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On the atomic resolution of a field ion microscope
SURFACE
SCIENCE 26 (1971) 61-84 0 North-Holland
ON THE ATOMIC
RESOLUTION Y. C. CHEN
Publishing Co.
OF A FIELD ION MICROSCOPE*
and D. N. SEIDMAN
Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14850, U.S.A.
Received 17 August 1970 The effects of tip temperature ( TT), imaging gas mixture, and electric field strength on the resolution of the field ion microscope have been measured quantitatively. The image diameter of individual tungsten atoms had a quadratic dependenceon TT between 11 “K and 62°K. This result was in agreement with the theory of resolution which predicted that the transverse component of the velocity of the imaging ion limited the resolution of the FIM. The data also indicated that the imaging gas atom was not accommodated fully to TT when it was ionized. The degree of thermal accommodation of the imaging gas atom prior to its ionization was increased as a result of imaging in a 10% neon-helium gas mixture as opposed to imaging with pure helium gas. The extrapolated value of the experimental contribution to the image size at 0°K was larger than the value predicted by the existing theory. This discrepancy was attributed to the neglect, in the existing model of resolution, of the nature of the local electric field strength in the region of space where the imaging atom was ionized. The dependence of image size of an atom at constant TT exhibited a maxi~lum. This result was at variance with the existing theory of atomic resolution. An explanation of this effect in terms of the field dependence of the equilibrium concentration of gas atoms at the surface of the tip was given.
1. Introduction
the most crucial steps in the development of the field ion microscope (FIM) occurred when Mfiller l,z) realized that the high atomic resolution of the FIM was dependent strongly upon accommodating thermally the imaging gas atoms to the temperature of a specimen which was maintained at a value of 78°K or less. The existing calculation of the atomic resolution [see section 2, eq. (1)] of a FXM indicated that the resolution was a function of 6 physical parameters. To date, there have been no measurements of the atomic resoiution where these parameters were varied under controlled experimental conditions. Thus, the present paper represents the first quantitative measurements of the effect of the tip temperature, imaging gas mixture, and electric field strength on the resolution of a FIM3). One of
* Research supported by the Advanced Research Projects Agency through the Materials Science Center at Cornell University. 61
62
Y. C. CHEN
AND
D. N. SEIQMAN
2. Basic principles and experimental program The only calculation of the atomic resolution that has been performed is a measure of the “extent to which the spatial modulation of the ion current density above the tip is preserved in the ion beam which reaches the phosphorescent screen of the FIM” [Southon4*5)]. This calculation of atomic resolution has been carried out by a number of authors [e.g., see Good and Muhers), Mi.iller7) and Gomer *)]. The derivations were all based on an analogous calculation of the resolution of a field electron microscope image which was first considered by Benjamin and JenkinsQ) and eIaborated on by Richter lo). Miiller rr), Ashworth la), Gomer Ia) and RoseI”). The expression for the atomic resolution (6) is given* by [see Miiller and Tsong’s l5) eq. (2.91)]
where h (A= h/2x) is PIanck’s constant, /I is a compression factor which depends on the exact shape of the tip, r, is the local radius of curvature, nz is the mass of the imaging gas ion, e the charge on the ion, ET the electric field strength at the tip, i is a geometric factor, and .sT the transverse thermal energy associated with the imaging gas atom when it is ionized at the surface which is x, A above the real surface of the FIM specimen. The quantity S in eq. (I) is not the resolution as defined in the standard manner by the Rayleigh criterion, instead it is the minimum image diameter of an atom on the viewing screen divided by the magnification of the FIM. The first term inside the braces in eq. (1) was obtained by considering the broadening caused by the Heisenberg Uncertainty Principle, and the radial projection effect. Each of these two effects makes an equal contribution to this first term. The second term inside the braces resulted from a consideration of the spreading of the ion beam as a result of the transverse thermal energy (~.r) of the imaging gas ion. For the purpose of analysis (see section 5) it is useful to rewrite eq. (1) in terms of the voltage on the tip (VT). Since V, and &are related through the relationship (2) eq. (1) becomes (3)
* We have omitted the quantity SOfrom Miiller and Tsong’s eq. (2.91) temporarily as this quantity will be discussed in section 5.2.
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
63
It is important to note that the resolution as defined by eq. (1) is a function of 6 physical parameters [i.e., 6 = S(@, r,, ET, 5, m, sT)] and that eq. (2) does not remove the dependence of 6 on rT. The physical reason for this dependence of 6 on rT is the fact that the angular momentum (p,) of the imaging ion is a function of rT (i.e., p,=mr,v,). This is seen from the basic expression for resolution [Gomers)] 6 = (r&,/R) D = 4o,r#(m/2e&)“,
(4)
where D is the measured distance on the screen which corresponds to a distance 6 at a surface which is at x,, and v, is the transverse component of the velocity of the imaging ion. Rewriting eq. (4) in terms of p, one obtains the expression 6 = 4pJ(1/2emI’#.
(5)
Therefore, an increase in pw results in a larger value of 6, and hence a decrease in resolution. The present paper presents the results of an investigation of the dependence of atomic resolution of the FIM under the folIowing controlled experimental conditions: (1) Variable TT at constant values of the parameters jI, rT, ET, 5 and m employing helium as the imaging gas (PHe= 1.87 x 10e3 Torr). The variable T, implied that sy was the only physical variable in eq. (3). (2) The above experiment was repeated on the exact same specimen, but the imaging gas was changed to a 10% neon-helium mixture (PNe =0.21 x I 0m3 Torr and Pn,= 1.87 x lop3 Torr). (3) Variable ET and cT at constant values of the parameters p, rT, i and m. The quantity sT is a function of ET (see section 5.1) hence it is impossible to vary ET independent of E=. 3. Experimental
techniques
The FIM employed was a stainless steel microscope which was evacuated by a 15 liter set-’ Vacion pump and a titanium sublimation pump. The microscope was operated statically at a background pressure in the range (1 to 10) x 10-l’ Torr. The required pressure of helium gas was obtained by diffusing helium through a quartz membrane, while the neon was leaked into the FIM from a bottle of research grade (99.995 min. vol. “//,) gas. The temperature of the FIM specimen was controlled by a continuous transfer liquid helium cryostat [Seidman et al. ‘“)I. The temperature stability of the specimen using this technique was L-O.01 “K and the specimen
64
Y. C. CHEN
AND D.N.
SEIDMAN
temperature* was monitored continuously with a miniature platinum resistance thermometer which was mounted very close to the specimen. This temperature monitoring thermometer was calibrated against a second resistance thermometer which was placed at the position where the specimen resided normally. The specimen voltage was controlled with a zero to 30 kV Spellman power supply which was filtered by a O.Ol’A Sorensen ripple filter. The output voltage of the power supply was monitored with a Hewlett-Packard 4-place digital voltmeter used in conjunction with a specially constructed 100: 1 voltage dividing network. The specimens were field evaporated to a final end form employing a zero to 4 kV pulse amplifier [Robertson and Seidmanrs)]. The images were recorded on Kodak Tri-X film employing a f/O.87 Super-Farron lens. For a given experiment, all images were recorded on the same roll of film and developed simultaneously. Hence, the film shrinkage was identical for each frame employed for the atomic image diameter measurements reported in section 4. The image diameter measurements were made on the negatives with the aid of a Vanguard motion analyzer. This motion analyzer [Scanlan et al. ‘s)] enlarged a micrograph to a 14.6 cm x 19 cm image. In addition, the analyzer had X and Y cross hairs which were used for determining the image diameter in two different directions. The fractional standard deviation for the measurement of an image diameter was N 0.025 on the enlarged 35 mm negative. All specimens were prepared from Westinghouse or Sylvania 0.0127 cm diam tungsten wire. The tips were electro-polished in a 10 wt% NaOH solution at 1 to 3 V dc. 4. Experimental
results
4.1. THE EFFECT OF THE TIP TEMPERATURE (TT) ON THE ATOMIC RESOLUTION The decrease in the atomic resolution in preceding from a TT of 11 “K to 62”K, at a constant value of Vr, is apparent from a visual comparison of the 5 FlM micrographs shown in fig. 1. These 5 patterns are of a region near the (110) pole. The specimen used for this experiment was first field evaporated to a final end form? at 36°K and then imaged at 11 “K at a best image voltage (BIV) of 10.2 kV. Subsequently, T, was increased to 23, 36, 49 and 62°K without changing Vr. Next, the image was recorded photographically at each * Seidmanand ScanlanlT) haveshown that the heating effects due to thermal radiation, the imaging gas, and the thermoelastic effect were negligible, hence the measured temperatures should correspond to the tip temperatures rather well. t The final end form was determined by the condition that the field evaporation rate was constant over the entire surface of the tip.
ON THE ATOMIC
RESOLUTION
OF A FIELD
11” K
ION MICROSCOPE
65
23” K
36” K Fig. 1
of the above TT’s. Note that the increase in the image size of an atom was large enough such that 2 atoms which were distinguishable at 11 “K merged completely into one another at 23°K (see region 1 in fig. 1). It should also be noted that the decrease in resolution caused 3 well resolved atoms to almost become a continuous linear chain of contrast (see region 4 in fig. 1).
