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Charge and spin states of helium atoms approaching metal surfaces
Surface Science 224 (1989) 31-42 North-Holland. Amsterdam
CHARGE AND SPIN METAL SURFACES Hideaki Department Received
KASAI
31
STATES
OF HELIUM
ATOMS
APPROACHING
and Ayao OKI.
of Applied Physics, Osaka Uniwrsity, 31 May 1989; accepted
for publication
Suita. Osaka 565, Jupun 19 August
1989
Charge and spin states of met&able He atoms and positive He ions approaching metal surfaces are investigated with the aid of a simple time-dependent model Ha~itonian. With a simple decouphng approximation, a closed set of Heisenberg equations of motion is derived for the time-dependent quantities describing the charge and spin states of the He atoms. The numerical calculations in the case of simple time-dependence for the He atom-metal interaction. show that the main fraction of incident metastable He atoms changes into positive He ions near the metal surfaces if the work function of the surfaces is high and the Fermi level is located relatively far below the energy level for the 2s orbital of the He atoms. The .~j~~~~et-t~~~~er c~wwersion of incident metastable He atoms as well as the conversion of the incident ions into metastable He atoms occurs efficiently near the surfaces if the work function gets lower and the Fermi level comes nearer to the energy level for the 2s orbital. However, if the work function gets even low and the Fermi level is located relatively far above the 2s level, the characteristic time for the singlet-triplet conversion becomes considerably long.
1. Introduction When a metastable He atom or a positive He ion approaches a metal surface, interaction between the electrons of the He atom and of the metal surface occurs [l--6]. At the beginning, interaction between the 2s electron of the He atom and electrons in the metal dominates, since the wave function of the 2s state, #Zs, spreads as compared with that of the Is state, $is. The charge and spin states of the He atoms are modified in this region. Subsequently, when the He atoms come nearer to the surface, Auger processes [6,7] (Auger neutralization for the ionized He atom or Auger de-excitation for the metastable He atom) take place due to the interaction between electrons in the Is orbital and in the substrate, which leads to electron emission. The kinetic energy distributions of the ejected electrons, which are measured experimentally in ion neutralization spectroscopy (INS) and metastable de-excitation spectroscopy (MDS) [l-6], contain information on the electronic structure of the outermost surface layers and also on the charge and spin states of the He atoms at the time when the Auger process occurs. Hence, in order to reveal the 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
electronic states of the outermost surface Iaver:, preciselv from the l\~nctic energy distributions of the ejected electrons. it is necessarv to clarifv the time-dependence of the charge and spin states of the Hc atoms coming towards the surfaces besides the Auger process itself. As for the Auger procex. detailed investigations have been done already [6.7]. However. the in\,ebtigation of the time-dependence of the charge and spin states of Hc atoms approaching the surfaces seems to be insufficient. in our lien Furtherm<)l-e. III recent experiments [1,2] a phenomenon called “vinglrt -triple1 u~tifw\iot1 ‘* 01 metastable He atoms has been observed in the de-excitation spectra of mct;~l surfaces covered with alkali metal atoms ( K/N i( 111) and (‘~,/C‘LI( I 10) 1. .41\0. the conversion of ions into metastable atoms has been found experimentall! [4]. Hence we consider it to be necessary to investigate the charge and hpin states of He atoms approaching metal surfaces more preciscl!. This i\ the purpose of the present paper. First of all. with a classical trajectory approximation for the motion of He atoms. a simple time-dependent model Hamiltonian is introduced in order to describe the electronic system in question in section 2. Subsequently. an approximate calculation is made for the timc-dcpendence of the charge and spin states for He atoms approaching metal surfaces in section 3. A summary is given in section 4.
2. Model The model Hamiltonian, H=
&plrn
which we investigate,
+h,(t)&,“+
c "
h.0 +u(t)nl,p?,.i
[c;,,(r)C;,C,,,+
is given h\; ll.C.1
h.n
ltJ(r)S,*S,.
(1)
where CL” (Cc,) is the electron creation operator for the state / X) with spin (T and energy eI of the conduction electron in the metal (for the 2s orbital of ;I He atom with spin u and energy cl, (r)), II = C’ C, and k’,\?,(t) is the matri\ element of hybridization between the conduction electron and the electron in the 2s orbital of the He atom. The fourth term represents in the intra-atomic Coulomb interaction for two electrons in the 2s orbital of the He atom. The last term represents the exchange interaction between electrons in the 7\ orbital and in the Is orbital of the He atom. and S, *S, can he rewritten :IS s,
*s,
= & 1 “.O’
C;,“C,\n’C~“.C7\n
-
d c Il,,“HJ,“,. l?.(T’
(7)
where CT,, is the electron creation operator for the 1s orbital of the He atom. We consider the case where a He atom approaches the surface and ~novcx in the region where the interaction between electrons in the 2s orbital of the Hc atom and in the substrate is still appreciable but the interaction hetwccn
H. Kasai, A. Okljl / Helium atoms approachrng
metal surfaces
33
electrons in the Is orbital of the He atom and in the substrate is not. The length of the region, with which we are concerned here, may be roughly estimated as the difference of the spatial extension of the radial parts of the wave function between the 2s orbital and the Is orbital, - 2 A. Since the trajectory approximation is introduced for the motion of the atom, cZs, U, J and VL2, have an explicit time-dependence. It is assumed that at the initial time t = t,,, the conduction electrons in the metal are in thermal equilibrium and that the atom is far from the surface, i.e. I’, zs( to) = 0. In the following we introduce operators such as C:,(t) and C&,(t) in the Heisenberg representation which obey the Heisenberg equation of motion, t) = [C’(t),
i(d/dt)C’(
H(t)].
