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Estimation of Requirements for the Formation ofNanocrystalline Diamond Driven by

Electron Beam AblationRedhouane Henda and Omar Alshekhli

Abstract— Conditions for the formation of thin films ofnanocrystalline diamond by electron-beam ablation of a graphitetarget are studied. The analysis is based on the predictions ofthe analytical solutions of two models describing the expansion ofthe plasma plume into a background gas and toward a substrate.The models allow the calculation of the pressure and temperatureof ablated nanoparticles upon impact with the substrate wherethe film is deposited. The calculation data are reported on thephase diagram of carbon and are used to assess conditions underwhich the diamond phase is likely to form on the substrate. Theresults show that decreasing gas pressure and target to substratedistance, over the practical range of accelerating voltage, isconducive to diamond formation.

Index Terms— Carbon, plasma properties, thin films.

I. INTRODUCTION

PULSED electron-beam ablation (PEBA) has beenincreasingly employed for the growth of thin films of

simple and multicomponent materials with a wide range ofproperties [1]–[4]. This method is experimentally simple andvery versatile and has many other advantages over otherphysical vapor-deposition techniques such as pulsed laserablation [4], [5]. The electron beam is characterized by shortpulse durations (∼100 ns), high electric current (∼1 kA),high fluence (∼10 J/cm2), and relatively low-energy electrons(∼10–20 keV), and is self-propagated to a target materialthrough a hollow cathode and dielectric capillary tube. Beampenetration (∼1 μm) into the target is accompanied by thehighly nonequilibrium process of ablation [4], whereby atomsare effectively excited and ionized for any type of material,even widegap semiconductors. The propagation of the ablationplasma into a background gas is a rather complex yet funda-mental problem in PEBA because the quality of grown films interms of growth rate, composition, structure, and morphologydepends on the kinetic energy density of the species emittedfrom the target and their interactions with the background gas.During expansion away from the target toward a substrate,the generated plasma is attenuated and thermalized bythe gas environment. Even though extensive and ongoing

Manuscript received July 11, 2014; revised October 21, 2014; acceptedNovember 15, 2014. This work was supported in part by the CanadaFoundation for Innovation and in part by the Natural Sciences and EngineeringResearch Council of Canada.

The authors are with the School of Engineering, Laurentian University,Greater Sudbury, ON P3E 2C6, Canada (e-mail: rhenda@laurentian.ca;).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TPS.2014.2373974

experimental work has been devoted to the study of thephenomena involved in PEBA plasma dynamics, manyquestions remain unanswered. Whereas PEBA has the abilityof achieving desired film quality, process conditions andparameters must be judiciously selected for a given materialsystem [5].

Diamond, the sp3-bonded allotrope of carbon, is a specialmaterial because of its wide range of attractive properties.It is the hardest known material, exhibits low friction, is wearresistant, has the lowest thermal expansion coefficient,is chemically inert, has high-electrical resistivity and thermalconductivity, and is optically transparent throughout thevisible-spectrum range. Given its many unique properties,diamond finds use in a wide range of applications, includingtribology, optical devices, biocompatible implants, andcatalysis [6]–[8]. The high cost and scarcity of diamond havehampered its widespread use in such applications, and, hence,research on the preparation of synthetic diamond has beenrelentless for many decades. Such efforts have shown thatthe properties of diamond films can be very close to thoseof natural diamond. The graphite-to-diamond transformationis difficult to achieve mainly due to the considerable kineticbarrier (activation energy) of the process [9]. Further kineticconsiderations also indicate that the rate of formation ofdiamond increases at high pressure. These observations are thefundamental basis for the high-pressure high-temperature syn-thesis routes of diamond including physical vapor-depositionmethods. In PEBA, the internal thermal and ionization energiesof target-carbon atoms during ablation are transformed intokinetic energy of the carbon particles (neutrals, ions) ejectedfrom the target. The energy of the plasma particles on impactwith the substrate will determine the carbon phase of the filmson the substrate surface. High-energy particles may end up inthe diamond phase upon cooling and condensation, whereaslow-energy particles may end up in the graphite phase.

