- Email: [email protected]
The role of frenkel excitons in low-energy electron interaction with molecular adfilms
Surface Science 120 (1982) L468-L476 North-Holland Publishing Company
L468
SURFACE SCIENCE LETTERS
THE ROLE OF FRENKEL EXCITONS IN LOW-ENERGY ELECTRON INTERACTION W I T H M O L E C U L A R A D F I L M S F.I. D A L I D C H I K a n d V.Z. S L O N I M Institute of Chemical Physics, Vorobevskoe Chaussee 26, Moscow 117334, USSR
Received 12 June 1981; accepted for publication 29 March 1982
The problem of the interaction of low-energy electrons with molecular films near a solid is discussed. It is shown that, as a result of the electron-metal interaction, the new "nonelastic" channel of electron-exciton complex decay is possible, during which the electron-exciton pair dissociates and the electron transmits into the metal. We have analysed the possible shapes of the resonance features observed in the experiments on the transmission and the reflection of the low-energy electrons from the surface of a solid covered by a monolayer molecular film.
Low-energy electron scattering at certain energies is k n o w n to occur through the f o r m a t i o n of metastable negative ions (see, e.g., refs. [1,2]). There are r e s o n a n c e features in the energy d e p e n d e n c e of the differential a n d total cross-sections in this case. Their shape is connected with the m e c h a n i s m responsible for the f o r m a t i o n a n d decay of the transient quasistationary state. It was f o u n d that for F e s h b a c h type of r e s o n a n c e * (temporary negative ion states associated with electronically excited states of atoms), the elastic crosssection could be described by the F a n o relation [3]:
o(E)
(~+y)2 1+~
E _ _ Er
,---
F
(1)
where o0 is the n o n - r e s o n a n t part of the cross-section, E the electron energy, E r the resonance level energy a n d y the "profile index", and F is inversely p r o p o r t i o n a l to the m e a n life time of the transient negative ion. A n a l o g o u s experiments performed recently [4,5] with solid targets (in particular, the investigations of the interaction of low m o n o e n e r g e t i c electrons with molecular adfilms) were most useful a n d informative. The high resolution studies by Sanche [4] have d e m o n s t r a t e d resonance structures, correlated in energy with gas-phase values, b u t their shape differ sufficiently. The symmetric r e s o n a n c e m a x i m a were observed in all k n o w n experiments in c o n t r a d i c t i o n
* The classification of the electron-atomic (molecular) resonances is given, e.g., in the review by Schulz [2]. 0 0 3 9 - 6 0 2 8 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L469
with formula (1). It was proposed in ref. [4] that the Fano line shape change can be explained in terms of the different decay mode of a transient negative ion in solid. It was supposed that the autodetaching electron does not decay into vacuum as it does in the gas phase, but returns to the conduction band where it has a high probability of being recaptured in a nearby molecule. This interpretation is not complete, because the coherent electron-atom interaction in the lattice is not taken into account. It is known that the monolayers of adsorbed atomic paraticles form - at low temperatures - regular structures, called two-dimensional crystals [6,7]. (The regular structure of admolecules in the systems studied by Sanche [4] is confirmed by recent experiments [8]). All states, including the quasistationary ones, in the sytems with translational symmetry are collectivized over centres. The weakly decaying states (so-called autoionizing bands, AIB) emerging in the case of quasistationary states obey the complex nonanalytical dispersion law [9-12]:
E( q) = Eo(q) - i F ( q ) ,
(2)
where q is the electron quasimomentum in AIB. In particular the Feshbach type resonances form the AIB of the bound.electron-exciton pair (electron-exciton complex, EEC), and its dispersion law imaginary part (F(q)) defines the mean EEC life time to electron detachment after its absorption of the exciton energy [11]. The resonance features due to formation and decay of such states which could be detected in electron scattering experiments must be described b y formula (1) with parameters ~ and y depending on q [12]. These results were found [11,12] while neglecting the electron-substrate interaction. This is justified only if the distance between the adfilm and the metal is large enough. The purpose of this paper is to investigate the effect of the substrate and its influence on the change of dispersion relation of the EEC and on the resonance line shapes. The algebraic equations system describing an electron interaction with a two-dimensional ordered lattice of two-level atoms in the presence of the external field was obtained in ref. [12]. In low-energy approximation it is: [x 0 - 2¢rG(v+)(E, Rs, Rs)] Ts(0 ) - 27r ~ G(v+)(E, Rs, Rs,)Ts,(O ) s'4=s
(2~-) 2
fdp exp(it,o~)~(t,) = 2~x(E, k, R~),
Rs, Rs,)Ts,(p)
(3a)
(K'--~°)a°fdp'exp(i(p' plo, lT~(p')
+x0, e x p ( - - i p p s ) r s ( o ) = 0.
