THEORETICAL STUDY OF THE FIELD INDUCED DESORPTION

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THEORETICAL STUDY OF THE FIELD INDUCED DESORPTION

N. Shima, M. Tsukada

To cite this version:

N. Shima, M. Tsukada. THEORETICAL STUDY OF THE FIELD INDUCED DESORPTION.

Journal de Physique Colloques, 1987, 48 (C6), pp.C6-221-C6-226. �10.1051/jphyscol:1987636�. �jpa-

00226841�

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Colioque C6, suppl6ment au no 11, Tome 48, novembre 1987

THEORETICAL STUDY OF THE FIELD INDUCED DESORPTION

N. Shima and M. Tsukada

Department of Physics, Faculty of Science, University of Tokyo, Hongo 7-3-1, l3unkyo-ku, Tokyo 113, Japan

A theory of the field induced desorption is proposed, which is valid for general cases between the adiabatic and the diabatic limit. The kinetic equation of the population is derived incorporating the mean first passage time to the ionized state. The rate of the field induced desorption is given by the sum of the relative level population devided by the mean first passage time. The mean first passage time is determined by the quantum mechanical procedure. The theory is applied to a simple one- dimensional model. The temperature dependence of the desorption rate showed a slight deviation from the Arrhenius like behavior, with the activation energy lying between the adiabatic and the diabatic limit.

1. Introduction.

Field induced desorption (FID) of the particles from surfaces affords direct microscopic information about surface atoms [1,2]. In order to investigate surface quantitatively with this phenomenon, the theory of its microscopic mechanism is needed, which has not been fully investigated. In fact the dynamical processes of FID present us some basically difficult and complicate problems of quantum mechanics and non-equilibrium statistical physics, although this seems not to have been paid due attention.

Considering the mechanism of FID, we replace the real adparticle-surface system by a one-dimensional diatomic like model for the simplicity. In the absence of the field, the ionic branch of the potential energy surface stays a t high-energy position. Under the high electric field, the ionic branch crosses the neutral branch a t Kc and the ionic state becomes the most

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987636

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E

,-...--- h,,

^neutral

^1p-y

Figure 1 Diabatic (a) and adiabatic (b) potential surfaces under the electric field.

stable for the region distant from the surface (Fig.1 a). Owing to the coupling between the ionic and neutral states, the lowest adiabatic potential surface has a peak around

R,

(Fig.1 b). If the energy splitting between the lower and upper adiabatic potential surface is large, namely in the case of strong field, the process follows along the adiabatic potential surface.

In this case, FID may be considered as a thermally activated process over the potential surface barrier. On the other hand, if the energy splitting is small, i.e. the case of weak field, the process proceeds not necessarily along the adiabatic potential surface. From the stand point of this picture, FID is not a simple activation process over the definite potential barrier. Thus if one expresses the desorption rate by the Arrhrenius relation, the activation energy Q and the prefactor k, depends considerably on temperature. In the real case, we must consider the caae of intermediate field strength taking into account of both of the adiabatic and the non-adiabatic effects. We try in this paper to propose a theory of FID based on our previous work 131, which is applicable for the intermediate coupling regions covering the adiabatic and non-adiabatic limit.

2. Kinetics of the level population

There are two different transition processes which change the each level population of the neutral state. One is the transition due to the substrate phonons whose rate is time independent. The other is the transition to the ionized state due to the electronic interaction whose rate depends on how long a particle stays at the level of the neutral branch. Taking into account of these two processes, the kinetic equation of the level population is derived as follows.

In the above ni(t) ia the i-th level population and Rij is the j-i transition rate due to the substrate phonons. Pi(t) is the escape rate to the i o n i d branch at the elapsed time t after the state generation at time t=O, i.e.,

where Ci(t) is the quantum amplitude of the state i determined in section 3. Then the equation to determine ni(-) of the stationary state can be obtained, with the aid of the final value theorem of the Laplace transformation, as follows.

where pi(RiJ) is defined by

l/fl(RiJ) can be expanded in powers of RJi and the first term of it is the inverse of the mean first passage time f l i which is def~ned by

If we approximate

the relation,

is obtained. In the model of section 4, it is numerically verified that the above relation holds within the error of about 10 %.

It can be shown that the stationary population obeys the Boltsmann distribution for the limit of

bi-.-.

The level population, however, is significantly surpressed than that expected by the Boltzmann distribution, if the escape rate is finite (l/fli>O). The population n,(-) sharply decreases above the energy of the crossing point E,, while the value of is only appreciable in that energy region. Thus the ionic deeorption rate defined by

is almost determined by only those levels located near the crossing point. The desorption rate k, is determined by a delicate competition of two kind of parameters, i.e., the mean f i t passage time {l/pi)

,

and the transition rates {Rij) among the vibrational levels by substrate phonons.

3. Microscopic theory of u,(R)

The wave function of the total electron and nucleus system Hamiltonian is expanded as,

ycr, ~ , t )

⁼

CC

0

CHi(t) e'

0t.l 1 ? H i (PI +4 (h W , ⁽¹¹⁾

where 6 ,(r,R) is the electronic wave function in the diabatic state a (=1,2), x JR) is the nucleus wave function for the i- th vibrational or continuum state over the diabatic potential surface a, and E ,i is the corresponding energy. In the above, r represents the electron coordinate and R represents nucleus coordinate respectively. The diabatic state a = l and 2 represent the neutral and the ionimd state, respectively (see Fig.la). These diabatic states are

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obtained by neglecting the coupling between the neutral ground state and the repulsive ionic state.

