the right hand side (RHS) of equation (2).

For polycrystalline films, Mayadas and Shatzkes [22]developed a simple model where grain boundaries are treatedas δ-function potential and the ratio of the grain boundaryresistivity to the bulk resistivity is given by the formula,

ρGB

ρ0=

(1 − 3

2α + 3α2 − 3α3 ln

(1 +

1

α

))−1

, (4)

where α = λ0/d × (R/(1 − R)), R = grain boundaryreflection coefficient. Therefore equation (4), which representsthe second term on the RHS of equation (2), may be used toevaluate the resistivity of pure polycrystalline films as it agreesreasonably well with experimental results. A comparison ofdifferent experimental results with equations (3) and (4) isshown in the appendix.

The third and fourth terms on the RHS of equation (2)are the changes in resistivity due to defects and impurities,which arise due to ion beam irradiation, and are determinedexperimentally from the change in sheet resistance (RRS).For that temperature variation of RS of the irradiated filmsis performed so that the phonon contribution can be subtractedout and change in RRS can be obtained.

The temperature dependence of the resistivity ρ of a metalis given by the Bloch–Gruneisen formula [23]

ρ(T ) = ρ0 + A

(T

�D

)5 ∫ �D/T

0

x5

(ex − 1)(1 − e−x)dx,

(5)

where ρ0 is the residual resistivity, A is the prefactor of theBloch–Gruneisen formula and �D is the Debye temperatureof the metal. �D is a characteristic property of material andarises from the electron–phonon interaction. With increasingfluence and mass, �D decreases from the pristine value. Here�D is given by

�D = �ωD

kB= �v

kB

(6π2N

V

)1/3

,

where �, kB and v are reduced Planck’s constant, Boltzmannconstant and velocity of sound in the sample, respectively, andN is total number of phonons in the volume V and ωD is the cut-off frequency associated with �D. At frequencies higher thanωD, the lattice becomes independent of the incoming vibration.Assuming that the volume of the nano-film does not changeduring irradiation and N is a constant for a particular sample

2

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

the decrease in �D is due to decrease in v and v is linearlyproportional to �D.

The velocity of sound v in solids depends on the bulkmodulus and the density of the material by the relation v =√

E/δ, where E is the bulk modulus and δ is the densityof material. We estimated the change in density of the filmafter irradiation with the assumption that upon irradiationvolume of the film does not change. The mass of 80 nm thicksilver film having an area of 1 cm2 is 8.44 × 10−5 g. If weirradiate it with Kr+ (mass no. 84) having maximum fluenceof 1.38 × 1016 ions cm−2, the extra mass added to the filmis 1.93 × 10−6 g. Therefore the change in density is 2.3%.This is the maximum expected change as it is estimated for Kr(heaviest ion) with maximum fluence and for other mass andfluence combinations the change in density will be smaller than2.3%. Therefore the contribution coming from density is quitesmall and the main contribution in the decrease of v comesfrom the change in the elastic modulus E of the sample.

3. Experimental details

A microwave generated multicusp plasma ion source capableof generating ion beams of gaseous elements is employed inthe experiment [24–26]. The irradiation system is capableof delivering broad ion beams as well as ion beamlets ofmuch smaller size ∼40 µm for localized irradiation [26].The beam fluence can be varied in the range (∼4.4 × 1015–7 × 1015 ions cm−2). Details of the ion source and irradiationsystem can be found in [24–26].

Ion beams of ∼20 mm diameter are utilized to uniformlyirradiate a sample of size 10 mm × 10 mm and thickness80–200 nm. The metallic nano-films are prepared on glasssubstrates using thermal evaporation and a slow deposition rateof ∼0.4 Å s−1 to have a uniform deposition. The thicknesses ofthe films are confirmed in a profilometer. The nano-films areirradiated for 2 s in all the experiments. It is seen that for longerirradiation times the nano-film is either etched out or blackeneddue to burning by excessive heat generated during irradiation.RS is measured using the van der Pauw method [27, 28]. Theresistivity (ρ) can be obtained from ρ = (RS × d). Two setsof experiments are carried out. In set one, various fluences ofargon ion beam of energy 0.3 keV have been used to irradiatethe nano-films. And in set two, the films are irradiated withdifferent ionic species (H2, N2, Ne, Ar, Kr) (mass number1–84) at a fixed fluence (9.6×1015 ions cm−2).