(38)
F.I. Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L470
R , = (p,, zA) are the radius vectors of adatoms, oa(p) is the exciton dispersion law, and p = (p<,p~., 0) its two-dimensional quasimomentum; the integration is restricted to the first Brillouin zone. G~v+) is the electron Green function in the external field V(r),
(
C'~v+)(E'r'R):,-klim G~v+)(E,r,R)
,
2rrlr_Ri
,
(4)
where ~¢0, ~l and ~¢01 are the parameters characterizing the low-energy electron scattering by an isolated centre [1], ~20 is the area of a unit cell of the adatom lattice, and x(E, k, r) is the electron wave function corresponding to a wave incident on the metal surface from outside. The values ~ ( n ) in (3) are the amplitudes of the spherical diverging waves in corresponding channels. The sytem (3) could be easily analyzed. The two first terms of (3a) describe an electron elastic scattering by the system of unexcited atoms. The term proportional to ~0, describes the exciton creation possibility. The first two terms of (3b) describe scattering of the electron by the lattice with a free exciton. The term proportional to (K I - x 0 ) describes the electron-exciton scattering. The EEC formation is due to this very interaction. The last term of (3b) takes into account the EEC annihilation possibility. The electron-substrate interaction is supposed to be potential and described by the function V(r). One can neglect the dependence of V(r) on transverse coordinates x and y in the low-energy electron case, which is usual for the EEC states. Then: 0, V(r)=
V(z),
if if
z--+-~, z>0, V(oc)=V0<0.
(5)
The crystal boundary z = 0 is assumed to be so selected that for z > 0 the potential coincides with that of the substrate. We can write two sets of wave functions: x ( E , k, r) = ~ e x p ( i k p ) c p ( E ,
(6)
kll, z),
ffl(+'( E, kbl, z) -- S l l ( E , k~, )/i<-)(E, kii, z),
~( e, k,,, z)= ~[s,2(e, w(E,kll,Z ) =
k,,)/; + >( e,k,,,z )
#-'(E,kii,z)
z>0,
- S22( E,k,i) #+'( E,kii,z ),
S2,(E,k )i,<-'(E,k~,z),
lim Z ~
-I- ¢,O
i2<+ )( E,kll,
z)exp( z ) e x p [ -7-i~/2(EV0)-/~;'-
z>0, z~0,
i,--(i,,/,.), k,,--(k~,kv),
,Lrn J,<
z<0,
k~z]
= 1.
(7)
(8)
F.I. Dalidehik, V.Z. Slonim / Role of Frenkel excitons
L471
The reflection and transmission coefficients S,j can be defined by characteristic values of the surface and volume bands of the semiinfinite crystals [13,15]. The Green function G(v+) in (3) is
G~v+)(E,r,r,)_
_
1 fdqeiq(o-°"G(v° E - - ~ , z , z ' . (2~) 2 _
(9)
The method of the one-dimensional Green function construction is well known:
G(1) -
2 [ep(z)z(z')O(z-z')+~p(z')z(z)O(z'-z)], w[~,~] where O(x) is the step function, and W[¢, ~] = ~ d ~ / d z - ~ d ~ / d z
(10)
is the Wronskian of the functions ¢p an ~-. The translation symmetry of interactions permits to solve the system (3) in the form: T~(0) = T(0) exp(ik r,Ps),
(1 l a)
rs(p)= T(p)exp[i(k,,-p,,lPs ] .
(llb)
By means of (11) the system (3) can be transformed into the inhomogeneous integral equation for T(0), which can be solved simply:
V(O) =
~( E'kl"zA)[1-~- (Kl -- X°) O( E'kll)]
(12)
z0( E.. ){ l + [~,- ~0- ~0~,/Zo( E. k,,/] D(~. k , ) } ' T(p)
:
-
~o, r ( o )
Zo( E_c,.o(p),kl _ p H ) [ i ..~ (Kl _Ko) O( E,kfl)] ,
(13)
where
Zo(E,q)=~o-2~+~(E,z~,z~)-2~ ~, G~+~(E,R,,0)exp(iqR,), (14) R,v~0
D(E,q)-
~-~0
(2~r)2
fBzdp { Z o [ E - - w ( p ) , p - - q ] )
- I
(15)
The spectrum of EEC states in regular adlattice is defined by roots of the dispersion equations:
1 + [1¢, -- x o - x~./Zo(E,q) ] D(E,q) = 0.
(16)
The series included in Z o and D can be calculated by the method described in refs. [9,12]. Inserting (7)-(10) into (14) and taking into account the formula 2 exp(ik',P~) S
(2rr)2 ~ 8 ( k l l - - g ) , no
(.17)
F.1. Dalidehik, V.Z. Slonim / Role of Frenkel excitons
L472
where g is a two-dimensional reciprocal wave-vector we have: Z o ( E , q) = Zo'°)( E, q) + Zo~"(E, q).
