Applied Surface Science 67 (1993) 481-486 North-Holland applied

surface science

Monte Carlo simulation of energy deposition and scattering by fast electrons in a field emission tip

P.P. Camus a, M.A. Turows k i b.c and T.F. Kel ly a,b,c

" Applied Superconductivity Center, b Materials Science Program, ~ Department of Materials Science and Engineering, Unit:ersity of Wisconsin, Madison, WI 53706, USA

Received 10 August 1992; accepted for publication 11 September 1992

In order to evaluate the efficiency of pulse heating a field emission specimen by an electron beam, it is necessary to know the total amount of energy deposited as a function of incident electron energy. The total energy deposited, however, also depends on the geometry of the specimen which in the case of F1M is approximately a parabola of revolution. Monte Carlo simulations were performed to model this interaction and to determine, principally, the amount of energy deposited into the sample per incident beam electron. By determining this information for incident beam locations along the specimen axis, it is possible to evaluate the effectiveness of specimen heating by high energy electrons. A detailed mapping of energy loss for variations in incident electron energy and beam position was determined.

1. Background

Historically, voltage pulsing has been the pri- mary means of pulsing the evaporation rate in atom probe analysis. For materials with low elec- trical conductivity, pulsed lasers have been used to momentarily heat the surface of the specimen to induce field evaporation. Another method of pulsed field evaporation was proposed whereby an electron beam impacting the near-apex region of the specimen causes volume heating and which in turn leads to field evaporation at the apex [1]. To investigate the feasibility of this proposal, a Monte Carlo simulation program was used to determine the amount of energy deposited into a specimen by an electron beam. This information was then used to determine the resultant heat flow characteristics [2].

General trends of the deposited energy were obtained as a function of specimen material, ini- tial beam energy, and specimen thickness. A rep- resentative value was then used to calculate a single average deposited power density across the whole paraboloidal specimen for use in the heat flow calculations. The most recent heat-flow cal-

culations [3] have looked in greater detail at the effects of a wider range of electron beam param- eters on the temporal temperature distribution. A more accurate simulation of the heat flow would be obtained if the value of the deposited power density would take into account a more realistic pulse shape and a bet ter estimate of the de- posited energy per electron at a given specimen position. In this paper, a thorough mapping of the energy of deposition as a function of position and incident electron energy will be obtained in order to optimize the operating conditions of an electron beam pulsed atom probe as suggested by the heat-flow calculations.

2. Procedure

The Monte Carlo program, which was de- scribed previously [2], is similar to that used to model e lect ron-specimen interactions for elec- tron microscopy [4]. The specimen is described as a monatomic amorphous sample with uniform density, atomic number, and atomic weight. Spec- imens containing non-uniform compositions (seg-

0169-4332/93/$06.00 (c3 1993 - Elsevier Science Publishers B.V. All rights reserved

482 P.P. Camus et al, / Energy deposition by fast electrons in a ]Teld emission tip

regation or clustering) would introduce computa- tional complications that could be taken into ac- count if required. Such effects are not important here, however, since the purpose of this paper is to obtain general operating guidelines. The Monte Carlo program is written for a paraboloidal speci- men geometry instead of the more typical semi- infinite or parallel plate geometry of electron microscopy samples. The incident electron energy was varied between 1 and 30 keV. The specimen shape was defined as a paraboloid having a shape factor of B = 1 , 2 7 × 1 0 - 2 ram-1 where z = B r ~ and z is the direction along the axis of the specimen and r is any direction perpendicular to z. The angle of the incident electrons was normal to the specimen axis, unless noted otherwise, with the initial axial beam position varying from 1 to 1000 nm from the specimen apex. The specimen material was iron. The depth position and ampli- tude of each energy loss event was recorded to determine the energy loss per electron as a func- tion of depth, total energy loss distribution, and the average total energy loss per electron.

3. Results

3. 1. Monte Carlo electron trciiectories

The trajectories that the electrons traverse from the exit aperture of the electron optical column to the specimen follow curved paths in the presence of an electric field [5]. The impact angle of the electrons with the specimen is there- fore expected to vary with the operating condi- tions of the electron gun and the atom probe. To examine this effect, Monte Carlo electron trajec- tory calculations were performed on paraboloidal specimens at normal and tilted incident electron angles, fig. 1, and the average energy loss per electron was determined. The incident angle was provided by a calculation of the electron trajec- tory from an electron optical column toward an FIM specimen for typical operating conditions [5]. For one specific set of conditions, the de- posited energy increased by a factor of 3 by tilting the incident beam by 23 ° toward the negative z axis. Even though this is a significant increase

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0 10 20 30 40 50

~ r

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L ~

~ ~ " 23 ° Tilted ence

10 2o 30 4o 50

Distance from Apex (nm) Fig. l. Monte Carlo simulation of the trajectories of 10 keV electrons inside an iron FIM specimen. The average energy deposited per electron is reduced for electron beams impact ing the specimen normal to the specimen axis as compared to those impacting at small angles. The dots indicate the position

of each e lect ron-specimen interaction.

and should be employed for rigorous heat flow calculations, the complexity of determining the incidence angle at the various impact positions for all operating conditions is so prohibitive that normal incidence to the specimen axis was used for all subsequent calculations.