66
Y. C. CHEN
49” K
AND D. N. SEIDMAN
62” K
Fig. 1. A series of five FIM patterns of the (011) region taken at 11, 23, 36, 49, and 62°K respectively. The numberedregions show the effect of increasing temperature on resolution. The voltage on the specimen was 10.2 kV (the BIV at 11 “K). The image at 36°K was accidentally recorded at a voltage which was - 200 V greater than 10.2 kV.
The decrease in resolution with increasing TT was also apparent in regions 2 and 3 offig. 1. The above observations were made quantitative by measuring the image diameter* of an atom (2) as a function of TT with all the five remaining variables held at fixed values. In order to distinguish among 3 possibilities for the dependence of I on T, (i.e., I versus TT,l2versus TT,and I3 versus T,.) the data were processed in the following manner: (1) The data were fitted to each of the above 3 possibilities by the method of least squares; (2) The goodness of fit in each of the above cases was tested using the Coefficient of Multiple Determinationao). The latter test indicated that for the 30 atoms measured in the vicinity of the (011) pole, 70% fitted a l2 versus T, plot, while 18% fitted a 1 versus TT plot, and the remainder a l3 versus TT plot. The results for 5 different atoms near the central (011) pole imaged with helium gas are shown in figs. 2 and 3. The numbered curves in fig. 3 correspond to the numbered atoms shown in the 5 micrographs in fig. 2. Note that the curves in fig. 3 were displaced from one another for clarity, hence the scale of I2 is arbitrary * The image diameter (I) was the measured image size of an atom as observed on the motion analyzer. The quantity I was used in the analysis rather than 6, since the fractional error in I is smaller than the fractional error in S (see Appendix A2).
67
3
5
1
2
4
4
11” K
23” K Fig. 2
Y. C. CHEN
4
AND D. N. SEIDMAN
36” K
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
I59
Fig. 2. A second series of fiveFIM patterns taken at If, 23,36,49, and 62%. and at a tip v&age of 10.2 kV. The centrat pole is the (011) pole. The numbered atoms are the ones whose image diameters were measured as a function of temperature {see fig. 3).
in this figure. The above results demonstrated that I had a quadratic dependence on Tr between I1 “K and 62°K. 4.2. THE EFFECTOF IMAGING GAS MIXTURE ON ATOMIC RESOLUTION The experiment described in section 4.1 was performed on 2 additional specimens which were field evaporated initially at 36°K. For each specimen the images were fust recorded with helium as the imaging gas, and then repeated with a 10% neon-helium gas mixture at identical vaiues of the parameters p, r,, PM,VT, and [. The graph of 1’ versus TT for some of the atoms examined on specimen number 2 in a 10% neon-heIium gas mixture is shown in fig. 4. Once again it was found that f had a quadratic dependence on Tf *. It was also found that the slope of the E2versus TTgraph was less in the 10% neon-helium gas mixture than it was in the pure helium gas at identical values of the parameters /I, r,, VT and 5. The quantity 6 for 15 atoms at 1I “K and 62°K is given in table 1 along with the intercept at TT=O"K [(C,),,,] and the slope (Cl) of the 6’ versus T*curve for each atom. It can be seen that at 11 “K the value of 6 ranged from (2.3 to 3.3) A, whiie at 62°K the range has increased to (2.6 to 3.8) A. The range in 6 for these same atoms imaged in * These data were also tested for goodness of fit employing the CoefEcient of Multiple Determination. This test indicated that of the 30 atoms examined, 86x, fitted a i2 versus TT plot, while 6 ?< fitted a I3 versus T;f plot, and 8 % a fversus TT plot.
-JO
Y. C. CHEN
AND D. N. SEIDMAN
3
Fig. 3. The square of the image diameter {P) of an atom versus the tip temperature (TT). The scale of I” is arbitrary since the curves were displaced from one another for clarity. The numbered curves correspond to the numbered atoms in fig. 2.
helium gas at 11 “K was (1.9 to 2.9) A, and at 62°K it was (2.7 to 3.4) ,&. Thus, the range of image sizes in helium gas was smaller, but within our experimental error for determining fi (see Appendix A). 4.3. THE EFFECT OF ELECTRIC
FIEfA
STRENGTH (VT) ON THE ATOMfC
RESOLUTION
The effect of & on the atomic resolution was determined for a specimen which was initially field evaporated to a final end form at 36°K and then examined at 11 “K. The quantity V, was varied from 7.9 kV to 9.6 kV in 6 approximately equal increments while the parameters p, r,, m and r, were maintained invariant. The series of 6 micrographs in fig. 5 shows the results of the above experiment. The image at the 9.6 kV is at BIV. The value of I for 25 atoms in this region was measured,9 and the results for 5 of these atoms are shown in fig_ 6 where the value of l2 (arbitrary units) is plotted as a function of Kr. The atoms numbered 1 to 5 in fig. 5 correspond to the labeled
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION
MICROSCOPE
71
8’
vs TT (OK) 10% Ne-He Imaging Gas Mixture
v, = 14.3kV
I IO
0
Fig. 4.
I
I
I
I
30 40 50 20 Tip Temperature (OK)
I
60
The square of the image diameter (12) of an atom versus the tip temperature constant voltage of 14.3 kV in a 10% neon-helium gas mixture.
at a
curves in fig. 6. It is seen that the dependence of Z2 on VT has no simple functional form. Instead, the image size exhibited a maximum value and then decreased with increasing V, up to the BIV. It is also noted that the maximum value of l2 occurred at different values of VT for different atoms. 5. Discussion 5.1. DEPENDENCE
OF
THE
ATOMIC
RESOLUTION
ON
TIP TEMPERATURE
(TT)
The principal experimental result obtained (see section 4.1) was that the quantity 1 had a quadratic dependence on TT. The expression for resolution [eq. (3)] under the condition of variable TT and constant values of rT, VT, p and m reduced to P=C1+C2ET, (6) where Ci = [4@rT/(2 mevT>‘] , (7) and C, = [16,G2r~/eVT]. (8)
Y. C. CHEN AND D. N. SEIDMAN
72
TABLE Pertinent
parameters
for temperature imaged
(Cl),,,
4 5 6 7 8 9 IO II 12 13 14 15
Range
x 101”
(cm”)
number
of
parameter
in
I
dependence of resolution a 10 “ANeeHe gas mixture (Cz’) x 101s (cm”cK-l)
for specimen
6(ll”K)* (A)
number
two
6(62”K) * (A)
7.3 7.8 8.3 5.1 8.3 6.6 9.6 7.1 8.3 6.2 6.5 5.3 10.4 6.7 6.0
4.9 5.6 2.7 2.5 3.6 6.6 7.1 4.4 6.4 4.1 7.5 6.6 6.9 7.9 7.2
2.8 2.9 2.9 2.3 2.9 2.7 3.2 2.8 3.0 2.6 2.7 2.5 3.3 2.7 2.6
3.2 3.4 3.2 2.6 3.2 3.3 3.7 3.1 3.5 3.0 3.3 3.1 3.8 3.4 3.2
5.1 to 10.4
2.5 to 7.9
2.3 to 3.3
2.6 to 3.8
* The absolute values of 6 were determined by dividing I by MT (total magnification of system). The fractional error in 6 had an upper limit of 0.25 in the present experiments (see Appendix
A).
If the imaging atoms were accommodated fully to the surface of the specimen when they were ionized at x, then sT would be equal to kT, from the equipartition theorem of thermal energy. Since the degree of thermal accommodation was unknown, we employed the following relationship between cT. and Tl xT = y(kT,) = + mu:, (9) where y was experimental it was ionized fully when it above model
a dimensionless parameter which was determined from the data. If the imaging atom was accommodated fully to TT when at X_ then y would be equal to one. If it was not accommodated was ionized, then y would be greater than unity. Employing the and eq. (9) the expression for the resolution [eq. (6)] became 6’ = C, + C;T,,
(10)
where C; = y (kc,). Hence, eq. (IO) predicted a quadratic temperature dependence for the quantity 6. This was in agreement with our experimental
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
73
2
7.9 kV 3
8.2 kV
2
Fig. 5
results (see figs. 3 and 4). The linearity of the J2 versus TT plot also indicated that y was independent of TT at constant values of V,, r,, p and m. If y were temperature dependent then the 6’ versus TT curves would have exhibited curvature. The parameter y was given by the expression
Y. C. CHEN
AND D. N. SEIDMAN
8.8 kV
2
Fig. 5 where Cl is the value of the slope determined from the a2 versus T, plot. Physically, the quantity C; is a measure of the temperature dependence of the resolution of the FIM. The method for obtaining y from the experimental data is given in Appendix A where it was shown that dy/y was -7 x 10e2. There are two points to be made with respect to the values obtained for y. The first point is that y was greater than unity. This fact implied that the
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION
9.2 kV
2
9.6 kV
2
MICROSCOPE
75
3
Fig. 5. This series of six FIM micrographs shows the effect of increasing electric field on resolution. The specimen was held at a constant tip temperature of 11 “K and the voltage increased to BIV (9.6 kV).