(lS)
Z~o°)(E,q) is given by (28a) of ref. [12], the second term is due to electron-substrate interaction:
(2E(q + g) E ( q + g ) = E - l ( q + g ) 2, ZA
D(E, q) in
8V(E,q) ~ [ SI{-)(E - ~min(q), 2"A)] 2,
(20)
and it can be neglected while calculating the real part of the EEC dispersion law. In this case the small parameter is the ratio of the inverse EEC life time to the width of AIB. Calculation of the damping, i.e. F(q), requires the imaginary part of D(E, q) to be taken into account although it is exponentionally small. Actually, if the electron energy is insufficient for the exciton creation the dispersion function D(E,q) is real only if the substrate is not taken into account or Im S~ ~(E, q) ~ 0 * (valid for some dielectric substrates if the energy corresponds to the band gap). In this case the EEC decay can take place only as a result of the absorption by an electron of the exciton energy and following detachment of the electron from the structure. If ImStl(E,q)d=O the new "nonelastic" channel of EEC decay is possible, during which the electron-exciton pair dissociates and the electron transmits into the metal by tunneling. In this case the exciton does not annihilate and remains in the lattice. Taking into account all these considerations, we find, if E < ~mm(P):
ImD(E,q) 1
, Im Sl, [E(q + g) - w( p)] { f l ( - ) [ E ( q
fop
+ g ) -- ~( p)]
)2
.......................... V2[,o(p)- E( q + g)] [Zo[E( q + g ) - ,o( p),q + g - p ] l
(21) Iterating (16) by the imaginary terms, we find: EEEC(q) =
Eo(q) -- i F ( q ) ,
(22)
* It should be emphasized that the regular adsorption on the dielectric substrate (NaCI) was observed by LEED in ref. [14]. The energy and angular dependence forefeld for a monolayer in ref. [12], most probably could be verified for this very system.
F.L Dalidchik, V.Z. Slonim / Role of Frenkelexcitons
where
L473
Eo(q) is the zero of the equation:
1+(x l-x0)ReD(E,q)=0, F(q) --- VAi(q ) +
Ft)(q),
(23)
FAt(q ) and I'D(q) are defined by
(x, •21 --Ko) 2 ( A R e D d
-"'e=Eo
iZo(Eo(q),q)lZ '
-1
FD(q) : ( - ~ R e D )e=eo(q)Im D(E0(q),q).
(24) (25)
Thus the parameter defining the competition of dissociative and autoionizing decay channels is the ratio:
#(q) = VD(q)/rAt(q ),
(26)
which can be more than unity (see below). Let us discuss the importance of the "dissociative" channel of EEC decay in experiments on the interaction of the low-energy electrons with a solid surface, covered by a monolayer of the regular structure of the atomic particles. It is clear that any additional imaginary part of F(q) changes the resonance feature shape of curves of the energy and angular dependences of the visible values. The electron-lattice system wave function is
~p= IO)¢~o(E,k,r) +
dPlP)~p(E--o~(p),k--p,r ),
(27)
where J0) corresponds to the nonexcited lattice and ]p) to the lattice with an exciton with a quasimomentum p. For our model the electron wave functions are
*o(E,k,r)=eikO(rp(E,kl, ' z)
2~rT(0) ~
~ e -i(k-g)p g
G(I) v [E(kll+g),ZA,Z]),r:(p,z),
p:(x,y),
(28)
dPp(E,k,r)-- 2"rrT(p) 2ei'k-g-t"r'Gg'[E(kl, + g ) - w ( p l , z A , z ] ~"~0
(29)
K
It is supposed that the lattice was in the ground state 10) before the scattering of incident electrons with momentum k. The transmission current found with (27) is ~20 J , r ( E , k ) = J t ° r ( E , k ) + ( - ~ ) 2 f , zdpS, r(p - - ,k, , E - -
J,°(e,,): 2rc,[el,,, g
8o, i %[e(,,,
~(p)),
(30)
(31)
L474
F.L Dalidehik, V.Z. Slonirn / Role of Frenkel excitons
i = ~/"2[ E(k,, + g ) -
1/Io] $12 [ E(k,, + g)]l 2,
(32)
The partial inelastic transmission current Jtr(P - kit) can be described by the expression (31) if E(kll + g ) , kll is substituted by E(kl~ + g ) - - o a ( p ) , k l l - p and 80s is omitted. It can be shown that the transmission N(E,k) and reflection R(E,k) coefficients calculated by (27) satisfy the unitary condition:
N( E,k ) + R( E,k ) = 1,
(34)
