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Electron-hole pair versus phonon excitation in molecule-surface collisions
Chemical Physics 116 (1987) 269-282 North-Holland,Amsterdam
269
Gert Due BILLING
Received 6 April 19217
A previouslydevelopedsemiclassicaldescriptionof atom-molecule s&ace scattering has been extended such that energy transfer to electron-hole pair excitation in the solid is included.Resultswith and withoutelectron-hale pair excitation for CO scattered from a P@ll) surface are presentedand discussed.
1. Introduction
The importance of electron-hole pair v&us phoaon excitation in s&ids as being responsible for energy-loss mechanisms in mol~~e-surface collisions and thereby being responsible for adso~tion has been the subject of some recent theoretical ~ves~~tions [1,2], however, with apparently conflicting results. In ref. [I] it was found that the inclusion of electronic friction (i.e. energy loss by electron-hole pair excitation) yielded sticking probabilities identical to unity* i.e. unrealistically high. Almost the opposite result was found in ref. [2] where a CO molecule bound to a Pt surface was considered. Here it was found that the damping rate of excited CO was almost entirely due to phonon coupling. Adding to this that some theoretical models [3] only consider and include electron damping and others only phonon coupling [4,5], in order to assess the problem it appears‘to be worthwhile to try to include both processes. As in some of our previous [6] ~~~a~ons we consider the CO-Pt(lll) system, but contrary to what has been done previously we include the effect by both mechanisms along the classical trajectories of the CO molecule, The trajectories are obtained by solving Hamilton’s equations of motion in an effective potential. This potential includes the “frictional” effect arising from both phonon and electron-hole pair excitation. In our previous publications [5,7] the relevant algebraic equations for a time-dependent forcing of bosons, as, e.g., the phonons, are presented and sulved. However, since the electrons are fermions we have to reformulate the approach given in ref. [7] in order to describe electronic excitation. The present model couples the phonon and the electronic excitation self-~nsist~tly to the motion of the incoming particle through an effective potential. In this respect it is related to the time-dependent self-consistent field approbation which has been used to study electron-hole pair excitation in atom-surface collisions [8]. 2. The model We consider a particle (ion or molecule> with a charge Se hitting a surface (see fig. I). The particle with electrons in the solid through the Coulomb interaction, term Se’/ 1R - q f where R = R(t) depends upon time through the classical trajectory of the incoming particle. For the electrons we assume a single particle approach, i.e. the one-electron hamiltotian is interacts
Hi”= -(A”/z@z)v;
-I- Y(q),
0301-01~/87/$03.~0 8 Blsevier Science Publishers B.V. ~o~h”HoB~d Physics Publishing Division)
0)
270
G.D. Billing / Electron - hole pair excitation in soli&
Fig. 1. The incoming particle (atom or molecule) with an electronic charge Se interacts with an electron in the solid. A simple periodic potential is assumed for the latter.
where m is the mass of the electron, 4 its position and V(c) the potential in which it moves. Simple model potentials have been used for V(c), i.e. square well or spherically symmetric potentials assuming a semi-infinite “jellium” model where the valence electrons are smeared out as an electron gas moving freely in the metal. Although we do not believe that the qualitative question concerning the importance of electron-hole pair excitation will depend critically upon the assumed model for the electron potential, in the present paper we shall use a Bloch potential which does explicitly include the corrugation of the surface (see below). With this approach is it furthermore possible to include another elementary excitation process, namely the one arising by phonon-electron coupling [9]. The total hamiltonian is now for one electron (2) and for N electrons #i=
2 Hi,
(3)
i-l
where we have omitted spin indices and where “1” indicates that the operator is expressed in terms of single particle operators. Introducing the creation/annihilation operators we have [9]: 15’=~(ilH(r)lj)Ci+Cj,
(4)
(i I HI i) = /$T(r)H(r)Gj(r) dr.
(5)
ij
where
Introducing
4(r)
HO(t)+j(r) we
as the eigenfunctions =ej+j(r)
to &O(r):
(6)
get: (7)
271
G.D. Billing / Electron -hole pair excitation in s&a%
where:
~j(f)=JIpf(r)[6e2/Ir-R(r)I]+(r) dr-
(8)
The state vectors are 1n, . . . n, . . . nN) and for fermions we have: ci 1n, . . . nj *. . nN) = fi(
- l)“j”“’
(9)
1nl . . . n, - 1.. . nN>
and nN)=Ji-Pli(-l)Cl~i”i~n,...ni+l...n,),
c+ lni...nj._.
00)
where n, = 0 or 1. The operators obey the following commutation relations: {Ci, q} = {ci+, cj”} =o
(II)
(Ci, q?} = s,,
Cl21
and
where {a, b) =ab+bu. 3. The the-dependent SHnger
equation
We now consider the time-dependent Schrijdinger equation for the system, i.e. iMU/& = 2r(t)U,
(13)
where 8, is defined by eq. (7). From an operator point of view the operator @i is identical to the W molecules. Thus we introduce the notation:
problem for polyatomic
oij = ci+q..
(14)
The operators Ojj form. a closed set with respect to commutations, i.e. [Oij,
Oke]
=
OiJjk
-
okjGieT
(15)
where we have used eqs. (11), (12). Thus the ~~utator is a linear comb~ation of the operators. In general the evolution operator U may be expressed in terms of these operators, i.e. U=n
eXP(lrijOij)s
06)
ij
If we as described in ref. [7] introduce a matrix representation of the operators Oij such that: To,*= ah?’
(17)
where S&!@ is an M x M matrix (M = N’) with zeros in all places except the ijth place which is unity, then the matrix representation of the original equation (13) becomes itisr,, = BTU
(18)
or i#=
BR,
(19)
G. D.
272
Billing / Electron - hole pair excitation in solids
where Bij= Fj(t)
Bii = ci
(i#j),
(20)
and I?( - co) = I (unit matrix). Using the representation (16) we get
(21) where we have used that exp( crijg:,!)) = I +fij8j,!‘,
fij=aij
(i#j),
fii=exp(oii)-1.
(22)
The evaluation of the product (21) may now be carried out using the graphic procedure given in ref. [7]. Once the product is evaluated the function fij (or equivalently aii) is given in terms of the solution to the matrix equation (19). The explicit value of fij is obtained either iteratively [7] or by solving the so-called inversion problem [7]. We notice that we to first order in the elements fij have f.c?)=Q..=R..-a.. IJ
‘J
IJ
(23)
‘I
and that any order of the solution to the inversion problem secures unitarity if the equations (19) are solved either exactly or by unitary methods as, e.g., the infinite-order sudden method (see below).
4. Transition probabilities In order to obtain the transition probability from a given state vector { n } to { m } we have to evaluate the matrix element (24) We now use eq. (16) and expand the exponential operators: eXP( ClijOij)
Furthermore
=
kg0 (oLfil/k!)
06.
(25)
we have
oijpz,...n i...
nM)=JI--n,~~(-l)S”]n,...ni+l...nj-l
...
where Sij= C n,+ k
C nk. k
Since ni is either 0 or 1 we see that: Oi:_In,... nM> =o
(k>,2)
(28)
and for i =j we get: exp( fxiiOii) 1n, . . . nM> = exP(niaii>
I nl
.nbf>.