Here we use units of A = 1. In order to describe the time-dependence of the charge and spin states of the He atom, the following probabilities are considered : P_(t)
= C(~2~“w~*s~o(f)[~ 0
P’(t)
= C([l ”
-n ,&)1[~
P:‘Jt)=C([l-n
,,&>I
n p:’
- ~,s~,wlhAt)>~
- ~;s-owl
%-&)[l
(34
[1 - ~,s-owl
- %-0(t)]
n,,,(t)>>
n,,“(t)>?
TCl)= C(%\&)[l - %~&)I 11- %&)I fh,(tb
G(t)= C;c,“~t~c,,~“~r)c.,l-O~t~(l_R(i~~. n
(3b)
(3c)
(34 04
Here P-(t), P’(t) and Py -1(t) (Py ?(t)) are the fractions, at time t, of metastable negative ions, positive ions and metastable neutral atoms, of which two electrons have antiparallel (parallel) spin with each other. The quantity G(t) represents the “spin flip” probability at t. It is noted that the sum over four kinds of P gives C(n,,,(r)[l 0
-%&>I>
= I.
Since we consider the case where a metastable He atom or a positive He ion approaches the surface and still moves in the region where the Auger process does not take place, the electron number of the 1s orbital is always one. The average value of S, *S, can be written as (S,(t).S,(t))=~G(t)+t[P!:,(t)-P:,(t)].
(4)
With the use of the equation equations can be derived, $F(t)=
of motion
method
[8], the following
CV~lh(r)[FZ;(r)+t~~(r)] x
-4A(t)F(t)-21m
(53)
Xexp[ic,(t-Z,,)], ;Il;P;&)=
-2A(t)P:,(t)+2A(r)Pm(r) -2Im
$P:Jr)=
C~z,(r)[F,,(r)-F,:(t)]
++(r)
C~2,(f)[~~(~)-G(~)]
=2A(r)P!
1
(5b)
exp[ic,(r-r,,)].
(SC)
t (r)
(t) +2A(t)P:'
i-2113-1 C~2,(r)[~~(~)+t~(r)]
&G(r)=
-2A(t)G(t)-2Im
Here F;;(t),
F,;(t)
ew[ic,(r-r,,)].
CCVk2,(t)F;1k(f) 0 h
and F;‘,(t)
(a = k) are defined
= C(c:“(t,,)n,,,~“(t)c,.~“(t)c,‘,
The quantity ,l,,n([)
exp[i~L(r--t(j)].
-2A(t)P’,‘,(t)+2A(t)Pp(r) -2Im
F;;(r)
set ol
=
n_(t)
b .(f)
\l
-n,,_,(t)
is independent
of 6.
(5~)
by
.(~Kk”(r)).
(ha)
is given by for cy =
+,
for(Y=
-.
It is assumed that Vkz,(t) is independent width A(t) for the 2s orbital defined by ~(r)=~Civ~?\(f)126(~-c~).
exp[ie,(t-r,,)].
(5d)
(7) of k and that the resonance
level
(8)
35
H. Kasai, A. Okiji / Helium atoms approaching metal .surfaces
With the use of a decoupling c ([l
- %“(4]
0
the following :F,(f)
k,(4[1
equation
-
approximation,
~hW1 G&0)G”(f0))
can be derived
= D(t)F,(t)
for FP, (t),
FFk (t) and
FTk(t ),
exp[i~A(~-4dlQW.
-iIL(f)f(ek)
(9)
Here, (F:,(t)
F,(t) =
&i(t) F&(t>
(loa)
&Y(t)
F;;(t) G (t> \ -G(t) G(t) p-(t) + py (t> 1
Q(t)
=
P+(t)+Pj:T(t)
(lob)
’
P’(f) + p’: (t> p-(t) + py T (4/ 1
-
i[e’(t)
A(t),
-i[e-(t)
- iiJ(t), D(t)
+ aJ(t)]
A(t),
A(t),
- +.~(t)] -2A(t),
0, -i[e’(t)
= 0, 0.
-2A(t),
+ aJ(t)]
-i[c-(t)
0, -i:J(t),
0, 0, -i[EC(t)
0, 0,
0, 0, A(t),
0, 0, 0’
0, 0, -i:J(t), -2A(t),
+ aJ(t)] - :J(t)] -i[c’(t)
-iiJ(t),
A(t),
-2A(t), -2A(t). - +~
0 0, 0 0. 0
’
(1Oc)
A(t) -2A(t)
1
r’(t) = czS(t) + U(t) (CC(t) = ez,(t)) and f(ek) is the Fermi function. Eq. (5) together with eq. (9) constructs a closed set of equations which can be solved numerically with the appropriate initial conditions for P-(t,), P+(to), I’: L (to), Py ? (to) and G(t,). In the case of U(t) = J(t) = 0 the above set of equations can be solved analytically and one can get the exact results for P-(t), P’(t), Py 1 (t) and Py T (t) for the decoupled equations.
Although eqs. (5) and (9) can be solved numerically for any time-dependence of the parameters involved. the results of numerical calculations arc shown in this paper for a simple time-dependence of a He aton~~metal interaction [8.9]; the interaction is limited to a region of finite distance from the surface mentioned above, where Vn?,( t ). c?,( t ). U( t ) and J( t ) are assumed to be constant. Then it is enough to follow the time-dependent probahilitie\ after sudden switching-on of VL2,( t) and hence 3( t ):
-l(t)JO
forr
(11)
fort>O.
In this case one can derive manipulation of eq. (9).