imaging ion was not accommodated fully to the temperature of the tip when it was ionized at x,. The published calculated curves of atomic resolution [e.g., Mtiller’), Mtiller and Tsongls)] as a function of tip radius assumed a priori that the imaging ion was accommodated to the temperature of tip [i.e., y= 1 in eq. (lo)], hence the calculated values of 6 were smaller than
Y.C.CHEN
AND D.N.SEIDMAN
Atom 3
Atom 5
I
8.0
I
8.2
I
I
6.4 8.6 Specimen
I
I
I
6.6 9.0 9.2 Voltage (kV)
I
I
9.4
9.E
Fig. 6. The square of the image diameter (P) of an atom as a function of the applied voltage at a constant tip temperature of 11 “K. The numbered atoms correspond to those indicated in fig. 5.
the measured ones. The second point is that the values of y were lower for the specimens imaged in the 10% neon-helium imaging mixture by 18% and 14% for specimens 2 and 3 respectively (see table 2). This decrease in y indicated that the imaging helium atoms were better accommodated to TT when they were ionized at X, in the presence of neon, as opposed to the case when pure helium was used as the imaging gas. McKinney and Brennerzl), Miillera2), and Mtiller, McLane and Panitz2s) have demonstrated that in the presence of the imaging field, both neon and helium are adsorbed on tungsten surfaces. Thus, in some manner the adsorbed neon increases the ability of helium to accommodate to TT. It is not clear that the adsorbed neon acted simply as an intermediate collision partner 24) in this process, for if this were the case then one would expect helium to be better accommodated in the presence of pure helium since helium is also adsorbed at BIV. From eq. (9) we can interpret physically y to be a measure of the average
ON THE ATOMIC
Summary
RESOLUTION
of relevant parameters
Average slope (C’S) x 10z8 (cmY”K-x) Range of (C’Z) x Ws (cmVK-.‘) Average value of 2/ (CI),,~ (A) Range of 1/ (Cl),,, (A) Average tip radius * rT (A) VT(kK) 1? -Tatal magnification (MT)
ION MICROSCOPE
77
dstermined from atom diameter measurements
._ Parameter
OF A FIELD
Specimen 2
PUlX helium 6.8 4.6 to 1O 2.2 I .6 to 2.x
Specimen 3
10% Neon-helium 5.6 2.5 to 7.8
2.1 2.3 to 3.2 307 307 14.3 14.3 6.5 5.3 ---.1. .--. -.. 3,s x 10” 3.5 x 106
Pure helium 7.2 3.6 to II 2.3 1.7 to 2.8 281 11.5 1.3 ~--, 4.1 x 106
10% Neon-helium 6.2 2.9 to 8.9 3.0 2.5 to 3.6 281 11.5 6.2 .---.-- ll-“. I 4.1 x 106
* This tip radius was determined by averaging the radii as measured from the (011) to the (112),(121),andthe(I11),andfromthe(101)tothe(112),(111),and the(211).
temperature (T,) of an imaging atom when it is ionized at x,. If this interpretation of y is correct, then we may use y to determine an integrated value of the accommodation coefficient (d) of the imaging gas atoms. That is, this value of ti represents an integration over any hopping motion of the gas atom that may have occurred on the surface of the metal prior to its ionizatian at x,. The classical value of the thermal accommodation coefficient (a) [e.g., see Kaminsky25)], in terms of temperature, is given by the expression
where T& is the equivalent mean temperature of the gas atoms just before they strike the surface of the specimen. The kinetic energy of a gas atom (molecule) in a field ET is expressed by (13) where a is the polarizability, 11the permanent dipole moment, and Tg is the average thermal temperature of the gas atom (molecule). The quantity Tes was found by equating eq. (13) to + kT,,. For the case of helium (or neon) where p is zero, and under the condition that we were dealing with an integrated value of the quantity u, eq, (12) became 04)
Y. CCFIEN
78
Ah’D D. N. SEIDMAN
This equation implied that at constant values of TF and y the value of d decreased with increasing tip temperature. Fig. 7 shows a plot of B feq. (f4)] as a functions of IT; for two different values ofy and T,, with E,=4.5 V (Al-” and a=02 x 10Mz4cm3 (polarizability of helium)*. This figure shows clearly that the value of d was controlled mainly by y and not Tg. For example, when T,=20”K and T,=70”K the value of d decreased from 0.92 to 0.89 when y
cl.41 0
1
’
IO 20
’
30
’
40
’
’
50 60
Tip Temperature
’
70
’
30
t
90
f*K)
Fig. 7. The calculated mean accommodation coefficient (8) as a function of tip temperature for two different values of y. See section 5.1 for a discussion of the model on which the calculations were based.
was increased from 5.3 to 7.3. This change is large compared with the effect of changing T, at constant y and T,. (e.g., compare curve IA with IB or IIA with IIB). It is clear that eq. (14) breaks down when ‘CT= (T, + (xE2/3k)ly (i.e., when &=O), and therefore is not applicabfe when T, approaches req. Hence, the absolute values of S predicted by this model may be incorrect, but the model does have the feature that it explains in a reasonable physical manner the quantities that controt the temperature dependence of the resotution. 5.2. THE LIMITING
RESOLUTION
AT 0°K
Experimental values of 6’ at T,=O”K
were obtained by a linear extra-
* The values of Tprr were very high; e.g., for r, =- 35”K, the value of Teq was 1122°K. Thus, even though y was greater than unity theequivalent temperature of the gas atom had decreased by ?OOO”Kprior to its ionization at x,~.
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
19
polation of the 6’ versus TT curves (see figs. 3 and 4) to T,=O”K. The values of this parameter are listed as (C,),,, in table 1 for the atoms measured on specimen 2. An examination of table 2 shows that the resolution at 0°K a a range of values. For example, the quantity 6 (0°K) [S (0°K) = (C, )&I h d ranged from 1.7 A to 3.6 A for atoms on specimen 3. These measured values of 6 (0°K) were large compared with the calculated quantity {[4hljrT/ (2~zel/,))]~}, which was 0.64 A in this particular case. Hence, in this example the experimental contribution to 6 at TT = O’Kwas 2.6 to 5.6 times greater than what should be obtained if only the Heisenberg Uncertainty Principle, and the broadening due to the radial projection limited the resolution at 0°K. It is shown in Appendix B that the zero point energy of an atom was not large enough for the root mean square displacement of an atom from its equilibrium position to make a significant contribution to 6 at 0°K. Therefore, the origin of the discrepancy between the observed and the calculated value of 6(O”K) must lie with another source. An obvious weakness of the present model for atomic resolution is the neglect of the exact electric field distribution above a surface atom in the region of space at x, and beyond, where the imaging atom is ionized. The dimensions of this volume are determined by the rate at which the electric field strength at x, falls off with distance relative to the electric field strength required for ionization of the imaging gas atom. Miiller and Tsongls) have made a similar point, and state simply that ionization may occur from a disk of diameter 6,, which is determined by the diameter of the imaging gas atom (molecule) and the diameter of the region at x, from which electron transfer to a surface atom is possible. The quantity 6, is statistical*, and hence must be added to eq. (1) quadratically. Thus, we obtain the expression for resolution (15) Eq. (15) implied that 6, was in the range - (1.5 to 3.5) A for the image diameters measured on specimen 3, which meant that 6, made a more significant contribution to 6 than did the quantity [4kfir,/(meVT)“]“. Hence, any further developments in the theory of resolution will require an accurate knowledge of the electric field distribution above a given atom. 5.3. THE EFFECT OF ELECTRIC FIELD STRENGTH ON ATOMICRESOLUTION The most striking result of this experiment (section 4.3) was the fact that the image diameter of an atom (see fig. 6) went through a maximum with * Miiller and I’son@) have added incorrectly 60 to the expression for 6 in a linear fashion.
PO
Y.C.CREN
AND
D.N.SiifDMAH
increasing ET fYT>_The generat shape of the cumes in fig_ 6 is similar to the ion current versus V, curves for ~~d~vjd~al atomic planes of tungsten cunsisting of -20
atoms {Chen and Seidmarta”)].