3.2. Specimen radius effect

Thc parabolic shape factor defines the average radius of the specimen. The larger this factor, the smaller the radius and the thinner the specimen. The energy deposited per electron is known to increase as the thickness increases, therefore, an increasing shape factor, B, decreases the energy deposited, fig. 2. For "thick" sections, as defined by the incident electron energy, the amount of energy deposited is nearly uniform with axial position for a given shape. For "thin" sections, the energy deposited depends upon the shape factor and the specimen geometry (proximity to the apex). When the data in fig. 2 arc plotted using log-log axes, the slope gives the functional- ity of the energy deposited per axial distance. This functionality does not change as the shape

P.P. C a m u s et al. / Energy deposition by fas t electrons in a f ie ld emission tip 483

7 > o

6 c o *6 5 w

o_ 4

o .3 3

~- 2 W o o~ 0 1 o

< 0 100 200

z = B * x 2

Shape Factor B=1200000

I B--400000 ~ _ B=127000

. . . . . . i . . . . . . . . i , , . i . . . . . . . 300 400 500 600 700 800 900 1000

Distance from A p e x , z ( n m )

Fig. 2. Effect of specimen radius of curvature on deposited energy in an iron FIM specimen. A larger parabolic shape factor (i.e., smaller radius of curvature of the specimen and thinner specimen thickness) leads to less energy deposited in the specimen. The

axial dependence of the energy deposited is uniform for various shape factors.

fac tor is var ied , a l though it is expec ted to be a d i f fe ren t va lue for a d i f fe ren t spec imen shape.

3.3. Electron energy loss profile with penetration depth

In the or iginal work [2], the energy depos i t ed per e lec t ron was assumed to be cons tan t th rough the th ickness of the spec imen. This implies that volume hea t ing of the spec imen would occur and subsequen t hea t flow calcula t ions took this as- sumpt ion into account . To test the val idi ty of this assumpt ion , a sca t te r ing event energy loss distr i- bu t ion was ob t a ined th rough the spec imen thick- n c s s .

Fig. 3 i l lus t ra tes the average energy loss pe r e lec t ron which occurs at each dep th pos i t ion while the e lec t ron is t ravers ing the spec imen. F o r most of the depth , the energy depos i t ed is re la t ively uniform. Only at the end of the e lec t ron pa th is the energy r educed to the 60% level. This de- c rease is p robab ly due to the r educed n u m b e r of in te rac t ions at large p e n e t r a t i o n d is tances and the reduc t ion of energy loss pe r in te rac t ion at lower energies , a l though this l a t t e r desc r ip t ion is only val id for this specific ca lcula t ion and is ex- pec ted to vary with inc ident energy and posi t ion. F r o m this analysis, the e lec t ron energy is de- pos i t ed near ly uni formly th rough the th ickness of

5

[ ]

3

o 2

m t

0

Incident Surface

-25 -20 -15 -10 -5 0 5 10 15 20 25

Radial Distance (nm I

Fig. 3. Variation of energy deposition per electron as a function of penetration into an iron FIM specimen by 10 keV electrons. Although there is a small variation in the deposited energy, it is relatively uniform for most of the penetration

indicating uniform volume heating.

the spec imen and the or iginal a s sumpt ion of vol- ume hea t ing is valid.

3.4. Total electron energy loss distribution

Tota l e lec t ron energy loss d is t r ibut ions were ob t a ined as a funct ion of d is tance f rom the apex and inc ident e lec t ron energy, fig. 4. The genera l shape of the d is t r ibut ions was very s imilar to the

484 P.P. Camus et al. / Ener,~,;v deposition t~y Jhs't electrons in a field emission tip

predicted Poisson distribution [4,6], which indi- cates that the specimen shape does not modify the distribution markedly. As expected, the maxi- mum in the spectrum increases as well as moves to lower energy loss as the specimen thickness decreases [3,4]. The variation of the peaks in the distributions with incident energy was complex and will be discussed in more detail below.

3.5. Al,erage total electron enerj,~ loss

The average total electron energy loss was determined from the total energy loss distribution as a function of both incident energy and axial position, fig. 5. It should be noted that the aver- age total energy loss is not similar to the maxi- mum occurrence (mode) nor the median value (50% integration) in the total energy loss distri- bution, but is significantly higher because of the very long tail in the energy loss distribution.

3.6. Effect o f energy o f incident electrons

At a given distance from the apex for a nomi- nal specimen thickness, the energy loss per elec- tron is found to go through a maximum as a function of incident electron energy. This func- tionality derives from the known dependence of energy loss on incident energy [4]. At low ener-

14000

12000

10000 o

8000

"6 6000

z 4000

2000

' lOnm Energy 30kV 25kV 20kV

100 nm

0.5 1

Total Energy Loss (keV)

1.5

Fig. 4. Total electron energy loss distributions as a function of incident electron energy (20 30 keV) at 10 and 100 nm from the apex of a paraboloidal specimen. Thinner specimens and

higher energies typically shift peaks to lower energy losses.

gies, the mean free path between inelastic events is small but the total energy deposited is limited by the low energy of the incident electrons. At high energies, the incident electrons have morc energy to deposit but the mean free path between collisions is much larger.