(29)
G.D. Billing / Electron -hole pair excitation in dids
213
Thus the transition amplitude (24) is given by:
(30)
Due to the truncation (28) the amplitude for fermions is considerably simpler than that for bosons [7]. To first order in aji we have
Usually one will not be interested in the transition probabilities between various electronic states, but rather in the overall effect of the electrons upon the incoming particle, i.e. we want to evaluate the effective potential which the molecule feels when interacting with the electron gas. In order to do that we first have to solve eq. (19).
5. An approximate solution to eq. (19) If in eq. (19) we introduce the following transformation
(32)
R(t) = R,(OS(t), where [R~(t)l
ke=
exd(ck/ih)t]
6ke9
(33)
we get iS=WS
(34)
where [Wlke=~-lVke(f)
exP[(i/h)(Ek-ce)tl
(35)
and w,, = 0. We shall below consider transitions from states at or just below the Fermi level, i.e. from In) = ]111...1,0...0) to states as: Im)=
]ll...oj...l,oo...li...o),
where some ordering of the states has been introduced. Notice that the above transition is induced by the operator Ci+Cj where i denote a level above, and j a level below the Fermi level. Due to the selection rules implied by the Bloch potential a certain number of states above the Fermi level are coupled to a given state
G.D. Billing. / Electron -hole pait excitarion in solids
274
j. If we now introduce a labeling of the j-states, i.e. j, where I = 1,2,. . . , F is the number of states below the Fermi level. The structure of the matrix W becomes:
(36)
where i(j,) denote those levels above the Fermi level which by the potential is coupled to j,. The simple structure of the coupling matrix W,, in eq. (36) yields an analytical solution in the time-dependent sudden limit, where: S(t) = exp (-i&‘di’W(t’)).
(37)
Here we have omitted the labelling (ee). Due to the simple structure expression (37) may easily be evaluated. We get:
[Sll, =
cos[ ( AILI’y2],
(38)
IS],, = ( A$,~~,z,‘/A$)) cos[ (AI:))~“]k [S]i= = [S]zi = -i~~~~~~~))-“’
of the coupling matrix W the
A
e > 1,
sin[ (A{:))l’*]
(39) e > 1,
(401
where
(41) (42) and
k
k
(43)
6. An effective potential An approach which has been rather successful in molecular dynamics calculations of inelastic transitions in molecules and which has also been adopted in molecule surface scattering [5], is to use the effective Ehrenfest potential when coupling the quantum (phonons/electrons) and the classical (the incoming molecule) subsystems. We shall therefore also use this approach in order to determine the effect of electron-hole pair excitation upon the trajectory. Thus we have
eff=w~I$),
(4)
where e indicates that it is only that part of the effective potential arising from coupling to the electrons
G. D. Billing / Electron - hole pair excitation in sold
215
which is included in (44). A similar procedure for the phonon coupling has previously been given [5]. Introducing the evolution operator (16) we see that: H&=
t, k@Wl
(G(#J+(
(45)
t,)( W,)).
The quantity U+#iU may now be evaluated using that: [C,, , exp( a,&)]
= Sikf;.j
[Ck+‘, exp(cuijOij)]
= -S,,fij
eXP(
(XijOij)
(46)
cj
and exp(cuijOij)Ci+.
(47)
Thus the commutation relations (46), (47) are the same as those for bosons [7]. Thus we get [7] u+c,‘c,u=
c,‘ce+
E
(Qzjcj”ce+
Qejc;cj)
+ CQ,:.Q,,C&
(48)
ij
j=l
where Qkj is defined by
Q/cj = Rtcj - S,j
(49)
and Rkj obtained from the solution to (19). Using now: (&J~cX&fJ)=O>
(50)
k+e
and (51)
(k~~c,‘~/c~k~) = nk, we have
KG=
Ccknk+
Cnkvkk+
k
k
C
he(Qk*ne+
Qeknk) + C CQk*iQeinivtclke, k,e
k,e
(52)
i
where the first term is the initial electron energy and the remaining part defines the effective potential which we shall use to obtain the “electronic friction”. The energy absorbed by the electronic degrees of freedom due to the interaction obtains as dt av,,,/i3t.
hE = -It’
(53)
fl
The expression for l!,, may be reduced by noting that v,, = 0 (see also below). Furthermore we will consider the situation where all states up to the Fermi level are occupied, i.e. the indices in the operator C,‘C, and thereby vk, denote a state e below e E [l; 3’1 and k a state above k E [F + 1; N], the Fermi level F. Thus the effective potential may be written as: Gr=
i k=l
E I=F+l
vkL
i
Q;“k+ E QZQri i=l
1
+
2 k=F+l
5
vk,
I=1
i
Qk: + IE QZQri 9 i=l I
(54)
where we have used that nk = 1 (k d F) and nk = 0 otherwise. Eq. (54) may also be written as: I’%=
i
:
k=l
f=F+l
Vk,%kQ,k + &kQ:k&
=2 Re k k-l
: I=F+l
&,RkkQ:k.
(55)
G.D. Billing / Electron-hole pair excitation in soiidr
276
The expression (55) is general if the initial state is as described above. Here it should be noticed that a temperature effect could be introduced by using in eq. (52) the Fermi dist~bution: (ui> = (1
+eXp((Ci-p)/kTIJ-‘.
(56)
Here however we shall just use eq. (55) and furthermore introduce the sudden solution given in section (5) for the functions R,, and Qek. Thus we get: K%= 2~+l(2@~?)1’2) where j,(x) Ai!= A$$ =
;
V,$-‘J
dt’P$(R(t’))
sin[~~,(f’-t)],
(57)
I-F-t-1
is a Bessel function, ~[Af$, e
(58)
’ dt’ IV&‘),
(59)
J to
wke = ti-‘Vke( R( t)) exp(iw,,t)
(60)
and ake = tt- ‘( E, - +).
7. One-electron model In order to determine the one-electron wavefunctions we assume that the potential is periodic according to the lattice geometry. For simplicity we shall for the “electronic part” consider a 1D lattice (see fig. 1) such that U(x) -2
c
U, cos Gx
(61)
GwO
and furthermore introduce a one-term approbation, lattice constant. More sophisticated potentials may elaborate bandstructure calculations but rather at excitation where the present simple approach suffices. the electron may be expanded as:
i.e. U(x) = 2Uo ws Gx where G = 2-/a and CIthe easily be introduced. However, we do not aim at the qualitative importance of electron-hole pair With a periodic potential (61) the wavefunction for
&(~)=$c(k-G)exp[i(k-G)x]. Thus from the oneparticle
(62)
Schr&l.inger equation (1) we get: (63)
Inserting (62) in this equation and introducing A, = h2k2/2m we get a set of equations for the expansion coefficients
’ uo b-2, UO
Qk
LG-.9 UO
t
\
4
&(k-2G)\
uo b-Q u,
C(k-
uo _.., \
G)
C(k)
. ..
