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Quantitative determinations of the temperature dependence of diffusion phenomena in the FIM
43
Surface Science 246 (1991) 43-49 North-Holland
Quantitative of diffusion Marcello
determinations of the temperature phenomena in the FIM *
dependence
F. Lovisa and Gert Ehrlich * *
Materials Research L-aboratory and Department Urbana, IL 61801, USA
of Materials
Received
6 August
29 July 1990; accepted
for publication
Science and Engineering,
University
of Illinois at Urbana-Champaign,
1990
In quantitative studies with the field ion microscope of atomic processes in an ordinary thermal environment, observations must be made intermittently, after cooling the sample to a low temperature, in order to avoid possible perturbation by the image field. Such measurements are bedeviled by a well known problem - changes in the atomic configurations under study during the transient periods, while the surface is either cooling down from or heating up to the desired temperature. Techniques for quantitatively accounting for such transient effects have been devised, and will be presented for two types of measurements - the determination of surface diffusivities, and also the distance distribution function of adatoms. For the former, the standard procedure is to measure the temperature profile of the sample support, and then to calculate corrections by an iterative method drawing on the mean-square displacements actually observed. A more reliable technique is to separately determine displacements as the diffusion interval is made infinitesimally small, and then to subtract this correction term from the measured values. This method is also appropriate for corrections to the observed distance distribution function. The calculational procedures required are outlined and then tested using Monte Carlo simulations designed to establish the contributions of multiple jumps in atomic diffusion on W(110).
1. Introduction Over the last few decades, the field ion microscope (FIM) has yielded considerable qualitative information about atomic behavior at surfaces in an ordinary thermal environment. Quantitatively reliable information, however, has been more difficult to attain. Imaging in the FIM is done at low temperatures, T = 20 K, to achieve the best resolution, and more importantly, to minimize any effects of the strenuous imaging conditions upon the surface under study. The standard procedure for examining phenomena occurring at elevated temperatures has therefore been to heat the sample in the absence of imaging fields. At the end of this high temperature treatment the sample is al-
* Supported by the Department DE-AC02-76ER01198. * * Hallene University Scholar.
0039-6028/91/$03.50
of Energy
under
contract
0 1991 - Elsevier Science Publishers
lowed to cool, and the atomic arrangement achieved is then imaged under the normal low temperature conditions. Quenching experiments of this type have long been known to be subject to inherent limitations [l]. The structure examined at low temperature is not necessarily indicative of that produced at the high temperature, rather, it may reflect conditions at some lower temperature at which atomic motion has slowed sufficiently to freeze in the surface under study. This annealing can be of great importance in trying to establish the effect of temperature upon equilibrium phenomena [2]. In past measurements of cluster thermodynamics, for example, various indirect methods had to be employed to minimize annealing during the transient period [3]. Heating and cooling transients also enter quantitative measurements of diffusion phenomena, but there has been rather less emphasis upon this problem. Here we will therefore outline some techniques that are effective in correcting diffusion data, obtained on resistively heated sam-
B.V. (North-Holland)
M. F. Louisa. G. Ehriich / Temperarum dependence o~d~~usio~ phenomena m the FIM
44
pies, for changes that may take place during ing and cooling transients.
2. The experimental
heat-
situation
In an ideal diffusion experiment, the sample is heated instantaneously to the preset diffusion temperature T,, at time t,, and, at the end of the diffusion interval Aht, at t,, cools immediately to the imaging temperatures The diffusivity D of the atom can therefore be derived by repeated observation of the displacement Ax, using the standard relation (Ax’)
= 20 At,
(11
where the diffusivity D is a function of the temperature, given in terms of the prefactor L>, and the activation energy En by D = Do exp( - E,/kT).
(2)
The reality, however, may be rather different. In the standard diffusion experiment 141, the single crystal specimen is spot-welded onto a hairpin loop, equipped with potential leads, as illustrated in fig. 1. The temperature of the support loop is controlled by electronics based on a double Kelvin bridge, which compares the potential Vs across the potential leads with that across a
1
Timer start -
tTRA ----a
-
tTRD
_
standard resistor, and adjusts the heating current so as to bring the resistance R, of that section of the support loop sampled by the potential leads into agreement with a preset value. A schematic of the temperature at the sample surface during a measurement cycle is sketched in fig. 1. At time t, the heating cycle is initiated, and the heating current supplied by the control unit raises the temperature rapidly, until at t, the temperature 7;, set for the experiment is reached and the timer is started. There is still the possibility, of course, that, depending upon the detailed setting of the control unit, there may be some overshoot of the temperature. Once this ascending transient period, TRA, is over, the temperature of the sample is assumed to remain sensibly constant at its selected value. At time t,, the heating current is interrupted by the timer at the end of the diffusion interval At, and the sample cools, primarily by conduction through the leads, until at time t, it reaches the temperature set for imaging. For a fixed diffusion temperature, the duration of the cooling transient, TRD, depends entirely upon the mounting arrangement, that is, upon the diameter, length. as well as upon the material properties of the single crystal specimen and the support loop. The characteristics of the ascending transient TRA will, in addition, also depend upon the magnitude of the heating current. In such a cycle, an atom may already be able to make jumps at some temperature 7;) below the set diffusion value TD. The mean-square displacement actually derived by repeated observation of the atomic displacement is therefore related to the diffusivity through the more complicated equations ( Ax~)~~)’ = (A.x’)~~~ = 2 D,/*‘exp( lo
10
11
13
Time
keckds)
Fig. 1. Schematic of temperature cycle in diffusion experiments. During the diffusion interval At, from I, to f2, electronic controller maintains temperature constant. Starting transients are shown for two different heating currents I, and I,.
