SIZE DEPENDENCE OF THE C.E.S.R. g SHIFT IN A SMALL METAL PARTICLE : A SIMPLE MODEL APPROACH

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SIZE DEPENDENCE OF THE C.E.S.R. g SHIFT IN A

SMALL METAL PARTICLE : A SIMPLE MODEL

APPROACH

C. Myles, J. Buttet

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C2, supplkment au no 7 , Tome 38, Juillet 1977, page C2-133

SIZE DEPENDENCE OF THE C.E.S.R.

g

SHIFT

IN

A

SMALL METAL PARTICLE:

A

SIMPLE MODEL APPROACH

(*)

C. W. MYLES and J. BUTTET Laboratoire de Physique ExpCrimentale Ecole Polytechnique FCdCrale de Lausanne

CH-1007 Lausanne, Switzerland

R6sum6. - Nous avons calcul6, en fonction du nombre d'atomes, le facteur g de 1'6lectron non appari6 d'une chaine lin6aire dans l'approximation de Huckel. On tient compte des effets de surface, qui brisent les rkgles de selection du couplage spin-orbite, en modifiant les potentiels atomiques des atomes B l'extrCmit6 de la chaine. Les contributions du premier et du second ordre dans le couplage spin-orbite sont calcul6es numkriquement.

Abstract. - We have calculated the g shift as a function of the number of atoms in a linear chain in the Huckel approximation. To include surface effects, which break the wave vector selection rule of the spin-orbit matrix elements, we assume that the atomic potentials of the end atoms are different. Both the first and second order contributions in spin-orbit coupling are calculated.

1. Introduction. - It has been suggested [I, 21 that conduction electron spin resonance (C. E. S .R.) in small metal particles might be observable even for metals where relaxation by spin-orbit coupling is so large that the signal cannot be seen in the bulk. Such observations might be possible because pho- non and surface relaxation mechanisms are quench- ed in small particles, due to the discreteness of the conduction electron energy levels. We thus expect each particular metal particle to be associated with a sharp resonance line whose g value depends upon the structure of the energy levels of that particle. If it is assumed that, in an ensemble of metal clusters of a given size, surface irregularities cause the energy levels to be statistically distributed, the C.E.S.R. lineshape will then be a superposition of sharp lines whose envelope will represent the measured signal. Kawabata [3] has calculated the average position and average width of this envelope using linear response theory. He defines a quantum limit as fiwz 4 6 and h / r

<

6, where 6 is the average level spacing, Amz is the spin Zeeman energy and T is a quantity which reduces to the spin lattice relaxation time in the bulk metal. In this limit he predicts that both the line width and the change in the g shift with respect to the bulk value are proportional to the square of the particle diameter. Recent experimental results [4, 51 are not in agreement with all of Kawabata's conclusions. Thus, further theoretical study of this problem appears to be needed. It is therefore the purpose of this paper to try to obtain some physical insight into the origins of the size dependence of the g shift by investigating this dependence in a simple model.

2. g shiit formalism.

-

In our model, an open linear chain of identical' atoms in the Hiickel approximation, the wave functions and energies are known analytically even in the presence of spin- orbit coupling. A calculation of the g shift could thus be done [6,

71

using a formalism similar to that used in bulk metal calculations, where the theory is valid to all orders in spin-orbit coupling and the magnetic field is treated as a first order perturbation. In order to follow Kawabata's treatment more closely, we shall instead start with wave functions which are valid to all orders in the magnetic field and use the spin-orbit interaction as a perturbation. If we assume that the energy levels of the small particle are discrete and that there is no orbital degeneracy, it is easy to show that the energy splitting between the levels

I

km

+

) and ( km - ) is, to second order in spin-orbit coupling :

.In eq. (I) we have used Kawabata's notation [3] and have assumed that the magnetic field is parallel to the z axis. Hso is the one electron spin-orbit interaction, including the vector potential to take into account gauge problems.

I

km

+

) and

I

km - ) are eigenstates of the one electron Hamiltonian 'Xo

for particle k, where %Yo includes the kinetic energy (in the presence of a magnetic field), the lattice potential, the surface potential, and the spin Zeeman energy. E(k, m) ts defined by :

(*) This work was supported in part by the Swiss National

_xo

₁

_kkm_t)₌_{[ E ( k ,}_{m )}₂

fund for scientific Research under grant 2.377-0.75. 2

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C2-134 C.W. MYLES AND J. BUTTET Equation (1) is valid only under the conditions :

(km

+ I

Hso

I

km - ) 4 h w Z (2)

The second condition is always satisfied if the g shift is small, while the first and third inequalities reduce to the Kawabata conditions hwz 4 6 and h / ~ 4 6. The conditions under which eq. (1) is valid are thus equivalent to the quantum limit. Further- more, the first order term in Hso is the usual A g for molecules

183

and, apart from averages, the second order term in Hso is easily shown to be equal to the

, A g calculated by Kawabata in the quantum limit. In a crystal with periodic boundary conditions the second order term is certainly smaller than the first order one, since HSO then connects only states

which belong to the same wave vector and which are thus in different bands. In a small particle this selection rule will be broken by surface effects, since the wave functions will no longer be Bloch states and the spin-orbit interaction of the surface atoms will be different. The spin-orbit matrix elements between states

I

km ) and

I

kq ) separated by energies of the order of 6 will then be non-zero and the second order term could become larger than the first order one. In what follows we shall study the relative amplitudes of both contributions 'to A g

for the case of a linear chain.