Experimentally, the temperature variation of RS of thepristine film is expected to follow the behaviour of ρlattice. Weperformed temperature variation of RS of the irradiated filmsso that the phonon contribution can be subtracted out and thesum of ρGB + ρdefects + ρimpurity can be obtained. From the lowtemperature measurements we have evaluated the RRR (ratioof room temperature to residual resistance) ratio defined asthe ratio of RS at room temperature (295 K) to RS at a lowtemperature (15 K).

To investigate the effect of impurities we have performedEDX spectroscopy measurements on samples irradiated withargon. These experiments are performed in the cross-sectionalmode and line scan and map scan of the samples are acquired

Figure 1. Temperature variation of (Rs) of pristine and irradiatedfilms with varying fluence for (a) silver (b), copper and (c)aluminum thin films.

to know the depth profile and the elemental mapping of theelements present in the sample. From these one can infer howmuch the gaseous implanted ions penetrated inside the metallicnano-film. For this experiment we have used p-type siliconwafers (1 1 0) of thickness ∼250 µm on which the metallicnano-film was deposited.

4. Results and discussions

Figure 1 shows the temperature variation of RS of the pristineand irradiated nano-films with varying fluence for threedifferent metals: (a) silver, (b) copper and (c) aluminum.The value of RS of pristine silver film at room temperature

3

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

is ∼ 0.2 . The corresponding value of resistivity of thepristine film is 3.83×10−8 m, which is ∼2.38 times the bulkresistivity of silver and matches well with the theoretical value(3.79 × 10−8 m) for polycrystalline silver films. Similarlyfor aluminum, the value of resistivity at room temperature is16.8 × 10−8 m, which is ∼6 times higher than the bulkresistivity (2.74 × 10−8 m) value of aluminum and forcopper the film resistivity (15.09 × 10−8 m) is ∼8 timesthe bulk resistivity value (1.7 × 10−8 m). The variation ofRS of pristine films with temperature is nearly linear initially(300–70 K), whereas below 50 K it deviates from the linearbehaviour in accordance with the Bloch–Gruneisen formula,as shown in figure 2. Similar trends are observed for pristinecopper and aluminum films where the transition from thelinear nature happens at around 40–60 K. For the irradiatedfilms RRS increases with increasing fluence. These results aresummarized in table 1.

Figure 2 represents the fitting of the Bloch–Gruneisenformula with the experimental data for the pristine filmsof (a) silver, (b) copper and (c) aluminum, respectively.Bloch–Gruneisen formula fits very well with the experimentalobtained data for copper and silver films. For silverand copper films, the RMS deviation of the data pointsfrom the formula is (0.0026 × 10−8 m) and (0.0127 ×10−8 m), respectively, and for aluminum it is a little higher(0.0526 × 10−8 m).

Figure 3 shows the temperature variation of RS of thepristine and irradiated nano-films with temperature for varyingionic species (mass no) and for three different metallic films(a) silver (b) copper and (c) aluminum. The variation of RS inpristine films shows a similar nature to that observed in figure 1.For the irradiated films RRS (value of Rs at 15 K) increaseswith irradiation with ionic species having higher atomic mass.Results for irradiation with different masses are summarizedin table 2.

The variation of RRR ratio is shown together for three setsof samples (viz. silver, copper and aluminum) in figures 4(a)and (b) for varying fluence and varying mass number,respectively. The RRR ratio decreases with increasing fluenceas well as increasing mass number. In both the cases coppershows a higher RRR ratio. With fluence the decrease in theRRR ratio is less (7.6–1.6) as compared to the decrease withmass number (12.4–1). For copper, the percentage change inthe RRR ratio is 78.9% for variation of fluence and 83.8% forvariation of mass.