= -i+(m[Q(t)il)+
X
expression
for t > 0 after ;I simple
i f: (fi?IRII) /=I rZ=l
f dr,ATexp[h,(t /0
the following
- t')]
sinh[ rT( t’ ~
t
(IIR~ ’ In)(/?lu(t’)
)]
11).
where X, (I = 1,. . ,6) is the eigenvalue of D. which is independent of t now. and R is a matrix consisting of the corresponding eigenvector Y,. (I 1X 1HI) represents (1. m) element of matrix X and T is the surface temperature. Before leaving this section, the results of numerical calculations are shown in fig. 1 for the case of J(t) = 0. In this case, the present Hamiltonian ia the same as that for the time-dependent NewnssAnderson model. for which simple decoupling approximations were introduced [&lo]. With the aid of the timedependent Newns-Anderson model, one could investigate the time-dependence of the probability of charge states and of spin polarization for an atom moving near a metal surface [8.10,11]. From the comparison of the numerical results for the time-dependence of spin polarization M( t ) = (II ‘, , ( t )) (JZ?,I (t )). the present approximation can be considered to correspond to the improved one #’ for the time-dependent Newns-Anderson model.
3. Numerical
results
On the basis of the theoretical method developed in the previous xection. the time-dependence of charge and spin states of metastable He atoms (He* ) =’
In
a closed
model).
set
of
equations
for
three
kinds
of scattering
of the moving
atom
are taken
[R.lO]).
the
time-dependent
process into
account
prohahllitiea
of conduction within
electrons
a simple
of hy
decoupling
being the
charged
electron
approxitmrtion
in
(~rnpro\ud rhe
orhlt;ll
(WC
rcf\.
37
H. Kusai, A. Okqr / Helium atotm approuching metal surfuces
0.0
L
0
I
1
a
t (WA)
1
2
3
Fig. 1. Time-dependence of spin polarization M(t). The numerical calculations are made for M(0) =l by use of eqs. (5) and (12) (solid line). eqs. (46)-(48) in ref. [7] (dashed line) and the improved equations (see refs. [8,10] and #‘) (crosses). The parameter used is U/A = 4 (6 2S= - U/2 measured from the Fermi level) at the absolute zero of temperature.
and positive He ions (He+) interacting with metal surfaces is investigated numerically. The quantities involved in the model for a He atom in an isolated state are given as czs = -4.6 eV (measured from the vacuum level), J = -0.8 eV (ferromagnetic coupling) and U = 4.5 eV (cZs + U corresponds to the these quantities are affinity level for a metastable He atom *2). Although modified as the atom approaches the metal surface, the modifications for J and U are neglected for simplicity in the following. 3. I. He * approaching
a surface
The time-dependence of the charge and spin states is shown for a metastable He atom initially in the single state (‘S He*) at the absolute zero of temperature in fig. 2. In the region near the surface we take A = 0.2 eV as an average value [3,4]. According to the experimental situation [1,2], the case is considered where the Fermi level cr. (taken as the origin of energy) is located between the effective 2s level of eZs - :J + + (for ‘S He*) and of c2\ + +J + $ (for “SHe”, a metastable in the triplet state). Here + is the work function of the metal surface. As one can see in the figure, P,(t), the fraction of ‘S He*, increases (P,(t), the fraction of ‘S He*, decreases) and reaches gradually its equilibrium value. It can be considered that the singlet-triplet conversion *’ The binding energy of the (1~2~2~) state of He- relative to the binding chosen as the value of the electron affinity of He* (see ref. (121).
energy
of He*
is
.6
Fig. 2. Time-dependence of Pm. P+. P, and zero of temperature. In the region near the surface we take c2, = -0.25 eV measured from the Fermi level as an average value [3.4]. The case of ‘SHe* approaching a metal surface is considered. The distance between positions of the atom at tWo different times, / and I + A/d, becomes 0.056 I\ (0.79 k) for E,,,, = 60 meV (12 eV). where E k,n is the kinetic energy of the atom normal to the surface. P, at the absolute
Fig. 3. Same as fig. 2 hut for a high-lctwkfunction metal (the work function is 0.75 eV higher than that in the case of fig. 2).
occurs efficiently in this case [1,2]. Here, P,( t ) and P,( t ) are given as ,\ ,\ ,> P,(t)=P;‘,(t)+[P;‘,(t)+G(t)]/2 and P,(r)=[P;‘,(t)-G(r)]/2, respectively. As soon as the interaction is switched-on and hence the 2s level of the He atom has a finite width of - 26, the electron in the He atom having an energy greater than eF starts to transfer to the metal surface. Hence the charge and spin states of the He atom are modified and the He atom reaches gradually its equilibrium charge and spin states. It can be considered that the memory for the initial spin state as well as for the initial charge state of the He atom practically vanishes near the surface if the velocity of the He atom is slow. By the numerical results shown in fig. 3 for the case where the work function of the metal surface is high as compared with that in the case of fig. 2 and cF IS located below the effective 2s level of ‘SHe” atom, it is confirmed that charge transfer occurs efficiently from the He atom to the metal surface and hence the main fraction of the ‘S He * atoms changes into He’ ions near the surface. The time-dependence of charge and spin states is shown in fig. 4 for a ‘S He* atom interacting with a metal surface of low work function as compared with that in the case of fig. 2. Since the number of electrons in the He atom having an energy higher than cF is small as compared with that in the case of fig. 2. little charge is transferred from the He atom to the metal
H. Kasal. A. Oki/i / Helium
Fig. 4. Same as fig. 2 but for a low-workfunction metal (the work function is 1.2 eV lower than that in the case of fig. 2).
atoms approachmg
metal surfaces
Fig. 5. Same as fig. 2 but proaching a metal surface. Py t ( t), the fraction of 3S He is shown as a function of t jSHe* initially with Sz = f S;=O(.““).