+ 20 At+
+ (Ax”)‘l, - E,/kT
20,
+ (A.x~)~~” ) dt
‘“exp( - E,,‘kT)
/ ‘2
(31
dt.
(4)
At low diffusion temperatures Tt,, the diffusion interval will generally be of such long duration that contributions from the transients (Ax*)~~~
M. F. Lovisa, G. Ehrlich / Temperature dependence of diffwion phenomena in the FIM
and will and then
( Ax*)~~” can be neglected. This happy state not prevail at high diffusion temperatures, the observed mean-square displacements must be corrected in some way.
n
45
~~
$$~~
~' 4-
::
ii I
3. Determination
of diffusivities
There is a standard technique, devised for correcting diffusivities for contributions from transients, provided these contributions are not too large. From the measured value of (Ax*)‘~~ we derive an effective diffusivity D(O), and then from measurements at different temperatures, an effective activation energy E$‘. The temperature profile of the loop is then measured, and from this, a quantitative plot of the sample temperature, such as sketched in fig. 1, is derived. A corrected value of the diffusivity is then obtained from eq. (4), by numerically evaluating the contribution of the transients TRA and TRD, using the preliminary measurevalues 0:“’ and E,(‘) . After correcting ments at other temperatures, a new set of diffusion parameters L$” and EM’ is derived, and this procedure is continued until the estimated parameters converge. This technique has been used successfully to arrive at quantitative values for the diffusivities of various metal atoms on W(211) [4], but is, nevertheless, limited. It is safe only if the transient corrections are small. Even then, the actual temperature profile of the surface during the cycle must be known, and such determinations are not easy. Finally, more information than just the diffusivities may be desired. For example, in order to properly explore the role of different atomic jump processes contributing to diffusion, it is often desirable to measure the displacement distribution function P,~, which gives the probability that after a diffusion interval At an atom will end up at a distance x from its origin [5]. Correcting experimentally obtained estimates of this distribution for transient effects in the manner outlined above for D would be quite difficult and uncertain. We have therefore devised a very direct method to compensate for heating and cooling transients in diffusion measurements. Atomic displacements are measured, as usual, at the end of a heating
;’
,^ x3a v
I
1 I
t
a
2-
l-
o-
Fig. 2. Effect of different temperature transients upon the mean-square displacement ( Ax~)~“’ - ( Ax*)~~ estimated from zero-time experiments. Monte Carlo simulations of 100000 observations compare these estimates with the true value (Ax*)~D, and that measured at the end of the cycle, (Ax*)~~‘.
cycle, and these displa~ments may include contributions that arise during both ascending and descending transients. In addition to these standard observations of atomic displacements, however, an equal number of observations is made under exactly the same conditions, except that the diffusion interval in the electronic timer is set to zero. In these zero-time determinations, the electronics begin heating the sample at time to; as soon as the set temperature TD of the diffusion experiment is reached, however, the heating current is interrupted, and the cooling transient begins. The displacement observed after imaging therefore very directly gives the contribution of atomic movement during the interval while the temperature is not at its set value. From repetition of this transient measurement we therefore obtain the sum (Ax*)~~
= (Ax~)~~
+ (Ax~)~~~,
(5)
which is needed to deduce the true mean-square displacement (Ax”)‘~ from the mean-square displacement (Ax~}~O~ observed in the usual diffusion experiment. In order to demonstrate that once observations
M. F. Louisa, G. Ehrlich / Temperature dependence of diffusion phenomena in the FIM
46
of transient contributions are in hand, reliable values of the diffusivity can be derived in this way, we have done extensive Monte Carlo simulations [6). Results obtained for diffusion in one-dimensional motion are shown in fig. 2. In this case the mean-square displacement (Ax2)r1, for the true distribution is kept constant ((Ax’)~~ = 2), but the transient effects are varied until (Ax*)~~ becomes comparable with (Ax’)‘“. The latter situation is extreme unlikely; in a real experiment, values of ( Ax’)~~/( Ax*)r~ > 0.3 are rare. Nevertheless, the mean-square displacement ( Ax’)~“’ (A.x2}rR derived by subtracting transient measured in a zero-time experiment agree well with the input values, and this procedure appears reliable for determining diffusivities.
4. Determination displacements
8 N 4
True Distribution
Transient Distribution
of the distribution of
The overall procedure for measuring the distribution of displacements in an ideal experiment is straightforward. Just as when ascertaining the diffusivity, the sample is heated to the predetermined diffusion temperature T,, for a time interval At. After the temperature has fallen and imaging conditions are established, the displacement of the atom is measured. This sequence is repeated to obtain a plot of the number of observations in which an atom is found at different positions on the surface. To ascertain the corrections required under actual conditions due to diffusion during the heating transients, these measurements are repeated, but with the diffusion interval set to zero. The methods for actually correcting the observed dist~bution for transients are somewhat complicated, and are most easily demonstrated for diffusion in one dimension. 4.1. One-dimensional
r
5 x104 4
Combined Distribution
3 MX
2 1 0 i ~rprrlm -6 -4
m
-2
0 x
2
munlmuma 4
6
Fig. 3. Monte Carlo simulation of the distance distribution at different stages of the temperature cycle, obtained by 1500 observations. At the end of the complete cycle, the true distribution has been broadened considerably by redistribution during transient.