In this paper, we analyze the effects of the main parameters,viz., beam accelerating voltage, target-substrate distance, andbackground gas pressure, on the deposition of diamond filmsby the channel-spark discharge PEBA technique from agraphite target. We use two models to calculate the pressureand temperature of ejected carbon particles from the targetinto the background gas (argon) upon impact with a substrate.

II. MODELS

The first model is due to [10] and [11], and is basedon the macroscopic conservation equations of momentum,

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2 IEEE TRANSACTIONS ON PLASMA SCIENCE

given in (1), and energy, given in (2), for an adiabatic processof an ensemble of No atoms expanding into a backgroundgas. The ensemble is considered as a piston grabbing, incor-porating and driving the gas atoms as it propagates toward thesubstrate. The equations accounting for the deceleration of theensemble of atoms and thermalization of the atoms and gascloud are

M NoVo = (M No + m N)V (1)

M NoV 2

o

2= (M No + m N)

V 2

2+ (No + N)

3

2kT . (2)

Here, V is the velocity of ions at a distance L from thetarget surface, Vo is the initial velocity of the atoms of atomicmass M , T is the temperature of the ensemble, and k isBoltzmann’s constant. No is calculated as the number of atomswithin a cone whose angle is conveniently chosen such thatits volume is L3/3, and N = ng(L3/3) is the number of thegas atoms (molecules) of mass m grabbed by the flux overthe distance L, where ng = Pg/kT is the background gasconcentration at pressure Pg .

Rearranging (1) and (2), the following expressions can beobtained as:

V

Vo= (1 + x3)

−1(3)

3kT

MV 2o

= x3

(1 + x3)(1 + μx3)(4)

where x is the normalized distance (= L/R), μ = M/m, andR = (3μNo/ng)1/3. R corresponds to the distance, whichthe ensemble has covered, whereby the total mass of grabbedgas atoms becomes equal to the mass of ejected No atoms.When the velocity of forward movement, V , is equal to thethermal velocity, (kT/M)1/2, the value of x can be determinedaccordingly from (3) and (4). This corresponds to a certaindistance, Lo, from the target toward the substrate. Next, thetemperature of the particles can be calculated given the valueof the initial velocity of the ensemble of particles. This willconstitute the temperature of particles on impact with thesubstrate assuming Lo to be the target-to-substrate distance.Finally, assuming the kinetic theory of gases to apply to theensemble of particles, it is possible to estimate the pressureon impact as

P = N∗kT (5)

where N∗ is the number of particles per unit of interacting vol-ume. The volume of substrate interacting with the impingingparticles is assumed to be the same as the volume of impingingparticles [12].

The second model, developed in [13] and [14] describesplasma plume expansion into vacuum by means of gas-dynamics equations. The analysis is based on a special solutionof the equations assuming an adiabatic expansion of theplasma and when the flow is self-similar. The special solutionhas been extended to plasma expansion into a background gasprovided that the background gas pressure is low, i.e., at mosta few 10−1 Pa [14], [15]. This is the case of PEBA, wherethe gas pressure is within the range of ∼0.10–2.5 Pa. Themodel assumes that the formation time of the initial vapor

Fig. 1. Schematic of plume expansion at the end of the pulse when thedimensions are Xo, Yo, and Zo, and after a time t , when the dimensions areX (t), Y (t), and Z(t) (with permissions from [15], 2011 American Instituteof Physics).

cloud is much shorter than its expansion time. The expansion ismodeled as a triaxial gaseous semiellipsoid, as shown in Fig. 1,whose semiaxes are initially equal to Xo, Yo, and Zo ≈ csτ ,where τ is the duration of the electron-beam pulse and cs isthe speed of sound in the vaporized material and given bycs = [γ (γ − 1)ε]1/2, where γ is the ratio of the usual specificheats. The expression of the pressure and temperature profilesat the top of the plasma plume (x = 0, y = 0, z = z) can beexpressed as [13]–[15]

P = E p

I2 (γ ) XY Z

(XoYo Zo

XY Z

)γ−1 [1 −

( z

Z

)2] γ

γ−1

(6)