The time developement of the expansion coefficient Cai is determined by

'&c,i(t)=&,f Cpj(t) e

^{i (Ed;}

-

E ~ j ) t <'XdiI

V=ipIXej),

^{(I Z)}

In the above

is the matrix element of the electronic Hamiltonian H,,(r,R) for the fixed nucleus coordinate R. The solution of eq.(12) for the initial condition that the state is occupied at the i-th level at t=O is given by

where GJw) is the Green's function, G i t [ w > s [ I N - E l i

- E

The above is obtained approximately by neglecting the coupling to the other vibrational states.

Substituting the right hand side of eq.(14) for the Ci(t) in eq.(S), we found fii(R) is obtained

The vibrational level is also shifted by .the electronic interaction with the ionic branch, which is determined by the real part of the pole of the Green's function. The self-energy part of the Green's function is evaluated by the use of Airy function representing approximately the nucleus wave functions along R direction around the crossing point. As the final result, Bi(R) is evaluated by

In the above f and ^{E ,} are defined as

with the gradients

F,

and

F,

of the neutral and ionic potential surface at the crossing point Rc, respectively. ^E

,,.

is the order of the energy width of the region below the crossing point, where the tunneling process is significant. This value is the energy scale of our theory. The coefficient V is defined by

with the vibration frequency over the neutral potential well, a,, and the electronic coupling constant V,=V,,(R=R,), between the neutral and the ionic states. The quantity V

corresponds to the adiabatic parameter in Landau-Zener theory. Z ( Q ) in eq.(18) is the universal function defined by

4. Numerical results and Discusaion

In this section, we will apply the method proposed in previous sections to a simple model system of the hydrogen desorption from Ni surface. The diabatic potential surface of the neutral state is approximated by the Morse potential as

un(R)= D (e-

^{2a (R- Re)}

- 2e-"R-R")

^{, (23)}

The set of the parameters are chosen as (R,, D, a)=(1.5K, 1.56eV, 2.52A-I). Theee valuea correspond to the bridge site chemisorption of a hydrogen atom on Ni(001) surface [4]. The diabatic potential surface of the ionic state under the electric field F is conventionally represented by

e2

UCCR) =

I- +- e F ( R + A ) -

^{-) ,}

The values of ionization enetgy minus work function I- 0 and metal screening length I are chosen aa 9.1 eV and 0.97A, respectively. The electronic coupling energy between the two diabatic states is assumed in the form of

V(R) - ^{e F R} e % ~ ( - b R ) ,

^{(2 5 )}

with the value of b equals 1.26i-'.

0 0.5 1.0

I " " " " " ' . "

^"?/T

Figure 2 The calculated ionic desorption rate kd for various electric fields.

Figure 3 The activation energy Q and the pre-exponential factor k, for various electric fields.

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Figure 4 Schematic potential surfaces under the electric field (left) and the renormalized vibrational level positions for various electric fields (right).The crossing energy E,, the top of the adiabatic potential barrier Ead, and the effective barrier height

E ,

are also shown indotted lines.

For the neutral potential surface, there are eleven vibrational states. Since the separations between the vibrational levels are considerably larger than the order of the phonon energy, the transition rate between the vibrational levels is essentially determined by the multi- phonon process. We use the similar multi-phonon treatment by Efrima, Jedrzejek, Freed, Hood, and Metieu [5].

Assuming a constant supply of the particles from the lowest state, ni(-) is numerically obtained by the kinetic equation derived in section 2. The ionic desorption rate kd, which is defined by eq.(lO) is shown in Fig.2 for various field strengths. The temperature dependence of it deviates slightly from the Arrhenius law, especially for the stronger field. The activati~n energy Q and the pre-exponental factor k, around the room temperature are shown in Fig.3.

Q decreases monotonically as the field strength F increases. T o understand the change of Q qualitatively, the crossing energy E,, the top energy of the adiabatic potential, Ed, and the effective barrier height E,z E,+Q are shown in Fig.4. E, changes monotonically from Ec for the weaker fields to Ead with the stronger field. This shows that the activation energy Q is determined kinematically and not by the potential information only. The pre-exponential factor k,, as seen from Fig.4, shows strong dependence on the field strength. It increases as the field is incresed, and then decreases as the field is increased from 2 . 7 ~ / A to 2 . 9 5 ~ / i . This is due to the drastic change of the character of the vibrational wave function of the most active level. Thus it' is important to consider the system quantum mechanically to determine the pre-exponential factor k,. In order to compare the value of kd quantitativly with the experiment, reliable self-consistent calculations of the diabatic surfaces and their coupling in the presence of the field are required, which is a future problem.

References

1. E.W.Miiller and T.T.Tsong, "Field Ion Microscopy, Principles and Applications" (Elsevier, New York, 1969)

2. E.W.Mii1ler and T.T.Tsong, in "Progress in Surface Science", S.G.Davison ed. Vo1.4, Part 1, (Pergamon, New York, 1973)

3 N.Shima and M.Tsukada, in "Dynamical Processes and Ordering on Solid Surfaces", A.Yoshimori and M.Tsukada ed. (Springer 1985)

4. T.H.Upton and W.A.Goddard 111, Phys. Rev. Lett. 42 472 (1979)

5. S.Efrima, C.Jedrzejek, K.F.Freed, E.Hood, and H.Metieu, J. Chem. Phys. 79 2436 (1983)