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Analysis of fractional order thermal desorption
Surface Science 187 (1987) 445-462 North-Holland, Amsterdam
445
ANALYSIS OF FRACTIONAL ORDER THERMAL DESORPTION Michael V O L L M E R and Frank T R , ~ G E R Physikalisches Institut der Unioersitiit Heidelber~ Philosophenweg 12, D-6900 Heidelberg 1, Fed. Rep. of Germany
Received 23 February 1987; accepted for publication 11 May 1987
Sodium on a LiF(100) single crystal surface has been studied by thermal desorption. With increasing coverage the spectra exhibit a pronounced shift to higher desorption temperatures. We find that the results can be explained by fractional order desorption kinetics where the order of desorption is x = 0.79+0.08. In addition, activation energies have been extracted. They vary between 0.55 and 0.8 eV for coverages ranging from 5 x 1013 to 1017 atoms/cm2. An analysis of fractional order thermal desorption is presented and the desorption energies are discussed. The results are interpreted in the framework of a microscopic model where sodium atoms desorb from the edges of Na,-clusters formed by adatom diffusion on the LiF insulator surface.
1. Introduction In thermal desorption, also called flash desorption or temperature programmed desorption [1-3], a sample consisting of a substrate with one or more adsorbates on the surface is heated uniformly with constant rate. The rate of desorbing particles is monitored with a mass spectrometer as a function of the surface temperature T. Depending on the combination of substrate and adsorbate, at least one pronounced m a x i m u m of the desorption rate is observed. The position and shape of such a spectrum contain information on the kinetics of the desorption and on the binding energies of the adsorhate. Thermal desorption processes are often distinguished by the formal order x, a parameter which characterizes that fraction of particles on the surface which participate in the critical step of the desorption. The value of x usually depends on the adsorbate/substrate combination and on the coverage. F o r m a n y systems first and second order kinetics are observed, i.e. with x = 1 and x = 2. The order of x = 1 corresponds to the desorption of independent particles with no lateral interaction. Consequently, the desorption rate dn/dt is directly proportional to the particle density on the surface, x - 2 can be observed, for example, if recombination of two species A and B takes place by surface diffusion before molecules of the type AB escape the surface potential, 0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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M. Vollmer, F. Triiger / Fractional order thermal desorption
and recombination is the rate limiting step. Zeroth order thermal desorption can also occur [4-6]. In this case the evaporation of, for example, bulk solid or liquid material takes place and the desorption rate is independent of coverage. First and second order processes occur primarily for thermal desorption of atoms or molecules from metal or semiconductor surfaces at submonolayer coverages. Such systems have been studied extensively, and the results as well as their interpretation compiled in a number of review articles [3,7-9]. However, only few publications can be found in the literature where thermal desorption from insulator surfaces was investigated, and a thorough analysis of such processes is not available at all. One of the reasons for this may be the experimental obstacle that insulator surfaces are more difficult to heat uniformly with constant rate as compared to metals. Nevertheless, it is surprising that only little information on such systems is available, particularly because molecules or atoms on weakly adsorbing non-metal substrates may exhibit interesting fractional order desorption kinetics, i.e. 0 < x < 1. Especially for metal deposits this can result from adatom diffusion and cluster formation on the insulator surface. Therefore, thermal desorption of such a system could be a useful technique to study the size dependence of cluster binding energies. For this purpose, however, the detailed understanding of fractional order desorption kinetics is an important prerequisite. In this paper we present an experimental investigation of fractional order thermal desorption and a detailed analysis of the spectra. For the measurements sodium on a LiF(100) single crystal surface was chosen. Evidence for cluster formation in this system is provided by a large number of experiments. It is well known, for example, from electron microscopy studies of metal deposits on alkali halide surfaces that nucleation takes place [10,11]. In addition, the presence of clusters on such a surface follows from our previous scattering experiments [12] and from measurements of optical absorption using lasers before and during the thermal desorption [13]. Also, as will be discussed further in the text, thermal desorption spectra of sodium on lithium fluoride can be understood in a consistent way if the results are interpreted in terms of a microscopic cluster model. This work therefore demonstrates that thermal desorption is a versatile tool to investigate clusters and to extract metal cluster binding energies as a function of cluster size.
2. Experimental procedures Since the experimental arrangement is discussed in more detail elsewhere [12], only a brief description of the set-up will be given here. It consists of an ultrahigh vacuum system with a base pressure of several 10 -1~ mbar. It is equipped with a number of components to generate a thermal atomic beam of sodium. The atoms pass several liquid nitrogen cooled diaphragms for collima-
M. Vollmer, F. Trtiger / Fractional order thermal desorption
447
tion of the beam and finally impinge on the LiF(100) single crystal, which is attached to a manipulator. Alternatively, the sodium atoms can be collected on a quartz crystal microbalance, which gives accurate measurement of the beam flux. The LiF substrate can be heated at a rate of 1 - 2 K / s with a small tantalum oven. There is also provision for cooling by liquid nitrogen. The surface temperature of the crystal is measured with a thermocouple. Both are in good mechanical and thermal contact. The reading of the thermocouple at room temperature, which is also measured accurately, serves as a reference. We estimate that the surface temperature is determined with an accuracy of + 1 K. Desorbing N a atoms are detected with a quadrupole mass spectrometer operating in single ion counting mode. The microbalance frequency, crystal temperature and desorption rate are stored in a microcomputer. Before taking thermal desorption spectra a number of test measurements were performed. Firstly, the temperature rise of the substrate surface was investigated. It is linear in the temperature range between 200 and 350 K which is of interest here. Secondly, flux measurements of the atomic beam have been performed with the microbalance [14]. They permit us to calculate the surface coverage for a given deposition time. This determination takes into account that part of the impinging N a atoms are not permantly adsorbed but inelastically scattered and that the sticking coefficients are different for the LiF crystal and the gold coated microbalance. The coverage of one equivalent monolayer with 3.7 x 1014 atoms on the LiF(100) surface corresponds to a frequency change of 3.5 Hz. This value was calculated with an interatomic distance for sodium of 3.7 ,~. The surface coverage could be determined with an accuracy of better than 10%. Before starting an experiment the sample was baked at 700 K for about 2 h. This removes contamination and active sites for adsorption [15]. The surface was also investigated by Auger electron spectroscopy [16]. This shows that the crystal remains clean after the heat treatment for the duration of at least one desorption experiment. In order to ensure reproducible conditions for the substrate surface the crystal was cleaned every time between two subsequent runs by heating to 700 K for more than 30 rain. Thermal desorption experiments were performed as follows. The sample was cleaned by heat treatment and cooled to 90 K. It was then exposed to the sodium atomic beam for a chosen period of time ranging from ten to several hundred seconds. Instantaneously after the deposition the crystal was rotated to face the mass spectrometer. It was heated and the desorption rate was recorded as a function of the surface temperature. Spectra for different initial coverages are depicted in fig. 1. With increasing coverage there is a pronounced shift to larger desorption temperatures. This may be regarded as a first indication of a fractional order process [3,17].
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M. Vollmer,F. Triiger / Fractionalorderthermaldesorption
counting rate arb. units
no=
2 . 3 . 1016 7 . 6 . 1015 4.3 9 1 0 1 5 ~ 1.5 9 1015
t
temperature (K) I
I
counting rate arb. units
no=
~"~'~'~"
1.3 9 1015
7.4 9 1014 3 . 8 . 1014 1.5 9 1014 7.1 * 1013 ~ " ' " ' ' ~ "
~ " ~ " ~
I
200
temperature (K)
300
I
i
400
Fig. 1. Thermal desorption spectra of sodium from LiF(100) for different initial coverages n 0-
3. Analysis of the spectra
3.1. General considerations Thermal desorption processes are c o m m o n l y described with a n A r r h e n i u s equation
-(dn/dt)(T)
= [n(T)] x
exp(-E/kT),
(1)
M. Vollmer, F. Triiger / Fractional order thermal desorption
449
where ( d n / d t ) ( T ) stands for the rate of desorbing particles and n ( T ) for the coverage at the temperature T. ~ is the frequency factor of typically 1012/s for vibration in the surface potential, E the activation energy for desorption and x the formal order. It is assumed that E and u are fixed for each spectrum. In the following the initial coverage, i.e. the deposited amount of adsorbate, will be denoted by n 0. In addition to eq. (1) a second approach for describing thermal desorption spectra can be found (among others) in the literature [7,18]. It assumes that both the vibrational frequency factor p and the activation energy are coverage dependent:
- (dn/dt)(T) = n(T)p(n) exp(-E(n)/kT).