=*-
f
(64)
211
G.D. BilIing / Electron -hole pair excitation in solids
w
fFmax
EminFig. 2. The electronic energy as a functionof wavevectorfor a two-band model obtained with the potential shown in fig. 1. The Fermi energyis cF and the baud gap 2U0 = U, - U,.
n a
now a simple two-band model [lo] we get:
Introducing
&-G i
ck
(65) xk(i”6k)(
v,
‘:&;))
=
(i)’
Eq. (65) determines the eigenvalues and vectors, i.e. f; = $(&+hk-cf
(&--
A,-,)[I
+ 4~,1/(~,-~,_,)2]1’2}
(66)
and the eigenvectors are: C(k)=
fUO/‘[U,2+(h,-r~)2]*‘2
(67)
and C(k-
G) = k[l-
C(k)2]1’2.
(68)
Consistent with the 1D approximation for the electronic wavefunctions we introduce a lD charge-electron interaction term, i.e. V( R, x, 8) = Se2// R - t 1 = 6e2/(R2 + x2 - ~R.x cos S)‘l”,
(69)
where B is the angle between the x and R axes (see fig. 2). The potential (69) may now be expanded in a Fourier series, i.e. (70)
v(R, x, 0) = c K,,(& 0) exp(imy), m
where y = x,/u and a is the lattice constant. Using the wavefunctions (62) we get the following coupling elements: {#klexp(imy)~~~~)=C(k)C(k‘)+C(k-G)C(k’-G),
k’=k-m/a,
= C(k - G)C(k’),
k’=k-G-m/a,
= C( k’ - G)C( k),
k’=k+G-m/a.
(71)
For the potential (69) the Fourier expansion may be carried out analytically. Thus for R 2 x we have: v(R,
X,
8) = T
g (a/R)‘y’P,(cos I--O
e)
(72)
G.D. Billing / Electron-hole
278
pair excitation in solids
and m
y’=
C CA,‘)exp(imy) , ?n=--m
(73)
where [ll] C~‘=i(-l)m/m, Co”’= 0
m#O
(74) (75)
and C,“,=2(-l)m/m2,
m#O,
(76)
cd” = n2/3 ,
(77)
etc. In the present calculations we shall only include the first three terms of eq. (72). In order to evaluate the effective potential (57) we introduce the free electron distribution such that i*
-t’;
de P(++).
density of states
(78)
At this point also the spin multiparty is introduced. The function h(r) is zero for Emin sg z G Em, (see fig. 2) and unity otherwise. Within the model we can astonish two situations: namely the case where EF’ E-9 which character&es a typical metal, and cF = Emin, the insulator case, In the latter situation the electrons have to be excited over the band gap 2U,. 8. The CO-Pt(ll1)
system
We have previously investigated energy transfer, sticking probab~ties and scattering angles for CO hitting a Pt surface [6]. In our previous calculations only phonon excitation was included in the model. It was found that a large amount of energy was transferred to the phonons caused by the large binding energy between CO and the Pt surface. This binding energy, about 1.2 eV at a distance of 2 A from the surface, was determined experimentally [12] and the atom-atom pair potential was determined in ref. [6] so as to fit experimental data as binding sites, frequencies and distances. The potential showed that CO is bound to the surface in a perpendicular configuration with the C end downwards and with a stretched equilibrium geometry. Also the dissociation energy for surface bound CO was diminished considerably as compared to CO in the gas phase. This behaviour may be explained by electron transfer from the metal to CO into an antibonding orbital. The amount of net electronic charge on CO is determined experimentally [13] to be about S = 0.3e for surface-bound CO. However, it is reasonable to assume that the charge increases gradually as the CO molecule approaches the surface. We therefore assume that 6 increases as 6 = 0.3e( Z,/2)3,
(79) i.e. according to a charge-image-charge interaction. Z, is the distance at which the CO is bound to the surface, i.e. 2, was set to 2 A. The classical trajectories for the incoming CO molecule were integrated with and without the additions potential arising from electron-hole pair excitation. Thus the total effective potential is: &=
v,({x;:,
F, Zi>)+
5
@“({X;:,
KY Zi))%k,
T,)
k=l
+<&({Xi,
q, Zi})
(i=lforCand2forO),
(80)
Table 1 Final kinetic energies, vibrational and rotational angular momenta of CO scattered from a Pt(ll1) surface. The runs marked by (* ) include electron-hole pair excitation Run
Final
Initid n
1 2 2* d) 3 3* 4 4* 5 5+ 6 6* 6** e) 6*** f)
:y
1.13 1.39
1.00 1.00
0 1
0 a
2.54
2.41
0
0
1.67
1.00
0
50
3.10
2.41
0
50
4.%
2.41
10
0
P, c,
co
’ iff”ti
U> CA)
R
A
0.75 0.89 0.94 1.82 1.67 0.98 0.98 1.87 1.95 2.05 2.27 2.24 2.06
0 0.5 O.!i 0.9 0.3 0 0 05 0.x 5.3 5.6 4.6 6.0
23 18 18 33 24 49 49 46 47 37 30 35 31
0.12 0.13 0.13 0.40 0.53 0.033 0.033 0.42 0.29 1.10 0.95 1.06 1.04
1.02 1.18 1.17 2.39 2.39
j
tp
t Eiat) bf
3*38 3.07 3.45 4.45 3.5 3.59
0.73 0.67 0.67 0.70 0.57 0.00 0.00 0.31 0.40 0.44 0.19 0.38 0.44
‘I 1 E -100 kJ/mol. bf (E,,) is average energy transferred to surface phonons for the reflected trajectories (IZ) and adsorped trajectories (A). ‘) Adsorption probabiIity. dj Electron-hole pair excitation included with c F = 9.41 E, band gap 2U0 =l 6, Emin = 4.21 z and E,, = 5.21 E (fig. 2). ‘I cp = 4.21 E, 2U0 = 1 c, Emin = 4.21 E and E,, = 5.21 c. 0, F = 3.75 E, 2U0 = 2 E, E,,= 3.75 t and E,, = 5.75 c.
