On the mechanism of chemisorption on magnetic semiconductors

Surface Science 54 (1976) 101-l 10 0 North-Hol~nd Pub~shing Company

ON THE MEC~NISM

OF C~MISO~~ON

ON MAGNETIC

SEMICONDUCTOR E.L. NAGAEV and G.L. LAZAREW Ministry of Electrotechnical

Industry of the U.S.S.R., Pr. Kalinina 19, Moscow, U.S.S.R.

Received 30 September 1974; revised manuscript received 3 June 1975 A mechanism of chemisorption on magnetic semiconductors is considered at T = 0. The adatoms behave with respect to the electronic spectrum of a crystal as surface donors. Interaction of the valence electron of the atom with the spins of the crystal (s-d interaction) results in the formation of a ferromagnetic microregion in the vicinity of the adsorbed atom. The one- and two-electron bond of the adatom with the crystal is considered. The tendency for the adatom ferromagnetic regions to merge results in an effective attraction between

the adatoms; moreover many adatoms can attract each other simultaneo~ly.

I. Introduction The electronic approach to ~hem~sorption and catalysis applied to some reactions on the surface of a solid was developed by Roginsky [I], Vol’kenstein [2] and others. The theory of phenomena on semiconductor surfaces was developed in ref. [2], and it became possible to explain some experimentally observed effects, when the manyelectron correlations are negligible. However, the correlation effects are most important in many cases and it is necessary to take account of these for a consistent theory of chemisorption and catalysis on such materials as metals or magnetic ~miconductors to be developed. (The latter are semiconductor compounds of transitional or rare earth elements which are known to belong to the most efficient catalysts.) Inclusion of correlation effects in a theory of chemisorption and catalysis may be achieved by making use of the dielectr$ and magnetic wave number and frequency dependent susceptibility of the catalyst . In the present paper chemisorption on magnetic semiconductors with allowance for correlation effects is investigated theoretically. These manifest themselves in the exchange interaction of valence electrons between adsorbed atoms and localized magnetic moments of partially filled d or f levels of magnetic ions. This interaction may produce a change in magnetic ordering in the vicinity of the adatom, which is equivalent to a local change in the magnetic susceptibility.

* This idea was expressed by N.S. Lidorenko.

The mechanism investigated here is based on the properties of charge carriers established earlier by one of the authors [3,4]. Namely, in a nonferromagnetic semiconductor with any type of magnetic ordering or without it at all, generally speaking the conduction electron may create a ferromagnetic microregion with radius of several lattice parameters. The conduction electron gets autolocalized in the microregion, so that it can move through the crystal only together with the m~~roregio~. Such a quasiparticle - electron plus ferromagnetic microregion - was named a ferron. For the realization of such states it is necessary that certain conditions would be satisfied. In particular, these may be met in the compounds of rare earth metals, where such quasiparticles were discovered experimentally [5] . Ferron states are possible for electrons at the local levels too. Moreover, the conditions for their realization in this case are more favourable than for free charge carriers 141. One can easily conjecture that chemisorptio~ may lead to the formation of surface ferrons, for adatoms (or admolecules) behave with regard to the electronic spectrum of the crystal like surface donors or acceptors. Their formation should result in some specific features of chemisorption on magnetic semiconductors, stated below: (1) The adatom possesses a giant magnetic moment, which is virtually equal to the moment of the ferromagnetic microregion formed around it. (2) The ferromagnetic microregions of neighbouring atoms tend to merge with each other, which results in a specific mechanism of the attraction between adatoms. This is not bound up with the usual exchange interaction in the sense that there is no spin pairing for the valence electrons of adatoms. On the contrary, these spins should be parallel. It entails, that many atoms can be attracted to each other simultaneously; in contrast to the case of the usual exchange interaction with atoms attracting each other only pairwise. In its essence, this is a new ~lechanism of the chemical bond differing from the usual exchange mechanism. (3) In close analogy to it, chemisorption is especially energetically favoured in the vicinity of such surface defects of the crystalline structure, which are centers of ferromagnetic microregions in the absence of adatoms (for example, donor impurities on the surface). In terms of the inhomogeneous surface theory of Roginsky [I], it means that the surface becomes i~omogeneous because of the appearance of microregio~s with ferromagnetic ordering on it. In the vicinity of each defect of this type several atoms can be adsorbed simultaneously. (4) The adatom can capture one or several conduction electrons, which increases its binding energy to the crystal and the size of the ferromagnetic microregion. For the case of one electron captured such a state differs from the analogous one treated by Vol’kenstein 121 in that the spins of the electrons on the adatom are parallel. Summarizing, in what follows a new model for chemisorption of atoms on magnetic semiconductors will be developed which differs from the previous one in that account is taken of a ferromagnetic region arising in the vicinity of an adatom.