These curves are also similar
to the computed curves of Van Eekelen”?) for the ion current from a single atom as a function of ET. The physical origin of the maximum in Van Eekelen”s model lies with the fact that the equilibrium concentration of imaging gas atoms at the tip surface (cT) goes through a maximum as ET is increased toward the best image field. The ion current is proportional to the product of c=, and the probability of ionization’“), and since the latter quantity is an increasing func.t~on of E, the ion currentfrcrm a singfe atom as a function of i5& exhibits a max~~~um* The image diameter of an atom shoufd, by similar physical reasoning, therefore also exhibit a maximum in its size as a function of E,. That is, the: image size of an atom will increase initially as both ET and cT increase, This will happen because of the coupling betwee* the maximum in the quantity c”~ and the probability of ionization of a gas atom. This can be visualized readily by considering the shape of an equipotential above an atom. The effect of an increase in ET is to make this equipotential more sharply peaked, and since the probability of ionization increases as & increases, the diameter of the region (~ar~~le~ to the surface of the tip) above an atom in which ionization can occur is increased_ ~~mu~taneous~y, c, is increasing so that the ion current from each annufar element of area centered about the center of the image of an atom wifl increase as & increases, A further increase of ET resutts in a decrease of +, and therefore the ion current in each annu&r element of area will decrease, and thus the apparent image diameter wili decrease. As examination of eq. (1) under the condition of variable ET (and therefore variable cT) shows that the existing theory of resolution is not capable of explaining the above effect since it neglects the dependence of cT on Er. Hence, this tatter dependence must be kept in mind if one tries to make any conclusions regarding the effect of E, on S,
(1) The temperature dependence of rhe atomic resolution of a FM was measured between f 1 “K and 62°K at constant ET. The image size of a~1atom in this temperature range had a quadratic dependence on the tip temperature, (2) A simple analysis of the data showed that the imaging gas atoms were nor fully accommodated thermally to ‘-r, when they were ionized at x,. (3) The addition of 10% neon to the helium imaging gas mixture increased the degree of thermal accommodation of a helium atom prior to its ionization at x,.
ON THE ATOMIC RESOLUTION OF A FIELD ION MICROSCOPE
(4) The experimental
contribution
to the atomic
resolution
81
at 0°K
was
larger than the value predicted by the contributions from the Heisenberg Uncertainty Principle, and the broadening due to the radial projection effect. This disparity was attributed to the nature of the local electric field distribution at x, above an individual atom, which has not been accounted for in a quantitative form in the existing theory of resolution. (5) The image diameter of an atom as a function of ET (at constant TT) exhibited a maximum at a value of ET which was lower than the best image field. The presence of this maximum was at variance with the existing theory of atomic resolution. An explanation of this effect was given in terms of the field dependence of the equilibrium concentration of imaging gas atoms at the tip surface computed by Van Eekelenz7). Acknowledgments We wish to thank Mr. are also due to Professor useful discussions, to Drs. useful comments regarding for pointing out an error
R. Whitmarsh for his technical assistance. Thanks R. W. Balluffi for continuous encouragement and S. S. Brenner and J. T. McKinney for a number of the manuscript, and to Dr. H. A. M. van Eekelen in section 5.3 of the original manuscript. Appendix A
ON
THE FRACTIONAL
A.l.
The fractional
ERRORS
INVOLVED
IN DETERMINING
7 AND 6
error in y
The method for determining the quantity y from the temperature dependence of the image size of an atom is given in this section. The quantity I as measured on the motion analyzer was subject to three different magnification factors. Thus, the slope C; of eq. (7) is given by Cl; /M: where Ci is the slope of the curves presented in figs. 3 and 4. The overall magnification factor M: is given by M; = M;M;M;
,
WI
where MI is the linear magnification factor (R/PrT) of the FIM, M, is the linear magnification factor (i) of the Super-Farron lens, and M3 is the linear magnification factor (7.52) of the motion analyzer. Therefore, the quantity y is given by the expression
(AZ)
x2
Y, C. CHEN
AND D. N. SEIDMAN
where we have written y in terms of both V, and ET to indicate the dependence of y on the relevant physical parameters. In terms of the magnification (n/l:) the following rather simple practical expression for y was obtained (A31 The fractionat
error in y is shown readily
to be given by the expression
which was=7 x 10e2 for the experiments reported in this paper. The slopes (Ci) of the I2 versus T, curves were determined by the standard least squares method.
The error involved in the determination of the absoiute (S) was shown to be given by the expression
image diameter
where ali the terms have been defined previously. For the data in tables I and 2 the upper limit to (dfi/S) was 0.25. The term which made the largest contribution to the fractional error in 6 was, of course, (dr,/r,). For the specimens used in the present experiments this quantity was -0.20. Appendix B THE EFFECT OF THE ZERO POINT ENERGY AND THERMAL ~IBRAT~~~S
ON
RESOL.UTION
The effect of the zero point energy of the lattice on the root mean square displacement of an atom from its equi~ibr~llm position at T,=O”K is calculated in this section. The model chosen for the solid was a Debyc model. For this model it can be shown [e.g., see Wertheim’s)] that the root mean square displacement ofan atom from its equilibrium position (at T@ 0,) is given by
where M is the mass of an atom in the specimen, and 0, is the Debye temperature (379°K for tungsten}. At 7’=O”K this equation yielded a value of
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
83
[(AX)‘]’ of 0.04 A. It is also interesting to note that the temperature dependence predicted by the Debye model was rather weak. For example, at T=8O”K the value of [(Ax)‘]+ was only - 1.2 times greater than the value of 0°K. Hence, the thermal vibrations of the lattice did not affect the image diameter in the entire temperature range of our observations. The above calculations only apply rigorously to the atoms in the bulk of a crystal. Both theory and experiment indicate that the root mean square displacement of an atom at the surface of a crystal is larger than in the interior, and that [(AX)‘]” IS ’ greater in a direction normal to the surface than it is in a direction parallel to the surface. Fortunately, the existing experimental data in the literature indicated that this effect was not large enough to make a significant contribution to 6 at T,=O”K or any temperature of interest to the field ion microscopist. For example, Jones, McKinney and Webb2a) found that for atoms on the (111) surface of silver, for T$ O,, the quantity [(Ax)*]* was between 1.8 and 2.2 larger than for atoms in the bulk. For this particular plane the mean square displacements parallel and perpendicular to the surface were equal to within 25%.
References 1) E. W. Mtiller, Z. Naturforsch. lla (1956) 87. 2) E. W. Miiller, J. Appl. Phys. 27 (1956) 474. 3) Y. C. Chen and D. N. Seidman, in: 16th Field Emission Symposium, Pittsburgh, Pennsylvania, 1969 ; Y. C. Chen and D. N. Seidman, Cornell University Materials Science Center Report #1406(1970). 4) M. J. Southon, Ph. D. Thesis, University of Cambridge (1963). 5) M. J. Southon, in: Field Ion Microscopy (Plenum Press, New York, 1968) pp. 24-26. 6) R. H. Good and E. W. Mtiller, in: Handbuch derPhysik, Vol. 21 (Springer, Berlin, 1956) pp. 176-231. 7) E. W. Mtiller, in: Advances in Electronics and Electron Physics (Academic Press, New York, 1960) pp. 116-123. 8) R. Gomer, in: Field Emission and Field Ionization (Harvard Univ. Press, Cambridge, Mass., 1961) pp. 3742, p. 95. 9) M. Benjamin and R. 0. Jenkins, Proc. Roy. Sot. (London) A 176 (1940) 262. 10) G. Richter, Z. Physik 119 (1942) 406. 11) E. W. Mtiller, Z. Physik 120 (1943) 270. 12) F. Ashworth, in: Advances in Electronics (Academic Press, New York, 1951) p. 1. 13) R. Gomer, J. Chem. Phys. 20 (1952) 1772. 14) D. J. Rose, J. Appl. Phys. 27 (1956) 215. 15) E. W. Mtiller and T. T. Tsong, in: Field Zon Microscopy (American Elsevier, New York, 1969) pp. 4&48. 16) D. N. Seidman, R. M. Scanlan, D. L. Styris and J. W. Bohlen, J. Sci. Instr. 2 (1969) 473. 17) D. N. Seidman and R. M. Scanlan, Cornell University Materials Science Center Report #1376 (1970). 18) S. H. Robertson and D. N. Seidman, J. Sci. Instr. 1(1968) 1244. 19) R. M. Scanlan, D. L. Styris, D. N. Seidman and D. G. Ast, Cornell University Materials Science Center Report #1159 (1969).
84
Y.C.CHEN
AND D.N.SEIDMAN
20) N. R. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966) p. 62. 21) J. T. McKinney and S. S. Brenner, in: 16th Field Emission Symposium, Pittsburgh, Pennsylvania, 1969. 22) E. W. Miiller, Quart Rev. (Chem. Sot. London) 23 (1969) 177. 23) E. W. Miiller, S. B. McLane and J. A. Panitz, Surface Sci. 17 (1969) 430. 24) 0. Nishikawa and E. W. Miiller, J. Appl. Phys. 35 (1964) 2806. 25) M. Kaminsky, Atomic and ionic Impact Phenomena on Metal Surfaces (Academic Press, New York, 1965) p. 56. 26) Y. C. Chen and D. N. Seidman, in: 16th Field Emission Symposium. Pittsburgh, Pennsylvania, 1969; Y. C. Chen and D. N. Seidman, Cornell Materials Science Center Report 1490 (1971). 27) H. A. M. van Eekelen, Surface Sci. 21(1970) p. 21. 28) G. K. Wertheim, Miissbauer Effeect: Principles and Applications (Academic Press, New York, 1964) ch. 5. 29) E. R. Jones, J. T. McKinney and M. B. Webb, Phys. Rev. 151(1966) 476.
SCIENCE 26 (1971) 61-84 0 North-Holland
ON THE ATOMIC
RESOLUTION Y. C. CHEN
Publishing Co.