the origin
after a diffusion
interval
At is given by
(51
distribution functions x
In diffusion along a line, in which atoms can jump to a nearest-neighbor site at the rate (Y, to a second-nearest neighbor site at the rate 8, and to a third-nearest neighbor site at the rate y, the probability p, of finding the atom at a distance x from
j=
E
-*
I,(2Pt)1.~-,,-,k(2at),
(6)
where Z,,(u) is the modified Bessel function of order n and argument U. To ascertain the extent to which these different types of jumps contribute, the distribution of displacements measured for
M. F. Louisa, G. Ehrlich / Temperature dependence of dtj”ion phenomena in the FIM
atoms at the temperature T,,is compared with the distribution in eq. (6), and the various rates are adjusted until the best fit of the experiments is obtained. In an actual experiment, redistribution of the atom may occur during the heating transient. The extent to which this is significant is immediately evident from the zero-time experiment. Exactly how diffusion during the transient period affects the distribution of displacements is seen from the simulated experiment in fig. 3. The true distribution gives the probability of finding an atom at the distance x from the origin. It is obtained in these simulations by noting the position of the atom at the moment the temperature reaches T,,, and again after the time At, when the heating current is interrupted. Since we are always measuring displacements, we can consider diffusion at T,,as a process that redistributes the probability from its initial value of unity at x = 0, over all the available positions. The probability of a displacement w, measured in zero-time experiments can be viewed in much the same way as a redistribution of the probability from the origin over the other sites. In the analogue of a real experiment, when the position of the atom is noted only at the beginning and end of a cycle, the distribution comes about through the combination of diffusive motions in both intervals. The number N, at a displacement Table 1 Effect of transients upon probabilities (Ax*)~~
Nk =pkN. This number Nk will be redistributed during transients, so that at the end of the cycle actual number M, found at a displacement x units of the nearest-neighbour spacing) will given by linear equations of the form M_,= . . . +wrN_, + w,JV_, + IV-&,
M_, = .a.+w,N_,+w,N_,+w,,N_,+w_,N,
+w_,N,+w_,N,+w_,N,+ ---, M,,= ...+w3N_, +w~N_~+w~N-~ +w&, +w_,N, +w_,N,+w_,N,+ ***, Ml= ...+w~N_~+w&~+w~N_,+w~N~ +w,,N,+w_,N~+w_~N~+ .... I$= ... +w,N_,+w,N_,+w,N_, +w,No +w,N,+w,,N,+w_,N,+ ..-, Q-9
Ax=2
Ax=3
Ax=4
True
Estim.
True
Estim.
True x10
Estim. x10
True x10
Estim. x10
True x10
Estim. x10
0.507 0.507 0.505 0.506 0.503
0.509 0.504 0.505 0.508 0.500
0.148 0.146 0.150 0.149 0.148
0.147 0.147 0.149 0.148 0.152
0.54 0.54 0.53 0.53 0.54
0.53 0.54 0.52 0.54 0.52
0.26 0.27 0.26 0.26 0.26
0.26 0.27 0.26 0.24 0.24
0.13 0.14 0.13 0.14 0.13
0.13 0.14 0.13 0.13 0.15
0.451 0.421 0.360 0.316 0.279
probabilities 0.162 0.166 0.177 0.179 0.178
(7) the the (in be
+ w_,N, + w_~N~ + w_JVz + w-& + ***,
in 1D
Ax=1
Uncorrected 0.3 0.5 1.0 1.5 2.0
x after diffusion at the set temperature T,,will be redistributed during the transients in accord with the probability measured during the zero-time experiments, to give the combined distribution in fig. 3. If the total number of observations in an experiment is N, then the number of times an atom is found at a displacement k after diffusion at TD is just
Probabilities Ax=0
0.3 0.5 1.0 1.5 2.0
41
0.61 0.65 0.75 0.84 0.90
0.30 0.32 0.37 0.41 0.46
0.15 0.17 0.19 0.14 0.25
M. F. Louisa, G. Ehrlich / Temperature
48
which can be written Mk=
5
more concisely
dependence of diffusion phenomena Table 2 Effect of transients
as
wk-JY?,.