39
for ‘SHe* apThe probability * with .S, = + 1, for the case of 1 ( -) and
surface. Hence the decay rate to the equilibrium charge state from the initial charge state becomes considerably fast as compared with that in the case of fig. 2. As for the decay rate for the spin state, the electron in the 2s orbital of the He atom having, say, up-spin (unfavourable for the exchange interaction with Is electron spin) continues interchanging with the electron in the metal, which has a favourable spin, say down-spin, even if the He atom has already reached its equilibrium charge state. Hence, as one can see in the figure, it can be said that the decay rate to the equilibrium spin state from its initial spin state is considerably slow as compared with that for the charge state. It can be considered that the main fractions of the incident He atoms would retain memory for the initial spin state near the surface in this case if the velocity of the He atoms is relatively fast. As one can see in figs. 2 and 4, P-(t) takes a finite value near the surface and increases as the work function of the surface decreases. In fig. 5 the time-dependence of charge and spin states is shown for ‘S He* at the absolute zero of temperature. The probability P,(t) decreases (P,(t) increases) from its initial value as the interaction between the He atom and the metal surface is switched-on and tends gradually to its equilibrium value. As for P’(t): at the beginning (0 < t -C h/A) it takes a value smaller than in the case of fig. 2, because the effective 2s level for 3S He * is lower than for ‘SHe*. However, it tends gradually to the equilibrium value which is the same as that in the case of fig. 2. It is noted that the quantities Z’,(t), P,(t), P-(t) and P’(t) do not show any dependence on the magnetic quantum number of an incident ‘S He* atom, S7 = k 1 or 0. Although PT T (t), the
40
t (h/A) Fig. 6. Same as fig. 2 hut for He+ approaching
a metal
surface
probability of ‘S He * with S= = + 1. depends strongly on the initial magnetic with the recent quantum number of ‘S He* as shown in fig. 5. In connection experiments on the surface magnetism with the use of spin-polarized ‘S He* atoms [5], it would be interesting to extend the present calculations to the cast of spin-polarized ‘S He * atoms interacting with the surface of the magnetic metals. 3.2. He + upproaching
a surface
The time-dependence of the charge and spin states is shown in fig. 6 for ;I positive He atom (He+) at the absolute zero of temperature. As soon as the interaction is switched-on, charge transfer occurs from the metal to He ’ and hence He+ converts into a neutral metastable atom with a finite probability. The main fraction of the metastables is ‘S He*. Nevertheless, a minute in accord with the experimental fractions are ‘S He *. This is qualitatively findings [4]: it is suggested that a minute fraction of incident He’ ions turns into ‘SHe* before electron ejection through the Auger process. although the electrons are ejected mainly by He+ converted into ‘SHe”. However. in the case where the work function of the metal surface is high and hence the Fermi level of the metal surface is located relatively far below the energy level for the 2s orbital of He, the conversion of He+ into He* is suppressed and the Auger neutralization process to the ground state of He occurs at distances close tt) the metal surface [4]. 4. Summary In this paper the time-dependence of the charge and spin states of ‘S Hc*. “SHe* and He+ approaching metal surfaces is investigated with the aid of a
H. Kusui, A. Okijr
/ Helrum
u1om.y approuchrng
merul .surfuces
41
simple time-dependent model Hamiltonian. The equation of motion method is used for the time-dependent quantities describing the charge and spin states of the He atom. A decoupling approximation is introduced in order to get a closed set of equations for the above quantities. The numerical calculations are performed for simple time-dependence of the parameters involved. It is shown, as has been expected, that the memory for the initial spin state as well as for the initial charge state of a He atom has a tendency to vanish if the interaction is appreciable between electrons in the 2s orbital of the He atom and in the conduction band of the metal surface. In the case where the work function of the metal surface is high and hence the Fermi level of the metal surface is located relatively far below the energy level for the 2s orbital of the He atoms, the main fraction of the incident metastable He atoms turns to positive He ions near the surface. If the work function decreases e.g. because of adsorption of alkali metal atoms on the surface and if the Fermi level comes nearer to the energy level for the 2s orbital, the singlet-triplet conversion occurs efficiently and the main fractions of the incident ‘SHe* atoms or He+ ions change into ‘S He* atoms near the surface. However, if the work function gets even lower and the Fermi level is located relatively far above the energy level for the 2s orbital, the characteristic time for the singlet-triplet conversion becomes considerably long. B;i the numerical calculations at finite temperatures, it turns out that there is no surface temperature effect on the charge and spin states of He atoms interacting with metal surfaces in the physically relevant temperature range.
Acknowledgement
This work was supported in part by the Grant-in-Aid for Scientific Research on Priority Areas and for General Scientific Research both from the Ministry of Education, Science and Culture of Japan.
References [I] J. Lee, C. Hanrahan. J. Arias. F. Bozso, R.M. Martin and H. Metiu. Phys. Rev. Letters 54 (1985) 1440. [2] B. Woratschek, W. Sesselmann. J. Kiippers, G. Ertl and H. Haberland, Phys. Rev. Letters 55 (1985) 1231. [31 U’. Sesaelmann, B. Woratchek, J. Kiippers and G. Ertl, Phys. Rev. B 35 (1987) 1547. [4] H.D. Hagstrum. P. Petrie and E.E. Chaban, Phys. Rev. B 38 (1988) 10264. [Sj M. Onellion, M.W. Hart. F.B. Dunning and G.W. Walters. Phys. Rev. Letters 52 (1984) 380; M.W. Hart. M.S. Hammond, F.B. Dunning and G.W. Walters. Phys. Rev. B 39 (1989) 5488. [6] H.D. Hagstrum. Phys. Rev. 91 (1953) 543; 96 (1954) 336. [7] H. Kasai, W. Brenig and W. Hiibner, Surface Sci. 208 (1989) 487; W. Hiibner, W. Brenig and H. Kasai, to be published.
42 [X] [9] [lo] [ll] [121
H. Kasal. A. Okr/l / Helum utmm approuc~hrngnw~l surfacer H. Kasai and A. Okiji. Surface Sci. 183 (1987) 147. K. Makoshi. H. Kawai and A. Yoshimori, J. Phys. Sot. Japan 53 (1984) 2441. H. Nakanishi. H. Kasai and A. Okiji. Surface Sci. 197 (1988) 515. A. Yoshimori and K. Makoshi, Progr. Surface Sci. 21 (1986) 251. A.V. Bunge and C.F. Bunge, Phys. Rev. A 30 (19X4)2 179.