The changes in residual sheet resistances (RRS =(RRS)irradiated − (RRS)pristine) for silver, copper and aluminumwith beam fluence and ion mass are plotted in same graph asshown figures 5(a) and (b). We tried to fit the data with ageneral empirical formula. With fluence RRS varies from0.05–2.2 for different metallic films and with varying massnumber RRS varies from 0.01–1.7 . We have found thatthe variation of RRS with fluence can be fitted with anexponential of the form RRS = C[exp(t/α) − 1], wheret represents beam fluence (ions cm−2). The values of theparameters C and α for different cases are tabulated in table 3.Similarly, the variation of RRS with ionic species has aparabolic nature and can be fitted with an equation of the form

Figure 2. Bloch–Gruneisen equation fitting of the experimental datafor pristine films of (a) silver, (b) copper and (c) aluminum thinfilms.

RRS = Am2 + Bm, where m is the mass number of the ionicspecies in amu. The values of the parameters A and B arepresented in table 4. Using the fitting equations one can see thatRRS can be predicted for argon for different fluences and fordifferent mass number at a fixed fluence (9.6×1015 ions cm−2).

From the measurement of RS we observe a remarkablechange in RS upon irradiation; e.g. at room temperature whencopper film is irradiated with krypton ion, RS changes from1.987 to 4.042 (∼2 times) and with argon ion, RS changesfrom 1.987 to 3.386 (∼1.7 times). To further investigatethe change of RS we wanted to look at the location of implantedions in the copper film to understand the extent of diffusionof the implanted ions inside the copper layer. We performed

4

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

Table 1. Summary of the results obtained in figure 1.

Silver Copper Aluminum

Fluence RS at 15 K RS at 295 K RS at 15 K RS at 295 K RS at 15 K RS at 295 K(ion cm−2) () () () () () ()

Pristine 0.014 0.298 0.112 1.887 0.309 2.1078.7 × 1015 0.191 0.475 0.27 2.046 0.519 2.317

9.69 × 1015 0.453 0.868 0.459 2.234 0.93 2.7281.12 × 1016 0.801 1.633 0.871 2.647 1.401 3.1991.24 × 1016 1.306 2.24 1.59 3.366 1.854 3.6521.38 × 1016 1.689 2.873 2.265 4.04 2.376 4.174

Figure 3. Temperature variation of (Rs) of pure and irradiated filmswith varying mass number for (a) silver, (b) copper and (c)aluminum thin films.

cross-sectional EDX experiment on samples irradiated withargon and took line and map scans of the samples. From theline scan the depth profile of copper film and implanted Ar+ isobtained and from the map scan exact location of the Ar+ ionscan be obtained.

Figure 6(a) shows the line scan results for the Ar irradiatedcopper film on silicon substrate. The Ar ion fluence is1.24 × 1016 ions cm−2. The scanned line is marked by theyellow line. The thickness of the copper film is ∼175 nm.The depth of penetration of the Ar+ ions in the copper filmis ∼75 nm which is much higher than the penetration depthobtained from SRIM simulation (∼2 nm). In figure 6(b) EDXmap scan data are shown for silicon K-α1 line, copper L-α1and argon K-α1 lines in different colours which also confirmsthe penetration of Ar ions. From these results we see thatimplanted ions diffuse deep inside the copper film that ismuch larger than their penetration depth predicted by SRIMsimulations.

Knowing the diffusion length (i.e. how much the argonions penetrated inside the copper layer) from the EDXmeasurements, we can calculate the diffusion coefficient usingFick’s law [29] of diffusion. According to this, the diffusionlength Ld is given by

Ld = 2√

Dt, (6)

whereD is the diffusion coefficient and t is the time of diffusionwhich is 2 s in our experiments as we irradiated the samplesfor 2 s. For Ld = 50 nm, D is found to be 3.12×10−16 m2 s−1.Since we could not find values of diffusion coefficient of argonions in copper substrate in the literature, we compared thevalues obtained with other experimental results on diffusion.For example, at room temperature the diffusion coefficientof hydrogen in nickel is ∼2 × 10−14 m2 s−1 [30] and thediffusion coefficient of gallium ions implanted in silicon layeris found to be 1.3 × 10−15 m2 s−1 at 400 ◦C [31]. For low-energy (300 eV) Ar+ irradiation in (In, Ga)Sb heterojunctionthe diffusion length Ld of the ion bombardment created defectsis [32] ∼100 nm. These results lie in a range that is comparableto the values obtained by us.

