Applied Surface Science 67 (1993) 467-472 North-Holland applied

surface science

Simulation of rapid thermal pulsing for field evaporation

P.P. Camus ~, D.J. La r son a,b and T.F. Kelly ~,b,c '~ Applied Superconductiuity Center, ~ Materials Science Program, and c Department of Materials Science and Engineering, University of Wisconsin, Madison, WI 53706, USA

Received 10 August 1992; accepted for publication 11 September 1992

Pulsed field evaporation should be achievable using an electron beam for thermal pulsing. This article explores the opt imum specimen heating configurations resulting from this pulsing. The subsequent evaporation rate and the evaporation probability per pulse are then calculated. The heat transfer due to volume heating of the specimen is calculated using a one-dimensional Crank-Nicholson method. The evaporation rate is calculated using an Arrhenius expression along with previously determined experimental data. The heat transfer results indicate that, for the conditions simulated, a heated width of 240 nm can have the largest range of heated offsets and still produce the desired temporal heating characteristics. The evaporation rate results indicate that an evaporation probability of one ion per 20 pulses should be obtainable. The timer-limited mass resolution for time-of-flight measurements is predicted to exceed 250 at full width tenth maximum, although experimental values are expected to be lower.

1. Introduct ion

1.1. Background

In a typical atom probe field ion microscope (APFIM), ions are removed from the surface of the specimen by field evaporation in short (nanosecond) pulses. In conventional atom probes, the specimen voltage is pulsed in order to achieve pulsed field evaporation. In the case of pulsed-laser atom probe (PLAP), pulsed field evaporation is produced by heating the specimen surface with short (nanosecond) laser pulses. Nu- merical calculations of the heat flow characteris- tics resulting from this heating [1] and an investi- gation of the subsequent evaporation pulse shapes [2] have been performed previously. The heated length of specimen was as small as 5 /zm with the assumption that surface heating occurs to a pene- tration depth of a few nanometers. The energy is deposited nearly uniformly along the heated re- gion. It was found that 300 ps photon pulses produce 5 ns apex temperature pulses. This low rate of cooling was attributed to the large irradi-

ated volume and resultant large deposited en- ergy.

Pulsed field evaporation can also be obtained using an electron beam for thermal pulsing [3]. Energy deposition by high energy (5-50 keV) electrons results in volume heating of the speci- men. The magnitude of the heating depends upon the material, initial beam energy, and current density of the beam [4]. Numerical simulations of e lec t ron/spec imen interactions [4,5] indicate that sufficient energy could be deposited by a conven- tional scanning electron microscope (SEM) to produce thermally-stimulated field evaporation. As a directed electron beam can be focused to heat lengths of the specimen shorter than 50 nm, it is hoped that a minimum volume of material must be electron irradiated to produce rapid ( < 1 ns) apex temperature pulses.

The range of parameters investigated in some initial heat flow calculations [4] has been ex- panded in this paper to determine an optimum specimen heating configuration. The sensitivity of the heating characteristics to electron beam size and position and an estimate of the acceptable

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468 P.P. Carnus et al. / Simtdation o f rapid therntal pulsing fi~r field et'aporation

spatial error in the location of the heated volume have been made. Spatial error in the electron- specimen impact position could be introduced due to the highly precise calibration data re- quired to direct the electrons to desired positions on the specimen [6]. Detailed heat flow calcula- tions employ a range of heated lengths (0-500 nm) and offsets from the apex (0-100 nm). A simple calculation of evaporation rates during a thermal pulse, evaporation pulse time, and the corresponding integrated evaporation probability per pulse are then presented.

1.2. Heat transfer process

l Electr°n Beam ii ~ r

i i 1( ,[_. . I

H 0 i ( L )

Fig. I. Schematic illustration of the parameters used in heat transfer calculations.

A model of the heat transfer in a FIM speci- men was assumed as in the prior work [4]. The equation which describes the unsteady state heat conduction is given by

aT V2[k(T)T] + g ( t ) = p C p ( T ) ~ t , ( ] )

where the thermal conductivity, k ( T ) , and the heat capacity, Cp(T) are functions of tempera- ture. g ( t ) is the temporal volumetric power gen- eration function and its maximum value is given by

I b A E

g . . . . eVo (2)

where I , is the electron beam current incident on the specimen, A E is the average electron energy loss per electron, V 0 is the heated volume, and e is the electron charge. A value of 10 ~s W / m 3 was used for g ( t ) for all calculations. This power density could be obtained from an electron beam of 1 /~A which deposits an average of 1 keV per electron for an irradiated volume of 106 nm 3.