OF A FIELD ION MICROSCOPE*
and D. N. SEIDMAN
Department of Materials Science and Engineering, Cornell University, Ithaca, New York 14850, U.S.A.
Received 17 August 1970 The effects of tip temperature ( TT), imaging gas mixture, and electric field strength on the resolution of the field ion microscope have been measured quantitatively. The image diameter of individual tungsten atoms had a quadratic dependenceon TT between 11 “K and 62°K. This result was in agreement with the theory of resolution which predicted that the transverse component of the velocity of the imaging ion limited the resolution of the FIM. The data also indicated that the imaging gas atom was not accommodated fully to TT when it was ionized. The degree of thermal accommodation of the imaging gas atom prior to its ionization was increased as a result of imaging in a 10% neon-helium gas mixture as opposed to imaging with pure helium gas. The extrapolated value of the experimental contribution to the image size at 0°K was larger than the value predicted by the existing theory. This discrepancy was attributed to the neglect, in the existing model of resolution, of the nature of the local electric field strength in the region of space where the imaging atom was ionized. The dependence of image size of an atom at constant TT exhibited a maxi~lum. This result was at variance with the existing theory of atomic resolution. An explanation of this effect in terms of the field dependence of the equilibrium concentration of gas atoms at the surface of the tip was given.
1. Introduction
the most crucial steps in the development of the field ion microscope (FIM) occurred when Mfiller l,z) realized that the high atomic resolution of the FIM was dependent strongly upon accommodating thermally the imaging gas atoms to the temperature of a specimen which was maintained at a value of 78°K or less. The existing calculation of the atomic resolution [see section 2, eq. (1)] of a FXM indicated that the resolution was a function of 6 physical parameters. To date, there have been no measurements of the atomic resoiution where these parameters were varied under controlled experimental conditions. Thus, the present paper represents the first quantitative measurements of the effect of the tip temperature, imaging gas mixture, and electric field strength on the resolution of a FIM3). One of
* Research supported by the Advanced Research Projects Agency through the Materials Science Center at Cornell University. 61
62
Y. C. CHEN
AND
D. N. SEIQMAN
2. Basic principles and experimental program The only calculation of the atomic resolution that has been performed is a measure of the “extent to which the spatial modulation of the ion current density above the tip is preserved in the ion beam which reaches the phosphorescent screen of the FIM” [Southon4*5)]. This calculation of atomic resolution has been carried out by a number of authors [e.g., see Good and Muhers), Mi.iller7) and Gomer *)]. The derivations were all based on an analogous calculation of the resolution of a field electron microscope image which was first considered by Benjamin and JenkinsQ) and eIaborated on by Richter lo). Miiller rr), Ashworth la), Gomer Ia) and RoseI”). The expression for the atomic resolution (6) is given* by [see Miiller and Tsong’s l5) eq. (2.91)]
where h (A= h/2x) is PIanck’s constant, /I is a compression factor which depends on the exact shape of the tip, r, is the local radius of curvature, nz is the mass of the imaging gas ion, e the charge on the ion, ET the electric field strength at the tip, i is a geometric factor, and .sT the transverse thermal energy associated with the imaging gas atom when it is ionized at the surface which is x, A above the real surface of the FIM specimen. The quantity S in eq. (I) is not the resolution as defined in the standard manner by the Rayleigh criterion, instead it is the minimum image diameter of an atom on the viewing screen divided by the magnification of the FIM. The first term inside the braces in eq. (1) was obtained by considering the broadening caused by the Heisenberg Uncertainty Principle, and the radial projection effect. Each of these two effects makes an equal contribution to this first term. The second term inside the braces resulted from a consideration of the spreading of the ion beam as a result of the transverse thermal energy (~.r) of the imaging gas ion. For the purpose of analysis (see section 5) it is useful to rewrite eq. (1) in terms of the voltage on the tip (VT). Since V, and &are related through the relationship (2) eq. (1) becomes (3)
* We have omitted the quantity SOfrom Miiller and Tsong’s eq. (2.91) temporarily as this quantity will be discussed in section 5.2.
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
63
It is important to note that the resolution as defined by eq. (1) is a function of 6 physical parameters [i.e., 6 = S(@, r,, ET, 5, m, sT)] and that eq. (2) does not remove the dependence of 6 on rT. The physical reason for this dependence of 6 on rT is the fact that the angular momentum (p,) of the imaging ion is a function of rT (i.e., p,=mr,v,). This is seen from the basic expression for resolution [Gomers)] 6 = (r&,/R) D = 4o,r#(m/2e&)“,
(4)
where D is the measured distance on the screen which corresponds to a distance 6 at a surface which is at x,, and v, is the transverse component of the velocity of the imaging ion. Rewriting eq. (4) in terms of p, one obtains the expression 6 = 4pJ(1/2emI’#.
(5)
Therefore, an increase in pw results in a larger value of 6, and hence a decrease in resolution. The present paper presents the results of an investigation of the dependence of atomic resolution of the FIM under the folIowing controlled experimental conditions: (1) Variable TT at constant values of the parameters jI, rT, ET, 5 and m employing helium as the imaging gas (PHe= 1.87 x 10e3 Torr). The variable T, implied that sy was the only physical variable in eq. (3). (2) The above experiment was repeated on the exact same specimen, but the imaging gas was changed to a 10% neon-helium mixture (PNe =0.21 x I 0m3 Torr and Pn,= 1.87 x lop3 Torr). (3) Variable ET and cT at constant values of the parameters p, rT, i and m. The quantity sT is a function of ET (see section 5.1) hence it is impossible to vary ET independent of E=. 3. Experimental
techniques
The FIM employed was a stainless steel microscope which was evacuated by a 15 liter set-’ Vacion pump and a titanium sublimation pump. The microscope was operated statically at a background pressure in the range (1 to 10) x 10-l’ Torr. The required pressure of helium gas was obtained by diffusing helium through a quartz membrane, while the neon was leaked into the FIM from a bottle of research grade (99.995 min. vol. “//,) gas. The temperature of the FIM specimen was controlled by a continuous transfer liquid helium cryostat [Seidman et al. ‘“)I. The temperature stability of the specimen using this technique was L-O.01 “K and the specimen
64
Y. C. CHEN
AND D.N.
SEIDMAN
temperature* was monitored continuously with a miniature platinum resistance thermometer which was mounted very close to the specimen. This temperature monitoring thermometer was calibrated against a second resistance thermometer which was placed at the position where the specimen resided normally. The specimen voltage was controlled with a zero to 30 kV Spellman power supply which was filtered by a O.Ol’A Sorensen ripple filter. The output voltage of the power supply was monitored with a Hewlett-Packard 4-place digital voltmeter used in conjunction with a specially constructed 100: 1 voltage dividing network. The specimens were field evaporated to a final end form employing a zero to 4 kV pulse amplifier [Robertson and Seidmanrs)]. The images were recorded on Kodak Tri-X film employing a f/O.87 Super-Farron lens. For a given experiment, all images were recorded on the same roll of film and developed simultaneously. Hence, the film shrinkage was identical for each frame employed for the atomic image diameter measurements reported in section 4. The image diameter measurements were made on the negatives with the aid of a Vanguard motion analyzer. This motion analyzer [Scanlan et al. ‘s)] enlarged a micrograph to a 14.6 cm x 19 cm image. In addition, the analyzer had X and Y cross hairs which were used for determining the image diameter in two different directions. The fractional standard deviation for the measurement of an image diameter was N 0.025 on the enlarged 35 mm negative. All specimens were prepared from Westinghouse or Sylvania 0.0127 cm diam tungsten wire. The tips were electro-polished in a 10 wt% NaOH solution at 1 to 3 V dc. 4. Experimental
results
4.1. THE EFFECT OF THE TIP TEMPERATURE (TT) ON THE ATOMIC RESOLUTION The decrease in the atomic resolution in preceding from a TT of 11 “K to 62”K, at a constant value of Vr, is apparent from a visual comparison of the 5 FlM micrographs shown in fig. 1. These 5 patterns are of a region near the (110) pole. The specimen used for this experiment was first field evaporated to a final end form? at 36°K and then imaged at 11 “K at a best image voltage (BIV) of 10.2 kV. Subsequently, T, was increased to 23, 36, 49 and 62°K without changing Vr. Next, the image was recorded photographically at each * Seidmanand ScanlanlT) haveshown that the heating effects due to thermal radiation, the imaging gas, and the thermoelastic effect were negligible, hence the measured temperatures should correspond to the tip temperatures rather well. t The final end form was determined by the condition that the field evaporation rate was constant over the entire surface of the tip.
ON THE ATOMIC
RESOLUTION
OF A FIELD
11” K
ION MICROSCOPE
65
23” K
36” K Fig. 1
of the above TT’s. Note that the increase in the image size of an atom was large enough such that 2 atoms which were distinguishable at 11 “K merged completely into one another at 23°K (see region 1 in fig. 1). It should also be noted that the decrease in resolution caused 3 well resolved atoms to almost become a continuous linear chain of contrast (see region 4 in fig. 1).