According to the Bloch–Gruneisen formula the modifiedresistivity leads to a change in �D which in turn changes theelectron–phonon interaction in the sample. The temperaturevariation of the experimental sheet resistance is fitted withequation (5), as shown in figure 2, to obtain the Debyetemperature of the sample. Figures 7(a) and (b) shows the

5

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

Table 2. Summary of the results obtained in figure 3.

Silver Copper Aluminum

RS at 15 K RS at 295 K RS at 15 K RS at 295 K RS at 15 K RS at 295 KMass no. () () () () () ()

Pristine 0.014 0.298 0.112 1.987 0.309 2.117H2(1) 0.191 0.475 0.27 2.032 0.519 2.323N2 (14) 0.453 0.868 0.459 2.123 0.93 2.718Ne (20) 0.801 1.633 0.871 2.475 1.401 3.198Ar (40) 1.306 2.24 1.59 3.386 1.854 3.602Kr (83.8) 1.689 2.873 2.265 4.042 2.376 4.274

Figure 4. Plot of RRR ratio of the irradiated nano-films with (a)varying fluence and (b) varying ionic species.

variation of �D with beam fluence and ionic mass. Debyetemperatures of the pristine samples are close to bulk valueof the metals. With increase in fluence and mass number,�D decreases, although the nature is quite different in the twocases. The decrease in the Debye temperature is due to thedecrease in electron–phonon coupling strength (also knownas phonon softening [33, 34]) in the presence of implantedimpurity.

As per the discussion in section 2, figures 8(a) and (b)shows the variation of bulk modulus E with fluence and massnumber, respectively. This has a similar trend to the variationof �D.

Figure 5. Change in sheet resistance of irradiated silver, copper andaluminum films with (a) varying fluence and (b) varying ionicspecies.

Table 3. Fitting parameters for the variation of RRS with fluence.Fitting equation: RRS = C[exp(t/α) − 1].

Substrate Value of C () Value of α (cm−2)

Silver 0.0275 3.34 × 1015

Copper 0.0137 2.72 × 1015

Aluminum 0.0511 3.69 × 1015

6

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

5. Discussions and conclusions

In this article low-energy ion irradiation induced changesin electrical properties of metallic thin films have beeninvestigated by varying the beam fluence and the ionic mass.The beam fluence has been varied from 8 × 1015 to 1.4 ×1016 ions cm−2 and different gaseous ions (H+, N+, Ne+, Ar+,Kr+) are used. Thin films of silver, copper and aluminum withdimensions (1 × 1) cm2 and thickness 80–200 nm have beenused in the experiments. We have used a bigger sample toget enough resolution for the sheet resistance measurements.To investigate the changes in ion irradiation induced sheetresistance because of the introduction of point defects in to thesample, temperature variation of sheet resistance is performed(300–15 K). At low temperatures, the phonon contribution tothe resistance almost vanishes, only temperature independentresistance due to static defects and impurities remains. Anempirical relation is proposed between the change in residual

Table 4. Fitting parameters for the variation of RRS with massnumber. Fitting equation:RRS = Am2 + Bm.

Substrate Value of A () Value of B ()

Silver 3.22 × 10−5 0.0091Copper 7.05 × 10−5 0.0129Aluminum 4.43 × 10−5 0.0162

Figure 6. (a) EDX line scan and (b) EDX map scan results for argon irradiated copper film on silicon substrate (beam fluence is1.24 × 1016 ions cm−2).

sheet resistance with beam fluence and mass of the ionicspecies. With varying fluence an exponential relation isfound, whereas for varying mass a parabolic fitting can closelyrepresent the experimentally obtained data. The correspondingfitting parameters are summarized in tables 3 and 4. The Debyetemperature (�D) is determined from Bloch–Gruneisen fittingof the temperature variation of sheet resistance data and it isfound that �D decreases with increase in both fluence andionic mass number arising primarily from the change in bulkmodulus of the nano-film. To explain the large change in sheetresistance upon irradiation, cross-sectional EDX experimentsare performed and the result reveals much larger penetrationof the impurities inside nano-films compared to the SRIMpredictions.