T = mC

k

ε (5γ − 3) (γ − 1)

2γ

(XoYo Zo

XY Z

)γ−1 [1 −

( z

Z

)2]

(7)

where I2(γ ) = (π3/2/2(γ − 1))(�(α + 2)/�(α + 7/2)),mC is the atomic mass of carbon, ε = E p/Mp is theratio of thermal energy of the initial plume, E p , per unitmass of ablated material, Mp , � is the Gamma-function, andα = (γ − 1)−1. A value of 1.25 is used for γ [15].In order to calculate the temperature and pressure, the values ofX , Y , and Z are needed. By setting the vertical dimension,Z , to the target to substrate distance L, X and Y havebeen estimated using plasma shape factors, kη and kζ [14].Under the conditions of PEBA deposition, Xo ≈ 0.14 cm,Yo ≈ 0.13 cm, and Zo ≈ 0.001 cm [4].

In both models, the temperature of the target surface atablation spot, i.e., upon electron-beam pulse bombardment,is required. The temperature of ablation spot can be obtainedfrom the simplified energy balance on the target as [5]

dT

dt= Q

CρSD(8)

where C is the specific heat capacity of the target, 710 J/kg · K,ρ is the target density, 2270 kg/m3, t is time, and S is the

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HENDA AND ALSHEKHLI: ESTIMATION OF REQUIREMENTS FOR THE FORMATION OF NANOCRYSTALLINE DIAMOND 3

Fig. 2. Calculated temperature and pressure of plasma particles fromthe models for different values of the electron-beam accelerating voltage(indicated in the inset in kilovolt). Strikovski’s model is indicated by solidtriangles and Anisimov’s model by solid squares. For both models, thebackground gas pressure Pg = 0.53 Pa and target-substrate distance is5 cm. The data show that particles cooling down in accordance with thegraphite–diamond equilibrium line will likely cross the liquid–solid line intothe diamond solid state.

beam cross section on the target surface, ∼0.05 × 10−4 m2,D is the absorption length (electron range), and Q is thebeam power density. In PEBA, both D and Q depend onthe electron-source voltage U , whereby QS ≈ I (U)U andD = 2.1 × 10−2 U2/ρ [1]. I (U) denotes the beam currentas a function of the accelerating voltage and characterizesPEBA electron gun (as a design parameter) [5]. It is clearthat the interplay between Q(U) and D(U) controls the targetsurface heating rate in PEBA. Given that the electron-beampulse energy can be expressed as E(U) = UI(U)τ (τ is thepulsewidth), available experimental data of Ehave been usedfor the integration of (8) [5].

III. RESULTS AND DISCUSSION

Fig. 2 represents the calculated temperature and pressure ofejected plasma clusters upon impact with the substrate usingStrikovski’s and Anisimov’s models described previously.The data are reported on the (T, P) phase diagram ofcarbon highlighting the stable regions of diamond (D) andgraphite (G). The solid lines delimit the stable forms of carbonand the dotted lines define regions, where one of the forms inthe metastable state (denoted by an asterisk) coexists with theother form in the stable state. The dashed line is the continua-tion of the graphite–diamond equilibrium line with no physicalbearing (a guide to the eye only). While it is not within thescope of this paper, the phase diagram of carbon has been thesubject of controversy for many years and has been updated asmore information has become available [16]. It is worth notingthat the position of the graphite–diamond equilibrium line,however, has been accurately established through experimentsand thermodynamic calculations [17], [18]. It is this regionof the phase diagram that is most relevant to this paper.