The boundary conditions assumed for the cal- culations as shown in fig. 1 are:

L = specimen length = l 0 / z m H = generator width O = generator offset T = 5 0 K, t = 0 for O < z < L ~tT/az=O, T = 5 0 K , z = L OT/afi = O, 0 <_ z <_ L

{go ..... O < z < O + H g = z < O , O + H < z < L

The power generation function was set at gm~,× until the specimen apex temperature reached 300 K and was then set to 0. Two-dimensional calcu- lations have shown that conduction of heat akmg the specimen axis predominates for volume heat- ing providing that cooling by radiation is ne- glected and the sample is under vacuum [4]. The results presented here for the heat transfer are due to a one-dimensional finite difference calcu- lation using a modified Crank-Nicholson method [7].

1.3. Euaporation rates and time resolution

After a temporal temperature profile is ob- tained for a specific heating geometry and pulse width, it is possible to estimate the absolute evap- oration rate that occurs at the specimen apex. From these pulse shapes, estimates of the total number of ions evaporated per pulse and the temporal width of the evaporation pulse may be determined. The uncertainty in the time of evap- oration will affect the final mass resolution ob- tainable in time-of-flight measurements.

For thermally-induced field evaporation, the rate of evaporation at a constant applied field follows an Arrhenius relation

K = K o e -e/k~r, (3)

where Q is the activation energy and K() is a pre-exponential constant. The dependence of these variables on the field for tungsten ions has been experimentally determined previously. Val-

P.P. Camus et al. / Simulation of rapid thermal pulsing for field e+,aporation 469

ues of Q = 0 . 2 0 + 0 . 0 0 4 eV and K 0 = 3 × 1 0 it with an uncertainty of one order of magnitude [8] have been used for the current work. These val- ues correspond to an evaporation field of 57 V / n m .

2. Results and discussion

2.1. Heat transfer

Since there are only small differences in the heat transfer behavior for aluminum, iron, and tungsten [4], the results presented here are for tungsten only. The thermal history of a tungsten specimen is shown in fig. 2 for various heating geometries. For H = 100 nm, fig. 2a, the offset cases heat to 300 K in 2-4 ns. For H = 200 and 300 nm, figs. 2b and 2c, the heating times are significantly shorter and the specimen reaches 300 K in less than 1 ns for many of the offset values. For any H value, the cases of large offset ( O = 5 0 - 1 0 0 nm) are found to heat and cool relatively slowly. This is due to the fact that the heat must flow to the specimen apex from the heated region during the pulse. This also leads to a slight overshoot of 300 K at the apex, as shown in fig. 2, because the temperature of the heated region must actually rise to greater than 300 K. During the cooling, the entire length O + H has been raised to at least 300 K and acts as a buffer to slow the heat flow away from the apex. All the cases examined in fig. 2 return to the base tem- perature of 50 K in less than 1 #s. This would permit pulsing the specimen at frequencies of up to 1 MHz.

As a basic criterion for optimizing the time resolution of the pulse, it is desirable that the sample reaches 300 K in less than 1 ns. This is chosen so that very slow initial heating does not occur which deposits a large total energy in the specimen. With regard to this, it is found that H = 100 nm is too small for all cases examined. A 200 nm width satisfies this condition for offsets up to ~ 20 nm and a 300 nm width does so for all offsets examined except 100 nm. A three-dimen- sional surface plot of the time for the specimen apex to reach 300 K as a function of H and O is

35O

v 3oo v

d3 ~ - 200 (1) O_ E ~so Q) p-

1o0

X Q) EL <

100 nrn Genera to r Wid th (a)

Of fse t : ~'i' Onm [.']'