(2)
Here, the order x is not included as a parameter. The kinetics of the process is contained implicitly in the coverage dependences of u(n) and E(n). However, in order to describe the experimental data consistently "unreasonably u values" of up to 102~ can be obtained [7]. These values are erraneous and cannot be interpreted as oscillation frequencies of the adsorbed particles in the surface potential. In the present paper the analysis of the measurements is based exclusively on the approach of eq. (1).
3.2. Determination of binding energies, kinetic order and preexponential factor The binding or activation energy, the order of desorption and the preexponential factor can be extracted from the spectra in a variety of ways [7,8,17,19,20]. The binding energies can be derived regardless of coverage and without any assumption on the preexponential factor if the analysis is restricted to the low temperature tail of the desorption spectra [21]. With the logarithm of eq. (1) one obtains
ln(dn/dt) = -E/kT+
In(v) + x In(n).
(3)
If the analysis only relies on the low temperature part of the spectra, i.e. on a decrease of the initial coverage of only about 4%, the variation of the term x In(n) in eq. (3) is negligibly small. It has no influence on the binding energy independent of the value of x and can therefore be dropped. Also, the value for the vibrational frequency u can safely assumed to be constant. Under such conditions an Arrhenius diagram, i.e. a plot of ln(dn/dt) as a function of 1 / T directly yields the activation energy [21]. However, only those desorption spectra of the present experiment can be analyzed along these lines for which the initial coverage is not too small, i.e. exceeds approximately 5 • 1014 particles/cm 2. If n 0 is smaller the low signal-to-noise ratio makes it necessary to include a decrease of 10% or even 40% in the Arrhenius diagram. In this case, however, x In(n) changes appreciably and can no longer be neglected. Contrary to the simple analysis of the spectra outlined above, the order of
450
M. Vollmer, F. Triiger / Fractional order thermal desorption
Iog(dn/dt) nrb. units
~
,
n o ~ 2 .3 ~ 10 16 "t
1015 n o = 1.5- 10"~
~
n =4 2 " 1014 O
.
[2%]
~
. ."~
"
~,.~
[4%]
o
[10%] n o =7.1 9 1013 ~
[40%]
1000(K-1) T I
I
I
l
3.4
3.6
3.8
4.0
Fig. 2. Arrhenius diagrams with log(dn/dt) versus 1/T. The spectra have been analysed by taking into account a decrease of 2, 4, 10 and 40%, respectively, of the initial coverage n o
desorption x must now be considered as an additional parameter. This has been done by fitting the thermal desorption spectra (see fig. 2) to the analytical expression of eq. (3). The activation energy E and the order of desorption x were varied independently as fitting parameters. It is assumed that both are reasonably constant for each individual spectrum in the temperature interval considered here, i.e. from the tail of the spectrum up to almost the maximum of the desorption rate. Also, the vibrational frequency ~ is treated as a parameter but kept constant for each spectrum. It should be mentioned that in many experiments the order of desorption is determined by isothermal reaction order plots [17]. This procedure, however, cannot be applied here since it requires constant values of the desorption energy and the vibrational frequency for all spectra under consideration.
M. Vollmer, F. Triiger / Fractional order thermal desorption
451
In (dn/dt) ,
8.0
%
~
6.0
(a)
(b(b) ~
o
.
4.0
=% 1000IT I
I
3.2
I
I
3.4
I
I
I
3.6
3.8
Fig. 3. Arrhenius diagram covering the low temperature tail up to the maximum of a desorption spectrum. The initial coverage was 2.4x 1015 atoms. The arrow indicates a 4% decrease of the initial coverage. The solid lines represent theoretical curves calculated according to eq. (3): (a) With ln(dn/dt) versus 1/T. Here, the order of desorption is not included as a parameter, i.e. x ln(n) is treated as constant. (b) Obtained by a fit, that also takes into account the variation of x ln(n), i.e. with the activation energy and the order of desorption as parameters. The best fit is obtained for x = 0.79+0.08.
It is interesting to compare the analyses of a desorption spectrum made by simply plotting the rate dn/dt as a function of 1/T (see fig. 3 curve a), or obtained with a fit that also takes into account the variation of x In(n) (see fig. 3 curve b). For the onset of the desorption signal there is no noticeable difference between the two approaches. This reflects the negligible change of
binding energy (eV) 0.8
iI
t
0.7
ti
9
i
0.6
0.5
log (coverage) [ atoms/cm 2] I
14
I
I
15
16
I
17
Fig. 4. Binding energies as a function of coverage.
M. VoHmer,F. Triiger / Fractional order thermal desorption
452
reaction order
I.O
TT
0.8
Ii
0.6
0.4
log (coverage) [ atoms/cm 2 ] t
I
t
14
I
t
I
n
16
15
Fig. 5. Reaction order as a function ofcoverage. In(n) at the tail of the desorption spectrum. As the m a x i m u m of the signal is approached, however, the difference becomes evident. The linear plot with d n / d t versus 1 / T exhibits a pronounced deviation whereas the fit with x = 0.79 reproduces the data accurately. This excellent agreement also confirms the above assumption that the desorption energy and the vibrational frequency can be regarded as constant for the rise of the desorption signal, i.e. for T < Tp. The decrease of the spectrum, however, cannot be reproduced with a fit based on eq. (1). This indicates that E and v change appreciably for T > Tp. The results for the energy values are plotted in fig. 4. Starting from 0.55 eV they increase as a function of coverage and finally approach a "saturation"
16
log ( v [stl
14
12 log (coverage) [ atoms/cm 2]
I0 t
|
I
14
15
t
I
i
16
Fig. 6. Dependence of the frequency factor r as a function of coverage.
M. Vollmer, F. Tri~'ger / Fractional order thermal desorption
453
value of about 0.8 eV. The large error bars at low coverages result primarily from the limited signal-to-noise ratio. In addition, there is the possibility of slightly different defect densities and therefore different nucleation conditions for subsequent experiments which also causes a scatter of the energy values. This is reflected in small variations of the low temperature part of the desorption spectra. The reaction order x is displayed in fig. 5 as a function of coverage. It is almost constant and has a mean value of x = 0.79 + 0.08. The vibrational frequency v has also been obtained, v is coverage dependent and varies between 6 • 1012 and 6 x 1013/s (see fig. 6). These values are in agreement with the vibrational frequencies for adsorbates that are usually cited in the literature.
3.3. Analysis of the desorption temperatures and the remaining coverage So far we have analyzed each desorption spectrum individually. In the following the characteristics of different spectra will be compared. As briefly stated above, an important feature of the signals is the shift of the spectra to higher desorption temperatures with increasing coverage. This dependence is displayed in fig. 7. The halfwidth of the signals remains practically constant. There are two more essential characteristics of the spectra. As can be seen from fig. 8, the counting rate at the maximum of the signal is almost proportional to the coverage. In addition, the coverage remaining on the surface after the signal maximum has been reached is a constant fraction of the initial coverage (see fig. 9). As will be explained below, these relations follow directly and quantitatively from the Arrhenius description of fractional
peak temperature (K) 330
310
290
270
/
log (coverage)
/
[ atoms/cm 2 ] i
I
14
,
I
15
*
I
,
16
Fig. 7. Dependence of the temperature Tp at the maximum of the desorption signal as a function of coverage.
M. Vollmer, F. Triiger / Fractionalorder thermaldesorption
454
dll log [~t- Tp)
]
log (coverage) [ atoms/cm 21 I
i
I
14
i
I
15
i
16
Fig. 8. Counting rate at the maximum of the desorption signal as a function of coverage. The slope of the line is 0.98. order thermal desorption with the energy values, vibrational frequencies and the constant order of desorption quoted above. They provide an alternative description of the spectra in terms of experimental parameters such as coverage, counting rate and desorption temperature. We first derive a relation between the remaining coverage at the m a x i m u m of the signal and the coverage initially deposited on the surface. Integrating eq. (1) gives
n ( T ) = [nlo-x + (1 - x ) ( v E / f l k )( El( b ) - e - b / b - E l ( a ) + e - a / a ) ] 1/0-~), (4)
n(Tp) no
0.8
0.6
0.4 0.2
tog (coverage) [ atoms/cm 2 ] m
I
14
,
I
15
i
I
16
I
I
17
Fig. 9. Coverage remaining on the surface at the maximum of the desorption signal normalized with respect to the initial coverage versus initial coverage.