It is true that SRIM is a very useful tool for evaluating‘implantation range’ and ‘damage distribution’ of an energeticion in a target material with a reasonable accuracy. Similarly,TRIM, a Monte Carlo computer program, calculates theinteractions of energetic ions with amorphous targets.However, even with such a reasonable accuracy in calculations,there are several limitations with this program. The reasonfor this mismatch is because of the fact that SRIM/TRIMsimulations are not really applicable at these energies andfluences. These programs are mainly based on ballisticrecoil mechanism in an amorphous target (atoms at randompositions) and consider individual ions suffering binary nuclear

7

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

Figure 7. Variation of Debye temperature for pure and irradiatedsamples as a function of (a) beam fluence and (b) ionic mass.

collisions with target atoms, neglecting many-body effectsand the amount of material already implanted which makesonly a rough estimate, particularly at low ion energies and forhigh ion fluences. In addition, it does not take into accountthe defects induced by inelastic processes (by ionizationand due to lattice phonons). Since thermally activatedand ion beam assisted relaxation effects are neglected, itdoes not correctly describe the number of point defects(vacancies, interstitials) and extended defects (e.g. dislocationloops). During the simulation, local atomic rearrangements(grain growth and crystalline quality) or lower energyconfigurations (alloys) are not considered. Furthermore, inTRIM simulations, ions always penetrate virgin targets, i.e.,accumulation of ion erosion; build-up of lattice damage andimpurity agglomerations are not taken into account. Todescribe the interactions at these energies and fluences, wehave also employed ‘ion implantation’ simulation software‘TRIDYN HZDR’ [35, 36]. The experimental condition asdescribed in figure 6 is simulated using ‘TRIDYN HZDR’.The simulation results indicate some improvement over theSRIM/TRIM results, but are still unable to explain the higherpenetration of argon ions into the Cu film (using SRIMsimulations we obtained the maximum penetration depth for

Figure 8. Variation of bulk modulus for pure and irradiatedsamples as a function of (a) beam fluence and (b) ionic mass.

argon ions to be approximately 2 nm, whereas using theTRIDYN HZDR simulation code we obtain approximately4.5 nm). The reason for the higher penetration depth observedin the experiments could be related to the high thermal diffusionof the implanted ions due to heat generated at the surface of thefilm during the high dose ion beam irradiation which can allowthe implanted ions to diffuse further deeper into the Cu film.

The diffusion coefficient D(T ) of any material has anexponentially increasing nature with temperature and may bewritten as, D(T ) = D0 exp(−(Qd/RT )) where Qd is theactivation energy for diffusion, R is the molar gas constantand T is the temperature. Therefore a temperature rise of 100 Kfrom room temperature increases the diffusion coefficient by 3orders of magnitude, which causes a higher diffusion range forimplanted argon ions. Similar phenomena have been reportedfor boron implantation in silicon where an initial rapid transientdiffusion occurs in crystal silicon samples [37].

Acknowledgments

The authors gratefully acknowledge the financial supportfrom the Department of Science and Technology (DST),Government of India, under the Center for Nanotechnologyat Indian Institute of Technology (IIT), Kanpur, for researchundertaken in the area of focused ion beam technologies. One

8

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

of the authors (AC) would like to acknowledge the Ministry ofHuman Resource and Development (MHRD), India for SeniorResearch Fellowship. We thank Professor Y N Mohapatrafor allowing us to use the closed cycle refrigerator for lowtemperature measurements and Professors H C Verma andAnjan K Gupta for fruitful discussions. We are also grateful toDepartment of Materials Science and Engineering, IIT Kanpurfor the SEM and EDX facility.