where V, contains a sum of atom-atom pair potentials between the C and 0 atoms and the N crystal atoms in their equilibrium position. The second term arises due to the coupling to the phonons and the last, given by eq. (57) due to the coupling to electron-hole pair excitation. The number of phonon modes is M = 3N - 6 and qk(f, T,) is a coupling coefficient derived previously [5,6] which depends explicitly upon time and surface temperature rS. To the effective potential we have to add the C-O Morse interaction potential in order to obtain the total potential which governs the motion of the atoms C and 0. The incident angles to the surface normal were chosen to be 8 = 45 O and (IP= 0 O. The surface temperature of the Pt(l11) surface was set to T, = 500 K and the initial vibrational/rotational angular momenta were (n, j) = (0, 0),(1, 0), (10, 0) or (0, 50) in units of A. Also the kinetic energy was varied, i.e. E,=lrorE, = 2.41 r (1 E = 100 kJ/mol) (see also table 1). Table 1 shows the results from 6 different runs. The runs marked by (* ) include. electron-hole pair excitation. In order to avoid influence from statistical fluctuations upon the comparison the same initial conditions (phase angles and aiming point) were chosen for the randomly selected trajectories in the two sets of runs n and n*. Due to the strong binding energy between CO and the Pt surface we may obtain fairly large sticking probabilities P, of 0.6-0.7. But table 1 shows that the initial condition plays an important role for the size of Pa.Thus increasing the initial kinetic energy decreases the adsorption probability from 0.73 at I&., = 1 E to 0.66 at 2.41 E. If however the energy is stored in the vibrational and, especially, in the rotational motions a much more pronounced effect is seen. Thus Pa= 0 if the initial rotational angular momentum is j = 50. The reason for this is to be found in the way CO is strongly bound to the surface, namely in the perpendicular configuration with the C end downwards. If the CO molecule is allowed to rotate rapidly there is a large tendency to rotate away from the favourable configuration and to hit the surface with the 0 end, and, since the 0-Pt interaction is repulsive [6], the molecule will be reflected rather than adsorbed, These trajectories will also transfer only little energy to the
280
G.4 &l&g / Electron - hole pair excitation in soIi&
I
I
10
20
0 3o
Fig. 3. Energy a~m~ation to the surface phoncms as a functionof scatteringangle B for CO scattered from a Pt(ll1) surface. The incident angles are B = 45” and $= O”. The surface temperatureis T, = 500 K, the kinetic energy 1 E (100 kJ/mol) and the initial vibrational and rotational angular momenta (n, j) pp(0,50) (in units of tt). The final scattering angle rpis 180 Ifr5 O.We notice that the trajectories(markedby 0) fall withina characteristicloop (run 3, table 1).
20
h0
60
80
6,,
20
10
60
80
e"
Fig. 4. Same as fig. 3 but for run 5 of table 1. Left panel shows the result obtained including phonon coupling only whereas the right panel shows the distribution if electron-hole pair excitation is included. The average energy transfer (.E& is also indicated. The energy-loss loops are less characteristic than in fig. 3 due to larger stickingprobabilities(see runs 5 and 5 * in table 1).
surface phonons (see table 1) and the scattering angle 9 will be close to 180°, i.e. along the negative x-axis (see fig. 3). The scattering angle Cp(see fig. 3) is correlated to the energy transfer Ei, to the surface phonons such that small B-valuesyield large energy transfer and vice versa. The results may be depicted by the so-called energy-loss loops as shown in fig. 3. For these trajectories which due to the initial condition do not come close to the surface for a long period there is no effect of electron-hole pair excitation. In cases where the adsorption probability is larger the energy accomodation increases also for the reflected trajectories as does the residence time at the surface. Thus, due to the form of s(t) and the interaction (72), we would expect an increasing importance of electron-hole pair excitation. Table 1 confirms this conjecture although the overall effect appears to be modest. The runs 6 * * and 6 * * * model a typical insulator case where the electrons have to be excited over the band gap of 2U, in the simple two-band model used here. We notice that the effect of electron-hole pair excitation diminishes with the size of the band gap and hence it may be neglected in these cases. The most sensitive probe on the importance of electronic friction comes from the comparison of the differential scattering picture as represented, e.g., by the energy-loss loops shown in fig. 4 for runs 5 and 5 *. We notice that the number of trajectories with large energy transfer to the phonons is somewhat smaller if the electron-hole pair excitation is included. Thus the existence of an ad~tion~ energy-loss channel tends to diminish the transfer of energy to the phonons and to increase the adsorption probability if the molecule is highly rotationally excited. If however the molecule is excited vibrationally the adsorption probability decreases slightly and the phonon excitation increases. If the initial conditions are such that the molecule interacts only little with the surface, i.e. if (Etit) is small, then there is little or no effect of electron-hole pair excitation. The results are probably very system dependent since we know that efficient interaction between the CO molecule and the surface takes place through the vibrational degree of freedom, due to the way CO is bound to the surface. Thus the sticking probability, energy accomodation, residence time at the surface, and importance of electron-hole pair excitation will increase simultaneously. For the runs 3, 5 and 6 the differences between
G.D. Billing / Electron-hole pair excitation in solids
281
Table 2 The five most excited phonon modes for the average runs 3, 5 and 6 (see table 1). The average excitation strength is (+) frequency wx in units of 1014 s-1 Run
Mode No.
(Pk)
Ok
3
9 10 8 16 40
7.37 4.52 4.51 4.16 3.33
0.037 0.039 0.027 0.064 0.128
3*
9 7 8 10 25
4.13 4.12 3.60 3.31 3.30
0.037 0.019 0.027 0.039 0.089
10 9 8 16 70
6.30 5.75 4.59 ‘3.81 2.89
0.039 0.037 0.027 0.064 0.179
5*
8 9 7 10 25
5.82 4.15 3.28 3.14 2.75
0.027 0.037 0.019 0.039 0.089
9 16 10 8 40
10.9 6.34 6.10 4.76 4.42
0.037 0.064 0.039 0.027 0.128
6*
9 10 16 118 8
5.76 3.67 2.98 2.73 2.55
0.037 0.039 0.064 0.22 0.027
6
(Pk)
*k
Run
Mode No.
and the
the two sets of results are important and we have shown in table 2 the excitation strength (pk) for phonon mode k as a function of frequency for the 5 most excited modes. The energy transfer to a given mode is approximately given by tide. Firstly we notice that it is mainly low-frequency (acoustical) modes which are excited, secondly that the excitation strength may change considerably (30-508) when electron-hole pair excitation is included. Despite this we find that the phonon coupling constitutes the dominating energy-loss chamrel, and that electron pair excitation may be neglected for metal atom/molecule scattering if only qualitative results are needed and may be completely neglected for insulators with band gaps of about 2 eV or larger.
This research was supported by the Danish Natural Science Council and the EEC scientific program under grant No. STI-06ZJ-C(CD).
References [l] C.-Y. Lee and A.E. De Pristo, J. Chem. Phys. 84 (1986) 485. [2] G.E. Korzeniewski, E. Hood and H. Metiu, J. Chem. Phys. 80 (1984) 6274. [3] B. Hellsing, M. Persson and B.I. Lundqvist, Surface Sci. 126 (1983) 147; J.W. Gadxuk and H. Metiu, Phys. Rev. B 22 (1980) 2603; M. Persson and B. Hellsing, Phys. Rev. Letters 49 (1982) 662; P.M. Ekhenique, R.M. Nieminen and R.H. Ritchie, Solid State Commun. 37 (1981) 779; J.W. Gad&, Phys. Rev. B 24 (1981) 1866; 2. Rirson, R.B. Gerber and A. Nitzan, Surface Sci. 124 (1983) 279. [4] J. Loremen and L.M. Raff, J. Chem. Phys. 52 (1970) 6134; J.C. TuBy, J. Chem. Phys. 73 (1980) 6333; P.M. AgrawaI and L.M. Raff, J. Chem. Phys. 77 (1982) 3946; A.E. De Pristo, Surface Sci. 141 (1984) 40.