4.2. Two-dimensional
motion:
distributions
on bee
upon estimated
diffusion
parameters
(Ax*)~~/(Ax’)~D
Estimated (Ax*)
P/a
v/a
0.15
2.00 2.29
0.21 0.19
0.10 0.09
Corrected Uncorrected
0.25
2.00 2.50
0.20 0.17
0.10 0.09
Corrected Uncorrected
0.50
2.07 3.00
0.22 0.16
0.10 0.08
Corrected Uncorrected
0.75
2.07 3.53
0.19 0.10
0.12 0.08
Corrected Uncorrected
1.00
2.00 3.95
0.15 0.10
0.09 0.07
Corrected Uncorrected
2.00 0.3-2.0
0.20 0.10
0.10 0.05
k=-m
In actual experiments, we have available from the normal observations the number of atoms M/, found at the different displacements k at the end of a cycle, and from the zero-time measurements we know the redistribution probabilities w, during the transients. Unknown is the true distribution, described by the number of atoms N,,, at a displacement m after diffusion entirely at the temperature T,. Because of the redistribution during the transients, the number of relations in eq. (9) will likely be larger than the number of unknowns, and the linear equations are therefore best solved for N, by singular value decomposition [7]. That this scheme works can be easily tested using simulated data for 100000 diffusion intervals. The results are shown in table 1. In the simulations the conditions during the actual diffusion interval At were kept constant, with (Ax’)‘” = 2, but the contributions of the transients were increased in steps until (Ax’)~“/(Ax~)‘~ = 1. In the simulations the true distribution at the end of the diffusion interval at TD is immediately available, and is listed in table 1 under the heading “true”. These values are compared with the estimated probabilities derived from eq. (9) using the distribution at the end of the complete cycle and also the zerotime correction. The importance of the transients effects can be appreciated by comparing the values of the uncorrected probabilities, from observations at the end of the complete temperature cycle with the true values. In table 2 we separately list the jump-rate ratios derived by fitting the displacement distribution function eq. (6) to the estimated displacement distribution. The results found in this way are in excellent agreement with the input values.
in the FIM
Input: Input:
true transient
in 1 D
able for such surfaces. Consider diffusion in which jumps take place at the rate OLbetween nearest neighbors and at the rate /3 between second-nearest neighbor sites along the close-packed directions; jumps along the Cartesian coordinates occur at the rates h and p. These processes contribute to the mean-square displacement along x (in units of u/2, where a is the lattice spacing) according to (Ax’)
= 4(” + 2*p + 2*h + 2*p)t.
(10)
The standard procedure in deriving values of the jump rates is to compare the distribution of displacements p,,,, with that predicted by a stochastic model involving the unknown jump rates as parameters, namely p,,,.=exp[-4(a+/I+h+p)t] x
E m=-m
x’(.\+,),*-
x
,)I
F I=
E n-pm
4wu.,-,,,2-
E ,=_m tz
1,(2/V
2,(24
nl+,l-2,(2~r)
-m
(110) xI,,(4Xt)I,,(W).
In extending these correction procedures to two dimensions we confine ourselves to the bee (I 10) plane, as there is considerable diffusion data avail-
(11)
Again it is the true distribution, unaffected by temperature transients, which is needed for this comparison, but it is MA,,, the number of times an
ki. F. Louisa, G. Ehrlich / Temperature dependence of diffuon Table 3 Effect of transients
upon estimated
diffusion
parameters
in 2D
(A~*)~~/(Ax~)ru
Estimated
0.15
1.98 2.23
0.21 0.19
0.10 0.09
0.05 0.04
Corrected Uncorrected
0.25
2.03 2.46
0.19 0.17
0.15 0.13
0.06 0.05
Corrected Uncorrected
0.50
2.01 2.97
0.21 0.12
0.13 0.05
0.04 0.03
Corrected Uncorrected
0.75
2.03 3.22
0.16 0.157
0.14 0.09
0.06 0.01
Corrected Uncorrected
1.00
2.08 4.02
0.26 0.17
0.00 0.06
0.06 0.04
Corrected Uncorrected
2.00 0.3-2.0
0.20 0.10
0.10 0.05
0.05 0.025
Input: Input:
true transient
atom is observed at a displacement k, 1 at the end of a temperature cycle, which is available from experiment. This is related to the true distribution values N,,,,n by the set of linear equations
wd=
c Cwk-m.l-nNm.n; m
(12)
phenomena in the FIM
49
ment, with 1500 observations. Just as in the onedimensional simulations, the conditions during the true diffusion interval are held constant while the contributions from transients are increased. The diffusion parameters derived from the different cycles are listed in table 3. It is clear that even under adverse conditions, when the mean-square displacement during the transients amounts to 3/4 the true value and the number of observations is limited, jump rates in good agreement with the input values can be obtained using the correction methods outlined. 5. Summary In designing a diffusion experiment it is best to make the diffusion intervals large by comparison with the duration of heating or cooling transients. When this is not possible, however, zero-time experiments very directly provide the information necessary to correct for the transients. These require some additional effort, but appreciably extend the range of diffusion conditions that can be successfully explored.
n
here w,,~ gives the redistribution probability to i, j; it is the probability that during the transients the atom will be moved to a position i, j from the origin. Both Mk,, and w,,~ are measurable quantities. The former are found from the observations after a complete heating cycle, the latter are determined from the redistribution during the zerotime experiments. The desired quantities, namely Nm.n can be derived in the usual way by solving these equations using singular value decomposition. Monte Carlo simulations have been done on bee (110) under conditions similar to a real experi-
References 111H. Mykura, in: Molecular
Processes on Solid Surfaces, Eds. E. Draughs, R.D. Gretz and R.I. Jaffee (McGraw-Hill, New York, 1969) p. 129. PI D.W. Bassett and D.R. Tice, Surf. Sci. 40 (1973) 499. 131 K. Stolt, J.D. Wrigley and G. Ehrlich, J. Chem. Phys. 69 (1978) 1151. 141 S.C. Wang and G. Ehrhch, Surf. Sci. 206 (1988) 451. 151 J.D. Wrigley, M.E. Twigg and G. Ehrlich, J. Chem. Phys. 93 (1990) 2885. of the Monte Carlo Method in 161 K. Binder, Applications Statistical Physics (Springer, Berlin, 1984). S.A. Teukolsky and W.T. 171 W.H. Press, B.P. Flannery, Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).