where the transmission and reflection coefficients are defined by
N(E,k)=Jtr(E,k)[2E(kl,)]
1/2 ] 1/2
R(E,K)=Jr(E,k)[2E(klI)
(35a)
E •/..0min(p)"
(35b)
The reflectivity has the simplest form for the case E < Wmm(P); it is T(0)
'~(e(kH), zA)
R(ek)=lS'l[E(k)] t211 ao Sll[e(kt]w[
,,] 12
+ IT(0)I 2 EO[E(k+ g ) ] ~ / E ( t , , + g ) a2
X
g
"
E(k,,)
cp[E(k"+g)'zA]
(36)
The results of our calculation of the reflection coefficient based on (36) are given in fig. 1. Let us consider the resonance feature shape in the neighbourhood of the resonance energy Eo(k11). Bearing in mind that
r(o)
[eo(kH) z ][e eo( t]
(37)
E - E o ( k ) + i[I'~(k ,) + rD(k,~)] ' we can write 2
R(E,k)~ISI,[E(k,,)] [ 1--~[E(k,,)]
E ~ Eo(k,~ ),
,(k,,) + y ( k , , ) +i,8(kl, ) 2, :~;)+i[l+/~(k,,)]
(38)
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L475
[tee. ax3 un
i
i
i
7,49
i
i
e, 5 0
z:~ev
g~i
Fig. 1. Energy dependence of the elastic reflection coefficient from a monolayer of Xe atoms physisorbed on a metal surface. Curves a and b correspond to incident angles 0 =0 ° and 0 =5 ° respectively. The beam orientation to the crystallographic axes of the Xe lattice is the same as in ref. [12]. The values x0=-0.17, Km=0.20, x021=1.36×10 2 0%=0.30 (au), II0=4.5 eV and IzAl=3.2 A were used in calculations according to eq. (6). The exciton dispersion law in the two-dimensional hexagonal closely packed lattice atoms was taken as: w ( p ) = w0 +2/3[cos(pxa) +2 cos(~pxa ) cos(½ V/3pva)], with/7 =0.056 eV, a = 5.5 ,~. The electron transmission coefficient Sml was taken as [
=
+
].
where
ReZo( E,kli) E- Eo(kl,) "Y(kiil=imZo(E,kH), E(kH)-- FAI(k,,) ' a n d (I)(E, k H) is a slow function of its a r g u m e n t s in the given interval of energy. I f / 3 << 1 (it means, e.g., that the s u b s t r a t e can be neglected) and
R(E,k) ~-IS,,[E(kr,)]l 2 ( 'l ++c
.y)X 2
(39)
T h e line shape suits the relation p r e d i c t e d b y F a n o . T h e r e is a shear r e s o n a n c e m i n i m u m of the reflection coefficient if /3 >> 1 (see fig. 1). It follows from eq. (38) that there is a r e s o n a n c e m a x i m u m in the t r a n s m i s s i o n coefficient i f / 3 >> 1. (It should be e m p h a s i z e d that high collimation of the b e a m is necessary to observe the fine structure p r e d i c t e d by this theory. T h a t is w h y the a n g u l a r divergence leads to the r e s o n a n c e structure averaging a n d o n l y m a x i m u m s survive for o b s e r v a t i o n . ) The shape of this m a x i m u m can be d e s c r i b e d b y a f o r m u l a of the B r e i t = W i g n e r type: r~i(ki,) rD(ki,)
N( E, k) ~ [E-
Eo(k,,)]~ + [rA,(k,, )+ rD(k,,)] ~'
(40)
L476
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
where FAl(kll ) is proportional to a partial width of the input channel, corresponding to the formation of the EEC by the electron travelling from the vacuum to the adlayer.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G.F. Drukarev, Stolknovenia Elektronov s Atomami i Moleculami (Moscow, 1978). G.J~ Schulz, Rev. Mod. Phys. 45 (1973) 378. U. Fano, Phys. Rev. 124 (1961) 1866. L. Sanche, J. Chem. Phys. 71 (1979) 4860. T. Huang and W.H. Hamill, J. Phys. Chem. Solids 36 (1975) 661 L.A. Bolshov, A.P. Napartovich, A.G. Naumovets and A.G. Fedorus, Usp. Fiz. Nauk 122 (1977) 125. J.P. Hobson, in: Surface Science, Recent Progress and Perspectives (CRC Press, Cleveland, OH, 1974). L.E. Firment and G.A. Somorjai, Surface Sci. 84 (1979) 275. F.I. Dalidchik, Zh. Eksperim. Teor. Fiz. 77 (1979) 2422. G.V. Golubkov, F.I. Dalidchik, and G.K. Ivanov, Surface Sci. 93 (1980) L89. F.I. Dalidchik and V.Z. Slonim, ZETF Pigma 31 (1980) 122. F.I. Dalidchik and V.Z. Slonim, Zh. Eksperim. Teor. Fiz. 79 (1980) 2126. A.M. Brodsky and M.I. Urbakh, Fiz. Tverd. Tela 17 (1975) 2669. V. Bardi, A. Glachant and M. Bienfait, Surface Sci. 97 (1980) 137. A.M. Brodskii and M.I. Urbakh, Phys. Status Solidi (b) 76 (1976) 93.