CHARGE AND SPIN METAL SURFACES Hideaki Department Received
KASAI
31
STATES
OF HELIUM
ATOMS
APPROACHING
and Ayao OKI.
of Applied Physics, Osaka Uniwrsity, 31 May 1989; accepted
for publication
Suita. Osaka 565, Jupun 19 August
1989
Charge and spin states of met&able He atoms and positive He ions approaching metal surfaces are investigated with the aid of a simple time-dependent model Ha~itonian. With a simple decouphng approximation, a closed set of Heisenberg equations of motion is derived for the time-dependent quantities describing the charge and spin states of the He atoms. The numerical calculations in the case of simple time-dependence for the He atom-metal interaction. show that the main fraction of incident metastable He atoms changes into positive He ions near the metal surfaces if the work function of the surfaces is high and the Fermi level is located relatively far below the energy level for the 2s orbital of the He atoms. The .~j~~~~et-t~~~~er c~wwersion of incident metastable He atoms as well as the conversion of the incident ions into metastable He atoms occurs efficiently near the surfaces if the work function gets lower and the Fermi level comes nearer to the energy level for the 2s orbital. However, if the work function gets even low and the Fermi level is located relatively far above the 2s level, the characteristic time for the singlet-triplet conversion becomes considerably long.
1. Introduction When a metastable He atom or a positive He ion approaches a metal surface, interaction between the electrons of the He atom and of the metal surface occurs [l--6]. At the beginning, interaction between the 2s electron of the He atom and electrons in the metal dominates, since the wave function of the 2s state, #Zs, spreads as compared with that of the Is state, $is. The charge and spin states of the He atoms are modified in this region. Subsequently, when the He atoms come nearer to the surface, Auger processes [6,7] (Auger neutralization for the ionized He atom or Auger de-excitation for the metastable He atom) take place due to the interaction between electrons in the Is orbital and in the substrate, which leads to electron emission. The kinetic energy distributions of the ejected electrons, which are measured experimentally in ion neutralization spectroscopy (INS) and metastable de-excitation spectroscopy (MDS) [l-6], contain information on the electronic structure of the outermost surface layers and also on the charge and spin states of the He atoms at the time when the Auger process occurs. Hence, in order to reveal the 0039-6028/89/$03.50 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)
B.V.
electronic states of the outermost surface Iaver:, preciselv from the l\~nctic energy distributions of the ejected electrons. it is necessarv to clarifv the time-dependence of the charge and spin states of the Hc atoms coming towards the surfaces besides the Auger process itself. As for the Auger procex. detailed investigations have been done already [6.7]. However. the in\,ebtigation of the time-dependence of the charge and spin states of Hc atoms approaching the surfaces seems to be insufficient. in our lien Furtherm<)l-e. III recent experiments [1,2] a phenomenon called “vinglrt -triple1 u~tifw\iot1 ‘* 01 metastable He atoms has been observed in the de-excitation spectra of mct;~l surfaces covered with alkali metal atoms ( K/N i( 111) and (‘~,/C‘LI( I 10) 1. .41\0. the conversion of ions into metastable atoms has been found experimentall! [4]. Hence we consider it to be necessary to investigate the charge and hpin states of He atoms approaching metal surfaces more preciscl!. This i\ the purpose of the present paper. First of all. with a classical trajectory approximation for the motion of He atoms. a simple time-dependent model Hamiltonian is introduced in order to describe the electronic system in question in section 2. Subsequently. an approximate calculation is made for the timc-dcpendence of the charge and spin states for He atoms approaching metal surfaces in section 3. A summary is given in section 4.
2. Model The model Hamiltonian, H=
&plrn
which we investigate,
+h,(t)&,“+
c "
h.0 +u(t)nl,p?,.i
[c;,,(r)C;,C,,,+
is given h\; ll.C.1
h.n
ltJ(r)S,*S,.
(1)
where CL” (Cc,) is the electron creation operator for the state / X) with spin (T and energy eI of the conduction electron in the metal (for the 2s orbital of ;I He atom with spin u and energy cl, (r)), II = C’ C, and k’,\?,(t) is the matri\ element of hybridization between the conduction electron and the electron in the 2s orbital of the He atom. The fourth term represents in the intra-atomic Coulomb interaction for two electrons in the 2s orbital of the He atom. The last term represents the exchange interaction between electrons in the 7\ orbital and in the Is orbital of the He atom. and S, *S, can he rewritten :IS s,
*s,
= & 1 “.O’
C;,“C,\n’C~“.C7\n
-
d c Il,,“HJ,“,. l?.(T’
(7)
where CT,, is the electron creation operator for the 1s orbital of the He atom. We consider the case where a He atom approaches the surface and ~novcx in the region where the interaction between electrons in the 2s orbital of the Hc atom and in the substrate is still appreciable but the interaction hetwccn
H. Kasai, A. Okljl / Helium atoms approachrng
metal surfaces
33
electrons in the Is orbital of the He atom and in the substrate is not. The length of the region, with which we are concerned here, may be roughly estimated as the difference of the spatial extension of the radial parts of the wave function between the 2s orbital and the Is orbital, - 2 A. Since the trajectory approximation is introduced for the motion of the atom, cZs, U, J and VL2, have an explicit time-dependence. It is assumed that at the initial time t = t,,, the conduction electrons in the metal are in thermal equilibrium and that the atom is far from the surface, i.e. I’, zs( to) = 0. In the following we introduce operators such as C:,(t) and C&,(t) in the Heisenberg representation which obey the Heisenberg equation of motion, t) = [C’(t),
i(d/dt)C’(
H(t)].