E. L. Nagaev, G. L. Luzarewlchemisorption

on magnetic semiconductors

103

2. Model In the present paper the chemisorption of a one-electron atom on the antiferromagnetic crystal will be treated for the sake of simplicity. The following model will be used: the crystal consists of cations and anions alternating with each other, only the former have a nonzero spin. The adsorbed atom is located near one of the anions and its valence electron is pulled into the crystal [6]. In this case the probability of the electron being on the adatom is considerably lower than in the case of adsorption near a cation considered by Vol’kenstein [2] . It will be neglected below and the valence electron will be treated as completely pulled into the crystal. The electron of an adatom as well as the conduction electrons belonging to the surface conduction band, can be located only on cations, lying on the surface of the crystal. The specific feature of adsorption on the surface of magnetic semiconductors is the strong exchange interaction of the adatom electrons with cation spins, with the maximum gain in their exchange energy being delivered by the ferromagnetic ordering. Therefore these electrons tend to establish ferromagnetic ordering in the vicinity of the adatom. Simultaneously the exchange interaction between cations which is nonferromagnetic hinders the ferromagnetic ordering establishing. As a result of the competition between these two factors ferromagnetic ordering will be established in some region around the adatom. The radius R of the region is determined from the condition of the minimum of total energy of the system, when we consider the ground state of the system, and from the minimum of its free energy, when the temperature is high enough. In this paper only the groun state of the system will be considered.

3. One-electron

binding of adatom with crystal

The Schrijdinger equation describing the states of the valence electron of the adatom on the crystal surface in the effective mass approximation takes the form: 9$$,

=

+9(r~F) -$I$0=(EO --~p)$~,(FT~).

The following notations are used: S is the magnitude of the spin of the cation;A is the s-d exchange integral, the energy of the s-d exchange being small in comparison with the width of the conduction band W; r is the radial distance in the surface measured from the adatom; 19(rE F) is equal to unity within the ferromagnetic microregion and zero outside it; 52 is the area of the ferromagnetic microregion; TN is the surface Ne’el temperature;a is the lattice constant; m* is the surface effective mass of the electron; E = (e. + 1)/2, where eO is the crystal dielectric constant. The energy E. is purely electronic, i.e. it does not include the energy of the Coulomb interaction between the adsorbed ion and the lattice (since the charge of the former

104

E.L. Nagaev, G.L. Lazarew/Chemisorption

on magnetic semiconductors

is positive it makes it possible for the adion to be located so that its nearest neighbour from the lattice is an anion [6]). The electron is assumed to be localized on the first surface layer. This assumption is based on the following two factors: (1) The surface conduction band lies below the bulk one by 0.1-l eV. This statement for ionic crystals may be easily proved making use of methods developed in ref. [I 11. (2) The energy of the Coulomb interaction of the adatom ion and the electron inside the bulk is co/e times more than that for the electron on the surface, the distances between the ion and the electron being the same. Eq. (1) differs from the equation used by Bench-Bruevich [7] in taking account of the formation of the ferromagnetic microregion with the area Q = BIRDaround the adatom on the surface of the antiferromagnet, which requires an energy *TN Q%r2. Within the microregion the electron energy is lowered by AS/2 because of the exchange interaction of the electron with localized spins of cations. The minimum energy of the system is determined making use of the variational principle. In the general case the solution to eq. (l), expressed in terms of the degenerate hypergeometric functions, makes it possible to find out the energy from a transcendental equation, presented in the Appendix. However, it is of a very complicated form and may be solved only by numerical methods. To make the results more obvious, an approximate solution of eq. (1) will be used instead of the exact one. A solution of the problem under consideration will be obtained in the two limiting cases e2 /ErOS AS/2 and e2/er0 GAS/2 (r. = @moaB/m* , ag is the Bohr radius and m. is the mass of the free electron). Each of them may be realized for real magnetic semiconductors. 3.1. The case ofthe weuk s-d exchange (e2/u0