M. Vollmer, F. Triiger / Fractional order thermal desorption
455
where p
El(x)
= fx ~ex-x ' dx'
for x > 0
is the real exponential integral, and/3 is the heating rate. a and b are defined by a = E / k T o and b = E/kT. The real exponential integral El(b ) cannot be expressed analytically. However, El( b ) - e-b/b -- El( a ) + e-a/a (see eq. (4)) can be approximated by a simple function ( - e - b / b ) f ( b ) in the interval which is of interest here. It turns out that fib)= A / b y does not deviate by more than one percent from the tabulated values (e.g. see ref. [22]) if the constant A and y are chosen appropriately. One gets
f ( E / k T ) -~ 0.7866/( E / k T )095~ We now combine eqs. (4) and (1) and use that the derivative of the desorption rate d ( d n / d t ) / d T is zero at the maximum of the signal. This gives
nl-X( Tp ) = ( xvk T~//3E ) e x p ( - E / k Tp ),
(5)
Tp is the temperature at the maximum desorption rate. If we use the above approximation for the real exponential integral and compare eq. (4) with eq. (5), a relation between the initial coverage n o and the coverage n(Tp), which remains on the surface at the maximum of the signal, is obtained: n(Tp) no
[1 - (1 +
x (1 - x ) ( E / k T p ) f ( E / k T p )
) - 1]l/(a--x)
(6)
Since (E/kTp)f(E/kTp) only varies between 0.9216 and 0.9297 for the (E, Tp) combinations considered here, eq. (6) can finally be expressed in a simple form
n(Tp) [ no
x
]a/O-x).
(7)
x + 0.9256(1 - x)
Experimentally (see fig. 9) we obtain n(Tp)/no=0.354+O.O1. With this constant ratio the order of desorption x = 0.8 + 0.05 satisfies eq. (7), a result which coincides with the value of x = 0.79 _+ 0.08 derived above. Obviously, the constant ratio of the remaining and the initial coverage directly follows from the constant order of desorption. In order to understand why the maximum desorption rate depends linearly on the initial coverage, we use again that the derivative of eq. (1) is zero for T = Tp. With this condition eq. (1) becomes
( dn/dt )( Tp ) = t e n ( Tp ) /xkTp 2.
(8)
Since the ratio of the remaining and the initial coverage is constant for
456
M. Vollmer, F. Triiger / Fractional order thermal desorption
maximum desorption temperature, i.e. n(Tp) = Cn o, eq. (8) can be expressed as follows: log[ (dn/dt)(Tp)] = log(n0) + l o g ( C f l E / x k T2 ).
(9)
Experimentally, we observe that log[(dn/dt)(Tp)] is proportional to log(n0) (see fig. 8). This is exactly what the above formula expresses since the second term K = log(CflE/(xkTp2)] is essentially constant. This can be seen easily by calculating the ratios E / T 2 for all energy values and the corresponding maximum desorption temperatures determined above. One finds E / T 2 = const. The simple linear dependence ln[(dn/dt)(Tp)] = l n ( n 0 ) + K between the maximum counting rate and the coverage therefore follows from the relation E / T 2 = const. As a further point, the above equations can also be used to explain that the peak temperatures of the desorption spectra increase linearly with the logarithm of the initial coverage, i.e. Tp cc ln(n0). With eq. (5) and n(Tp) = Cn o one finds
( Cno) '-x = ( xpkTp2/flE ) e x p ( - E/kTp).
(10)
Taking the logarithm of eq. (1) and using that E / k T 2 = const. = D we obtain Tp = [ln( n~-lu ) _ ln( flD/x ) ] / D .
(11)
At a first glance eq. (11) describes a decrease rather than an increase of Tp as a function of coverage n o, since (x - 1) < 0. We have to keep in mind, however, that p is different for each spectrum and increases from 6 • 1012 to 6 x 1013/s. This change can be approximated by the simple function ~ ( n o ) - n y with y = 0.25. Therefore, the negative exponent (x - 1) = - 0.21 is compensated by the positive value of y, so that we indeed obtain an increase of the peak temperatures with coverage.
4. Discussion
4.1. Binding energies and frequency factor For metal atoms on insulator substrates the energy barrier for surface diffusion is typically less than one half of the binding energy in the surface potential. Therefore, adatom diffusion can take place very easily. The metal atoms collide with each other or with surface defects and form clusters, a process well known in epitaxy as the Volmer-Weber growth mode. Direct experimental evidence for cluster nucleation in such a system is provided by in situ electron microscopy. Similarly, sodium clusters are formed on the LiF(100) surface in the present experiment. Their mean size and number density can be determined by a
M. Vollmer, F. Triiger / Fractional order thermal desorption
457
quantitative analysis of the rate of inelastically scattered Na atoms during the deposition [12]. The cluster size ranges from 10 to 150 nm for the coverages quoted above. The density is 5 • 108/cm 2. This number equals the typical defect density on alkali halide single crystals after annealing with the heating procedure applied here. A comparison of the mean cluster size and the density shows that the average cluster distance is still large enough to prevent coalescence. If the dusters are heated during a thermal desorption experiment, atoms with low coordination number, i.e. low binding energies, desorb preferably. This argument holds, for example, for atoms located on the surface or at the edges of the clusters. The coordination number and the related binding energy, however, depend on the cluster size. The size, on the other hand, grows with coverage since the total number of dusters is constant (see above). This explains the dependence of fig. 4, in which the increase of the experimental desorption energies with coverage is depicted. The maximum value of E = 0.8 eV is consistent with the bulk enthalpy of sublimation for edge or perimeter atoms of the sodium dusters [12]. During a desorption experiment each cluster becomes smaller and smaller and the desorption energy decreases (see fig. 4). Therefore, one would expect that the desorption energy changes as a function of temperature in each desorption spectrum. We observe, however, that the Arrhenius approach of eq. (1) with a constant value for E is consistent with the experimental data. As can be seen from fig. 3, E remains constant up to the maximum of the desorption rate. In order to explain this apparent contradiction, we have to consider that the clusters on the surface do not have a single well defined size but a certain size distribution. The vapour pressure p(r) of these small particles is size dependent [23] and can be calculated with the Kelvin equation. For a given temperature p(r) is proportional to e A/r where r stands for tile radius of the cluster. A is a constant. This relation implies that small clusters evaporate more quickly than larger ones. Therefore, a given cluster size distribution on the surface changes in such a way that the relative concentration of large clusters increases. They have larger binding energies and only evaporate at higher temperatures. This is reflected in an almost constant value of the binding energy which characterizes the desorption spectrum in a considerable temperature interval. With these arguments in mind it has to be emphasized that the desorption energies are average values for a certain cluster size distribution. Another interesting point is to compare the variation of the binding energy as a function of coverage with the change of the preexponential factor p (see fig. 6). Even though the error bars for u are relatively large, the results of our experiments indicate a slight increase of the frequency factor with coverage and therefore also with binding energy. This can be understood with .the simple assumption of the particle vibrating e.g. in a van der Waals or Morse
M. Vollmer, F. Trgiger / Fractional order thermal desorption
458
potential. As the depth increases, i, also increases. The analytical function v ( E ) is a power dependence where the exponent is given by the shape of the potential.
4.2. Order of desorption In eq. (1) [n(T)] x represents the number of particles which participate in the critical step of the desorption. If sodium atoms desorb from the clusters, [n(T)] x is given by the number of surface atoms or the number of atoms located at the perimeters or at the edges. It depends on the phase of the clusters, i.e. if they are liquid droplets or solid crystals, whether the perimeters or the edges, respectively, constitute the more relevant parameter. As already mentioned above, the coordination number for both is lower than for other surface atoms so that desorption should first occur from these binding sites. The order of the desorption process can be calculated if we assume, for example, that hemispherical clusters of one size ( R ) are formed on the surface. We obtain for the total number of atoms, the number of surface atoms and the number of atoms located at the perimeters of the clusters, respectively ntot = ~rNe, ( ( R ) /ro) 3, ns,, = 2 ~rNc,((R)/ro)
2, "per = 2~rNe,(( R ) /ro),
(12)
where No1 denotes the total number of clusters on the surface, r0 is the lattice xs.r and n per = n t~o~r for desorption of surface constant. Assuming nsur = ntot atoms and perimeter atoms, respectively, the order of desorption can be expressed as follows: Xsu r =
ln(nsur)/ln(ntot) ,
Xp= = In( n pe~)/ln(n tot ).
(13)
This simple way of calculating the order of desorption can be adapted to the situation here by taking into account that the clusters are not monodisperse with size ( R ) . Rather, a Gaussian size distribution with
f(R)=(Ncl/ 2~o)exp[-(R-(R))2/2a
2]
(14)
seems to be a reasonable approximation [11]. 2ox/2 is the width of the distribution at which f(R) has dropped to 1 / e of its maximum value, f(R) is normalized by +oo
.,_[~ f(R) dR = Uc,.