Here we use units of A = 1. In order to describe the time-dependence of the charge and spin states of the He atom, the following probabilities are considered : P_(t)
= C(~2~“w~*s~o(f)[~ 0
P’(t)
= C([l ”
-n ,&)1[~
P:‘Jt)=C([l-n
,,&>I
n p:’
- ~,s~,wlhAt)>~
- ~;s-owl
%-&)[l
(34
[1 - ~,s-owl
- %-0(t)]
n,,,(t)>>
n,,“(t)>?
TCl)= C(%\&)[l - %~&)I 11- %&)I fh,(tb
G(t)= C;c,“~t~c,,~“~r)c.,l-O~t~(l_R(i~~. n
(3b)
(3c)
(34 04
Here P-(t), P’(t) and Py -1(t) (Py ?(t)) are the fractions, at time t, of metastable negative ions, positive ions and metastable neutral atoms, of which two electrons have antiparallel (parallel) spin with each other. The quantity G(t) represents the “spin flip” probability at t. It is noted that the sum over four kinds of P gives C(n,,,(r)[l 0
-%&>I>
= I.
Since we consider the case where a metastable He atom or a positive He ion approaches the surface and still moves in the region where the Auger process does not take place, the electron number of the 1s orbital is always one. The average value of S, *S, can be written as (S,(t).S,(t))=~G(t)+t[P!:,(t)-P:,(t)].
(4)
With the use of the equation equations can be derived, $F(t)=
of motion
method
[8], the following
CV~lh(r)[FZ;(r)+t~~(r)] x
-4A(t)F(t)-21m
(53)
Xexp[ic,(t-Z,,)], ;Il;P;&)=
-2A(t)P:,(t)+2A(r)Pm(r) -2Im
$P:Jr)=
C~z,(r)[F,,(r)-F,:(t)]
++(r)
C~2,(f)[~~(~)-G(~)]
=2A(r)P!
1
(5b)
exp[ic,(r-r,,)].
(SC)
t (r)
(t) +2A(t)P:'
i-2113-1 C~2,(r)[~~(~)+t~(r)]
&G(r)=
-2A(t)G(t)-2Im
Here F;;(t),
F,;(t)
ew[ic,(r-r,,)].
CCVk2,(t)F;1k(f) 0 h
and F;‘,(t)
(a = k) are defined
= C(c:“(t,,)n,,,~“(t)c,.~“(t)c,‘,
The quantity ,l,,n([)
exp[i~L(r--t(j)].
-2A(t)P’,‘,(t)+2A(t)Pp(r) -2Im
F;;(r)
set ol
=
n_(t)
b .(f)
\l
-n,,_,(t)
is independent
of 6.
(5~)
by
.(~Kk”(r)).
(ha)
is given by for cy =
+,
for(Y=
-.
It is assumed that Vkz,(t) is independent width A(t) for the 2s orbital defined by ~(r)=~Civ~?\(f)126(~-c~).
exp[ie,(t-r,,)].
(5d)
(7) of k and that the resonance
level
(8)
35
H. Kasai, A. Okiji / Helium atoms approaching metal .surfaces
With the use of a decoupling c ([l
- %“(4]
0
the following :F,(f)
k,(4[1
equation
-
approximation,
~hW1 G&0)G”(f0))
can be derived
= D(t)F,(t)
for FP, (t),
FFk (t) and
FTk(t ),
exp[i~A(~-4dlQW.
-iIL(f)f(ek)
(9)
Here, (F:,(t)
F,(t) =
&i(t) F&(t>
(loa)
&Y(t)
F;;(t) G (t> \ -G(t) G(t) p-(t) + py (t> 1
Q(t)
=
P+(t)+Pj:T(t)
(lob)
’
P’(f) + p’: (t> p-(t) + py T (4/ 1
-
i[e’(t)
A(t),
-i[e-(t)
- iiJ(t), D(t)
+ aJ(t)]
A(t),
A(t),
- +.~(t)] -2A(t),
0, -i[e’(t)
= 0, 0.
-2A(t),
+ aJ(t)]
-i[c-(t)
0, -i:J(t),
0, 0, -i[EC(t)
0, 0,
0, 0, A(t),
0, 0, 0’
0, 0, -i:J(t), -2A(t),
+ aJ(t)] - :J(t)] -i[c’(t)
-iiJ(t),
A(t),
-2A(t), -2A(t). - +~
0 0, 0 0. 0
’
(1Oc)
A(t) -2A(t)
1
r’(t) = czS(t) + U(t) (CC(t) = ez,(t)) and f(ek) is the Fermi function. Eq. (5) together with eq. (9) constructs a closed set of equations which can be solved numerically with the appropriate initial conditions for P-(t,), P+(to), I’: L (to), Py ? (to) and G(t,). In the case of U(t) = J(t) = 0 the above set of equations can be solved analytically and one can get the exact results for P-(t), P’(t), Py 1 (t) and Py T (t) for the decoupled equations.
Although eqs. (5) and (9) can be solved numerically for any time-dependence of the parameters involved. the results of numerical calculations arc shown in this paper for a simple time-dependence of a He aton~~metal interaction [8.9]; the interaction is limited to a region of finite distance from the surface mentioned above, where Vn?,( t ). c?,( t ). U( t ) and J( t ) are assumed to be constant. Then it is enough to follow the time-dependent probahilitie\ after sudden switching-on of VL2,( t) and hence 3( t ):
-l(t)JO
forr
(11)
fort>O.
In this case one can derive manipulation of eq. (9).
= -i+(m[Q(t)il)+
X
expression
for t > 0 after ;I simple
i f: (fi?IRII) /=I rZ=l
f dr,ATexp[h,(t /0
the following
- t')]
sinh[ rT( t’ ~
t
(IIR~ ’ In)(/?lu(t’)
)]
11).
where X, (I = 1,. . ,6) is the eigenvalue of D. which is independent of t now. and R is a matrix consisting of the corresponding eigenvector Y,. (I 1X 1HI) represents (1. m) element of matrix X and T is the surface temperature. Before leaving this section, the results of numerical calculations are shown in fig. 1 for the case of J(t) = 0. In this case, the present Hamiltonian ia the same as that for the time-dependent NewnssAnderson model. for which simple decoupling approximations were introduced [&lo]. With the aid of the timedependent Newns-Anderson model, one could investigate the time-dependence of the probability of charge states and of spin polarization for an atom moving near a metal surface [8.10,11]. From the comparison of the numerical results for the time-dependence of spin polarization M( t ) = (II ‘, , ( t )) (JZ?,I (t )). the present approximation can be considered to correspond to the improved one #’ for the time-dependent Newns-Anderson model.