Appendix

To determine the sheet resistance of single-crystal nano-films,the electron transport governed by the Boltzmann transportequation [20] given by,

∂f

∂t+

∂f

∂ �x · �pm

+∂f

∂ �p · �F = ∂f

∂t

∣∣∣∣coll

, (A1)

needs to be solved, where f is the electron distribution functioninside the metallic nano-film. �F is the external force onelectrons which can be due to any applied electric field, �pis the electron momentum and (∂f /∂t) is the rate of changeof the distribution function with time which is zero at steadystate. (∂f /∂x) is the change of the distribution function withposition. The collisional term (∂f /∂t) can be defined as(

∂f

∂t

)coll

= − (f − f0)

τ= −f1

τ,

where f0 is the unperturbed electron distribution function inthe absence of the external force (i.e. electric field); f1 is thechange in unperturbed electron distribution function in time τ

which is the electron relaxation time. At steady state the aboveequation (A1) reduces to

∂f

∂ �x · �pm

+∂f

∂ �p · �F = −f − fo

τ. (A2)

Assuming the external electric field Eex to be applied is in thex direction, and the film is in the x–y plane, and the thicknessis along z, the above equation reduces to a scalar equationgiven by

Vz

∂f1

∂z+

f1

τ= eEex

m

∂f0

∂Vx

. (A3)

where Vx and Vz are the components of electron velocity alongx and z directions, respectively. f1 is obtained by solvingequation (A3) and is given by

f1 = eEexτ

m

∂f0

∂Vx

[1 + φ exp

(− z

τνx

)](A4)

where φ is integration constant, to be determined from theboundary conditions of f1. The current density can be obtainedfrom J = 2e(m/h)3

∫(Vxf1) dv. The resistivity of the system

is defined as ρ = E/J .For single-crystal films, the theory of surface scattering is

given by Fuchs [21]. The ratio of thin-film resistivity to thebulk resistivity is given by the formula,ρlattice

ρ0=

[1 −

(3

2k

)(1 − p)

∫ ∞

d

(1

t3− 1

t5

)

× 1 − e−kt

1 − pe−ktdt

]−1

, (A5)

Figure A1. Thickness dependent resistivity for single-crystal silverfilm. The solid lines are the plot of equation (A4) for different pvalues. The experimental data points are obtained from [38].

Figure A2. Thickness dependent resistivity for polycrystallinecopper film. The experimental data points are obtained from [39].

where k = d/λ0, p is the coefficient of specular reflection.p = 1 indicates all the scattering is specular and the resistivityis equal to that of the bulk material, whereas p = 0 meanscompletely diffused scattering.

Figure A1 shows the experimental data obtained forsingle-crystal silver films [38]. Experimental data agree withtheoretical prediction of equation (A5) with p = 0.5.

Therefore, equation (A5) may be used to evaluate theresistivity of pure single-crystal films as it agrees well withthe experimental results.

In general, thin metal films are not approximate tosingle crystal. It is more like consisting of an array ofrandomly oriented polycrystalline films or grains. Therefore,additional resistance is introduced due to scattering at thegrain boundaries. To see the effect of these grain boundaries,Mayadas and Shatzkes [22] developed a simple model wheregrain boundaries are treated as δ-function potential. The ratioof the grain boundary resistivity to the bulk resistivity is given

9

J. Phys. D: Appl. Phys. 46 (2013) 435304 A Chowdhury and S Bhattacharjee

by the formula,

ρg

ρ0=

(1 − 3

2α + 3α2 − 3α3 ln

(1 +

1

α

))−1

, (A6)

where

α = λ0

d× R

1 − R,

R = grain boundary reflection coefficient. Figure A2shows the variation of resistivity with film thickness for apolycrystalline copper film. The experimental data are takenfrom [39]. Theoretical lines are drawn for R = 0.4.

Therefore equation (A6) may be used to evaluate theresistivity of pure polycrystalline films as it agrees reasonablywith experimental results.

References

[1] Vauth S and Mayr S G 2008 Phys. Rev. B 77 155406[2] Frost F, Schindler A and Bigl F 2000 Phys. Rev. Lett. 85 4116[3] Gayone J E, Pregliasco R G, Gomez G R, Sanchez E A and

Grizzi O 1997 Phys. Rev. B 56 4186[4] Steiner C, Bolliger B, Erbudak M, Hong M, Kortan A R,

Kwo J and Mannaerts J P 2000 Phys. Rev. B 62 R10614[5] Macko S, Frost F, Ziberi B, Forster D and Michely T 2010

Nanotechnology 21 085301[6] Ziberi B, Frost F, Hoche Th and Rauschenbach B 2005 Phys.