(15)
M. Vollmer, F. Triiger / Fractional order thermal desorption
459
Table 1 Order of desorption calculated according to eq. (13) for evaporation of individual sodium atoms a)
(R) (A)
x~,, b)
Xpo, b)
100 500 1000 1500
0.93 0.89 0.88 0.87
0.82 0.75 0.73 0.72
( D ) (A)
xsur c)
x~g c)
100 500 1000 1500
0.94 0.90 0.89 0.88
0.85 0.78 0.75 0.74
a) the number density of clusters was N o = 5 • 108/cm 2 and the lattice constant r0 = 3.7 h, [12]. b) Evaporation from the surface and the perimeter of hemispherical sodium clusters with mean radius ( R ). c) Evaporation from the surface and the edges of cubic sodium clusters with average length of the side ( D ) .
An integration gives the total number of atoms in the clusters, the number of surface atoms and the number of perimeter atoms nto t = {~rUc,((R)/ro) 3 + 2rrNc,((R)e2/ro3),
(16a)
nsur = 27rNd ( ( R ) / r o)z + 2~rNd (o/ro)2, ripe r = 2~rUd((R)/ro).
(16b)
(16c)
Very similar expressions are obtained if the clusters are not treated as hemispherical particles but as cubes. With these formulae we have calculated x for desorption processes of surface atoms, perimeter atoms and edge atoms. The width of the cluster size distribution was assumed to be one half of the average cluster size, i.e. o = (R)/4vc2 (see e.g. ref. [11]). The results are compiled in table 1. There is almost no difference between the values for the order of desorption if the clusters are treated as hemispherical or cubic particles, even though the number of desorption sites is quite different. This implies that backreactions during the desorption, i.e. recapture of desorbed adatoms diffusing on the LiF surface, can hardly affect the value for the order of desorption. Such reactions would cause an effective change of the number of desorption sites. As can be seen from table 1, this modifies the x-values only slightly. However, one can clearly distinguish between desorption from the surface or from the perimeters or edges, respectively. In addition, the order of desorption slightly decreases with cluster size. If o is increased by a factor of two the values for the order of desorption change by less than one percent. A comparison of the results of table 1 with the experimental value of x = 0.79 + 0.08 gives very good agreement with Xper and X~dg. This indicates
460
M. Vollmer, F. Triiger / Fractional order thermal desorption
that desorption either from the perimeters or the edges of the clusters takes place and confirms the picture of the desorption process already outlined above in connection with the discussion of the activation energies. Unfortunately, due to the error bars of + 0.08, the expected slight decrease of x with cluster size cannot be verified experimentally. Similar to the desorption energies, the order of desorption constitutes an average value for a certain cluster size distribution. This does not only follow from the presence of a size distribution on the surface but also from the decreasing size of each cluster during a desorption experiment. For a given coverage the order also depends on the cluster density. Therefore, for fractional order processes, x does not simply characterize a certain adsorbate/substrate system with a given number of atoms on the surface. In order to interpret the physical meaning of x correctly, it must also be stated in how many clusters the atoms are distributed. In addition to the above analysis of fractional order desorption one may ask whether the results can also be explained consistently if the order changes within a spectrum for example from x = 0 to x = 1. Such a process could take place if the surface would first be totally covered with sodium. As already mentioned above, our previous scattering measurements [12] and electron microscopy work [1031] give unambiguous indication that this can be ruled out. On the other hand, a change of the desorption order during the temperature rise would be difficult to confirm experimentally. This is due to the fact that the low temperature tail of the desorption spectrum is very insensitive to the order of desorption because the change of x In(n) is negligible (compare eq. (1)). As can be seen from fig. 3, the tail of the signal is described equally well with x = 0.79 or by totally omitting the term x In(n), i.e. choosing an arbitrary value of x. Only as the maximum of the signal is approached the determination of the order becomes feasible.
4.3. Comparison to related work Arthur and Cho have studied thermal desorption of Cu and Au from graphite [24]. They argue that metal atom aggregates are formed on the surface as two-dimensional islands and suggest that desorption is energetically most favourable for edge atoms. In their analysis they make the questionable assumption that the number of edge atoms is proportional to N 1/2, where N denotes the total number of atoms. A correct procedure, however, does require to take into account the total number of clusters which has a pronounced influence on x (see above). Therefore, it is not surprising that the two values for the order of desorption, namely x = 1 / 3 and x = 1/2, which Arthur and Cho consider, do not give satisfactory agreement with the experimental data. Nevertheless, they propose a fractional order process with x = 1/2. We have calculated the order of desorption for the coverage of n o = 1.8 • 1014
M. Vollmer, F. Triiger / Fractional order thermal desorption
461
a t o m s / c m 2 quoted in [24] under the assumption of desorption from the edges of two-dimensional islands. One gets, for example, x = 0.85 if the total number of clusters is Nd = 109. Still assuming the same value of n o = 1.8 x 10 a4 for the coverage, x decreases slowly as the number of clusters is diminished. It reaches the value of x = 0.64 for Nd = 103. For desorption from three-dimensional islands slightly lower values for x are obtained. However, no matter if two or three dimensional clusters are considered, the value of x = 1 / 2 can be obtained only for unreasonably low cluster densities. We have also roughly analyzed the spectra of Arthur and Cho according to the procedure presented in the present paper. We find that the order of x -~ 0.75 would be appropriate. It permits us to explain the spectra consistently if u = 1012/s is chosen. This value can more easily be interpreted as an oscillation frequency of an atom in the surface potential as compared to p = 5 x 1015/s assumed in [24]. Similarly, a microscopic model for isothermal desorption spectroscopy of Ag on Si(111) was developed by Le Lay et al. [25]. Again, the total number of islands was not included as a parameter. Contrary to the present experiment, three different phases of the adsorbate may coexist on the Si(111) surface resulting in a zeroth order process. Fractional order kinetics has also been observed for hydrogen on Zn(0001) [17]. In this system the activation energy does not change with coverage. Based on isothermal reaction order plots the order of desorption of x = 1 / 2 was determined. The authors cautiously suggest desorption of edge atoms from condensed adsorbate islands. The existence of these islands is assumed to be due to attractive interactions caused by "through substrate effects" [17], but is not directly confirmed by experimental evidence. In contrast to the present case of sodium on LiF, the island growth for hydrogen on Zn(0001) is not attributed to strong adsorbate-adsorbate interactions. Therefore, it seems that the extracted energies should not be regarded as cluster or island binding energies and rather reflect a special case of adsorbate-substrate interactions. Other experiments where a fractional order desorption process was considered concern Ni, Cu, Ag and Au on W surfaces [7,18]. An important difference as compared to our experiments is that metal adsorbates on metal surfaces were investigated in the submonolayer regime. For these systems coverage dependent activation energies were observed. This dependence was attributed to phase transitions between vapor and condensed regions of the adsorbate. Especially for Cu it was argued that two-dimensional islands were formed on the surface and that desorption takes place either from their edges or interior. These two possibilities could give rise to the observed dependence of the activation energy on the coverage. However, the order of desorption was not determined since the spectra were analyzed according to eq. (la). Therefore, it is difficult to say if the desorption mechanism is similar to that of N a from LiF(IO0).
462
M. Vollmer, F. Trtiger / Fractional order thermal desorption
5. Conclusions
In the present paper we have discussed an analysis of fractional order thermal desorption and an interpretation in terms of a simple, plausible, cluster model. For future work, experiments on smaller dusters with radii below 100 ,~ would be very interesting and are presently in preparation. Possibly, the analysis applied here may have to be modified in order to describe their thermal desorption spectra consistently. Of course it would be most interesting to study monodisperse clusters by thermal desorption instead of clusters with a relatively broad size distribution. However, no preparation technique for such samples is presently in sight, although evidence for beams with neutral dusters of a single size has been obtained just recently [26].