L468
SURFACE SCIENCE LETTERS
THE ROLE OF FRENKEL EXCITONS IN LOW-ENERGY ELECTRON INTERACTION W I T H M O L E C U L A R A D F I L M S F.I. D A L I D C H I K a n d V.Z. S L O N I M Institute of Chemical Physics, Vorobevskoe Chaussee 26, Moscow 117334, USSR
Received 12 June 1981; accepted for publication 29 March 1982
The problem of the interaction of low-energy electrons with molecular films near a solid is discussed. It is shown that, as a result of the electron-metal interaction, the new "nonelastic" channel of electron-exciton complex decay is possible, during which the electron-exciton pair dissociates and the electron transmits into the metal. We have analysed the possible shapes of the resonance features observed in the experiments on the transmission and the reflection of the low-energy electrons from the surface of a solid covered by a monolayer molecular film.
Low-energy electron scattering at certain energies is k n o w n to occur through the f o r m a t i o n of metastable negative ions (see, e.g., refs. [1,2]). There are r e s o n a n c e features in the energy d e p e n d e n c e of the differential a n d total cross-sections in this case. Their shape is connected with the m e c h a n i s m responsible for the f o r m a t i o n a n d decay of the transient quasistationary state. It was f o u n d that for F e s h b a c h type of r e s o n a n c e * (temporary negative ion states associated with electronically excited states of atoms), the elastic crosssection could be described by the F a n o relation [3]:
o(E)
(~+y)2 1+~
E _ _ Er
,---
F
(1)
where o0 is the n o n - r e s o n a n t part of the cross-section, E the electron energy, E r the resonance level energy a n d y the "profile index", and F is inversely p r o p o r t i o n a l to the m e a n life time of the transient negative ion. A n a l o g o u s experiments performed recently [4,5] with solid targets (in particular, the investigations of the interaction of low m o n o e n e r g e t i c electrons with molecular adfilms) were most useful a n d informative. The high resolution studies by Sanche [4] have d e m o n s t r a t e d resonance structures, correlated in energy with gas-phase values, b u t their shape differ sufficiently. The symmetric r e s o n a n c e m a x i m a were observed in all k n o w n experiments in c o n t r a d i c t i o n
* The classification of the electron-atomic (molecular) resonances is given, e.g., in the review by Schulz [2]. 0 0 3 9 - 6 0 2 8 / 8 2 / 0 0 0 0 - 0 0 0 0 / $ 0 2 . 7 5 © 1982 N o r t h - H o l l a n d
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L469
with formula (1). It was proposed in ref. [4] that the Fano line shape change can be explained in terms of the different decay mode of a transient negative ion in solid. It was supposed that the autodetaching electron does not decay into vacuum as it does in the gas phase, but returns to the conduction band where it has a high probability of being recaptured in a nearby molecule. This interpretation is not complete, because the coherent electron-atom interaction in the lattice is not taken into account. It is known that the monolayers of adsorbed atomic paraticles form - at low temperatures - regular structures, called two-dimensional crystals [6,7]. (The regular structure of admolecules in the systems studied by Sanche [4] is confirmed by recent experiments [8]). All states, including the quasistationary ones, in the sytems with translational symmetry are collectivized over centres. The weakly decaying states (so-called autoionizing bands, AIB) emerging in the case of quasistationary states obey the complex nonanalytical dispersion law [9-12]:
E( q) = Eo(q) - i F ( q ) ,
(2)
where q is the electron quasimomentum in AIB. In particular the Feshbach type resonances form the AIB of the bound.electron-exciton pair (electron-exciton complex, EEC), and its dispersion law imaginary part (F(q)) defines the mean EEC life time to electron detachment after its absorption of the exciton energy [11]. The resonance features due to formation and decay of such states which could be detected in electron scattering experiments must be described b y formula (1) with parameters ~ and y depending on q [12]. These results were found [11,12] while neglecting the electron-substrate interaction. This is justified only if the distance between the adfilm and the metal is large enough. The purpose of this paper is to investigate the effect of the substrate and its influence on the change of dispersion relation of the EEC and on the resonance line shapes. The algebraic equations system describing an electron interaction with a two-dimensional ordered lattice of two-level atoms in the presence of the external field was obtained in ref. [12]. In low-energy approximation it is: [x 0 - 2¢rG(v+)(E, Rs, Rs)] Ts(0 ) - 27r ~ G(v+)(E, Rs, Rs,)Ts,(O ) s'4=s
(2~-) 2
fdp exp(it,o~)~(t,) = 2~x(E, k, R~),
Rs, Rs,)Ts,(p)
(3a)
(K'--~°)a°fdp'exp(i(p' plo, lT~(p')
+x0, e x p ( - - i p p s ) r s ( o ) = 0.