Rev. B 72 235310[7] Alkemade P F A 2006 Phys. Rev. Lett. 96 107602[8] Hansen H, Redinger A, Messlinger S, Stoian G, Rosandi Y

and Urbassek H M 2006 Phys. Rev. B 73 235414[9] Ray M P, Lake R E, Sosolik C E, Thomsen L B, Nielsen G,

Chorkendorff I and Hansen O 2009 Phys. Rev. B80 161405(R)

[10] Martin D S, Cole R J and Weightman P 2005 Phys. Rev. B72 035408

[11] Khakshouri S and Duffy D M 2009 Phys. Rev. B 80 035415[12] Schmuki P, Erickson L E and Lockwood D J 1998 Phys. Rev.

Lett. 80 4060[13] Panigrahy B, Aslam M and Bahadur D 2011 Appl. Phys. Lett.

98 183109

[14] Everts F, Wormeester H and Poelsema B 2008 Phys. Rev. B78 155419

[15] Goh X M, Dragomir N M, Jamieson D N, Roberts A andBelton D X 2007 Appl. Phys. Lett. 91 181102

[16] Soltan A S, Vassen R and Jung P 1991 J. Appl. Phys.70 793

[17] Soltan A S, Jung P and Gadallah A A 1992 J. Phys.:Condens. Matter 4 9573

[18] Mills D L, Fert A and Campbell I A 1971 Phys. Rev. B4 196

[19] Dunlop J B, Gruner G and Napoli F 1974 Solid StateCommun. 15 13

[20] Coutts T J 1974 Electrical Conduction in Thin Metal Films(Amsterdam: Elsevier)

[21] Fuchs K 1938 Proc. Camb. Phil. Soc. 34 100[22] Mayadas A F and Shatzkes M 1970 Phys. Rev. B 1 1382[23] Ziman J M 1960 Electrons and Phonons: The Theory of

Transport Phenomena in Solids (Oxford: Clarendon)[24] Mathew J V, Chowdhury A and Bhattacharjee S 2008 Rev. Sci.

Instrum. 79 063504[25] Chowdhury A and Bhattacharjee S 2010 J. Appl. Phys.

107 093307[26] Chowdhury A and Bhattacharjee S 2011 AIP Adv. 1 042150[27] van der Pauw L J 1958 Philips Res. Rep. 13 1[28] Kim G T, Park J G, Park Y W, Muller-Schwanneke C,

Wagenhals M and Roth S 1999 Rev. Sci. Instrum. 70 2177[29] Bird R B, Stewart W E and Lightfoot E N 1976 Transport

Phenomena (New York: Wiley)[30] Wimmer E, Wolf W, Sticht J, Saxe P, Geller C B, Najafabadi R

and Young G A 2008 Phys. Rev. B 77 134305[31] Batabyal R, Patra S, Roy A, Roy S, Bischoff L and Dev B N

2009 Appl. Surf. Sci. 256 536[32] Eltoukhy A H and Greene J E 1980 J. Appl. Phys. 51 4444[33] Kastle G, Muller T, Boyen H -G, Klimmer A and Ziemann P

2004 J. Appl. Phys. 96 7272[34] Kastle G, Boyen H -G, Schroder A, Plettl A and Ziemann P

2004 Phys. Rev. B 70 165414[35] Moller W, Eckstein W and Biersack J P 1988 Comput. Phys.

Commun. 55 355[36] www.hzdr.de/db/Cms?pOid=21578&pNid=0[37] Sedgwick T O, Michel A E, Deline V R, Cohen S A and

Lasky J B 1988 J. Appl. Phys. 63 1452[38] Larson D C and Boiko B T 1964 Appl. Phys. Lett. 5 155[39] Lim J-W, Mimura K and Isshiki M 2003 Appl. Surf. Sci.

217 95

10