66
Y. C. CHEN
49” K
AND D. N. SEIDMAN
62” K
Fig. 1. A series of five FIM patterns of the (011) region taken at 11, 23, 36, 49, and 62°K respectively. The numberedregions show the effect of increasing temperature on resolution. The voltage on the specimen was 10.2 kV (the BIV at 11 “K). The image at 36°K was accidentally recorded at a voltage which was - 200 V greater than 10.2 kV.
The decrease in resolution with increasing TT was also apparent in regions 2 and 3 offig. 1. The above observations were made quantitative by measuring the image diameter* of an atom (2) as a function of TT with all the five remaining variables held at fixed values. In order to distinguish among 3 possibilities for the dependence of I on T, (i.e., I versus TT,l2versus TT,and I3 versus T,.) the data were processed in the following manner: (1) The data were fitted to each of the above 3 possibilities by the method of least squares; (2) The goodness of fit in each of the above cases was tested using the Coefficient of Multiple Determinationao). The latter test indicated that for the 30 atoms measured in the vicinity of the (011) pole, 70% fitted a l2 versus T, plot, while 18% fitted a 1 versus TT plot, and the remainder a l3 versus TT plot. The results for 5 different atoms near the central (011) pole imaged with helium gas are shown in figs. 2 and 3. The numbered curves in fig. 3 correspond to the numbered atoms shown in the 5 micrographs in fig. 2. Note that the curves in fig. 3 were displaced from one another for clarity, hence the scale of I2 is arbitrary * The image diameter (I) was the measured image size of an atom as observed on the motion analyzer. The quantity I was used in the analysis rather than 6, since the fractional error in I is smaller than the fractional error in S (see Appendix A2).
67
3
5
1
2
4
4
11” K
23” K Fig. 2
Y. C. CHEN
4
AND D. N. SEIDMAN
36” K
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
I59
Fig. 2. A second series of fiveFIM patterns taken at If, 23,36,49, and 62%. and at a tip v&age of 10.2 kV. The centrat pole is the (011) pole. The numbered atoms are the ones whose image diameters were measured as a function of temperature {see fig. 3).
in this figure. The above results demonstrated that I had a quadratic dependence on Tr between I1 “K and 62°K. 4.2. THE EFFECTOF IMAGING GAS MIXTURE ON ATOMIC RESOLUTION The experiment described in section 4.1 was performed on 2 additional specimens which were field evaporated initially at 36°K. For each specimen the images were fust recorded with helium as the imaging gas, and then repeated with a 10% neon-helium gas mixture at identical vaiues of the parameters p, r,, PM,VT, and [. The graph of 1’ versus TT for some of the atoms examined on specimen number 2 in a 10% neon-heIium gas mixture is shown in fig. 4. Once again it was found that f had a quadratic dependence on Tf *. It was also found that the slope of the E2versus TTgraph was less in the 10% neon-helium gas mixture than it was in the pure helium gas at identical values of the parameters /I, r,, VT and 5. The quantity 6 for 15 atoms at 1I “K and 62°K is given in table 1 along with the intercept at TT=O"K [(C,),,,] and the slope (Cl) of the 6’ versus T*curve for each atom. It can be seen that at 11 “K the value of 6 ranged from (2.3 to 3.3) A, whiie at 62°K the range has increased to (2.6 to 3.8) A. The range in 6 for these same atoms imaged in * These data were also tested for goodness of fit employing the CoefEcient of Multiple Determination. This test indicated that of the 30 atoms examined, 86x, fitted a i2 versus TT plot, while 6 ?< fitted a I3 versus T;f plot, and 8 % a fversus TT plot.
-JO
Y. C. CHEN
AND D. N. SEIDMAN
3
Fig. 3. The square of the image diameter {P) of an atom versus the tip temperature (TT). The scale of I” is arbitrary since the curves were displaced from one another for clarity. The numbered curves correspond to the numbered atoms in fig. 2.
helium gas at 11 “K was (1.9 to 2.9) A, and at 62°K it was (2.7 to 3.4) ,&. Thus, the range of image sizes in helium gas was smaller, but within our experimental error for determining fi (see Appendix A). 4.3. THE EFFECT OF ELECTRIC
FIEfA
STRENGTH (VT) ON THE ATOMfC
RESOLUTION
The effect of & on the atomic resolution was determined for a specimen which was initially field evaporated to a final end form at 36°K and then examined at 11 “K. The quantity V, was varied from 7.9 kV to 9.6 kV in 6 approximately equal increments while the parameters p, r,, m and r, were maintained invariant. The series of 6 micrographs in fig. 5 shows the results of the above experiment. The image at the 9.6 kV is at BIV. The value of I for 25 atoms in this region was measured,9 and the results for 5 of these atoms are shown in fig_ 6 where the value of l2 (arbitrary units) is plotted as a function of Kr. The atoms numbered 1 to 5 in fig. 5 correspond to the labeled
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION
MICROSCOPE
71
8’
vs TT (OK) 10% Ne-He Imaging Gas Mixture
v, = 14.3kV
I IO
0
Fig. 4.
I
I
I
I
30 40 50 20 Tip Temperature (OK)
I
60
The square of the image diameter (12) of an atom versus the tip temperature constant voltage of 14.3 kV in a 10% neon-helium gas mixture.
at a
curves in fig. 6. It is seen that the dependence of Z2 on VT has no simple functional form. Instead, the image size exhibited a maximum value and then decreased with increasing V, up to the BIV. It is also noted that the maximum value of l2 occurred at different values of VT for different atoms. 5. Discussion 5.1. DEPENDENCE
OF
THE
ATOMIC
RESOLUTION
ON
TIP TEMPERATURE
(TT)
The principal experimental result obtained (see section 4.1) was that the quantity 1 had a quadratic dependence on TT. The expression for resolution [eq. (3)] under the condition of variable TT and constant values of rT, VT, p and m reduced to P=C1+C2ET, (6) where Ci = [4@rT/(2 mevT>‘] , (7) and C, = [16,G2r~/eVT]. (8)
Y. C. CHEN AND D. N. SEIDMAN
72
TABLE Pertinent
parameters
for temperature imaged
(Cl),,,
4 5 6 7 8 9 IO II 12 13 14 15
Range
x 101”
(cm”)
number
of
parameter
in
I
dependence of resolution a 10 “ANeeHe gas mixture (Cz’) x 101s (cm”cK-l)
for specimen
6(ll”K)* (A)
number
two
6(62”K) * (A)
7.3 7.8 8.3 5.1 8.3 6.6 9.6 7.1 8.3 6.2 6.5 5.3 10.4 6.7 6.0
4.9 5.6 2.7 2.5 3.6 6.6 7.1 4.4 6.4 4.1 7.5 6.6 6.9 7.9 7.2
2.8 2.9 2.9 2.3 2.9 2.7 3.2 2.8 3.0 2.6 2.7 2.5 3.3 2.7 2.6
3.2 3.4 3.2 2.6 3.2 3.3 3.7 3.1 3.5 3.0 3.3 3.1 3.8 3.4 3.2
5.1 to 10.4
2.5 to 7.9
2.3 to 3.3
2.6 to 3.8
* The absolute values of 6 were determined by dividing I by MT (total magnification of system). The fractional error in 6 had an upper limit of 0.25 in the present experiments (see Appendix
A).
If the imaging atoms were accommodated fully to the surface of the specimen when they were ionized at x, then sT would be equal to kT, from the equipartition theorem of thermal energy. Since the degree of thermal accommodation was unknown, we employed the following relationship between cT. and Tl xT = y(kT,) = + mu:, (9) where y was experimental it was ionized fully when it above model
a dimensionless parameter which was determined from the data. If the imaging atom was accommodated fully to TT when at X_ then y would be equal to one. If it was not accommodated was ionized, then y would be greater than unity. Employing the and eq. (9) the expression for the resolution [eq. (6)] became 6’ = C, + C;T,,
(10)
where C; = y (kc,). Hence, eq. (IO) predicted a quadratic temperature dependence for the quantity 6. This was in agreement with our experimental
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
73
2
7.9 kV 3
8.2 kV
2
Fig. 5
results (see figs. 3 and 4). The linearity of the J2 versus TT plot also indicated that y was independent of TT at constant values of V,, r,, p and m. If y were temperature dependent then the 6’ versus TT curves would have exhibited curvature. The parameter y was given by the expression
Y. C. CHEN
AND D. N. SEIDMAN
8.8 kV
2
Fig. 5 where Cl is the value of the slope determined from the a2 versus T, plot. Physically, the quantity C; is a measure of the temperature dependence of the resolution of the FIM. The method for obtaining y from the experimental data is given in Appendix A where it was shown that dy/y was -7 x 10e2. There are two points to be made with respect to the values obtained for y. The first point is that y was greater than unity. This fact implied that the
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION
9.2 kV
2
9.6 kV
2
MICROSCOPE
75
3
Fig. 5. This series of six FIM micrographs shows the effect of increasing electric field on resolution. The specimen was held at a constant tip temperature of 11 “K and the voltage increased to BIV (9.6 kV).