> AS/Z)

In this case the trial wave function is represented in the form (Z), which in the limit A + 0 gives the exact solution of the problem, provided the parameter b is equal to 2: 1 2b GO=---exp fir

ro

Otherwise the parameter h is treated as a variational one as well as R. Minimizing the total energy of the system, with respect to them, we obtain the folIowing expression for the equilibrium value of the radius R of the ferromagnetic microregion R =!$-ln

[:E

and corresponding

($j2], energy E. (the bottom of the conduction

band is zero point)

E.L. Nagaev, G.L. Lazarew/Chemisorption on magnetic semiconductors

105

Inparticularwith~~5,A5/2~O.1eV,T~n.3x10~3eV,a”3~andm*~mo, R should be - 4 8, and E, N -2.3 eV. But this means R N u, so that these numerical results are not very reliable though the qualitative picture remains valid. 3.2. The case of strong s-d ex&ange (e2/ero @AS/2) This case in the zeroth approximation in the Coulomb interaction corresponds to the free surface ferron. Its wave functions in the two-dimensional case are expressed in terms of Bessel functions of an integer order. It should be pointed out that the first excited state unlike the ground state is of the p-type [S] . Therefore the trial wave function is represented in the form going over into the exact solution of the problem fore’=0 + Cz KO(KO1) [l -6(PER)],

Go = C1~o(ko’) e(rER)

(9

where fi2k2

-=-_-o 2m*

AS

h2Kti

2

2m*

and &(kr), K,(K~) are the Bessel functions of the first and second kind of imaginary arguments (m = O,l,. . .) correspondingly. The quantities k. and R play the role of the variational parameters here. C1 and C2 are determined from the conditions of the joining at Y =R and of the normalization of the wave function (5). The radius R of the ferromagnetic microregion expressed in terms of the Bloch’s integral iS I (IB I = R2/2m*a2) is

where R, is the radius of the free surface ferron (i.e. for e = 0) in the two-dimensional case. The expression for its energy is as follows: RoP-A++5.76]B]

0

2

;

For the values of the parameters E ‘u 12, AS/2 N 1 eV, TN N 3 x 10m3 eV, a N 3 8, m *IN Eqs!t6) and (7) giveR - 12 A and E 2c -0.96 eV i e. the moment of the adsorbed atom is equal approximately to fifty ato?nic moments’&S (pn is the Bohr magneton). The values indicated above are typical, in particular, for compounds of rare-earth elements (EuTe, EuSe).