References [1] P.A. Readhead, Vacuum 12 (1962) 203. [2] G. Ehrlich, J. Appl. Phys. 32 (1961) 4. [3] D. Menzel, in: Chemistry and Physics of Solid Surfaces IV, Vol. 20 of Springer Series Chem. Phys. (Springer, Berlin, 1982). [4] J.A. Venables and M. Bienfait, Surface Sci. 61 (1976) 667. [5] K. Nagai, T. Shibanuma and M. Hashirnoto, Surface Sci. 145 (1984) L459. [6] R. Opila and R. Gomer, Surface Sci. 112 (1981) 1. [7] E. Bauer, F. Bonczek, H. Poppa and G. Todd, Surface Sci. 53 (1975) 87. [8] C.M. Chan, R. Aris and W.H. Weinberg, Appl. Surface Sci. 1 (1978) 360. [9] D.A. King, Surface Sci. 47 (1975) 384. [10] J.A. Venables, G.D.T. Spiller and M. Hanbiicken, Rept. Progr. Phys. 47 (1984) 399. [11] H. Schmeisser, Thin Solid Films 22 (1974) 83. [12] M. Vollmer and F. Triiger, Z. Phys. D3 (1986) 291. [13] W. Hoheisel, M. VoUmer and F. Trager, to be published. [14] H.K. Pulker, E. Benes, D. Hammer and E. SiSllner, Thin Solid Films 32 (1976) 27. [15] J. Estel, H. Hoinkes, H. Kaarmann, H. Nahr and H. Wilsch, Surface Sci. 54 (1976) 393. [16] W. Weibler, PhD Thesis, Heidelberg, 1982, unpublished. [17] L. Chan and G.L. Griffin, Surface Sci. 145 (1984) 185. [18] J. Kolaczkiewicz and E. Bauer, Surface Sci. 175 (1986) 508. [19] J.L. Falconer and R.J. Madix, Surface Sci. 48 (1975) 393. [20] Z. Knor, Surface Sci. 154 (1985) L233. [21] E. Habenschaden and J. Kiippers, Surface Sci. 138 (1984) L147. [22] M. Abramowitz and I.A. Stegun, Eds., Handbook of Mathematical Functions, 5th ed. (Dover, New York, 1968). [23] J.R. Sambles, L.M. Skinner and N.D. Lisgarten, Proc. Roy. Soc. (London) A318 (1970) 507. [24] J.R. Arthur and A.Y. Cho, Surface Sci. 36 (1973) 641. [25] G. Le Lay, M. Manneville and R. Kern, Surface Sci. 72 (1978) 405. [26] M. Abshagen, J. Kowalski, M. Meyberg, G. zu Putlitz, F. Tr~iger and J. Well, to be published.
445
ANALYSIS OF FRACTIONAL ORDER THERMAL DESORPTION Michael V O L L M E R and Frank T R , ~ G E R Physikalisches Institut der Unioersitiit Heidelber~ Philosophenweg 12, D-6900 Heidelberg 1, Fed. Rep. of Germany
Received 23 February 1987; accepted for publication 11 May 1987
Sodium on a LiF(100) single crystal surface has been studied by thermal desorption. With increasing coverage the spectra exhibit a pronounced shift to higher desorption temperatures. We find that the results can be explained by fractional order desorption kinetics where the order of desorption is x = 0.79+0.08. In addition, activation energies have been extracted. They vary between 0.55 and 0.8 eV for coverages ranging from 5 x 1013 to 1017 atoms/cm2. An analysis of fractional order thermal desorption is presented and the desorption energies are discussed. The results are interpreted in the framework of a microscopic model where sodium atoms desorb from the edges of Na,-clusters formed by adatom diffusion on the LiF insulator surface.
1. Introduction In thermal desorption, also called flash desorption or temperature programmed desorption [1-3], a sample consisting of a substrate with one or more adsorbates on the surface is heated uniformly with constant rate. The rate of desorbing particles is monitored with a mass spectrometer as a function of the surface temperature T. Depending on the combination of substrate and adsorbate, at least one pronounced m a x i m u m of the desorption rate is observed. The position and shape of such a spectrum contain information on the kinetics of the desorption and on the binding energies of the adsorhate. Thermal desorption processes are often distinguished by the formal order x, a parameter which characterizes that fraction of particles on the surface which participate in the critical step of the desorption. The value of x usually depends on the adsorbate/substrate combination and on the coverage. F o r m a n y systems first and second order kinetics are observed, i.e. with x = 1 and x = 2. The order of x = 1 corresponds to the desorption of independent particles with no lateral interaction. Consequently, the desorption rate dn/dt is directly proportional to the particle density on the surface, x - 2 can be observed, for example, if recombination of two species A and B takes place by surface diffusion before molecules of the type AB escape the surface potential, 0039-6028/87/$03.50 9 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
446
M. Vollmer, F. Triiger / Fractional order thermal desorption
and recombination is the rate limiting step. Zeroth order thermal desorption can also occur [4-6]. In this case the evaporation of, for example, bulk solid or liquid material takes place and the desorption rate is independent of coverage. First and second order processes occur primarily for thermal desorption of atoms or molecules from metal or semiconductor surfaces at submonolayer coverages. Such systems have been studied extensively, and the results as well as their interpretation compiled in a number of review articles [3,7-9]. However, only few publications can be found in the literature where thermal desorption from insulator surfaces was investigated, and a thorough analysis of such processes is not available at all. One of the reasons for this may be the experimental obstacle that insulator surfaces are more difficult to heat uniformly with constant rate as compared to metals. Nevertheless, it is surprising that only little information on such systems is available, particularly because molecules or atoms on weakly adsorbing non-metal substrates may exhibit interesting fractional order desorption kinetics, i.e. 0 < x < 1. Especially for metal deposits this can result from adatom diffusion and cluster formation on the insulator surface. Therefore, thermal desorption of such a system could be a useful technique to study the size dependence of cluster binding energies. For this purpose, however, the detailed understanding of fractional order desorption kinetics is an important prerequisite. In this paper we present an experimental investigation of fractional order thermal desorption and a detailed analysis of the spectra. For the measurements sodium on a LiF(100) single crystal surface was chosen. Evidence for cluster formation in this system is provided by a large number of experiments. It is well known, for example, from electron microscopy studies of metal deposits on alkali halide surfaces that nucleation takes place [10,11]. In addition, the presence of clusters on such a surface follows from our previous scattering experiments [12] and from measurements of optical absorption using lasers before and during the thermal desorption [13]. Also, as will be discussed further in the text, thermal desorption spectra of sodium on lithium fluoride can be understood in a consistent way if the results are interpreted in terms of a microscopic cluster model. This work therefore demonstrates that thermal desorption is a versatile tool to investigate clusters and to extract metal cluster binding energies as a function of cluster size.
2. Experimental procedures Since the experimental arrangement is discussed in more detail elsewhere [12], only a brief description of the set-up will be given here. It consists of an ultrahigh vacuum system with a base pressure of several 10 -1~ mbar. It is equipped with a number of components to generate a thermal atomic beam of sodium. The atoms pass several liquid nitrogen cooled diaphragms for collima-
M. Vollmer, F. Trtiger / Fractional order thermal desorption
447
tion of the beam and finally impinge on the LiF(100) single crystal, which is attached to a manipulator. Alternatively, the sodium atoms can be collected on a quartz crystal microbalance, which gives accurate measurement of the beam flux. The LiF substrate can be heated at a rate of 1 - 2 K / s with a small tantalum oven. There is also provision for cooling by liquid nitrogen. The surface temperature of the crystal is measured with a thermocouple. Both are in good mechanical and thermal contact. The reading of the thermocouple at room temperature, which is also measured accurately, serves as a reference. We estimate that the surface temperature is determined with an accuracy of + 1 K. Desorbing N a atoms are detected with a quadrupole mass spectrometer operating in single ion counting mode. The microbalance frequency, crystal temperature and desorption rate are stored in a microcomputer. Before taking thermal desorption spectra a number of test measurements were performed. Firstly, the temperature rise of the substrate surface was investigated. It is linear in the temperature range between 200 and 350 K which is of interest here. Secondly, flux measurements of the atomic beam have been performed with the microbalance [14]. They permit us to calculate the surface coverage for a given deposition time. This determination takes into account that part of the impinging N a atoms are not permantly adsorbed but inelastically scattered and that the sticking coefficients are different for the LiF crystal and the gold coated microbalance. The coverage of one equivalent monolayer with 3.7 x 1014 atoms on the LiF(100) surface corresponds to a frequency change of 3.5 Hz. This value was calculated with an interatomic distance for sodium of 3.7 ,~. The surface coverage could be determined with an accuracy of better than 10%. Before starting an experiment the sample was baked at 700 K for about 2 h. This removes contamination and active sites for adsorption [15]. The surface was also investigated by Auger electron spectroscopy [16]. This shows that the crystal remains clean after the heat treatment for the duration of at least one desorption experiment. In order to ensure reproducible conditions for the substrate surface the crystal was cleaned every time between two subsequent runs by heating to 700 K for more than 30 rain. Thermal desorption experiments were performed as follows. The sample was cleaned by heat treatment and cooled to 90 K. It was then exposed to the sodium atomic beam for a chosen period of time ranging from ten to several hundred seconds. Instantaneously after the deposition the crystal was rotated to face the mass spectrometer. It was heated and the desorption rate was recorded as a function of the surface temperature. Spectra for different initial coverages are depicted in fig. 1. With increasing coverage there is a pronounced shift to larger desorption temperatures. This may be regarded as a first indication of a fractional order process [3,17].