(38)
F.I. Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L470
R , = (p,, zA) are the radius vectors of adatoms, oa(p) is the exciton dispersion law, and p = (p<,p~., 0) its two-dimensional quasimomentum; the integration is restricted to the first Brillouin zone. G~v+) is the electron Green function in the external field V(r),
(
C'~v+)(E'r'R):,-klim G~v+)(E,r,R)
,
2rrlr_Ri
,
(4)
where ~¢0, ~l and ~¢01 are the parameters characterizing the low-energy electron scattering by an isolated centre [1], ~20 is the area of a unit cell of the adatom lattice, and x(E, k, r) is the electron wave function corresponding to a wave incident on the metal surface from outside. The values ~ ( n ) in (3) are the amplitudes of the spherical diverging waves in corresponding channels. The sytem (3) could be easily analyzed. The two first terms of (3a) describe an electron elastic scattering by the system of unexcited atoms. The term proportional to ~0, describes the exciton creation possibility. The first two terms of (3b) describe scattering of the electron by the lattice with a free exciton. The term proportional to (K I - x 0 ) describes the electron-exciton scattering. The EEC formation is due to this very interaction. The last term of (3b) takes into account the EEC annihilation possibility. The electron-substrate interaction is supposed to be potential and described by the function V(r). One can neglect the dependence of V(r) on transverse coordinates x and y in the low-energy electron case, which is usual for the EEC states. Then: 0, V(r)=
V(z),
if if
z--+-~, z>0, V(oc)=V0<0.
(5)
The crystal boundary z = 0 is assumed to be so selected that for z > 0 the potential coincides with that of the substrate. We can write two sets of wave functions: x ( E , k, r) = ~ e x p ( i k p ) c p ( E ,
(6)
kll, z),
ffl(+'( E, kbl, z) -- S l l ( E , k~, )/i<-)(E, kii, z),
~( e, k,,, z)= ~[s,2(e, w(E,kll,Z ) =
k,,)/; + >( e,k,,,z )
#-'(E,kii,z)
z>0,
- S22( E,k,i) #+'( E,kii,z ),
S2,(E,k )i,<-'(E,k~,z),
lim Z ~
-I- ¢,O
i2<+ )( E,kll,
z)exp( z ) e x p [ -7-i~/2(EV0)-/~;'-
z>0, z~0,
i,--(i,,/,.), k,,--(k~,kv),
,Lrn J,<
z<0,
k~z]
= 1.
(7)
(8)
F.I. Dalidehik, V.Z. Slonim / Role of Frenkel excitons
L471
The reflection and transmission coefficients S,j can be defined by characteristic values of the surface and volume bands of the semiinfinite crystals [13,15]. The Green function G(v+) in (3) is
G~v+)(E,r,r,)_
_
1 fdqeiq(o-°"G(v° E - - ~ , z , z ' . (2~) 2 _
(9)
The method of the one-dimensional Green function construction is well known:
G(1) -
2 [ep(z)z(z')O(z-z')+~p(z')z(z)O(z'-z)], w[~,~] where O(x) is the step function, and W[¢, ~] = ~ d ~ / d z - ~ d ~ / d z
(10)
is the Wronskian of the functions ¢p an ~-. The translation symmetry of interactions permits to solve the system (3) in the form: T~(0) = T(0) exp(ik r,Ps),
(1 l a)
rs(p)= T(p)exp[i(k,,-p,,lPs ] .
(llb)
By means of (11) the system (3) can be transformed into the inhomogeneous integral equation for T(0), which can be solved simply:
V(O) =
~( E'kl"zA)[1-~- (Kl -- X°) O( E'kll)]
(12)
z0( E.. ){ l + [~,- ~0- ~0~,/Zo( E. k,,/] D(~. k , ) } ' T(p)
:
-
~o, r ( o )
Zo( E_c,.o(p),kl _ p H ) [ i ..~ (Kl _Ko) O( E,kfl)] ,
(13)
where
Zo(E,q)=~o-2~+~(E,z~,z~)-2~ ~, G~+~(E,R,,0)exp(iqR,), (14) R,v~0
D(E,q)-
~-~0
(2~r)2
fBzdp { Z o [ E - - w ( p ) , p - - q ] )
- I
(15)
The spectrum of EEC states in regular adlattice is defined by roots of the dispersion equations:
1 + [1¢, -- x o - x~./Zo(E,q) ] D(E,q) = 0.
(16)
The series included in Z o and D can be calculated by the method described in refs. [9,12]. Inserting (7)-(10) into (14) and taking into account the formula 2 exp(ik',P~) S
(2rr)2 ~ 8 ( k l l - - g ) , no
(.17)
F.1. Dalidehik, V.Z. Slonim / Role of Frenkel excitons
L472
where g is a two-dimensional reciprocal wave-vector we have: Z o ( E , q) = Zo'°)( E, q) + Zo~"(E, q).