4. Two-electron

binding of adatom with crystal

If as a result of the adsorption

the ferromagnetic

microregion

was formed around

106

E.L. Nagaev, G.L. Lazarewfchemisorption

on magnetic semiconductors

the adatom, the microregion may capture an electron belonging to the crystal, for example, an electron from the surface conduction band. As before, the reason for this is the gain in the s-d exchange energyAS/2, which an electron acquires, going over from the antiferromagnetic region to the ferromagnetic one. Since the spin of the second electron must be parallel to that of the first one, the second electron is prohibited to be in the same orbital state with the minimum energy as the first electron. Therefore, if the radius of the ferromagnetic region had not changed after the second electron capturing, the energy of the second electron inside the potential well would have exceeded substantially that of the first. Indeed, the conditions for the second electron capturing are more favourable as this leads to an increase of the ferromagnetic microregion radius, which in turn lowers the electron levels in the well. Generally speaking, in this ferromagnetic microregion not necessarily one but more crystal electrons of the crystal may get localized, as each of them acquires the gain in the s-d exchange energy. However, the increase in the kinetic energy for each subsequently trapped electron and the Coulomb repulsion between them restricts their number. Nevertheless, in the analogy with the bulk properties of the magnetic semiconductors [9] it may be inferred that the three, four and more electron binding of the adatom with the crystal is possible. Only the two-electron binding will be considered in what follows. It should be noted that the two-electron binding is possible in nonmagnetic crystals too; in which case the spins of both electrons are antiparallel [2] . This implies that depending on the s-d interaction being strong or weak, the electron spins should be parallel or antiparallel. In the first case the ferromagnetic cluster near the adatom is formed, in the second it does not occur. The latter means that the gain in the s-d exchange energy (-AS) is considerably smaller than the loss in energy caused by the transition of the electron from the ground state to the first excited one (- e2/er0). (This transition is necessary for the realization of the triplet state.) We shall consider only the case of the strong s-d bond, where the electron spins are parallel, and in the region of the radius R around the adatom where the antiferromagnetic ordering is replaced by the ferromagnetic one. In the case considered the wave equation can be written as follows:

t? e2 er+Er 2

12 I

$=

E-J%+. a2

Since the distance between adion and its nearest neighbour is small compared to the radius of the electron state, the former is put equal to zero in eq. (8). This assumption is very convenient in many respects, but it fails to take into account the dipoledipole interaction between adatoms. The trial wave function is taken in the form:

E.L. Nagaev, G. L. Lazarewlchemisorption

on magnetic semiconductors

107

Here \c’,, and $I play the role of the wave functions of the ground and the first excited state of the one-electron Hamiltonian H,, respectively. The former is given by eq. (S), i.e. it is independent of the angular variable up.The corresponding energy is given by the expression (7) apart from the term proportional to TN. The wave function $, is represented with the aid of two variational parameters kl and R in the form Q,(T,(P)=

{c,Jl(klr)8(r~R)tCqK1(KIr) [l -WER)IItexp(*id

The energy E, corresponding E,

to it is given by the expression 2

d/t

14.67 IBt ; 0

(11)

Using eqs. (Q (9), (3, (7), (10) and (11) we obtain the next expression for the energy of the two-electron state of the adatom 2

2

E”-ASf20.43IBj

+ 0

The radius of the ferromagnetic minimum of the energy is

-3;+nTN

microregion

f 0

.

determined

(12) from the condition

&, = 1.38R,,.

R=a=I?,[L#(;$

of the

(13)

Thus, should the second electron be trapped. by the adatom, the area of the ferromagnetic region will increase by a factor of two.

5. Ferron mechanism of attraction

between adsorbed atoms

Using the results obtained above, we proceed to discuss the question of the effective attaction between two adsorbed atoms. We shall treat only the case of one-electron chemisorption, since in the case of two-electron chemisorption the Coulomb repulsion between adatoms dominates. We assume that two adsorbed atoms A and B are separated from each other not very strongly, so that the ferromagnetic regions formed by each of the adatoms overlap. Then, if the overlap is not very large, the problem may be solved by means of perturbation theory. The change in the energy of the system under consideration when the atoms approach each other from infinity to the distance z is given to the first order in the perturbation theory by the expression (z=2l-d, rn+, =rno +mmagn

TRACY>>).

(14)

108

E.L. Nagaev, G.L. LazarewfChemisorption

on magnetic semiconductors

The symbol Q in (14) characterizes the overlap between the ferromagnetic regions. The index (tt) indicates that the spins of both electrons are parallel to each other. The terms entering (14) have the following meaning. The term AEo corresponds to the well known expression for the energy of interaction in the problem of two hydrogen atoms. Apart from the form of the atomic wave functions, the difference between them consists in taking account of the s-d exchange energy in the expression for AEo : rn0=-=

-

Co-*o l-rr

[J+,

dr, ii(r,)

V(rl, r2>

@r,)

Jdrl dr, tiA(rl) J/A(r2)GB(rl) GB(r2)Vrl ,r2)1

x 11- Sb, h2 G,(r,) JIA(r2)JIB(rl>iB(r,

9

(15)

where V(rl,r2)=+

(

-&

2

-<

e2

2 t-

2 tz

‘12

1

- $AS [O(rl E FB) + e(r, E FA)] .