Surface Science 246 (1991) 43-49 North-Holland
Quantitative of diffusion Marcello
determinations of the temperature phenomena in the FIM *
dependence
F. Lovisa and Gert Ehrlich * *
Materials Research L-aboratory and Department Urbana, IL 61801, USA
of Materials
Received
6 August
29 July 1990; accepted
for publication
Science and Engineering,
University
of Illinois at Urbana-Champaign,
1990
In quantitative studies with the field ion microscope of atomic processes in an ordinary thermal environment, observations must be made intermittently, after cooling the sample to a low temperature, in order to avoid possible perturbation by the image field. Such measurements are bedeviled by a well known problem - changes in the atomic configurations under study during the transient periods, while the surface is either cooling down from or heating up to the desired temperature. Techniques for quantitatively accounting for such transient effects have been devised, and will be presented for two types of measurements - the determination of surface diffusivities, and also the distance distribution function of adatoms. For the former, the standard procedure is to measure the temperature profile of the sample support, and then to calculate corrections by an iterative method drawing on the mean-square displacements actually observed. A more reliable technique is to separately determine displacements as the diffusion interval is made infinitesimally small, and then to subtract this correction term from the measured values. This method is also appropriate for corrections to the observed distance distribution function. The calculational procedures required are outlined and then tested using Monte Carlo simulations designed to establish the contributions of multiple jumps in atomic diffusion on W(110).
1. Introduction Over the last few decades, the field ion microscope (FIM) has yielded considerable qualitative information about atomic behavior at surfaces in an ordinary thermal environment. Quantitatively reliable information, however, has been more difficult to attain. Imaging in the FIM is done at low temperatures, T = 20 K, to achieve the best resolution, and more importantly, to minimize any effects of the strenuous imaging conditions upon the surface under study. The standard procedure for examining phenomena occurring at elevated temperatures has therefore been to heat the sample in the absence of imaging fields. At the end of this high temperature treatment the sample is al-
* Supported by the Department DE-AC02-76ER01198. * * Hallene University Scholar.
0039-6028/91/$03.50
of Energy
under
contract
0 1991 - Elsevier Science Publishers
lowed to cool, and the atomic arrangement achieved is then imaged under the normal low temperature conditions. Quenching experiments of this type have long been known to be subject to inherent limitations [l]. The structure examined at low temperature is not necessarily indicative of that produced at the high temperature, rather, it may reflect conditions at some lower temperature at which atomic motion has slowed sufficiently to freeze in the surface under study. This annealing can be of great importance in trying to establish the effect of temperature upon equilibrium phenomena [2]. In past measurements of cluster thermodynamics, for example, various indirect methods had to be employed to minimize annealing during the transient period [3]. Heating and cooling transients also enter quantitative measurements of diffusion phenomena, but there has been rather less emphasis upon this problem. Here we will therefore outline some techniques that are effective in correcting diffusion data, obtained on resistively heated sam-
B.V. (North-Holland)
M. F. Louisa. G. Ehriich / Temperarum dependence o~d~~usio~ phenomena m the FIM
44
pies, for changes that may take place during ing and cooling transients.
2. The experimental
heat-
situation
In an ideal diffusion experiment, the sample is heated instantaneously to the preset diffusion temperature T,, at time t,, and, at the end of the diffusion interval Aht, at t,, cools immediately to the imaging temperatures The diffusivity D of the atom can therefore be derived by repeated observation of the displacement Ax, using the standard relation (Ax’)
= 20 At,
(11
where the diffusivity D is a function of the temperature, given in terms of the prefactor L>, and the activation energy En by D = Do exp( - E,/kT).
(2)
The reality, however, may be rather different. In the standard diffusion experiment 141, the single crystal specimen is spot-welded onto a hairpin loop, equipped with potential leads, as illustrated in fig. 1. The temperature of the support loop is controlled by electronics based on a double Kelvin bridge, which compares the potential Vs across the potential leads with that across a
1
Timer start -
tTRA ----a
-
tTRD
_
standard resistor, and adjusts the heating current so as to bring the resistance R, of that section of the support loop sampled by the potential leads into agreement with a preset value. A schematic of the temperature at the sample surface during a measurement cycle is sketched in fig. 1. At time t, the heating cycle is initiated, and the heating current supplied by the control unit raises the temperature rapidly, until at t, the temperature 7;, set for the experiment is reached and the timer is started. There is still the possibility, of course, that, depending upon the detailed setting of the control unit, there may be some overshoot of the temperature. Once this ascending transient period, TRA, is over, the temperature of the sample is assumed to remain sensibly constant at its selected value. At time t,, the heating current is interrupted by the timer at the end of the diffusion interval At, and the sample cools, primarily by conduction through the leads, until at time t, it reaches the temperature set for imaging. For a fixed diffusion temperature, the duration of the cooling transient, TRD, depends entirely upon the mounting arrangement, that is, upon the diameter, length. as well as upon the material properties of the single crystal specimen and the support loop. The characteristics of the ascending transient TRA will, in addition, also depend upon the magnitude of the heating current. In such a cycle, an atom may already be able to make jumps at some temperature 7;) below the set diffusion value TD. The mean-square displacement actually derived by repeated observation of the atomic displacement is therefore related to the diffusivity through the more complicated equations ( Ax~)~~)’ = (A.x’)~~~ = 2 D,/*‘exp( lo
10
11
13
Time
keckds)
Fig. 1. Schematic of temperature cycle in diffusion experiments. During the diffusion interval At, from I, to f2, electronic controller maintains temperature constant. Starting transients are shown for two different heating currents I, and I,.