282 [S] [6] [7] (8) [9] [lo] [ll] [12]
G.D. Billing / Electron -hole pair excitation in solidr
G.D. Bihing, Chem. Phys. 70 (1982) 223; 74 (1983) 143. G.D. Billing, Chem. Phys. 86 (1984) 349. G.D. Billing, Chem. Phys. 51 (1980) 417; Computer Phys. Rept. 1 (1984) 237. 2. Kirson, RB. Gerber, A. Nitran and M.A. Ratner, Surface Sci. 137 (1984) 527. D. Pines, Elementary excitations in solids (Benjamin Cummings, Reading, 1963). C. Kittel, Introduction to solid state physics (Wiley, New York, 1976). IS. Gradshteyn and I.M. Ryahik, Tables of integrals, series and products (Academic Press, New York, 1965). A.M. Baro and H. Ibach, J. Chem. Phys. 71 (1979) 4812; P.R. Norton, J.W. Goodale and E.B. Se&irk, Surface Sci. 83 (1979) 189; J.L. Glance and V.N. Korchan, Surface Sci. 75 (1978) 733; L.C. Isett and J.M. Blakely, Surface Sci. 58 (1976) 397. [13] H. Ibach, Surface Sci. 66 (1977) 56.
269
Gert Due BILLING
Received 6 April 19217
A previouslydevelopedsemiclassicaldescriptionof atom-molecule s&ace scattering has been extended such that energy transfer to electron-hole pair excitation in the solid is included.Resultswith and withoutelectron-hale pair excitation for CO scattered from a P@ll) surface are presentedand discussed.
1. Introduction
The importance of electron-hole pair v&us phoaon excitation in s&ids as being responsible for energy-loss mechanisms in mol~~e-surface collisions and thereby being responsible for adso~tion has been the subject of some recent theoretical ~ves~~tions [1,2], however, with apparently conflicting results. In ref. [I] it was found that the inclusion of electronic friction (i.e. energy loss by electron-hole pair excitation) yielded sticking probabilities identical to unity* i.e. unrealistically high. Almost the opposite result was found in ref. [2] where a CO molecule bound to a Pt surface was considered. Here it was found that the damping rate of excited CO was almost entirely due to phonon coupling. Adding to this that some theoretical models [3] only consider and include electron damping and others only phonon coupling [4,5], in order to assess the problem it appears‘to be worthwhile to try to include both processes. As in some of our previous [6] ~~~a~ons we consider the CO-Pt(lll) system, but contrary to what has been done previously we include the effect by both mechanisms along the classical trajectories of the CO molecule, The trajectories are obtained by solving Hamilton’s equations of motion in an effective potential. This potential includes the “frictional” effect arising from both phonon and electron-hole pair excitation. In our previous publications [5,7] the relevant algebraic equations for a time-dependent forcing of bosons, as, e.g., the phonons, are presented and sulved. However, since the electrons are fermions we have to reformulate the approach given in ref. [7] in order to describe electronic excitation. The present model couples the phonon and the electronic excitation self-~nsist~tly to the motion of the incoming particle through an effective potential. In this respect it is related to the time-dependent self-consistent field approbation which has been used to study electron-hole pair excitation in atom-surface collisions [8]. 2. The model We consider a particle (ion or molecule> with a charge Se hitting a surface (see fig. I). The particle with electrons in the solid through the Coulomb interaction, term Se’/ 1R - q f where R = R(t) depends upon time through the classical trajectory of the incoming particle. For the electrons we assume a single particle approach, i.e. the one-electron hamiltotian is interacts
Hi”= -(A”/z@z)v;
-I- Y(q),
0301-01~/87/$03.~0 8 Blsevier Science Publishers B.V. ~o~h”HoB~d Physics Publishing Division)
0)
270
G.D. Billing / Electron - hole pair excitation in soli&
Fig. 1. The incoming particle (atom or molecule) with an electronic charge Se interacts with an electron in the solid. A simple periodic potential is assumed for the latter.
where m is the mass of the electron, 4 its position and V(c) the potential in which it moves. Simple model potentials have been used for V(c), i.e. square well or spherically symmetric potentials assuming a semi-infinite “jellium” model where the valence electrons are smeared out as an electron gas moving freely in the metal. Although we do not believe that the qualitative question concerning the importance of electron-hole pair excitation will depend critically upon the assumed model for the electron potential, in the present paper we shall use a Bloch potential which does explicitly include the corrugation of the surface (see below). With this approach is it furthermore possible to include another elementary excitation process, namely the one arising by phonon-electron coupling [9]. The total hamiltonian is now for one electron (2) and for N electrons #i=
2 Hi,
(3)
i-l
where we have omitted spin indices and where “1” indicates that the operator is expressed in terms of single particle operators. Introducing the creation/annihilation operators we have [9]: 15’=~(ilH(r)lj)Ci+Cj,
(4)
(i I HI i) = /$T(r)H(r)Gj(r) dr.
(5)
ij
where
Introducing
4(r)
HO(t)+j(r) we
as the eigenfunctions =ej+j(r)
to &O(r):
(6)
get: (7)
271
G.D. Billing / Electron -hole pair excitation in s&a%
where:
~j(f)=JIpf(r)[6e2/Ir-R(r)I]+(r) dr-
(8)
The state vectors are 1n, . . . n, . . . nN) and for fermions we have: ci 1n, . . . nj *. . nN) = fi(
- l)“j”“’
(9)
1nl . . . n, - 1.. . nN>
and nN)=Ji-Pli(-l)Cl~i”i~n,...ni+l...n,),
c+ lni...nj._.
00)
where n, = 0 or 1. The operators obey the following commutation relations: {Ci, q} = {ci+, cj”} =o
(II)
(Ci, q?} = s,,
Cl21
and
where {a, b) =ab+bu. 3. The the-dependent SHnger
equation
We now consider the time-dependent Schrijdinger equation for the system, i.e. iMU/& = 2r(t)U,
(13)
where 8, is defined by eq. (7). From an operator point of view the operator @i is identical to the W molecules. Thus we introduce the notation:
problem for polyatomic
oij = ci+q..
(14)
The operators Ojj form. a closed set with respect to commutations, i.e. [Oij,
Oke]
=
OiJjk
-
okjGieT
(15)
where we have used eqs. (11), (12). Thus the ~~utator is a linear comb~ation of the operators. In general the evolution operator U may be expressed in terms of these operators, i.e. U=n
eXP(lrijOij)s
06)
ij
If we as described in ref. [7] introduce a matrix representation of the operators Oij such that: To,*= ah?’
(17)
where S&!@ is an M x M matrix (M = N’) with zeros in all places except the ijth place which is unity, then the matrix representation of the original equation (13) becomes itisr,, = BTU
(18)
or i#=
BR,
(19)
G. D.
272
Billing / Electron - hole pair excitation in solids
where Bij= Fj(t)
Bii = ci
(i#j),
(20)
and I?( - co) = I (unit matrix). Using the representation (16) we get
(21) where we have used that exp( crijg:,!)) = I +fij8j,!‘,
fij=aij
(i#j),
fii=exp(oii)-1.