--~50 nm / ~ , ,

10 12 10.11 10 lo 10-~ 10 ~ 107 10 a

Time (s)

O_ E

X

O_ <

350

300

250

2OO

150

100

50

200 nrn Genera to r Width

Of fset : 0 nm

- 10 nm - -20 nm -.50 nm

100 nm

(bi

0 .... ' . . . . . . . . ' . . . . . . . i , lO "12 lO t~ lO4O 10-9 104 107

Time (s) 10 +

350

300 Q)

Q.) I--

lOO

<

300 n m Genera to r Width (cl

Of fse t : ~, 0 nm ],~',

- ---10 nm (,' '~'~ --20 nm ~,+ '~', :~ 50 nm_ ,~,, ,x,

I ........ ' ........ ' ........ ' ..... o-12 lO.n 10-1o lO-O lO ~ 1o + 1o +

Time (s)

Fig. 2. T h e r m a l history of F IM spec imen for var ious g e n e r a t o r

offsets. (a) W = 100 nm, (b) W = 200 nm, (e) W = 300 nm.

shown in fig. 3. The shaded regions are for heat- ing times of less than 1 ns and 2 ns. It is clear that a wide variety of heating widths and offsets will heat the sample to 300 K in 1 ns or less as desired.

It is also necessary to examine the temporal width of the pulse and to limit the permissible

470 P,P. Camus et al. / Simulation of rapid thermal pulsing )'or field euaporation

.-4 ~(

Fig. 3. Time for specimen apex temperature to reach 300 K.

5(3 ~

E

2O

~ - - 2 s

r" 10

~ K i 100

Generator Offset (nm)

Fig. 5. Contour plot containing time to 300 K and time above 250 K data which shows optimum range of values.

heating geometries based on these results. The total time that the specimen spends above 250 K is a good indication of the potential evaporation pulse width, fig. 4. Since the evaporation rate varies exponentially with temperature, a change in temperature from 250 K to 300 K results in nearly an order of magnitude change in evapora- tion rate. Thus, below 250 K the probability of an evaporation event is low and may be ignored for the purposes of determining an approximate evaporation pulse width. Also, this parameter, "Time above 250 K", has been used in previous work [4] and the present results may be therefore directly compared.

For any constant O value, there is a minimum in the time above 250 K values as H increases

Fig. 4. Estimate of potential evaporation pulse width as indi- cated by time specimen spends above 250 K.

from 0. At the larger widths the specimen re- ceives too much total energy and subsequently spends a long time above 250 K. At very small widths the specimen heats very slowly; this leads to near thermal equilibration of a large portion of the specimen, and thus a large amount of energy deposited. Increasing the offset is found to in- crease the minimum in the time above 250 K due to the same reasons stated previously. The results presented here agree with previous work for the zero offset axis [4] and reveal a small area of widths and offsets for which the time spent above 250 K is 1 ns or less. As in the previous figure, the shaded regions in fig. 4 correspond to times of less than 1 ns and 2 ns.

By combining the criteria of time to 300 K and time above 250 K of less than 1 ns, an acceptable range of widths and offsets can be illustrated on a contour plot, fig. 5. The shaded area shows this region. For the conditions of the simulation (material, generator power density, etc.), a heat- ing width of ~ 240 nm can have the largest range of offsets and still produce acceptable heating performance. This figure provides an estimate of the allowable errors in positioning the electron beam on the specimen. This contrasts with calcu- lations of thermal transients produced by laser pulsing where a 5 # m length irradiated for 300 ps produced 5 ns wide apex temperature pulses [1]. The greater temporal width of the pulse in this case is due to the larger irradiated volume during laser pulsing which requires nearly 4 orders of

P.P. Camus et al. / Simulation of rapid thermal pulsing for field evaporation 471

350

3 0 0 . , .. , - ' / ~ '

m ~ - 200 ID

~ 1 5 0

10c x Q ~ 5Q - - t O O n m h e a t e d f o r 2.5ns

- -2OOnrn heated for O.94ns 30Onto hea ted for 0.67ns

T i m e (ns)

Fig. 6. Thermal history of FIM specimen for three different heating configurations.

0.12 - - 1OOnm heated for 2.sns - -2OOnm heated for 0.94he

.~ 3OOnm heated for 0.67ns . . . . . . . . . . - . . . . 03 / " / " " ~ 0.0£

O O_ 05 >

LIJ O.OE ,'

r" o

~ 0.0~

I'-"

1 5 4 - Time ns)

Fig. 8. Estimate of evaporation probability per pulse as given by the total ions evaporated during the heating pulse.

magnitude more energy to be deposited during heating and the cooling rate is slowed accord- ingly.