+ 20 At+
+ (Ax”)‘l, - E,/kT
20,
+ (A.x~)~~” ) dt
‘“exp( - E,,‘kT)
/ ‘2
(31
dt.
(4)
At low diffusion temperatures Tt,, the diffusion interval will generally be of such long duration that contributions from the transients (Ax*)~~~
M. F. Lovisa, G. Ehrlich / Temperature dependence of diffwion phenomena in the FIM
and will and then
( Ax*)~~” can be neglected. This happy state not prevail at high diffusion temperatures, the observed mean-square displacements must be corrected in some way.
n
45
$$
::
ii I
3. Determination
of diffusivities
There is a standard technique, devised for correcting diffusivities for contributions from transients, provided these contributions are not too large. From the measured value of (Ax*)‘~~ we derive an effective diffusivity D(O), and then from measurements at different temperatures, an effective activation energy E$‘. The temperature profile of the loop is then measured, and from this, a quantitative plot of the sample temperature, such as sketched in fig. 1, is derived. A corrected value of the diffusivity is then obtained from eq. (4), by numerically evaluating the contribution of the transients TRA and TRD, using the preliminary measurevalues 0:“’ and E,(‘) . After correcting ments at other temperatures, a new set of diffusion parameters L$” and EM’ is derived, and this procedure is continued until the estimated parameters converge. This technique has been used successfully to arrive at quantitative values for the diffusivities of various metal atoms on W(211) [4], but is, nevertheless, limited. It is safe only if the transient corrections are small. Even then, the actual temperature profile of the surface during the cycle must be known, and such determinations are not easy. Finally, more information than just the diffusivities may be desired. For example, in order to properly explore the role of different atomic jump processes contributing to diffusion, it is often desirable to measure the displacement distribution function P,~, which gives the probability that after a diffusion interval At an atom will end up at a distance x from its origin [5]. Correcting experimentally obtained estimates of this distribution for transient effects in the manner outlined above for D would be quite difficult and uncertain. We have therefore devised a very direct method to compensate for heating and cooling transients in diffusion measurements. Atomic displacements are measured, as usual, at the end of a heating
;’
,^ x3a v
I
1 I
t
a
2-
l-
o-
Fig. 2. Effect of different temperature transients upon the mean-square displacement ( Ax~)~“’ - ( Ax*)~~ estimated from zero-time experiments. Monte Carlo simulations of 100000 observations compare these estimates with the true value (Ax*)~D, and that measured at the end of the cycle, (Ax*)~~‘.
cycle, and these displa~ments may include contributions that arise during both ascending and descending transients. In addition to these standard observations of atomic displacements, however, an equal number of observations is made under exactly the same conditions, except that the diffusion interval in the electronic timer is set to zero. In these zero-time determinations, the electronics begin heating the sample at time to; as soon as the set temperature TD of the diffusion experiment is reached, however, the heating current is interrupted, and the cooling transient begins. The displacement observed after imaging therefore very directly gives the contribution of atomic movement during the interval while the temperature is not at its set value. From repetition of this transient measurement we therefore obtain the sum (Ax*)~~
= (Ax~)~~
+ (Ax~)~~~,
(5)
which is needed to deduce the true mean-square displacement (Ax”)‘~ from the mean-square displacement (Ax~}~O~ observed in the usual diffusion experiment. In order to demonstrate that once observations
M. F. Louisa, G. Ehrlich / Temperature dependence of diffusion phenomena in the FIM
46
of transient contributions are in hand, reliable values of the diffusivity can be derived in this way, we have done extensive Monte Carlo simulations [6). Results obtained for diffusion in one-dimensional motion are shown in fig. 2. In this case the mean-square displacement (Ax2)r1, for the true distribution is kept constant ((Ax’)~~ = 2), but the transient effects are varied until (Ax*)~~ becomes comparable with (Ax’)‘“. The latter situation is extreme unlikely; in a real experiment, values of ( Ax’)~~/( Ax*)r~ > 0.3 are rare. Nevertheless, the mean-square displacement ( Ax’)~“’ (A.x2}rR derived by subtracting transient measured in a zero-time experiment agree well with the input values, and this procedure appears reliable for determining diffusivities.
4. Determination displacements
8 N 4
True Distribution
Transient Distribution
of the distribution of
The overall procedure for measuring the distribution of displacements in an ideal experiment is straightforward. Just as when ascertaining the diffusivity, the sample is heated to the predetermined diffusion temperature T,, for a time interval At. After the temperature has fallen and imaging conditions are established, the displacement of the atom is measured. This sequence is repeated to obtain a plot of the number of observations in which an atom is found at different positions on the surface. To ascertain the corrections required under actual conditions due to diffusion during the heating transients, these measurements are repeated, but with the diffusion interval set to zero. The methods for actually correcting the observed dist~bution for transients are somewhat complicated, and are most easily demonstrated for diffusion in one dimension. 4.1. One-dimensional
r
5 x104 4
Combined Distribution
3 MX
2 1 0 i ~rprrlm -6 -4
m
-2
0 x
2
munlmuma 4
6
Fig. 3. Monte Carlo simulation of the distance distribution at different stages of the temperature cycle, obtained by 1500 observations. At the end of the complete cycle, the true distribution has been broadened considerably by redistribution during transient.