(lS)
Z~o°)(E,q) is given by (28a) of ref. [12], the second term is due to electron-substrate interaction:
(2E(q + g) E ( q + g ) = E - l ( q + g ) 2, ZA
D(E, q) in
8V(E,q) ~ [ SI{-)(E - ~min(q), 2"A)] 2,
(20)
and it can be neglected while calculating the real part of the EEC dispersion law. In this case the small parameter is the ratio of the inverse EEC life time to the width of AIB. Calculation of the damping, i.e. F(q), requires the imaginary part of D(E, q) to be taken into account although it is exponentionally small. Actually, if the electron energy is insufficient for the exciton creation the dispersion function D(E,q) is real only if the substrate is not taken into account or Im S~ ~(E, q) ~ 0 * (valid for some dielectric substrates if the energy corresponds to the band gap). In this case the EEC decay can take place only as a result of the absorption by an electron of the exciton energy and following detachment of the electron from the structure. If ImStl(E,q)d=O the new "nonelastic" channel of EEC decay is possible, during which the electron-exciton pair dissociates and the electron transmits into the metal by tunneling. In this case the exciton does not annihilate and remains in the lattice. Taking into account all these considerations, we find, if E < ~mm(P):
ImD(E,q) 1
, Im Sl, [E(q + g) - w( p)] { f l ( - ) [ E ( q
fop
+ g ) -- ~( p)]
)2
.......................... V2[,o(p)- E( q + g)] [Zo[E( q + g ) - ,o( p),q + g - p ] l
(21) Iterating (16) by the imaginary terms, we find: EEEC(q) =
Eo(q) -- i F ( q ) ,
(22)
* It should be emphasized that the regular adsorption on the dielectric substrate (NaCI) was observed by LEED in ref. [14]. The energy and angular dependence forefeld for a monolayer in ref. [12], most probably could be verified for this very system.
F.L Dalidchik, V.Z. Slonim / Role of Frenkelexcitons
where
L473
Eo(q) is the zero of the equation:
1+(x l-x0)ReD(E,q)=0, F(q) --- VAi(q ) +
Ft)(q),
(23)
FAt(q ) and I'D(q) are defined by
(x, •21 --Ko) 2 ( A R e D d
-"'e=Eo
iZo(Eo(q),q)lZ '
-1
FD(q) : ( - ~ R e D )e=eo(q)Im D(E0(q),q).
(24) (25)
Thus the parameter defining the competition of dissociative and autoionizing decay channels is the ratio:
#(q) = VD(q)/rAt(q ),
(26)
which can be more than unity (see below). Let us discuss the importance of the "dissociative" channel of EEC decay in experiments on the interaction of the low-energy electrons with a solid surface, covered by a monolayer of the regular structure of the atomic particles. It is clear that any additional imaginary part of F(q) changes the resonance feature shape of curves of the energy and angular dependences of the visible values. The electron-lattice system wave function is
~p= IO)¢~o(E,k,r) +
dPlP)~p(E--o~(p),k--p,r ),
(27)
where J0) corresponds to the nonexcited lattice and ]p) to the lattice with an exciton with a quasimomentum p. For our model the electron wave functions are
*o(E,k,r)=eikO(rp(E,kl, ' z)
2~rT(0) ~
~ e -i(k-g)p g
G(I) v [E(kll+g),ZA,Z]),r:(p,z),
p:(x,y),
(28)
dPp(E,k,r)-- 2"rrT(p) 2ei'k-g-t"r'Gg'[E(kl, + g ) - w ( p l , z A , z ] ~"~0
(29)
K
It is supposed that the lattice was in the ground state 10) before the scattering of incident electrons with momentum k. The transmission current found with (27) is ~20 J , r ( E , k ) = J t ° r ( E , k ) + ( - ~ ) 2 f , zdpS, r(p - - ,k, , E - -
J,°(e,,): 2rc,[el,,, g
8o, i %[e(,,,
~(p)),
(30)
(31)
L474
F.L Dalidehik, V.Z. Slonirn / Role of Frenkel excitons
i = ~/"2[ E(k,, + g ) -
1/Io] $12 [ E(k,, + g)]l 2,
(32)
The partial inelastic transmission current Jtr(P - kit) can be described by the expression (31) if E(kll + g ) , kll is substituted by E(kl~ + g ) - - o a ( p ) , k l l - p and 80s is omitted. It can be shown that the transmission N(E,k) and reflection R(E,k) coefficients calculated by (27) satisfy the unitary condition:
N( E,k ) + R( E,k ) = 1,
(34)