(22)
The evaluation of the product (21) may now be carried out using the graphic procedure given in ref. [7]. Once the product is evaluated the function fij (or equivalently aii) is given in terms of the solution to the matrix equation (19). The explicit value of fij is obtained either iteratively [7] or by solving the so-called inversion problem [7]. We notice that we to first order in the elements fij have f.c?)=Q..=R..-a.. IJ
‘J
IJ
(23)
‘I
and that any order of the solution to the inversion problem secures unitarity if the equations (19) are solved either exactly or by unitary methods as, e.g., the infinite-order sudden method (see below).
4. Transition probabilities In order to obtain the transition probability from a given state vector { n } to { m } we have to evaluate the matrix element (24) We now use eq. (16) and expand the exponential operators: eXP( ClijOij)
Furthermore
=
kg0 (oLfil/k!)
06.
(25)
we have
oijpz,...n i...
nM)=JI--n,~~(-l)S”]n,...ni+l...nj-l
...
where Sij= C n,+ k
C nk. k
Since ni is either 0 or 1 we see that: Oi:_In,... nM> =o
(k>,2)
(28)
and for i =j we get: exp( fxiiOii) 1n, . . . nM> = exP(niaii>
I nl
.nbf>.
(29)
G.D. Billing / Electron -hole pair excitation in dids
213
Thus the transition amplitude (24) is given by:
(30)
Due to the truncation (28) the amplitude for fermions is considerably simpler than that for bosons [7]. To first order in aji we have
Usually one will not be interested in the transition probabilities between various electronic states, but rather in the overall effect of the electrons upon the incoming particle, i.e. we want to evaluate the effective potential which the molecule feels when interacting with the electron gas. In order to do that we first have to solve eq. (19).
5. An approximate solution to eq. (19) If in eq. (19) we introduce the following transformation
(32)
R(t) = R,(OS(t), where [R~(t)l
ke=
exd(ck/ih)t]
6ke9
(33)
we get iS=WS
(34)
where [Wlke=~-lVke(f)
exP[(i/h)(Ek-ce)tl
(35)
and w,, = 0. We shall below consider transitions from states at or just below the Fermi level, i.e. from In) = ]111...1,0...0) to states as: Im)=
]ll...oj...l,oo...li...o),
where some ordering of the states has been introduced. Notice that the above transition is induced by the operator Ci+Cj where i denote a level above, and j a level below the Fermi level. Due to the selection rules implied by the Bloch potential a certain number of states above the Fermi level are coupled to a given state
G.D. Billing. / Electron -hole pait excitarion in solids
274
j. If we now introduce a labeling of the j-states, i.e. j, where I = 1,2,. . . , F is the number of states below the Fermi level. The structure of the matrix W becomes:
(36)
where i(j,) denote those levels above the Fermi level which by the potential is coupled to j,. The simple structure of the coupling matrix W,, in eq. (36) yields an analytical solution in the time-dependent sudden limit, where: S(t) = exp (-i&‘di’W(t’)).
(37)
Here we have omitted the labelling (ee). Due to the simple structure expression (37) may easily be evaluated. We get:
[Sll, =
cos[ ( AILI’y2],
(38)
IS],, = ( A$,~~,z,‘/A$)) cos[ (AI:))~“]k [S]i= = [S]zi = -i~~~~~~~))-“’
of the coupling matrix W the
A
e > 1,
sin[ (A{:))l’*]
(39) e > 1,
(401
where
(41) (42) and
k
k
(43)
6. An effective potential An approach which has been rather successful in molecular dynamics calculations of inelastic transitions in molecules and which has also been adopted in molecule surface scattering [5], is to use the effective Ehrenfest potential when coupling the quantum (phonons/electrons) and the classical (the incoming molecule) subsystems. We shall therefore also use this approach in order to determine the effect of electron-hole pair excitation upon the trajectory. Thus we have
eff=w~I$),
(4)
where e indicates that it is only that part of the effective potential arising from coupling to the electrons
G. D. Billing / Electron - hole pair excitation in sold
215
which is included in (44). A similar procedure for the phonon coupling has previously been given [5]. Introducing the evolution operator (16) we see that: H&=
t, k@Wl
(G(#J+(
(45)
t,)( W,)).
The quantity U+#iU may now be evaluated using that: [C,, , exp( a,&)]
= Sikf;.j
[Ck+‘, exp(cuijOij)]
= -S,,fij
eXP(
(XijOij)
(46)
cj
and exp(cuijOij)Ci+.
(47)
Thus the commutation relations (46), (47) are the same as those for bosons [7]. Thus we get [7] u+c,‘c,u=
c,‘ce+
E
(Qzjcj”ce+
Qejc;cj)
+ CQ,:.Q,,C&
(48)
ij
j=l
where Qkj is defined by
Q/cj = Rtcj - S,j
(49)
and Rkj obtained from the solution to (19). Using now: (&J~cX&fJ)=O>
(50)
k+e
and (51)
(k~~c,‘~/c~k~) = nk, we have
KG=
Ccknk+
Cnkvkk+
k
k
C
he(Qk*ne+
Qeknk) + C CQk*iQeinivtclke, k,e
k,e
(52)
i
where the first term is the initial electron energy and the remaining part defines the effective potential which we shall use to obtain the “electronic friction”. The energy absorbed by the electronic degrees of freedom due to the interaction obtains as dt av,,,/i3t.
hE = -It’
(53)
fl
The expression for l!,, may be reduced by noting that v,, = 0 (see also below). Furthermore we will consider the situation where all states up to the Fermi level are occupied, i.e. the indices in the operator C,‘C, and thereby vk, denote a state e below e E [l; 3’1 and k a state above k E [F + 1; N], the Fermi level F. Thus the effective potential may be written as: Gr=
i k=l
E I=F+l
vkL
i
Q;“k+ E QZQri i=l
1
+
2 k=F+l
5
vk,
I=1
i
Qk: + IE QZQri 9 i=l I
(54)
where we have used that nk = 1 (k d F) and nk = 0 otherwise. Eq. (54) may also be written as: I’%=
i
:
k=l
f=F+l
Vk,%kQ,k + &kQ:k&
=2 Re k k-l
: I=F+l
&,RkkQ:k.
(55)
G.D. Billing / Electron-hole pair excitation in soiidr
276
The expression (55) is general if the initial state is as described above. Here it should be noticed that a temperature effect could be introduced by using in eq. (52) the Fermi dist~bution: (ui> = (1
+eXp((Ci-p)/kTIJ-‘.
(56)
Here however we shall just use eq. (55) and furthermore introduce the sudden solution given in section (5) for the functions R,, and Qek. Thus we get: K%= 2~+l(2@~?)1’2) where j,(x) Ai!= A$$ =
;
V,$-‘J
dt’P$(R(t’))
sin[~~,(f’-t)],
(57)
I-F-t-1
is a Bessel function, ~[Af$, e
(58)
’ dt’ IV&‘),
(59)
J to
wke = ti-‘Vke( R( t)) exp(iw,,t)
(60)
and ake = tt- ‘( E, - +).