the origin
after a diffusion
interval
At is given by
(51
distribution functions x
In diffusion along a line, in which atoms can jump to a nearest-neighbor site at the rate (Y, to a second-nearest neighbor site at the rate 8, and to a third-nearest neighbor site at the rate y, the probability p, of finding the atom at a distance x from
j=
E
-*
I,(2Pt)1.~-,,-,k(2at),
(6)
where Z,,(u) is the modified Bessel function of order n and argument U. To ascertain the extent to which these different types of jumps contribute, the distribution of displacements measured for
M. F. Louisa, G. Ehrlich / Temperature dependence of dtj”ion phenomena in the FIM
atoms at the temperature T,,is compared with the distribution in eq. (6), and the various rates are adjusted until the best fit of the experiments is obtained. In an actual experiment, redistribution of the atom may occur during the heating transient. The extent to which this is significant is immediately evident from the zero-time experiment. Exactly how diffusion during the transient period affects the distribution of displacements is seen from the simulated experiment in fig. 3. The true distribution gives the probability of finding an atom at the distance x from the origin. It is obtained in these simulations by noting the position of the atom at the moment the temperature reaches T,,, and again after the time At, when the heating current is interrupted. Since we are always measuring displacements, we can consider diffusion at T,,as a process that redistributes the probability from its initial value of unity at x = 0, over all the available positions. The probability of a displacement w, measured in zero-time experiments can be viewed in much the same way as a redistribution of the probability from the origin over the other sites. In the analogue of a real experiment, when the position of the atom is noted only at the beginning and end of a cycle, the distribution comes about through the combination of diffusive motions in both intervals. The number N, at a displacement Table 1 Effect of transients upon probabilities (Ax*)~~
Nk =pkN. This number Nk will be redistributed during transients, so that at the end of the cycle actual number M, found at a displacement x units of the nearest-neighbour spacing) will given by linear equations of the form M_,= . . . +wrN_, + w,JV_, + IV-&,
M_, = .a.+w,N_,+w,N_,+w,,N_,+w_,N,
+w_,N,+w_,N,+w_,N,+ ---, M,,= ...+w3N_, +w~N_~+w~N-~ +w&, +w_,N, +w_,N,+w_,N,+ ***, Ml= ...+w~N_~+w&~+w~N_,+w~N~ +w,,N,+w_,N~+w_~N~+ .... I$= ... +w,N_,+w,N_,+w,N_, +w,No +w,N,+w,,N,+w_,N,+ ..-, Q-9
Ax=2
Ax=3
Ax=4
True
Estim.
True
Estim.
True x10
Estim. x10
True x10
Estim. x10
True x10
Estim. x10
0.507 0.507 0.505 0.506 0.503
0.509 0.504 0.505 0.508 0.500
0.148 0.146 0.150 0.149 0.148
0.147 0.147 0.149 0.148 0.152
0.54 0.54 0.53 0.53 0.54
0.53 0.54 0.52 0.54 0.52
0.26 0.27 0.26 0.26 0.26
0.26 0.27 0.26 0.24 0.24
0.13 0.14 0.13 0.14 0.13
0.13 0.14 0.13 0.13 0.15
0.451 0.421 0.360 0.316 0.279
probabilities 0.162 0.166 0.177 0.179 0.178
(7) the the (in be
+ w_,N, + w_~N~ + w_JVz + w-& + ***,
in 1D
Ax=1
Uncorrected 0.3 0.5 1.0 1.5 2.0
x after diffusion at the set temperature T,,will be redistributed during the transients in accord with the probability measured during the zero-time experiments, to give the combined distribution in fig. 3. If the total number of observations in an experiment is N, then the number of times an atom is found at a displacement k after diffusion at TD is just
Probabilities Ax=0
0.3 0.5 1.0 1.5 2.0
41
0.61 0.65 0.75 0.84 0.90
0.30 0.32 0.37 0.41 0.46
0.15 0.17 0.19 0.14 0.25
M. F. Louisa, G. Ehrlich / Temperature
48
which can be written Mk=
5
more concisely
dependence of diffusion phenomena Table 2 Effect of transients
as
wk-JY?,.
4.2. Two-dimensional
motion:
distributions
on bee
upon estimated
diffusion
parameters
(Ax*)~~/(Ax’)~D
Estimated (Ax*)
P/a
v/a
0.15
2.00 2.29
0.21 0.19
0.10 0.09
Corrected Uncorrected
0.25
2.00 2.50
0.20 0.17
0.10 0.09
Corrected Uncorrected
0.50
2.07 3.00
0.22 0.16
0.10 0.08
Corrected Uncorrected
0.75
2.07 3.53
0.19 0.10
0.12 0.08
Corrected Uncorrected
1.00
2.00 3.95
0.15 0.10
0.09 0.07
Corrected Uncorrected
2.00 0.3-2.0
0.20 0.10
0.10 0.05
k=-m
In actual experiments, we have available from the normal observations the number of atoms M/, found at the different displacements k at the end of a cycle, and from the zero-time measurements we know the redistribution probabilities w, during the transients. Unknown is the true distribution, described by the number of atoms N,,, at a displacement m after diffusion entirely at the temperature T,. Because of the redistribution during the transients, the number of relations in eq. (9) will likely be larger than the number of unknowns, and the linear equations are therefore best solved for N, by singular value decomposition [7]. That this scheme works can be easily tested using simulated data for 100000 diffusion intervals. The results are shown in table 1. In the simulations the conditions during the actual diffusion interval At were kept constant, with (Ax’)‘” = 2, but the contributions of the transients were increased in steps until (Ax’)~“/(Ax~)‘~ = 1. In the simulations the true distribution at the end of the diffusion interval at TD is immediately available, and is listed in table 1 under the heading “true”. These values are compared with the estimated probabilities derived from eq. (9) using the distribution at the end of the complete cycle and also the zerotime correction. The importance of the transients effects can be appreciated by comparing the values of the uncorrected probabilities, from observations at the end of the complete temperature cycle with the true values. In table 2 we separately list the jump-rate ratios derived by fitting the displacement distribution function eq. (6) to the estimated displacement distribution. The results found in this way are in excellent agreement with the input values.