3. Numerical
results
On the basis of the theoretical method developed in the previous xection. the time-dependence of charge and spin states of metastable He atoms (He* ) =’
In
a closed
model).
set
of
equations
for
three
kinds
of scattering
of the moving
atom
are taken
[R.lO]).
the
time-dependent
process into
account
prohahllitiea
of conduction within
electrons
a simple
of hy
decoupling
being the
charged
electron
approxitmrtion
in
(~rnpro\ud rhe
orhlt;ll
(WC
rcf\.
37
H. Kusai, A. Okqr / Helium atotm approuching metal surfuces
0.0
L
0
I
1
a
t (WA)
1
2
3
Fig. 1. Time-dependence of spin polarization M(t). The numerical calculations are made for M(0) =l by use of eqs. (5) and (12) (solid line). eqs. (46)-(48) in ref. [7] (dashed line) and the improved equations (see refs. [8,10] and #‘) (crosses). The parameter used is U/A = 4 (6 2S= - U/2 measured from the Fermi level) at the absolute zero of temperature.
and positive He ions (He+) interacting with metal surfaces is investigated numerically. The quantities involved in the model for a He atom in an isolated state are given as czs = -4.6 eV (measured from the vacuum level), J = -0.8 eV (ferromagnetic coupling) and U = 4.5 eV (cZs + U corresponds to the these quantities are affinity level for a metastable He atom *2). Although modified as the atom approaches the metal surface, the modifications for J and U are neglected for simplicity in the following. 3. I. He * approaching
a surface
The time-dependence of the charge and spin states is shown for a metastable He atom initially in the single state (‘S He*) at the absolute zero of temperature in fig. 2. In the region near the surface we take A = 0.2 eV as an average value [3,4]. According to the experimental situation [1,2], the case is considered where the Fermi level cr. (taken as the origin of energy) is located between the effective 2s level of eZs - :J + + (for ‘S He*) and of c2\ + +J + $ (for “SHe”, a metastable in the triplet state). Here + is the work function of the metal surface. As one can see in the figure, P,(t), the fraction of ‘S He*, increases (P,(t), the fraction of ‘S He*, decreases) and reaches gradually its equilibrium value. It can be considered that the singlet-triplet conversion *’ The binding energy of the (1~2~2~) state of He- relative to the binding chosen as the value of the electron affinity of He* (see ref. (121).
energy
of He*
is
.6
Fig. 2. Time-dependence of Pm. P+. P, and zero of temperature. In the region near the surface we take c2, = -0.25 eV measured from the Fermi level as an average value [3.4]. The case of ‘SHe* approaching a metal surface is considered. The distance between positions of the atom at tWo different times, / and I + A/d, becomes 0.056 I\ (0.79 k) for E,,,, = 60 meV (12 eV). where E k,n is the kinetic energy of the atom normal to the surface. P, at the absolute
Fig. 3. Same as fig. 2 hut for a high-lctwkfunction metal (the work function is 0.75 eV higher than that in the case of fig. 2).
occurs efficiently in this case [1,2]. Here, P,( t ) and P,( t ) are given as ,\ ,\ ,> P,(t)=P;‘,(t)+[P;‘,(t)+G(t)]/2 and P,(r)=[P;‘,(t)-G(r)]/2, respectively. As soon as the interaction is switched-on and hence the 2s level of the He atom has a finite width of - 26, the electron in the He atom having an energy greater than eF starts to transfer to the metal surface. Hence the charge and spin states of the He atom are modified and the He atom reaches gradually its equilibrium charge and spin states. It can be considered that the memory for the initial spin state as well as for the initial charge state of the He atom practically vanishes near the surface if the velocity of the He atom is slow. By the numerical results shown in fig. 3 for the case where the work function of the metal surface is high as compared with that in the case of fig. 2 and cF IS located below the effective 2s level of ‘SHe” atom, it is confirmed that charge transfer occurs efficiently from the He atom to the metal surface and hence the main fraction of the ‘S He * atoms changes into He’ ions near the surface. The time-dependence of charge and spin states is shown in fig. 4 for a ‘S He* atom interacting with a metal surface of low work function as compared with that in the case of fig. 2. Since the number of electrons in the He atom having an energy higher than cF is small as compared with that in the case of fig. 2. little charge is transferred from the He atom to the metal
H. Kasal. A. Oki/i / Helium
Fig. 4. Same as fig. 2 but for a low-workfunction metal (the work function is 1.2 eV lower than that in the case of fig. 2).
atoms approachmg
metal surfaces
Fig. 5. Same as fig. 2 but proaching a metal surface. Py t ( t), the fraction of 3S He is shown as a function of t jSHe* initially with Sz = f S;=O(.““).