448
M. Vollmer,F. Triiger / Fractionalorderthermaldesorption
counting rate arb. units
no=
2 . 3 . 1016 7 . 6 . 1015 4.3 9 1 0 1 5 ~ 1.5 9 1015
t
temperature (K) I
I
counting rate arb. units
no=
~"~'~'~"
1.3 9 1015
7.4 9 1014 3 . 8 . 1014 1.5 9 1014 7.1 * 1013 ~ " ' " ' ' ~ "
~ " ~ " ~
I
200
temperature (K)
300
I
i
400
Fig. 1. Thermal desorption spectra of sodium from LiF(100) for different initial coverages n 0-
3. Analysis of the spectra
3.1. General considerations Thermal desorption processes are c o m m o n l y described with a n A r r h e n i u s equation
-(dn/dt)(T)
= [n(T)] x
exp(-E/kT),
(1)
M. Vollmer, F. Triiger / Fractional order thermal desorption
449
where ( d n / d t ) ( T ) stands for the rate of desorbing particles and n ( T ) for the coverage at the temperature T. ~ is the frequency factor of typically 1012/s for vibration in the surface potential, E the activation energy for desorption and x the formal order. It is assumed that E and u are fixed for each spectrum. In the following the initial coverage, i.e. the deposited amount of adsorbate, will be denoted by n 0. In addition to eq. (1) a second approach for describing thermal desorption spectra can be found (among others) in the literature [7,18]. It assumes that both the vibrational frequency factor p and the activation energy are coverage dependent:
- (dn/dt)(T) = n(T)p(n) exp(-E(n)/kT).
(2)
Here, the order x is not included as a parameter. The kinetics of the process is contained implicitly in the coverage dependences of u(n) and E(n). However, in order to describe the experimental data consistently "unreasonably u values" of up to 102~ can be obtained [7]. These values are erraneous and cannot be interpreted as oscillation frequencies of the adsorbed particles in the surface potential. In the present paper the analysis of the measurements is based exclusively on the approach of eq. (1).
3.2. Determination of binding energies, kinetic order and preexponential factor The binding or activation energy, the order of desorption and the preexponential factor can be extracted from the spectra in a variety of ways [7,8,17,19,20]. The binding energies can be derived regardless of coverage and without any assumption on the preexponential factor if the analysis is restricted to the low temperature tail of the desorption spectra [21]. With the logarithm of eq. (1) one obtains
ln(dn/dt) = -E/kT+
In(v) + x In(n).
(3)
If the analysis only relies on the low temperature part of the spectra, i.e. on a decrease of the initial coverage of only about 4%, the variation of the term x In(n) in eq. (3) is negligibly small. It has no influence on the binding energy independent of the value of x and can therefore be dropped. Also, the value for the vibrational frequency u can safely assumed to be constant. Under such conditions an Arrhenius diagram, i.e. a plot of ln(dn/dt) as a function of 1 / T directly yields the activation energy [21]. However, only those desorption spectra of the present experiment can be analyzed along these lines for which the initial coverage is not too small, i.e. exceeds approximately 5 • 1014 particles/cm 2. If n 0 is smaller the low signal-to-noise ratio makes it necessary to include a decrease of 10% or even 40% in the Arrhenius diagram. In this case, however, x In(n) changes appreciably and can no longer be neglected. Contrary to the simple analysis of the spectra outlined above, the order of
450
M. Vollmer, F. Triiger / Fractional order thermal desorption
Iog(dn/dt) nrb. units
~
,
n o ~ 2 .3 ~ 10 16 "t
1015 n o = 1.5- 10"~
~
n =4 2 " 1014 O
.
[2%]
~
. ."~
"
~,.~
[4%]
o
[10%] n o =7.1 9 1013 ~
[40%]
1000(K-1) T I
I
I
l
3.4
3.6
3.8
4.0
Fig. 2. Arrhenius diagrams with log(dn/dt) versus 1/T. The spectra have been analysed by taking into account a decrease of 2, 4, 10 and 40%, respectively, of the initial coverage n o
desorption x must now be considered as an additional parameter. This has been done by fitting the thermal desorption spectra (see fig. 2) to the analytical expression of eq. (3). The activation energy E and the order of desorption x were varied independently as fitting parameters. It is assumed that both are reasonably constant for each individual spectrum in the temperature interval considered here, i.e. from the tail of the spectrum up to almost the maximum of the desorption rate. Also, the vibrational frequency ~ is treated as a parameter but kept constant for each spectrum. It should be mentioned that in many experiments the order of desorption is determined by isothermal reaction order plots [17]. This procedure, however, cannot be applied here since it requires constant values of the desorption energy and the vibrational frequency for all spectra under consideration.
M. Vollmer, F. Triiger / Fractional order thermal desorption
451
In (dn/dt) ,
8.0
%
~
6.0
(a)
(b(b) ~
o
.
4.0
=% 1000IT I
I
3.2
I
I
3.4
I
I
I
3.6
3.8
Fig. 3. Arrhenius diagram covering the low temperature tail up to the maximum of a desorption spectrum. The initial coverage was 2.4x 1015 atoms. The arrow indicates a 4% decrease of the initial coverage. The solid lines represent theoretical curves calculated according to eq. (3): (a) With ln(dn/dt) versus 1/T. Here, the order of desorption is not included as a parameter, i.e. x ln(n) is treated as constant. (b) Obtained by a fit, that also takes into account the variation of x ln(n), i.e. with the activation energy and the order of desorption as parameters. The best fit is obtained for x = 0.79+0.08.
It is interesting to compare the analyses of a desorption spectrum made by simply plotting the rate dn/dt as a function of 1/T (see fig. 3 curve a), or obtained with a fit that also takes into account the variation of x In(n) (see fig. 3 curve b). For the onset of the desorption signal there is no noticeable difference between the two approaches. This reflects the negligible change of
binding energy (eV) 0.8
iI
t
0.7
ti
9
i
0.6
0.5
log (coverage) [ atoms/cm 2] I
14
I
I
15
16
I
17
Fig. 4. Binding energies as a function of coverage.
M. VoHmer,F. Triiger / Fractional order thermal desorption
452
reaction order
I.O
TT
0.8
Ii
0.6
0.4
log (coverage) [ atoms/cm 2 ] t
I
t
14
I
t
I
n
16
15
Fig. 5. Reaction order as a function ofcoverage. In(n) at the tail of the desorption spectrum. As the m a x i m u m of the signal is approached, however, the difference becomes evident. The linear plot with d n / d t versus 1 / T exhibits a pronounced deviation whereas the fit with x = 0.79 reproduces the data accurately. This excellent agreement also confirms the above assumption that the desorption energy and the vibrational frequency can be regarded as constant for the rise of the desorption signal, i.e. for T < Tp. The decrease of the spectrum, however, cannot be reproduced with a fit based on eq. (1). This indicates that E and v change appreciably for T > Tp. The results for the energy values are plotted in fig. 4. Starting from 0.55 eV they increase as a function of coverage and finally approach a "saturation"
16
log ( v [stl
14
12 log (coverage) [ atoms/cm 2]
I0 t
|
I
14
15
t
I
i
16
Fig. 6. Dependence of the frequency factor r as a function of coverage.
M. Vollmer, F. Tri~'ger / Fractional order thermal desorption
453
value of about 0.8 eV. The large error bars at low coverages result primarily from the limited signal-to-noise ratio. In addition, there is the possibility of slightly different defect densities and therefore different nucleation conditions for subsequent experiments which also causes a scatter of the energy values. This is reflected in small variations of the low temperature part of the desorption spectra. The reaction order x is displayed in fig. 5 as a function of coverage. It is almost constant and has a mean value of x = 0.79 + 0.08. The vibrational frequency v has also been obtained, v is coverage dependent and varies between 6 • 1012 and 6 x 1013/s (see fig. 6). These values are in agreement with the vibrational frequencies for adsorbates that are usually cited in the literature.
3.3. Analysis of the desorption temperatures and the remaining coverage So far we have analyzed each desorption spectrum individually. In the following the characteristics of different spectra will be compared. As briefly stated above, an important feature of the signals is the shift of the spectra to higher desorption temperatures with increasing coverage. This dependence is displayed in fig. 7. The halfwidth of the signals remains practically constant. There are two more essential characteristics of the spectra. As can be seen from fig. 8, the counting rate at the maximum of the signal is almost proportional to the coverage. In addition, the coverage remaining on the surface after the signal maximum has been reached is a constant fraction of the initial coverage (see fig. 9). As will be explained below, these relations follow directly and quantitatively from the Arrhenius description of fractional
peak temperature (K) 330
310
290
270
/
log (coverage)
/
[ atoms/cm 2 ] i
I
14
,
I
15
*
I
,
16
Fig. 7. Dependence of the temperature Tp at the maximum of the desorption signal as a function of coverage.