in the FIM
Input: Input:
true transient
in 1 D
able for such surfaces. Consider diffusion in which jumps take place at the rate OLbetween nearest neighbors and at the rate /3 between second-nearest neighbor sites along the close-packed directions; jumps along the Cartesian coordinates occur at the rates h and p. These processes contribute to the mean-square displacement along x (in units of u/2, where a is the lattice spacing) according to (Ax’)
= 4(” + 2*p + 2*h + 2*p)t.
(10)
The standard procedure in deriving values of the jump rates is to compare the distribution of displacements p,,,, with that predicted by a stochastic model involving the unknown jump rates as parameters, namely p,,,.=exp[-4(a+/I+h+p)t] x
E m=-m
x’(.\+,),*-
x
,)I
F I=
E n-pm
4wu.,-,,,2-
E ,=_m tz
1,(2/V
2,(24
nl+,l-2,(2~r)
-m
(110) xI,,(4Xt)I,,(W).
In extending these correction procedures to two dimensions we confine ourselves to the bee (I 10) plane, as there is considerable diffusion data avail-
(11)
Again it is the true distribution, unaffected by temperature transients, which is needed for this comparison, but it is MA,,, the number of times an
ki. F. Louisa, G. Ehrlich / Temperature dependence of diffuon Table 3 Effect of transients
upon estimated
diffusion
parameters
in 2D
(A~*)~~/(Ax~)ru
Estimated
0.15
1.98 2.23
0.21 0.19
0.10 0.09
0.05 0.04
Corrected Uncorrected
0.25
2.03 2.46
0.19 0.17
0.15 0.13
0.06 0.05
Corrected Uncorrected
0.50
2.01 2.97
0.21 0.12
0.13 0.05
0.04 0.03
Corrected Uncorrected
0.75
2.03 3.22
0.16 0.157
0.14 0.09
0.06 0.01
Corrected Uncorrected
1.00
2.08 4.02
0.26 0.17
0.00 0.06
0.06 0.04
Corrected Uncorrected
2.00 0.3-2.0
0.20 0.10
0.10 0.05
0.05 0.025
Input: Input:
true transient
atom is observed at a displacement k, 1 at the end of a temperature cycle, which is available from experiment. This is related to the true distribution values N,,,,n by the set of linear equations
wd=
c Cwk-m.l-nNm.n; m
(12)
phenomena in the FIM
49
ment, with 1500 observations. Just as in the onedimensional simulations, the conditions during the true diffusion interval are held constant while the contributions from transients are increased. The diffusion parameters derived from the different cycles are listed in table 3. It is clear that even under adverse conditions, when the mean-square displacement during the transients amounts to 3/4 the true value and the number of observations is limited, jump rates in good agreement with the input values can be obtained using the correction methods outlined. 5. Summary In designing a diffusion experiment it is best to make the diffusion intervals large by comparison with the duration of heating or cooling transients. When this is not possible, however, zero-time experiments very directly provide the information necessary to correct for the transients. These require some additional effort, but appreciably extend the range of diffusion conditions that can be successfully explored.
n
here w,,~ gives the redistribution probability to i, j; it is the probability that during the transients the atom will be moved to a position i, j from the origin. Both Mk,, and w,,~ are measurable quantities. The former are found from the observations after a complete heating cycle, the latter are determined from the redistribution during the zerotime experiments. The desired quantities, namely Nm.n can be derived in the usual way by solving these equations using singular value decomposition. Monte Carlo simulations have been done on bee (110) under conditions similar to a real experi-
References 111H. Mykura, in: Molecular
Processes on Solid Surfaces, Eds. E. Draughs, R.D. Gretz and R.I. Jaffee (McGraw-Hill, New York, 1969) p. 129. PI D.W. Bassett and D.R. Tice, Surf. Sci. 40 (1973) 499. 131 K. Stolt, J.D. Wrigley and G. Ehrlich, J. Chem. Phys. 69 (1978) 1151. 141 S.C. Wang and G. Ehrhch, Surf. Sci. 206 (1988) 451. 151 J.D. Wrigley, M.E. Twigg and G. Ehrlich, J. Chem. Phys. 93 (1990) 2885. of the Monte Carlo Method in 161 K. Binder, Applications Statistical Physics (Springer, Berlin, 1984). S.A. Teukolsky and W.T. 171 W.H. Press, B.P. Flannery, Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).