39
for ‘SHe* apThe probability * with .S, = + 1, for the case of 1 ( -) and
surface. Hence the decay rate to the equilibrium charge state from the initial charge state becomes considerably fast as compared with that in the case of fig. 2. As for the decay rate for the spin state, the electron in the 2s orbital of the He atom having, say, up-spin (unfavourable for the exchange interaction with Is electron spin) continues interchanging with the electron in the metal, which has a favourable spin, say down-spin, even if the He atom has already reached its equilibrium charge state. Hence, as one can see in the figure, it can be said that the decay rate to the equilibrium spin state from its initial spin state is considerably slow as compared with that for the charge state. It can be considered that the main fractions of the incident He atoms would retain memory for the initial spin state near the surface in this case if the velocity of the He atoms is relatively fast. As one can see in figs. 2 and 4, P-(t) takes a finite value near the surface and increases as the work function of the surface decreases. In fig. 5 the time-dependence of charge and spin states is shown for ‘S He* at the absolute zero of temperature. The probability P,(t) decreases (P,(t) increases) from its initial value as the interaction between the He atom and the metal surface is switched-on and tends gradually to its equilibrium value. As for P’(t): at the beginning (0 < t -C h/A) it takes a value smaller than in the case of fig. 2, because the effective 2s level for 3S He * is lower than for ‘SHe*. However, it tends gradually to the equilibrium value which is the same as that in the case of fig. 2. It is noted that the quantities Z’,(t), P,(t), P-(t) and P’(t) do not show any dependence on the magnetic quantum number of an incident ‘S He* atom, S7 = k 1 or 0. Although PT T (t), the
40
t (h/A) Fig. 6. Same as fig. 2 hut for He+ approaching
a metal
surface
probability of ‘S He * with S= = + 1. depends strongly on the initial magnetic with the recent quantum number of ‘S He* as shown in fig. 5. In connection experiments on the surface magnetism with the use of spin-polarized ‘S He* atoms [5], it would be interesting to extend the present calculations to the cast of spin-polarized ‘S He * atoms interacting with the surface of the magnetic metals. 3.2. He + upproaching
a surface
The time-dependence of the charge and spin states is shown in fig. 6 for ;I positive He atom (He+) at the absolute zero of temperature. As soon as the interaction is switched-on, charge transfer occurs from the metal to He ’ and hence He+ converts into a neutral metastable atom with a finite probability. The main fraction of the metastables is ‘S He*. Nevertheless, a minute in accord with the experimental fractions are ‘S He *. This is qualitatively findings [4]: it is suggested that a minute fraction of incident He’ ions turns into ‘SHe* before electron ejection through the Auger process. although the electrons are ejected mainly by He+ converted into ‘SHe”. However. in the case where the work function of the metal surface is high and hence the Fermi level of the metal surface is located relatively far below the energy level for the 2s orbital of He, the conversion of He+ into He* is suppressed and the Auger neutralization process to the ground state of He occurs at distances close tt) the metal surface [4]. 4. Summary In this paper the time-dependence of the charge and spin states of ‘S Hc*. “SHe* and He+ approaching metal surfaces is investigated with the aid of a
H. Kusui, A. Okijr
/ Helrum
u1om.y approuchrng
merul .surfuces
41
simple time-dependent model Hamiltonian. The equation of motion method is used for the time-dependent quantities describing the charge and spin states of the He atom. A decoupling approximation is introduced in order to get a closed set of equations for the above quantities. The numerical calculations are performed for simple time-dependence of the parameters involved. It is shown, as has been expected, that the memory for the initial spin state as well as for the initial charge state of a He atom has a tendency to vanish if the interaction is appreciable between electrons in the 2s orbital of the He atom and in the conduction band of the metal surface. In the case where the work function of the metal surface is high and hence the Fermi level of the metal surface is located relatively far below the energy level for the 2s orbital of the He atoms, the main fraction of the incident metastable He atoms turns to positive He ions near the surface. If the work function decreases e.g. because of adsorption of alkali metal atoms on the surface and if the Fermi level comes nearer to the energy level for the 2s orbital, the singlet-triplet conversion occurs efficiently and the main fractions of the incident ‘SHe* atoms or He+ ions change into ‘S He* atoms near the surface. However, if the work function gets even lower and the Fermi level is located relatively far above the energy level for the 2s orbital, the characteristic time for the singlet-triplet conversion becomes considerably long. B;i the numerical calculations at finite temperatures, it turns out that there is no surface temperature effect on the charge and spin states of He atoms interacting with metal surfaces in the physically relevant temperature range.
Acknowledgement
This work was supported in part by the Grant-in-Aid for Scientific Research on Priority Areas and for General Scientific Research both from the Ministry of Education, Science and Culture of Japan.
References [I] J. Lee, C. Hanrahan. J. Arias. F. Bozso, R.M. Martin and H. Metiu. Phys. Rev. Letters 54 (1985) 1440. [2] B. Woratschek, W. Sesselmann. J. Kiippers, G. Ertl and H. Haberland, Phys. Rev. Letters 55 (1985) 1231. [31 U’. Sesaelmann, B. Woratchek, J. Kiippers and G. Ertl, Phys. Rev. B 35 (1987) 1547. [4] H.D. Hagstrum. P. Petrie and E.E. Chaban, Phys. Rev. B 38 (1988) 10264. [Sj M. Onellion, M.W. Hart. F.B. Dunning and G.W. Walters. Phys. Rev. Letters 52 (1984) 380; M.W. Hart. M.S. Hammond, F.B. Dunning and G.W. Walters. Phys. Rev. B 39 (1989) 5488. [6] H.D. Hagstrum. Phys. Rev. 91 (1953) 543; 96 (1954) 336. [7] H. Kasai, W. Brenig and W. Hiibner, Surface Sci. 208 (1989) 487; W. Hiibner, W. Brenig and H. Kasai, to be published.
42 [X] [9] [lo] [ll] [121
H. Kasal. A. Okr/l / Helum utmm approuc~hrngnw~l surfacer H. Kasai and A. Okiji. Surface Sci. 183 (1987) 147. K. Makoshi. H. Kawai and A. Yoshimori, J. Phys. Sot. Japan 53 (1984) 2441. H. Nakanishi. H. Kasai and A. Okiji. Surface Sci. 197 (1988) 515. A. Yoshimori and K. Makoshi, Progr. Surface Sci. 21 (1986) 251. A.V. Bunge and C.F. Bunge, Phys. Rev. A 30 (19X4)2 179.