The maximum of the energy loss distribution is observed to move to higher average energy loss values and to higher energies when the incident beam is displaced from the apex. This can bc explained by the increased number of interac- tions that occurs as the specimen thickness in- creases and the electrons are not "lost" so readily from the specimen, either through the apex or the opposite surface.

3. 7. Effect o f specimen position o f incident electrons

For a given incident electron energy, the en- ergy deposited increases with increasing distance from the apex (increasing specimen thickness). However, the functionality is different for low energy electrons than for high energy electrons. This is readily observed in fig. 6 where the effect of specimen shape is accentuated. At low ener- gies, the energy of deposition only increases slightly as the specimen thickness increases down the specimen shank. At intermediate energies, a large increase in the deposited energy is observed near the apex as the distance from the apex increases. Farther from the apex, the deposited energy once again becomes nearly constant. At the highest energies used for these calculations, the average energy loss is markedly increasing for impact positions up to 1000 nm from the apex.

The thickness dependence of the average en- ergy loss implies that the specimen shape is play- ing a major role in the amount of energy being deposited in the specimen. When the energy de- posited remains essentially constant, the majority of the deposited energy arises from interactions with a "thick" specimen, as defined by the initial energy, and electrons are not transmitted through the bottom surface of the sample. At the highest energies, the specimen is relatively "thinner", as defined by the initial energy, and the majority oI the electrons are lost as transmitted electrons.

P.P. Camus et al. / Energy deposition by fast electrons in a field emission tip 485

Heat flow calculations have been performed using the simplistic assumption that the energy deposited per electron was equal to 1 keV for the whole length of the heated volume [5]. The opti- mum shape of that volume was determined to have a length of 240 nm placed 0 nm from the apex. Assuming a typical initial electron beam energy of 15 keV and a specimen voltage of 10

kV (incident electron energy of 25 keV), this work predicts average energy losses that vary from 0 to 2 keV along the length of the heated volume. Clearly, the assumed value for the heat flow is in the range of calculated values, but detailed analysis of heat flow requires detailed calculations of the axial variation of the electron energy loss. The current work shows that the heat

12

Energy Loss per Electron

(kev)

10

8

6 Energy Loss per Electron

(key) 4

Distance from (rim)

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6

Energy Loss per Electron

(key)

6

5

4

3 Energy Loss per Electron

(key) 2

10(

Axial Distar 30 (nm)

r)

Fig. 5. Average energy deposited as a function of incident electron energy and incident electron position on an iron FIM specimen.

tO

486 P.P. Camus et al. / Energy deposition by fast electrons in a [~eM emission tip

10

Energy Loss per Electron

(key)

i

Energy Loss 1 per Electron

(kev)

1000

Axial Dist~ (nm)

1 0

Fig. 6. Average energy deposited as a function of incident electron energy and incident electron position on an iron FIM specimen emphasizing the effect of specimen geometry.

flow calculations overestimated the energy de- posited at the apex, and it is expected that the results of the Monte Carlo simulations may mod- ify those of the heat flow work by moving the optimum heated volume away from the apex.

4. Conclusions

The energy deposited by an electron beam in FIM specimens increases significantly with in- creasing angle from normal incidence to the spec- imen axis but is found to decrease markedly with a reduction in the radius of curvature of the specimen.

The previously held assumption that energy is deposited uniformly through the thickness of the specimen is valid.

The energy deposited by electrons is a highly non-uniform function of the incident electron en- ergy and incident specimen position and should be mapped completely for the most accurate heat flow calculations.

Acknowledgments

This work is sponsored by the National Sci- ence Foundation under grant #DMR-8911332 (Dr. John Hurt) and the Electric Power Research Institute under agreement #RP8009-5 (Dr. Thomas Schnieder). Helpful discussions with L.M. Holzman and D.J. Larson are gratefully acknowl- edged.

References

[1] T.F. Kelly, D.C. Mancini, J.J. McCarthy and N.A. Zreiba, Surf. Sci. 246 (1991) 396.

[2] T.F. Kelly, N.A. Zreiba, B.D. Howell and F.G. Bradley, Surf. Sci. 246 (1991) 377.

[3] P.P. Camus, D.J. Larson and T.F. Kelly, Appl. Surf. Sci. 67 (1993) 467.

[4] D.E. Newbury, in: Principles of Analytical Electron Mi- croscopy (Plenum, New York, 1989) p. 1.

[5] D.J. Larson, P.P. Camus and T.F. Kelly, Appl. Surf. Sci. 67 (1993) 473.

[6] R.F. Egerton, Electron Energy-Loss Spectroscopy (Plenum, New York, 1986).