2.2. Evaporation rates and time resolution

Fig. 6 shows the temperature profile for three different heating configurations at a constant generator offset of zero. An absolute evaporation rate at the specimen apex was calculated for the three heating pulses, fig. 7. The time resolution is determined as the full width at half maximum (FWHM) of the evaporation profile; 0.570, 0.276, and 0.272 ns, and as full width at tenth maximum (FWTM); 1.740, 0.849, and 0.929 ns for the 100, 200, and 300 nm width cases, respectively. The longer heated width for the 300 nm case slows

the heat flow away from the specimen apex (smaller temperature gradient) but makes it pos- sible to heat the specimen apex to 300 K faster than the other two eases. Note, however, the crossover point in the cooling curves for the 200 and 300 nm data in both figs. 6 and 7. This shows the effect of different heated geometries on the cooling characteristics of the specimen. The prod- uct of generator width and pulse time gives a relative value for the total energy deposited in the specimen. The total energy deposited in the 100 rim-2.5 ns and 300 nm-0 .67 ns cases is 1.33 and 1.07 times that deposited in the 200 nm-0 .94 ns case, which cools the fastest.

The data in fig. 7 can be integrated to obtain the total number of ions evaporated over the pulse, fig. 8. This gives an estimate of the evapo- ration probability per pulse for the various condi-

109 - - 1 0 0 n m heated for 2.Sns

~ " - - 2 0 0 n m heated for 0.94ns "~- .... 300nrn heated for 0.67ns 03 t--

.O 108 ~,~

(D / ~ * ~ \ / '~ K = KO exp(-Q/kT) "~ / . \ / ', Q = 0 2 0 e V rr'r._ ~ / \ \ / ' \,, Ko=Se11

¢-i : \ ' \

UJ

1 2 3 4 5

T i m e (ns) Fig. 7. Evaporation rate at specimen apex for three different

heating configurations.

¢/) t -

._o v 03

cr

c- O

0 r~

> UJ

10 g

10 B

107

106 Jif,

- - 2 0 0 n m heated for 0.94ns uncertainty in Ko and Q

2 3 Time (ns)

Fig. 9. Estimate of uncertainty in evaporation rate calculation.

472 P,P. Carnus et al. / Simulation o f rapid thermal pulsing.l'br field el,aporation

0.25

"(3 0.2

0

m°-0.15

LIJ 09 C 0.1 O

i~_. 0.0~

- - 2 0 0 n m heated for 0.94ns uncer ta in ty in Ko and Q

/

3 4

Time (ns) Fig. 10. Estimate of uncertainty in ew~poration probability.

3. Conclusions

A range of optimum SEM operating condi- tions and specimen heated volumes were ob- tained, based on heat flow calculations. Pulsed field evaporation probabilities and evaporated- ions-per-pulse data were calculated which indi- cate that an evaporation rate that exceeds that required for contemporary 3DAP experiments is obtainable, using optimized electron beam heat- ing conditions. Calculated timer-limited mass res- olutions typically exceed 250 for frequencies up to 1 MHz.

tions. The total number of ions evaporated per pulse scales with the relative total energy de- posited in the specimen. The shortest pulse, as defined by FWTM, is obtained with the 200 nm generator width and is 0.849 ns.

An investigation of the error margin inherent in the experimental data for the above rate calcu- lations was also performed. Figs. 9 and 10 show the maximum range of error introduced by vary- ing Q by +0.004 eV and K 0 over one order of magnitude [8]. For the absolute evaporation rate, the error range is 1.3 orders of magnitude at the peak. The error range for the total ions evapo- rated is 1.4 orders of magnitude for the final evaporation probability. Note that the magnitude of the minimum evaporation probability is higher than used for a conventional 3D atom probe (3DAP). Thus, a beam current density in the SEM that produces 10 ~ W / m 3 is sufficient for electron beam pulsed heating and can even be reduced if a single-hit detector is to be employed.

A mass resolution estimate for this pulsing scheme was obtained using a 500 ns flight time for W 3~ for a short time-of-flight instrument. Using the relation m / A m : t / ( 2 d t ) , the maxi- mum timer-limited mass resolution is predicted to exceed 900 at FWHM and 250 at FWTM, although experimental values are expected to be much lower.

Acknowledgements

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. Tho- mas Schnieder). Helpful discussions with L.M. Holzman and Dr. M.A. Turowski are gratefully acknowledged.

References

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

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[7] M.N. Ozisik, Heal Conduction (Wiley. New York, 1¢~80) pp. 493 496.

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