imaging ion was not accommodated fully to the temperature of the tip when it was ionized at x,. The published calculated curves of atomic resolution [e.g., Mtiller’), Mtiller and Tsongls)] as a function of tip radius assumed a priori that the imaging ion was accommodated to the temperature of tip [i.e., y= 1 in eq. (lo)], hence the calculated values of 6 were smaller than
Y.C.CHEN
AND D.N.SEIDMAN
Atom 3
Atom 5
I
8.0
I
8.2
I
I
6.4 8.6 Specimen
I
I
I
6.6 9.0 9.2 Voltage (kV)
I
I
9.4
9.E
Fig. 6. The square of the image diameter (P) of an atom as a function of the applied voltage at a constant tip temperature of 11 “K. The numbered atoms correspond to those indicated in fig. 5.
the measured ones. The second point is that the values of y were lower for the specimens imaged in the 10% neon-helium imaging mixture by 18% and 14% for specimens 2 and 3 respectively (see table 2). This decrease in y indicated that the imaging helium atoms were better accommodated to TT when they were ionized at X, in the presence of neon, as opposed to the case when pure helium was used as the imaging gas. McKinney and Brennerzl), Miillera2), and Mtiller, McLane and Panitz2s) have demonstrated that in the presence of the imaging field, both neon and helium are adsorbed on tungsten surfaces. Thus, in some manner the adsorbed neon increases the ability of helium to accommodate to TT. It is not clear that the adsorbed neon acted simply as an intermediate collision partner 24) in this process, for if this were the case then one would expect helium to be better accommodated in the presence of pure helium since helium is also adsorbed at BIV. From eq. (9) we can interpret physically y to be a measure of the average
ON THE ATOMIC
Summary
RESOLUTION
of relevant parameters
Average slope (C’S) x 10z8 (cmY”K-x) Range of (C’Z) x Ws (cmVK-.‘) Average value of 2/ (CI),,~ (A) Range of 1/ (Cl),,, (A) Average tip radius * rT (A) VT(kK) 1? -Tatal magnification (MT)
ION MICROSCOPE
77
dstermined from atom diameter measurements
._ Parameter
OF A FIELD
Specimen 2
PUlX helium 6.8 4.6 to 1O 2.2 I .6 to 2.x
Specimen 3
10% Neon-helium 5.6 2.5 to 7.8
2.1 2.3 to 3.2 307 307 14.3 14.3 6.5 5.3 ---.1. .--. -.. 3,s x 10” 3.5 x 106
Pure helium 7.2 3.6 to II 2.3 1.7 to 2.8 281 11.5 1.3 ~--, 4.1 x 106
10% Neon-helium 6.2 2.9 to 8.9 3.0 2.5 to 3.6 281 11.5 6.2 .---.-- ll-“. I 4.1 x 106
* This tip radius was determined by averaging the radii as measured from the (011) to the (112),(121),andthe(I11),andfromthe(101)tothe(112),(111),and the(211).
temperature (T,) of an imaging atom when it is ionized at x,. If this interpretation of y is correct, then we may use y to determine an integrated value of the accommodation coefficient (d) of the imaging gas atoms. That is, this value of ti represents an integration over any hopping motion of the gas atom that may have occurred on the surface of the metal prior to its ionizatian at x,. The classical value of the thermal accommodation coefficient (a) [e.g., see Kaminsky25)], in terms of temperature, is given by the expression
where T& is the equivalent mean temperature of the gas atoms just before they strike the surface of the specimen. The kinetic energy of a gas atom (molecule) in a field ET is expressed by (13) where a is the polarizability, 11the permanent dipole moment, and Tg is the average thermal temperature of the gas atom (molecule). The quantity Tes was found by equating eq. (13) to + kT,,. For the case of helium (or neon) where p is zero, and under the condition that we were dealing with an integrated value of the quantity u, eq, (12) became 04)
Y. CCFIEN
78
Ah’D D. N. SEIDMAN
This equation implied that at constant values of TF and y the value of d decreased with increasing tip temperature. Fig. 7 shows a plot of B feq. (f4)] as a functions of IT; for two different values ofy and T,, with E,=4.5 V (Al-” and a=02 x 10Mz4cm3 (polarizability of helium)*. This figure shows clearly that the value of d was controlled mainly by y and not Tg. For example, when T,=20”K and T,=70”K the value of d decreased from 0.92 to 0.89 when y
cl.41 0
1
’
IO 20
’
30
’
40
’
’
50 60
Tip Temperature
’
70
’
30
t
90
f*K)
Fig. 7. The calculated mean accommodation coefficient (8) as a function of tip temperature for two different values of y. See section 5.1 for a discussion of the model on which the calculations were based.
was increased from 5.3 to 7.3. This change is large compared with the effect of changing T, at constant y and T,. (e.g., compare curve IA with IB or IIA with IIB). It is clear that eq. (14) breaks down when ‘CT= (T, + (xE2/3k)ly (i.e., when &=O), and therefore is not applicabfe when T, approaches req. Hence, the absolute values of S predicted by this model may be incorrect, but the model does have the feature that it explains in a reasonable physical manner the quantities that controt the temperature dependence of the resotution. 5.2. THE LIMITING
RESOLUTION
AT 0°K
Experimental values of 6’ at T,=O”K
were obtained by a linear extra-
* The values of Tprr were very high; e.g., for r, =- 35”K, the value of Teq was 1122°K. Thus, even though y was greater than unity theequivalent temperature of the gas atom had decreased by ?OOO”Kprior to its ionization at x,~.
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
19
polation of the 6’ versus TT curves (see figs. 3 and 4) to T,=O”K. The values of this parameter are listed as (C,),,, in table 1 for the atoms measured on specimen 2. An examination of table 2 shows that the resolution at 0°K a a range of values. For example, the quantity 6 (0°K) [S (0°K) = (C, )&I h d ranged from 1.7 A to 3.6 A for atoms on specimen 3. These measured values of 6 (0°K) were large compared with the calculated quantity {[4hljrT/ (2~zel/,))]~}, which was 0.64 A in this particular case. Hence, in this example the experimental contribution to 6 at TT = O’Kwas 2.6 to 5.6 times greater than what should be obtained if only the Heisenberg Uncertainty Principle, and the broadening due to the radial projection limited the resolution at 0°K. It is shown in Appendix B that the zero point energy of an atom was not large enough for the root mean square displacement of an atom from its equilibrium position to make a significant contribution to 6 at 0°K. Therefore, the origin of the discrepancy between the observed and the calculated value of 6(O”K) must lie with another source. An obvious weakness of the present model for atomic resolution is the neglect of the exact electric field distribution above a surface atom in the region of space at x, and beyond, where the imaging atom is ionized. The dimensions of this volume are determined by the rate at which the electric field strength at x, falls off with distance relative to the electric field strength required for ionization of the imaging gas atom. Miiller and Tsongls) have made a similar point, and state simply that ionization may occur from a disk of diameter 6,, which is determined by the diameter of the imaging gas atom (molecule) and the diameter of the region at x, from which electron transfer to a surface atom is possible. The quantity 6, is statistical*, and hence must be added to eq. (1) quadratically. Thus, we obtain the expression for resolution (15) Eq. (15) implied that 6, was in the range - (1.5 to 3.5) A for the image diameters measured on specimen 3, which meant that 6, made a more significant contribution to 6 than did the quantity [4kfir,/(meVT)“]“. Hence, any further developments in the theory of resolution will require an accurate knowledge of the electric field distribution above a given atom. 5.3. THE EFFECT OF ELECTRIC FIELD STRENGTH ON ATOMICRESOLUTION The most striking result of this experiment (section 4.3) was the fact that the image diameter of an atom (see fig. 6) went through a maximum with * Miiller and I’son@) have added incorrectly 60 to the expression for 6 in a linear fashion.
PO
Y.C.CREN
AND
D.N.SiifDMAH
increasing ET fYT>_The generat shape of the cumes in fig_ 6 is similar to the ion current versus V, curves for ~~d~vjd~al atomic planes of tungsten cunsisting of -20
atoms {Chen and Seidmarta”)].