where the transmission and reflection coefficients are defined by
N(E,k)=Jtr(E,k)[2E(kl,)]
1/2 ] 1/2
R(E,K)=Jr(E,k)[2E(klI)
(35a)
E •/..0min(p)"
(35b)
The reflectivity has the simplest form for the case E < Wmm(P); it is T(0)
'~(e(kH), zA)
R(ek)=lS'l[E(k)] t211 ao Sll[e(kt]w[
,,] 12
+ IT(0)I 2 EO[E(k+ g ) ] ~ / E ( t , , + g ) a2
X
g
"
E(k,,)
cp[E(k"+g)'zA]
(36)
The results of our calculation of the reflection coefficient based on (36) are given in fig. 1. Let us consider the resonance feature shape in the neighbourhood of the resonance energy Eo(k11). Bearing in mind that
r(o)
[eo(kH) z ][e eo( t]
(37)
E - E o ( k ) + i[I'~(k ,) + rD(k,~)] ' we can write 2
R(E,k)~ISI,[E(k,,)] [ 1--~[E(k,,)]
E ~ Eo(k,~ ),
,(k,,) + y ( k , , ) +i,8(kl, ) 2, :~;)+i[l+/~(k,,)]
(38)
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
L475
[tee. ax3 un
i
i
i
7,49
i
i
e, 5 0
z:~ev
g~i
Fig. 1. Energy dependence of the elastic reflection coefficient from a monolayer of Xe atoms physisorbed on a metal surface. Curves a and b correspond to incident angles 0 =0 ° and 0 =5 ° respectively. The beam orientation to the crystallographic axes of the Xe lattice is the same as in ref. [12]. The values x0=-0.17, Km=0.20, x021=1.36×10 2 0%=0.30 (au), II0=4.5 eV and IzAl=3.2 A were used in calculations according to eq. (6). The exciton dispersion law in the two-dimensional hexagonal closely packed lattice atoms was taken as: w ( p ) = w0 +2/3[cos(pxa) +2 cos(~pxa ) cos(½ V/3pva)], with/7 =0.056 eV, a = 5.5 ,~. The electron transmission coefficient Sml was taken as [
=
+
].
where
ReZo( E,kli) E- Eo(kl,) "Y(kiil=imZo(E,kH), E(kH)-- FAI(k,,) ' a n d (I)(E, k H) is a slow function of its a r g u m e n t s in the given interval of energy. I f / 3 << 1 (it means, e.g., that the s u b s t r a t e can be neglected) and
R(E,k) ~-IS,,[E(kr,)]l 2 ( 'l ++c
.y)X 2
(39)
T h e line shape suits the relation p r e d i c t e d b y F a n o . T h e r e is a shear r e s o n a n c e m i n i m u m of the reflection coefficient if /3 >> 1 (see fig. 1). It follows from eq. (38) that there is a r e s o n a n c e m a x i m u m in the t r a n s m i s s i o n coefficient i f / 3 >> 1. (It should be e m p h a s i z e d that high collimation of the b e a m is necessary to observe the fine structure p r e d i c t e d by this theory. T h a t is w h y the a n g u l a r divergence leads to the r e s o n a n c e structure averaging a n d o n l y m a x i m u m s survive for o b s e r v a t i o n . ) The shape of this m a x i m u m can be d e s c r i b e d b y a f o r m u l a of the B r e i t = W i g n e r type: r~i(ki,) rD(ki,)
N( E, k) ~ [E-
Eo(k,,)]~ + [rA,(k,, )+ rD(k,,)] ~'
(40)
L476
F.L Dalidchik, V.Z. Slonim / Role of Frenkel excitons
where FAl(kll ) is proportional to a partial width of the input channel, corresponding to the formation of the EEC by the electron travelling from the vacuum to the adlayer.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
G.F. Drukarev, Stolknovenia Elektronov s Atomami i Moleculami (Moscow, 1978). G.J~ Schulz, Rev. Mod. Phys. 45 (1973) 378. U. Fano, Phys. Rev. 124 (1961) 1866. L. Sanche, J. Chem. Phys. 71 (1979) 4860. T. Huang and W.H. Hamill, J. Phys. Chem. Solids 36 (1975) 661 L.A. Bolshov, A.P. Napartovich, A.G. Naumovets and A.G. Fedorus, Usp. Fiz. Nauk 122 (1977) 125. J.P. Hobson, in: Surface Science, Recent Progress and Perspectives (CRC Press, Cleveland, OH, 1974). L.E. Firment and G.A. Somorjai, Surface Sci. 84 (1979) 275. F.I. Dalidchik, Zh. Eksperim. Teor. Fiz. 77 (1979) 2422. G.V. Golubkov, F.I. Dalidchik, and G.K. Ivanov, Surface Sci. 93 (1980) L89. F.I. Dalidchik and V.Z. Slonim, ZETF Pigma 31 (1980) 122. F.I. Dalidchik and V.Z. Slonim, Zh. Eksperim. Teor. Fiz. 79 (1980) 2126. A.M. Brodsky and M.I. Urbakh, Fiz. Tverd. Tela 17 (1975) 2669. V. Bardi, A. Glachant and M. Bienfait, Surface Sci. 97 (1980) 137. A.M. Brodskii and M.I. Urbakh, Phys. Status Solidi (b) 76 (1976) 93.