7. One-electron model In order to determine the one-electron wavefunctions we assume that the potential is periodic according to the lattice geometry. For simplicity we shall for the “electronic part” consider a 1D lattice (see fig. 1) such that U(x) -2
c
U, cos Gx
(61)
GwO
and furthermore introduce a one-term approbation, lattice constant. More sophisticated potentials may elaborate bandstructure calculations but rather at excitation where the present simple approach suffices. the electron may be expanded as:
i.e. U(x) = 2Uo ws Gx where G = 2-/a and CIthe easily be introduced. However, we do not aim at the qualitative importance of electron-hole pair With a periodic potential (61) the wavefunction for
&(~)=$c(k-G)exp[i(k-G)x]. Thus from the oneparticle
(62)
Schr&l.inger equation (1) we get: (63)
Inserting (62) in this equation and introducing A, = h2k2/2m we get a set of equations for the expansion coefficients
’ uo b-2, UO
Qk
LG-.9 UO
t
\
4
&(k-2G)\
uo b-Q u,
C(k-
uo _.., \
G)
C(k)
. ..
=*-
f
(64)
211
G.D. BilIing / Electron -hole pair excitation in solids
w
fFmax
EminFig. 2. The electronic energy as a functionof wavevectorfor a two-band model obtained with the potential shown in fig. 1. The Fermi energyis cF and the baud gap 2U0 = U, - U,.
n a
now a simple two-band model [lo] we get:
Introducing
&-G i
ck
(65) xk(i”6k)(
v,
‘:&;))
=
(i)’
Eq. (65) determines the eigenvalues and vectors, i.e. f; = $(&+hk-cf
(&--
A,-,)[I
+ 4~,1/(~,-~,_,)2]1’2}
(66)
and the eigenvectors are: C(k)=
fUO/‘[U,2+(h,-r~)2]*‘2
(67)
and C(k-
G) = k[l-
C(k)2]1’2.
(68)
Consistent with the 1D approximation for the electronic wavefunctions we introduce a lD charge-electron interaction term, i.e. V( R, x, 8) = Se2// R - t 1 = 6e2/(R2 + x2 - ~R.x cos S)‘l”,
(69)
where B is the angle between the x and R axes (see fig. 2). The potential (69) may now be expanded in a Fourier series, i.e. (70)
v(R, x, 0) = c K,,(& 0) exp(imy), m
where y = x,/u and a is the lattice constant. Using the wavefunctions (62) we get the following coupling elements: {#klexp(imy)~~~~)=C(k)C(k‘)+C(k-G)C(k’-G),
k’=k-m/a,
= C(k - G)C(k’),
k’=k-G-m/a,
= C( k’ - G)C( k),
k’=k+G-m/a.
(71)
For the potential (69) the Fourier expansion may be carried out analytically. Thus for R 2 x we have: v(R,
X,
8) = T
g (a/R)‘y’P,(cos I--O
e)
(72)
G.D. Billing / Electron-hole
278
pair excitation in solids
and m
y’=
C CA,‘)exp(imy) , ?n=--m
(73)
where [ll] C~‘=i(-l)m/m, Co”’= 0
m#O
(74) (75)
and C,“,=2(-l)m/m2,
m#O,
(76)
cd” = n2/3 ,
(77)
etc. In the present calculations we shall only include the first three terms of eq. (72). In order to evaluate the effective potential (57) we introduce the free electron distribution such that i*
-t’;
de P(++).
density of states
(78)
At this point also the spin multiparty is introduced. The function h(r) is zero for Emin sg z G Em, (see fig. 2) and unity otherwise. Within the model we can astonish two situations: namely the case where EF’ E-9 which character&es a typical metal, and cF = Emin, the insulator case, In the latter situation the electrons have to be excited over the band gap 2U,. 8. The CO-Pt(ll1)
system
We have previously investigated energy transfer, sticking probab~ties and scattering angles for CO hitting a Pt surface [6]. In our previous calculations only phonon excitation was included in the model. It was found that a large amount of energy was transferred to the phonons caused by the large binding energy between CO and the Pt surface. This binding energy, about 1.2 eV at a distance of 2 A from the surface, was determined experimentally [12] and the atom-atom pair potential was determined in ref. [6] so as to fit experimental data as binding sites, frequencies and distances. The potential showed that CO is bound to the surface in a perpendicular configuration with the C end downwards and with a stretched equilibrium geometry. Also the dissociation energy for surface bound CO was diminished considerably as compared to CO in the gas phase. This behaviour may be explained by electron transfer from the metal to CO into an antibonding orbital. The amount of net electronic charge on CO is determined experimentally [13] to be about S = 0.3e for surface-bound CO. However, it is reasonable to assume that the charge increases gradually as the CO molecule approaches the surface. We therefore assume that 6 increases as 6 = 0.3e( Z,/2)3,
(79) i.e. according to a charge-image-charge interaction. Z, is the distance at which the CO is bound to the surface, i.e. 2, was set to 2 A. The classical trajectories for the incoming CO molecule were integrated with and without the additions potential arising from electron-hole pair excitation. Thus the total effective potential is: &=
v,({x;:,
F, Zi>)+
5
@“({X;:,
KY Zi))%k,
T,)
k=l
+<&({Xi,
q, Zi})
(i=lforCand2forO),
(80)
Table 1 Final kinetic energies, vibrational and rotational angular momenta of CO scattered from a Pt(ll1) surface. The runs marked by (* ) include electron-hole pair excitation Run
Final
Initid n
1 2 2* d) 3 3* 4 4* 5 5+ 6 6* 6** e) 6*** f)
:y
1.13 1.39
1.00 1.00
0 1
0 a
2.54
2.41
0
0
1.67
1.00
0
50
3.10
2.41
0
50
4.%
2.41
10
0
P, c,
co
’ iff”ti
U> CA)
R
A
0.75 0.89 0.94 1.82 1.67 0.98 0.98 1.87 1.95 2.05 2.27 2.24 2.06
0 0.5 O.!i 0.9 0.3 0 0 05 0.x 5.3 5.6 4.6 6.0
23 18 18 33 24 49 49 46 47 37 30 35 31
0.12 0.13 0.13 0.40 0.53 0.033 0.033 0.42 0.29 1.10 0.95 1.06 1.04
1.02 1.18 1.17 2.39 2.39
j
tp
t Eiat) bf
3*38 3.07 3.45 4.45 3.5 3.59
0.73 0.67 0.67 0.70 0.57 0.00 0.00 0.31 0.40 0.44 0.19 0.38 0.44
‘I 1 E -100 kJ/mol. bf (E,,) is average energy transferred to surface phonons for the reflected trajectories (IZ) and adsorped trajectories (A). ‘) Adsorption probabiIity. dj Electron-hole pair excitation included with c F = 9.41 E, band gap 2U0 =l 6, Emin = 4.21 z and E,, = 5.21 E (fig. 2). ‘I cp = 4.21 E, 2U0 = 1 c, Emin = 4.21 E and E,, = 5.21 c. 0, F = 3.75 E, 2U0 = 2 E, E,,= 3.75 t and E,, = 5.75 c.