These curves are also similar
to the computed curves of Van Eekelen”?) for the ion current from a single atom as a function of ET. The physical origin of the maximum in Van Eekelen”s model lies with the fact that the equilibrium concentration of imaging gas atoms at the tip surface (cT) goes through a maximum as ET is increased toward the best image field. The ion current is proportional to the product of c=, and the probability of ionization’“), and since the latter quantity is an increasing func.t~on of E, the ion currentfrcrm a singfe atom as a function of i5& exhibits a max~~~um* The image diameter of an atom shoufd, by similar physical reasoning, therefore also exhibit a maximum in its size as a function of E,. That is, the: image size of an atom will increase initially as both ET and cT increase, This will happen because of the coupling betwee* the maximum in the quantity c”~ and the probability of ionization of a gas atom. This can be visualized readily by considering the shape of an equipotential above an atom. The effect of an increase in ET is to make this equipotential more sharply peaked, and since the probability of ionization increases as & increases, the diameter of the region (~ar~~le~ to the surface of the tip) above an atom in which ionization can occur is increased_ ~~mu~taneous~y, c, is increasing so that the ion current from each annufar element of area centered about the center of the image of an atom wifl increase as & increases, A further increase of ET resutts in a decrease of +, and therefore the ion current in each annu&r element of area will decrease, and thus the apparent image diameter wili decrease. As examination of eq. (1) under the condition of variable ET (and therefore variable cT) shows that the existing theory of resolution is not capable of explaining the above effect since it neglects the dependence of cT on Er. Hence, this tatter dependence must be kept in mind if one tries to make any conclusions regarding the effect of E, on S,
(1) The temperature dependence of rhe atomic resolution of a FM was measured between f 1 “K and 62°K at constant ET. The image size of a~1atom in this temperature range had a quadratic dependence on the tip temperature, (2) A simple analysis of the data showed that the imaging gas atoms were nor fully accommodated thermally to ‘-r, when they were ionized at x,. (3) The addition of 10% neon to the helium imaging gas mixture increased the degree of thermal accommodation of a helium atom prior to its ionization at x,.
ON THE ATOMIC RESOLUTION OF A FIELD ION MICROSCOPE
(4) The experimental
contribution
to the atomic
resolution
81
at 0°K
was
larger than the value predicted by the contributions from the Heisenberg Uncertainty Principle, and the broadening due to the radial projection effect. This disparity was attributed to the nature of the local electric field distribution at x, above an individual atom, which has not been accounted for in a quantitative form in the existing theory of resolution. (5) The image diameter of an atom as a function of ET (at constant TT) exhibited a maximum at a value of ET which was lower than the best image field. The presence of this maximum was at variance with the existing theory of atomic resolution. An explanation of this effect was given in terms of the field dependence of the equilibrium concentration of imaging gas atoms at the tip surface computed by Van Eekelenz7). Acknowledgments We wish to thank Mr. are also due to Professor useful discussions, to Drs. useful comments regarding for pointing out an error
R. Whitmarsh for his technical assistance. Thanks R. W. Balluffi for continuous encouragement and S. S. Brenner and J. T. McKinney for a number of the manuscript, and to Dr. H. A. M. van Eekelen in section 5.3 of the original manuscript. Appendix A
ON
THE FRACTIONAL
A.l.
The fractional
ERRORS
INVOLVED
IN DETERMINING
7 AND 6
error in y
The method for determining the quantity y from the temperature dependence of the image size of an atom is given in this section. The quantity I as measured on the motion analyzer was subject to three different magnification factors. Thus, the slope C; of eq. (7) is given by Cl; /M: where Ci is the slope of the curves presented in figs. 3 and 4. The overall magnification factor M: is given by M; = M;M;M;
,
WI
where MI is the linear magnification factor (R/PrT) of the FIM, M, is the linear magnification factor (i) of the Super-Farron lens, and M3 is the linear magnification factor (7.52) of the motion analyzer. Therefore, the quantity y is given by the expression
(AZ)
x2
Y, C. CHEN
AND D. N. SEIDMAN
where we have written y in terms of both V, and ET to indicate the dependence of y on the relevant physical parameters. In terms of the magnification (n/l:) the following rather simple practical expression for y was obtained (A31 The fractionat
error in y is shown readily
to be given by the expression
which was=7 x 10e2 for the experiments reported in this paper. The slopes (Ci) of the I2 versus T, curves were determined by the standard least squares method.
The error involved in the determination of the absoiute (S) was shown to be given by the expression
image diameter
where ali the terms have been defined previously. For the data in tables I and 2 the upper limit to (dfi/S) was 0.25. The term which made the largest contribution to the fractional error in 6 was, of course, (dr,/r,). For the specimens used in the present experiments this quantity was -0.20. Appendix B THE EFFECT OF THE ZERO POINT ENERGY AND THERMAL ~IBRAT~~~S
ON
RESOL.UTION
The effect of the zero point energy of the lattice on the root mean square displacement of an atom from its equi~ibr~llm position at T,=O”K is calculated in this section. The model chosen for the solid was a Debyc model. For this model it can be shown [e.g., see Wertheim’s)] that the root mean square displacement ofan atom from its equilibrium position (at T@ 0,) is given by
where M is the mass of an atom in the specimen, and 0, is the Debye temperature (379°K for tungsten}. At 7’=O”K this equation yielded a value of
ON THE ATOMIC
RESOLUTION
OF A FIELD
ION MICROSCOPE
83
[(AX)‘]’ of 0.04 A. It is also interesting to note that the temperature dependence predicted by the Debye model was rather weak. For example, at T=8O”K the value of [(Ax)‘]+ was only - 1.2 times greater than the value of 0°K. Hence, the thermal vibrations of the lattice did not affect the image diameter in the entire temperature range of our observations. The above calculations only apply rigorously to the atoms in the bulk of a crystal. Both theory and experiment indicate that the root mean square displacement of an atom at the surface of a crystal is larger than in the interior, and that [(AX)‘]” IS ’ greater in a direction normal to the surface than it is in a direction parallel to the surface. Fortunately, the existing experimental data in the literature indicated that this effect was not large enough to make a significant contribution to 6 at T,=O”K or any temperature of interest to the field ion microscopist. For example, Jones, McKinney and Webb2a) found that for atoms on the (111) surface of silver, for T$ O,, the quantity [(Ax)*]* was between 1.8 and 2.2 larger than for atoms in the bulk. For this particular plane the mean square displacements parallel and perpendicular to the surface were equal to within 25%.
References 1) E. W. Mtiller, Z. Naturforsch. lla (1956) 87. 2) E. W. Miiller, J. Appl. Phys. 27 (1956) 474. 3) Y. C. Chen and D. N. Seidman, in: 16th Field Emission Symposium, Pittsburgh, Pennsylvania, 1969 ; Y. C. Chen and D. N. Seidman, Cornell University Materials Science Center Report #1406(1970). 4) M. J. Southon, Ph. D. Thesis, University of Cambridge (1963). 5) M. J. Southon, in: Field Ion Microscopy (Plenum Press, New York, 1968) pp. 24-26. 6) R. H. Good and E. W. Mtiller, in: Handbuch derPhysik, Vol. 21 (Springer, Berlin, 1956) pp. 176-231. 7) E. W. Mtiller, in: Advances in Electronics and Electron Physics (Academic Press, New York, 1960) pp. 116-123. 8) R. Gomer, in: Field Emission and Field Ionization (Harvard Univ. Press, Cambridge, Mass., 1961) pp. 3742, p. 95. 9) M. Benjamin and R. 0. Jenkins, Proc. Roy. Sot. (London) A 176 (1940) 262. 10) G. Richter, Z. Physik 119 (1942) 406. 11) E. W. Mtiller, Z. Physik 120 (1943) 270. 12) F. Ashworth, in: Advances in Electronics (Academic Press, New York, 1951) p. 1. 13) R. Gomer, J. Chem. Phys. 20 (1952) 1772. 14) D. J. Rose, J. Appl. Phys. 27 (1956) 215. 15) E. W. Mtiller and T. T. Tsong, in: Field Zon Microscopy (American Elsevier, New York, 1969) pp. 4&48. 16) D. N. Seidman, R. M. Scanlan, D. L. Styris and J. W. Bohlen, J. Sci. Instr. 2 (1969) 473. 17) D. N. Seidman and R. M. Scanlan, Cornell University Materials Science Center Report #1376 (1970). 18) S. H. Robertson and D. N. Seidman, J. Sci. Instr. 1(1968) 1244. 19) R. M. Scanlan, D. L. Styris, D. N. Seidman and D. G. Ast, Cornell University Materials Science Center Report #1159 (1969).
84
Y.C.CHEN
AND D.N.SEIDMAN
20) N. R. Draper and H. Smith, Applied Regression Analysis (Wiley, New York, 1966) p. 62. 21) J. T. McKinney and S. S. Brenner, in: 16th Field Emission Symposium, Pittsburgh, Pennsylvania, 1969. 22) E. W. Miiller, Quart Rev. (Chem. Sot. London) 23 (1969) 177. 23) E. W. Miiller, S. B. McLane and J. A. Panitz, Surface Sci. 17 (1969) 430. 24) 0. Nishikawa and E. W. Miiller, J. Appl. Phys. 35 (1964) 2806. 25) M. Kaminsky, Atomic and ionic Impact Phenomena on Metal Surfaces (Academic Press, New York, 1965) p. 56. 26) Y. C. Chen and D. N. Seidman, in: 16th Field Emission Symposium. Pittsburgh, Pennsylvania, 1969; Y. C. Chen and D. N. Seidman, Cornell Materials Science Center Report 1490 (1971). 27) H. A. M. van Eekelen, Surface Sci. 21(1970) p. 21. 28) G. K. Wertheim, Miissbauer Effeect: Principles and Applications (Academic Press, New York, 1964) ch. 5. 29) E. R. Jones, J. T. McKinney and M. B. Webb, Phys. Rev. 151(1966) 476.