M. Vollmer, F. Triiger / Fractionalorder thermaldesorption
454
dll log [~t- Tp)
]
log (coverage) [ atoms/cm 21 I
i
I
14
i
I
15
i
16
Fig. 8. Counting rate at the maximum of the desorption signal as a function of coverage. The slope of the line is 0.98. order thermal desorption with the energy values, vibrational frequencies and the constant order of desorption quoted above. They provide an alternative description of the spectra in terms of experimental parameters such as coverage, counting rate and desorption temperature. We first derive a relation between the remaining coverage at the m a x i m u m of the signal and the coverage initially deposited on the surface. Integrating eq. (1) gives
n ( T ) = [nlo-x + (1 - x ) ( v E / f l k )( El( b ) - e - b / b - E l ( a ) + e - a / a ) ] 1/0-~), (4)
n(Tp) no
0.8
0.6
0.4 0.2
tog (coverage) [ atoms/cm 2 ] m
I
14
,
I
15
i
I
16
I
I
17
Fig. 9. Coverage remaining on the surface at the maximum of the desorption signal normalized with respect to the initial coverage versus initial coverage.
M. Vollmer, F. Triiger / Fractional order thermal desorption
455
where p
El(x)
= fx ~ex-x ' dx'
for x > 0
is the real exponential integral, and/3 is the heating rate. a and b are defined by a = E / k T o and b = E/kT. The real exponential integral El(b ) cannot be expressed analytically. However, El( b ) - e-b/b -- El( a ) + e-a/a (see eq. (4)) can be approximated by a simple function ( - e - b / b ) f ( b ) in the interval which is of interest here. It turns out that fib)= A / b y does not deviate by more than one percent from the tabulated values (e.g. see ref. [22]) if the constant A and y are chosen appropriately. One gets
f ( E / k T ) -~ 0.7866/( E / k T )095~ We now combine eqs. (4) and (1) and use that the derivative of the desorption rate d ( d n / d t ) / d T is zero at the maximum of the signal. This gives
nl-X( Tp ) = ( xvk T~//3E ) e x p ( - E / k Tp ),
(5)
Tp is the temperature at the maximum desorption rate. If we use the above approximation for the real exponential integral and compare eq. (4) with eq. (5), a relation between the initial coverage n o and the coverage n(Tp), which remains on the surface at the maximum of the signal, is obtained: n(Tp) no
[1 - (1 +
x (1 - x ) ( E / k T p ) f ( E / k T p )
) - 1]l/(a--x)
(6)
Since (E/kTp)f(E/kTp) only varies between 0.9216 and 0.9297 for the (E, Tp) combinations considered here, eq. (6) can finally be expressed in a simple form
n(Tp) [ no
x
]a/O-x).
(7)
x + 0.9256(1 - x)
Experimentally (see fig. 9) we obtain n(Tp)/no=0.354+O.O1. With this constant ratio the order of desorption x = 0.8 + 0.05 satisfies eq. (7), a result which coincides with the value of x = 0.79 _+ 0.08 derived above. Obviously, the constant ratio of the remaining and the initial coverage directly follows from the constant order of desorption. In order to understand why the maximum desorption rate depends linearly on the initial coverage, we use again that the derivative of eq. (1) is zero for T = Tp. With this condition eq. (1) becomes
( dn/dt )( Tp ) = t e n ( Tp ) /xkTp 2.
(8)
Since the ratio of the remaining and the initial coverage is constant for
456
M. Vollmer, F. Triiger / Fractional order thermal desorption
maximum desorption temperature, i.e. n(Tp) = Cn o, eq. (8) can be expressed as follows: log[ (dn/dt)(Tp)] = log(n0) + l o g ( C f l E / x k T2 ).
(9)
Experimentally, we observe that log[(dn/dt)(Tp)] is proportional to log(n0) (see fig. 8). This is exactly what the above formula expresses since the second term K = log(CflE/(xkTp2)] is essentially constant. This can be seen easily by calculating the ratios E / T 2 for all energy values and the corresponding maximum desorption temperatures determined above. One finds E / T 2 = const. The simple linear dependence ln[(dn/dt)(Tp)] = l n ( n 0 ) + K between the maximum counting rate and the coverage therefore follows from the relation E / T 2 = const. As a further point, the above equations can also be used to explain that the peak temperatures of the desorption spectra increase linearly with the logarithm of the initial coverage, i.e. Tp cc ln(n0). With eq. (5) and n(Tp) = Cn o one finds
( Cno) '-x = ( xpkTp2/flE ) e x p ( - E/kTp).
(10)
Taking the logarithm of eq. (1) and using that E / k T 2 = const. = D we obtain Tp = [ln( n~-lu ) _ ln( flD/x ) ] / D .
(11)
At a first glance eq. (11) describes a decrease rather than an increase of Tp as a function of coverage n o, since (x - 1) < 0. We have to keep in mind, however, that p is different for each spectrum and increases from 6 • 1012 to 6 x 1013/s. This change can be approximated by the simple function ~ ( n o ) - n y with y = 0.25. Therefore, the negative exponent (x - 1) = - 0.21 is compensated by the positive value of y, so that we indeed obtain an increase of the peak temperatures with coverage.
4. Discussion
4.1. Binding energies and frequency factor For metal atoms on insulator substrates the energy barrier for surface diffusion is typically less than one half of the binding energy in the surface potential. Therefore, adatom diffusion can take place very easily. The metal atoms collide with each other or with surface defects and form clusters, a process well known in epitaxy as the Volmer-Weber growth mode. Direct experimental evidence for cluster nucleation in such a system is provided by in situ electron microscopy. Similarly, sodium clusters are formed on the LiF(100) surface in the present experiment. Their mean size and number density can be determined by a
M. Vollmer, F. Triiger / Fractional order thermal desorption
457
quantitative analysis of the rate of inelastically scattered Na atoms during the deposition [12]. The cluster size ranges from 10 to 150 nm for the coverages quoted above. The density is 5 • 108/cm 2. This number equals the typical defect density on alkali halide single crystals after annealing with the heating procedure applied here. A comparison of the mean cluster size and the density shows that the average cluster distance is still large enough to prevent coalescence. If the dusters are heated during a thermal desorption experiment, atoms with low coordination number, i.e. low binding energies, desorb preferably. This argument holds, for example, for atoms located on the surface or at the edges of the clusters. The coordination number and the related binding energy, however, depend on the cluster size. The size, on the other hand, grows with coverage since the total number of dusters is constant (see above). This explains the dependence of fig. 4, in which the increase of the experimental desorption energies with coverage is depicted. The maximum value of E = 0.8 eV is consistent with the bulk enthalpy of sublimation for edge or perimeter atoms of the sodium dusters [12]. During a desorption experiment each cluster becomes smaller and smaller and the desorption energy decreases (see fig. 4). Therefore, one would expect that the desorption energy changes as a function of temperature in each desorption spectrum. We observe, however, that the Arrhenius approach of eq. (1) with a constant value for E is consistent with the experimental data. As can be seen from fig. 3, E remains constant up to the maximum of the desorption rate. In order to explain this apparent contradiction, we have to consider that the clusters on the surface do not have a single well defined size but a certain size distribution. The vapour pressure p(r) of these small particles is size dependent [23] and can be calculated with the Kelvin equation. For a given temperature p(r) is proportional to e A/r where r stands for tile radius of the cluster. A is a constant. This relation implies that small clusters evaporate more quickly than larger ones. Therefore, a given cluster size distribution on the surface changes in such a way that the relative concentration of large clusters increases. They have larger binding energies and only evaporate at higher temperatures. This is reflected in an almost constant value of the binding energy which characterizes the desorption spectrum in a considerable temperature interval. With these arguments in mind it has to be emphasized that the desorption energies are average values for a certain cluster size distribution. Another interesting point is to compare the variation of the binding energy as a function of coverage with the change of the preexponential factor p (see fig. 6). Even though the error bars for u are relatively large, the results of our experiments indicate a slight increase of the frequency factor with coverage and therefore also with binding energy. This can be understood with .the simple assumption of the particle vibrating e.g. in a van der Waals or Morse
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458
potential. As the depth increases, i, also increases. The analytical function v ( E ) is a power dependence where the exponent is given by the shape of the potential.