where V, contains a sum of atom-atom pair potentials between the C and 0 atoms and the N crystal atoms in their equilibrium position. The second term arises due to the coupling to the phonons and the last, given by eq. (57) due to the coupling to electron-hole pair excitation. The number of phonon modes is M = 3N - 6 and qk(f, T,) is a coupling coefficient derived previously [5,6] which depends explicitly upon time and surface temperature rS. To the effective potential we have to add the C-O Morse interaction potential in order to obtain the total potential which governs the motion of the atoms C and 0. The incident angles to the surface normal were chosen to be 8 = 45 O and (IP= 0 O. The surface temperature of the Pt(l11) surface was set to T, = 500 K and the initial vibrational/rotational angular momenta were (n, j) = (0, 0),(1, 0), (10, 0) or (0, 50) in units of A. Also the kinetic energy was varied, i.e. E,=lrorE, = 2.41 r (1 E = 100 kJ/mol) (see also table 1). Table 1 shows the results from 6 different runs. The runs marked by (* ) include. electron-hole pair excitation. In order to avoid influence from statistical fluctuations upon the comparison the same initial conditions (phase angles and aiming point) were chosen for the randomly selected trajectories in the two sets of runs n and n*. Due to the strong binding energy between CO and the Pt surface we may obtain fairly large sticking probabilities P, of 0.6-0.7. But table 1 shows that the initial condition plays an important role for the size of Pa.Thus increasing the initial kinetic energy decreases the adsorption probability from 0.73 at I&., = 1 E to 0.66 at 2.41 E. If however the energy is stored in the vibrational and, especially, in the rotational motions a much more pronounced effect is seen. Thus Pa= 0 if the initial rotational angular momentum is j = 50. The reason for this is to be found in the way CO is strongly bound to the surface, namely in the perpendicular configuration with the C end downwards. If the CO molecule is allowed to rotate rapidly there is a large tendency to rotate away from the favourable configuration and to hit the surface with the 0 end, and, since the 0-Pt interaction is repulsive [6], the molecule will be reflected rather than adsorbed, These trajectories will also transfer only little energy to the
280
G.4 &l&g / Electron - hole pair excitation in soIi&
I
I
10
20
0 3o
Fig. 3. Energy a~m~ation to the surface phoncms as a functionof scatteringangle B for CO scattered from a Pt(ll1) surface. The incident angles are B = 45” and $= O”. The surface temperatureis T, = 500 K, the kinetic energy 1 E (100 kJ/mol) and the initial vibrational and rotational angular momenta (n, j) pp(0,50) (in units of tt). The final scattering angle rpis 180 Ifr5 O.We notice that the trajectories(markedby 0) fall withina characteristicloop (run 3, table 1).
20
h0
60
80
6,,
20
10
60
80
e"
Fig. 4. Same as fig. 3 but for run 5 of table 1. Left panel shows the result obtained including phonon coupling only whereas the right panel shows the distribution if electron-hole pair excitation is included. The average energy transfer (.E& is also indicated. The energy-loss loops are less characteristic than in fig. 3 due to larger stickingprobabilities(see runs 5 and 5 * in table 1).
surface phonons (see table 1) and the scattering angle 9 will be close to 180°, i.e. along the negative x-axis (see fig. 3). The scattering angle Cp(see fig. 3) is correlated to the energy transfer Ei, to the surface phonons such that small B-valuesyield large energy transfer and vice versa. The results may be depicted by the so-called energy-loss loops as shown in fig. 3. For these trajectories which due to the initial condition do not come close to the surface for a long period there is no effect of electron-hole pair excitation. In cases where the adsorption probability is larger the energy accomodation increases also for the reflected trajectories as does the residence time at the surface. Thus, due to the form of s(t) and the interaction (72), we would expect an increasing importance of electron-hole pair excitation. Table 1 confirms this conjecture although the overall effect appears to be modest. The runs 6 * * and 6 * * * model a typical insulator case where the electrons have to be excited over the band gap of 2U, in the simple two-band model used here. We notice that the effect of electron-hole pair excitation diminishes with the size of the band gap and hence it may be neglected in these cases. The most sensitive probe on the importance of electronic friction comes from the comparison of the differential scattering picture as represented, e.g., by the energy-loss loops shown in fig. 4 for runs 5 and 5 *. We notice that the number of trajectories with large energy transfer to the phonons is somewhat smaller if the electron-hole pair excitation is included. Thus the existence of an ad~tion~ energy-loss channel tends to diminish the transfer of energy to the phonons and to increase the adsorption probability if the molecule is highly rotationally excited. If however the molecule is excited vibrationally the adsorption probability decreases slightly and the phonon excitation increases. If the initial conditions are such that the molecule interacts only little with the surface, i.e. if (Etit) is small, then there is little or no effect of electron-hole pair excitation. The results are probably very system dependent since we know that efficient interaction between the CO molecule and the surface takes place through the vibrational degree of freedom, due to the way CO is bound to the surface. Thus the sticking probability, energy accomodation, residence time at the surface, and importance of electron-hole pair excitation will increase simultaneously. For the runs 3, 5 and 6 the differences between
G.D. Billing / Electron-hole pair excitation in solids
281
Table 2 The five most excited phonon modes for the average runs 3, 5 and 6 (see table 1). The average excitation strength is (+) frequency wx in units of 1014 s-1 Run
Mode No.
(Pk)
Ok
3
9 10 8 16 40
7.37 4.52 4.51 4.16 3.33
0.037 0.039 0.027 0.064 0.128
3*
9 7 8 10 25
4.13 4.12 3.60 3.31 3.30
0.037 0.019 0.027 0.039 0.089
10 9 8 16 70
6.30 5.75 4.59 ‘3.81 2.89
0.039 0.037 0.027 0.064 0.179
5*
8 9 7 10 25
5.82 4.15 3.28 3.14 2.75
0.027 0.037 0.019 0.039 0.089
9 16 10 8 40
10.9 6.34 6.10 4.76 4.42
0.037 0.064 0.039 0.027 0.128
6*
9 10 16 118 8
5.76 3.67 2.98 2.73 2.55
0.037 0.039 0.064 0.22 0.027
6
(Pk)
*k
Run
Mode No.
and the
the two sets of results are important and we have shown in table 2 the excitation strength (pk) for phonon mode k as a function of frequency for the 5 most excited modes. The energy transfer to a given mode is approximately given by tide. Firstly we notice that it is mainly low-frequency (acoustical) modes which are excited, secondly that the excitation strength may change considerably (30-508) when electron-hole pair excitation is included. Despite this we find that the phonon coupling constitutes the dominating energy-loss chamrel, and that electron pair excitation may be neglected for metal atom/molecule scattering if only qualitative results are needed and may be completely neglected for insulators with band gaps of about 2 eV or larger.
This research was supported by the Danish Natural Science Council and the EEC scientific program under grant No. STI-06ZJ-C(CD).
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G.D. Billing / Electron -hole pair excitation in solidr
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