By inspection of the phase diagram, and in the case ofdeposition from a plasma state, diamond formation can beachieved when the decrease in the temperature (cooling downperiod) of the plasma clusters after impact with the substrate(which is kept at room temperature) is much rapid than thecorresponding decrease in pressure and in accordance withthe graphite–diamond equilibrium line (the pressure must be

Fig. 3. Calculated temperature and pressure of plasma particles, fromAnisimov’s model, for different values of the electron-beam acceleratingvoltage (indicated in the inset in kilovolt). Solid squares: Pg = 0.53 Pa.Solid diamonds: Pg = 1.1 Pa. Target-substrate distance is 5 cm. For thelatter pressure, the data show that the cooling down-path will likely cross theliquid–solid line into the graphite solid state.

higher relatively to the line) [18]. In this case, the stateof deposited film will end up in the diamond domain ofthe diagram. Under the aforementioned prerequisites and theprevailing deposition conditions of Fig. 2, films of diamond arelikely to be obtained. The calculated (T, P) points from bothmodels (Fig. 2) hint at such a likelihood for the investigatedrange of practical electron beam accelerating voltage. Ourexperimental results, based on the same conditions as thoseof Fig. 2, behave this way and confirm the models findings.High-quality (the percentage of sp3 carbon–carbon-bondedatoms is larger than 96%) nanocrystalline diamond films havebeen obtained. For more ample details on the experimentalconditions and analytical results, see [19] and [20]. As shownin Fig. 2 (and further figures), the temperature (and, hence,pressure) is not proportional to the voltage since the heatingrate of the target surface, as explained earlier in referenceto (8), is not monotonous with voltage but rather depends onthe interplay between the beam power density and electronrange, both functions of voltage. In all cases, the calculatedtemperature reaches a maximum value around an electron-beam accelerating voltage of ∼14.5 kV. It is worth notingthat there is a discrepancy between the estimated valuesfrom each model, whereas the P/T ratio is about the same.A plausible reason for the overestimated values obtained fromAnisimov’s model is that the latter has been developed forvacuum or near vacuum conditions, whereby thermalization isnegligible.

The effect of background gas (argon, in this case) pressureon cluster temperature and pressure has also been assessed.Fig. 3 depicts the calculated T and P when the gas pressureis set to 1.1 Pa. The data points are compared with thoseestimated at 0.53 Pa for the same target to substrate distanceof 5 cm. The results (Fig. 3) show that the temperature andpressure of the plasma clusters upon impact with the substrateare dramatically reduced, and indicate that the deposited filmsare very likely to consist mostly of graphite as the gas pressureis doubled from 0.53 to 1.1 Pa. In order for T and P to be inline with the prerequisites for diamond formation, the distancebetween the target and substrate must be reduced. This is not

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4 IEEE TRANSACTIONS ON PLASMA SCIENCE

Fig. 4. Calculated temperature and pressure of plasma particles, fromAnisimov’s model, for different values of the electron beam acceleratingvoltage (indicated in the inset in kilovolt). Solid squares: Pg = 0.53 Pa.Solid diamonds: Pg = 1.1 Pa. Target-substrate distance is 7 cm. For both setsof conditions, the data show that the cooling-down path will likely cross theliquid–solid line into the graphite solid state.

warranted from the practical viewpoint, as the minimum targetto substrate distance in the current PEBA system is 5 cm(or just slightly below).

The effect of target to substrate distance on the potentialof diamond deposition by PEBA is shown in Fig. 4. As canbe noticed, the increase in the distance, from 5 to 7 cm,is likely to be detrimental to diamond formation, if a reductionin T and P of the clusters is to behave according to thegraphite–diamond equilibrium relationship. While we havenot confirmed the veracity of these observations through ownexperimental efforts, the findings are in agreement with [21].In the latter study, the target-to-substrate distance was setto 12 cm (the pressure value was not given by the authors)and Raman spectroscopic measurements have not revealedany presence of the diamond phase (or even traces thereof).The chemical composition of the target was transferred tothe substrate during deposition, resulting in films of mainlygraphitic nature.

IV. CONCLUSION

We have studied pertinent process conditions for the for-mation of diamond via PEBA. Background gas pressure andtarget to substrate distance are critical parameters to achievefilms with the material in the desired form. The results arebased on two models accounting for plasma dynamics, anddiscussed with the aid of carbon (T, P) phase diagram. Thefindings have been crucial in guiding our experimental effortsto successfully deposit nanocrystalline diamond via PEBA.They also illustrate the power of well-grounded theoreticalmodels, and the potential of the latter in reducing tedioustrial and error experimental campaigns in the quest for thepreparation of desirable materials via physical vapor deposi-tion techniques like PEBA.

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