4.2. Order of desorption In eq. (1) [n(T)] x represents the number of particles which participate in the critical step of the desorption. If sodium atoms desorb from the clusters, [n(T)] x is given by the number of surface atoms or the number of atoms located at the perimeters or at the edges. It depends on the phase of the clusters, i.e. if they are liquid droplets or solid crystals, whether the perimeters or the edges, respectively, constitute the more relevant parameter. As already mentioned above, the coordination number for both is lower than for other surface atoms so that desorption should first occur from these binding sites. The order of the desorption process can be calculated if we assume, for example, that hemispherical clusters of one size ( R ) are formed on the surface. We obtain for the total number of atoms, the number of surface atoms and the number of atoms located at the perimeters of the clusters, respectively ntot = ~rNe, ( ( R ) /ro) 3, ns,, = 2 ~rNc,((R)/ro)
2, "per = 2~rNe,(( R ) /ro),
(12)
where No1 denotes the total number of clusters on the surface, r0 is the lattice xs.r and n per = n t~o~r for desorption of surface constant. Assuming nsur = ntot atoms and perimeter atoms, respectively, the order of desorption can be expressed as follows: Xsu r =
ln(nsur)/ln(ntot) ,
Xp= = In( n pe~)/ln(n tot ).
(13)
This simple way of calculating the order of desorption can be adapted to the situation here by taking into account that the clusters are not monodisperse with size ( R ) . Rather, a Gaussian size distribution with
f(R)=(Ncl/ 2~o)exp[-(R-(R))2/2a
2]
(14)
seems to be a reasonable approximation [11]. 2ox/2 is the width of the distribution at which f(R) has dropped to 1 / e of its maximum value, f(R) is normalized by +oo
.,_[~ f(R) dR = Uc,.
(15)
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459
Table 1 Order of desorption calculated according to eq. (13) for evaporation of individual sodium atoms a)
(R) (A)
x~,, b)
Xpo, b)
100 500 1000 1500
0.93 0.89 0.88 0.87
0.82 0.75 0.73 0.72
( D ) (A)
xsur c)
x~g c)
100 500 1000 1500
0.94 0.90 0.89 0.88
0.85 0.78 0.75 0.74
a) the number density of clusters was N o = 5 • 108/cm 2 and the lattice constant r0 = 3.7 h, [12]. b) Evaporation from the surface and the perimeter of hemispherical sodium clusters with mean radius ( R ). c) Evaporation from the surface and the edges of cubic sodium clusters with average length of the side ( D ) .
An integration gives the total number of atoms in the clusters, the number of surface atoms and the number of perimeter atoms nto t = {~rUc,((R)/ro) 3 + 2rrNc,((R)e2/ro3),
(16a)
nsur = 27rNd ( ( R ) / r o)z + 2~rNd (o/ro)2, ripe r = 2~rUd((R)/ro).
(16b)
(16c)
Very similar expressions are obtained if the clusters are not treated as hemispherical particles but as cubes. With these formulae we have calculated x for desorption processes of surface atoms, perimeter atoms and edge atoms. The width of the cluster size distribution was assumed to be one half of the average cluster size, i.e. o = (R)/4vc2 (see e.g. ref. [11]). The results are compiled in table 1. There is almost no difference between the values for the order of desorption if the clusters are treated as hemispherical or cubic particles, even though the number of desorption sites is quite different. This implies that backreactions during the desorption, i.e. recapture of desorbed adatoms diffusing on the LiF surface, can hardly affect the value for the order of desorption. Such reactions would cause an effective change of the number of desorption sites. As can be seen from table 1, this modifies the x-values only slightly. However, one can clearly distinguish between desorption from the surface or from the perimeters or edges, respectively. In addition, the order of desorption slightly decreases with cluster size. If o is increased by a factor of two the values for the order of desorption change by less than one percent. A comparison of the results of table 1 with the experimental value of x = 0.79 + 0.08 gives very good agreement with Xper and X~dg. This indicates
460
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that desorption either from the perimeters or the edges of the clusters takes place and confirms the picture of the desorption process already outlined above in connection with the discussion of the activation energies. Unfortunately, due to the error bars of + 0.08, the expected slight decrease of x with cluster size cannot be verified experimentally. Similar to the desorption energies, the order of desorption constitutes an average value for a certain cluster size distribution. This does not only follow from the presence of a size distribution on the surface but also from the decreasing size of each cluster during a desorption experiment. For a given coverage the order also depends on the cluster density. Therefore, for fractional order processes, x does not simply characterize a certain adsorbate/substrate system with a given number of atoms on the surface. In order to interpret the physical meaning of x correctly, it must also be stated in how many clusters the atoms are distributed. In addition to the above analysis of fractional order desorption one may ask whether the results can also be explained consistently if the order changes within a spectrum for example from x = 0 to x = 1. Such a process could take place if the surface would first be totally covered with sodium. As already mentioned above, our previous scattering measurements [12] and electron microscopy work [1031] give unambiguous indication that this can be ruled out. On the other hand, a change of the desorption order during the temperature rise would be difficult to confirm experimentally. This is due to the fact that the low temperature tail of the desorption spectrum is very insensitive to the order of desorption because the change of x In(n) is negligible (compare eq. (1)). As can be seen from fig. 3, the tail of the signal is described equally well with x = 0.79 or by totally omitting the term x In(n), i.e. choosing an arbitrary value of x. Only as the maximum of the signal is approached the determination of the order becomes feasible.
4.3. Comparison to related work Arthur and Cho have studied thermal desorption of Cu and Au from graphite [24]. They argue that metal atom aggregates are formed on the surface as two-dimensional islands and suggest that desorption is energetically most favourable for edge atoms. In their analysis they make the questionable assumption that the number of edge atoms is proportional to N 1/2, where N denotes the total number of atoms. A correct procedure, however, does require to take into account the total number of clusters which has a pronounced influence on x (see above). Therefore, it is not surprising that the two values for the order of desorption, namely x = 1 / 3 and x = 1/2, which Arthur and Cho consider, do not give satisfactory agreement with the experimental data. Nevertheless, they propose a fractional order process with x = 1/2. We have calculated the order of desorption for the coverage of n o = 1.8 • 1014
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461
a t o m s / c m 2 quoted in [24] under the assumption of desorption from the edges of two-dimensional islands. One gets, for example, x = 0.85 if the total number of clusters is Nd = 109. Still assuming the same value of n o = 1.8 x 10 a4 for the coverage, x decreases slowly as the number of clusters is diminished. It reaches the value of x = 0.64 for Nd = 103. For desorption from three-dimensional islands slightly lower values for x are obtained. However, no matter if two or three dimensional clusters are considered, the value of x = 1 / 2 can be obtained only for unreasonably low cluster densities. We have also roughly analyzed the spectra of Arthur and Cho according to the procedure presented in the present paper. We find that the order of x -~ 0.75 would be appropriate. It permits us to explain the spectra consistently if u = 1012/s is chosen. This value can more easily be interpreted as an oscillation frequency of an atom in the surface potential as compared to p = 5 x 1015/s assumed in [24]. Similarly, a microscopic model for isothermal desorption spectroscopy of Ag on Si(111) was developed by Le Lay et al. [25]. Again, the total number of islands was not included as a parameter. Contrary to the present experiment, three different phases of the adsorbate may coexist on the Si(111) surface resulting in a zeroth order process. Fractional order kinetics has also been observed for hydrogen on Zn(0001) [17]. In this system the activation energy does not change with coverage. Based on isothermal reaction order plots the order of desorption of x = 1 / 2 was determined. The authors cautiously suggest desorption of edge atoms from condensed adsorbate islands. The existence of these islands is assumed to be due to attractive interactions caused by "through substrate effects" [17], but is not directly confirmed by experimental evidence. In contrast to the present case of sodium on LiF, the island growth for hydrogen on Zn(0001) is not attributed to strong adsorbate-adsorbate interactions. Therefore, it seems that the extracted energies should not be regarded as cluster or island binding energies and rather reflect a special case of adsorbate-substrate interactions. Other experiments where a fractional order desorption process was considered concern Ni, Cu, Ag and Au on W surfaces [7,18]. An important difference as compared to our experiments is that metal adsorbates on metal surfaces were investigated in the submonolayer regime. For these systems coverage dependent activation energies were observed. This dependence was attributed to phase transitions between vapor and condensed regions of the adsorbate. Especially for Cu it was argued that two-dimensional islands were formed on the surface and that desorption takes place either from their edges or interior. These two possibilities could give rise to the observed dependence of the activation energy on the coverage. However, the order of desorption was not determined since the spectra were analyzed according to eq. (la). Therefore, it is difficult to say if the desorption mechanism is similar to that of N a from LiF(IO0).
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5. Conclusions
In the present paper we have discussed an analysis of fractional order thermal desorption and an interpretation in terms of a simple, plausible, cluster model. For future work, experiments on smaller dusters with radii below 100 ,~ would be very interesting and are presently in preparation. Possibly, the analysis applied here may have to be modified in order to describe their thermal desorption spectra consistently. Of course it would be most interesting to study monodisperse clusters by thermal desorption instead of clusters with a relatively broad size distribution. However, no preparation technique for such samples is presently in sight, although evidence for beams with neutral dusters of a single size has been obtained just recently [26].
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