(16)

The wave functions GA and $n centred around atoms A and B are given by eqs. (2) and (S), respectively. Substitution of eqs. (2), (5) and (16) into eq. (15) leads to the following results for AE, for the cases e2/u0 9 AS/2 and e2/ero <*S/2, correspondingly:

4 e2/ero

nT,

(17)

ASAS

for

e2 eAs fzrO 2 ’

(18)

As for hydrogenic atoms, the quantity AEo for parallel electron spins is positive. The eventual gain in the energy of two atoms is due to the second term in eq. (14), AEmagn describing the gain in the magnetic energy caused by the overlap of the two ferromagnetic regions. In fact, the total surface of the crystal in which the antiferromagnetic ordering is replaced by the ferromagnetic ordering, is decreased as a result of overlapping and so does the energy of the creation of the ferromagnetic region. This leads to the attraction between two atoms. The quantity AEmagn may be easily evaluated by expressing the overlap area of two ferromagnetic regions in terms of R and CLFor a small overlap this gives Lw

=

magn i.e. the attracting

-TN

.3i2R 1/2/a2

force is proportional

(1% to Q1/2. In reality the gain must be still larger,

as the ferromagnetic region changes its form too, in order ta provide the minimum of the energy. As is seen by comparing eq. {19) with eqs. (17) and (18) the gain in the magnetic - AR0 exceeds energy 1mEmap I exceeds the loss AR,. If the difference of lAEnr,,l repulsion omitted in previous calthe quantity Ed. _dip, describing the dipole-dipole c~Iations, the a dpsorbed atoms tend ta approach to each other to the distance of several lattice constants, which may be very important for processes of catalysis on the magnetic semiconductor. It is worth while noting that semiconductor mechanism just indicated makes it possible not only for two atoms, but for arbitrarily large numbers of them to attract each other simultaneo~ly, spins of all valence electrans being parallel to each other. Thus, ferromagnetic clusters, consisting of a considerable number of adatoms may form on the crystal surface. Although the calculation presented above was at T = 0, many qualitative results obtained here are correct for higher temperatures~ because the localized ferron states may exist to rather high temperatures.

Appendix The solution of eq. (1) is [lo] CA.11 64.2)

Were @and $ are the two~iffere~~ solutions of the degenerate h~~erge~metric equation, and eq. (1) is reduced to it. The solution (A.11 is nonsingular in the vicinity of zero and the solution (A.2) is nonsingular at infinity. From the conditions of the matching of the wave function following transcendental equation with respect to Et

at Y=R we obtain the

(A.31

E.L. Nagaev, G.L. LazarewfChemisorption

110

on magnetic semiconductors

References

[l] [2] [3] [4] [5] [6] [7] [8] [9]

[lo] [ll]

S.Z. Roginsky, Adsorption and Catalysis on Inhomogeneous Surfaces (Akad. Nauk SSSR, Moscow, 1948). F.F. Vol’kenstein, The Electronic Theory of Catalysis on Semiconductors (Fizmatgiz, Moscow, 1960). E.L. Nagaev, Zh. Eksperim. i Teor. Fiz. (Pisma) 6 (1967) 434. E.L. Nagaev, Zh. Eksperim. i Tear. Fiz. 54 (1968) 228. G. Busch, P. Streit and P. Wachter, Solid State Commun. 8 (1970) 1759. E.L. Nagaev, in: Proc. Conf. of Young Scientists, Moscow (Moscow, 1959). V.L. Bench-Bruevich, Zh. Fiz. Khim. 25 (1951) 1033. J. Mathews and R.L. Walker, Mathematical Methods of Physics (Benjamin, New York, 1964). E.L. Nagaev, Zh. Eksperim. i Teor. Fiz. (Pisma) 16 (1972) 558. H. Bateman and A. Erdelyi, Higher Transcendental Functions, Vol. 1 (McGraw-Hill, New York, 1953). V.V. Bryksin and Yu.A. Firsov, Fiz. Tverd. Tela 11 (1969) 2167.