3. g shift without surface effects.

-

If we assume that all atoms are identical and assume that there is no mixing between orbitals of different symmetry, the wave functions are known analytically [9] for both the closed and open linear chains in the Hiickel approximation. We have found that the open and closed chain models show no qualitative difference in their prediction of the N dependence of the g shift. This is related to the fact that the wave vector selection rule for the spin-orbit coupling still holds for an open linear chain in our approximation. The open chain model has the advantage that we can start from a situation similar to the bulk and then use perturbation theory to see qualitatively the effects of surfaces. If we choose p electrons as an atomic basis, the open chain wave functions are

191

:

I lua ) =

Jx

5

sin ["(I N + I + n

]

lnuo > ,

N + 1 n = ,

where overlap has been neglected to calculate the normalization, u is the spin index, a is the band index (p,, p,, p,) and

I

nua) is the atomic wave function at site n.

The corresponding energies are :

E, (a, N) = F ( a )

+

2 G ( a ) cos

1":

(4a)

where

F ( a ) = ( n u a I X I n u a ) (4b) G ( a ) = ( n % l , u a

I

X ( n u a ) (44 and % is the one electron Hamiltonian. If we assume that F(a) is independent of a , the energy bands appear as in figure 1. The degeneracy of the

x and y bands reflects the axial symmetry of the system.

FIG. 1.

-

Energy as a function of N in an open linear chain ( N = 11). The x, y band is doubly degenerate.

Formula (1) was obtained excluding the possibi- lity of orbital degeneracy, as is usual in A g

calculations. In order to compare our results to the usual case, in what follows we shall artificially lift the degeneracy and calculate only those A g values which do not depend on the lifting energy.

Stone [8] has shown that if we neglect the overlap between orbitals on different atoms, we can choose a local gauge to calculate the first order term in eq. (I), which then yields the formula :

( n = l , a I L q J n = l , c u ' ) ( n = l , c u ' ) L , I n = l , a )

El ( a , N ) - El ( a ' , N ) ( 5 )

where

I

lua) is the unpaired electron state, L, is the

qth cartegian component of the orbital angular momentum centered on atomic site 1 and h is the ,atomic spin-orbit coupling constant. In writing eq. (5) we have used the already mentioned selection rule :

(Iucu ( H S O I 1'(rJa')=SN,(1cra 1 H S O ) I ' u ' L Y ' ) . (6)

The main difference one expects to find between

Ag" (N) and Ag(O) ( N = a) will come from the N

dependence of the energy denominators in eq. (5). (Note also that the 1 of the unpaired level is a function of

N).

The degeneracy of the x and y bands prevents us from calculating the g shift parallel to the chain axis. Thus we will only be concerned with the g

shift Ag11 normal to the chain axis. If we assume one electron per atom and define

rg

= G(z) Ag/h,

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g SHIFT I N SMALL METAL PARTICLES C2-135

R

= G(x)/G(z). The numerical results for

R =

-

0.9 and N < 150 are shown in figure 2. For this case our model predicts two distinct g shifts, one corresponding to the case where the unpaired electron is in the

z

band and one corresponding to the case where it is in the x, y band. As shown in figure 2 we find that the Agy) for each band oscillates as a function of N about some average which falls to a bulk value following a 1/ N curve.

The true bulk value is a weighted average [lo]

between Agf') (x, N = m) and Agf) (z, N = a). This

model thus gives two lines situated on each side of the bulk value. At finite temperature the model predicts the two distinct lines merging into one when kT > S. Rather than discussing this effect, in what follows we shall focus on the surface effects

which allow transitions between levels which are near to each other in energy.

FIG. 2. Agll values in an open linear chain without suflace effects. Agy)(x, N) and rgy)(z, N) correspond to the cases where the unpaired electron is in the x, y or z band. The dashed lines

are

Fgy)(cu,

N) for an infinite chain. The smooth curves correspond to the change in rgyi(cu, N) which falls to

rgY(u,

N - *) itb 1/N.

4. g shift with surface effects.

-

It has been shown ''[ll'l that the Friedel charge oscillations which .appear near the surface of a metal are qualitatively obtained in a linear chain of atoms if the atomic potentials at the end atoms are different than those of the inner atoms. Following this idea we shall assume here that the surface effects can be

taken into account by replacing F(a) as defined in eq. (4b) by F(a)

+

AF(a) for the atoms n = 1 and

n = N and by then treating AF(a) for these end

atoms as a perturbation. The perturbed energies and wave functions are then (assuming that the surface energy will not mix bands) :

and

where El(a, N) is the unperturbed energy eq. (4a). In equations (7) and (8) we have defined :

and

c;;,

= b $,/[El (a, N ) - El,(% N)]

.

(9b) The perturbation expansion is valid if b;, < S. This

can be seen to be equivalent to A e a ) < G(a) if we use eq. (9a) and notice that N6

-

G(a).

The Ag formula one obtains using the wave functions and energies given in eqs. (7) and (8) can be decomposed into three terms : Agf"(N), Agy)(N) and Agi2)(N). Agf"(N) is the surface independent term given by eq. (5) ; A$ll(N) and Agi2)(N) are the

surface dependent terms which are respectively first

and second order in spin-orbit coupling. Notice that Ag$2j(N) can only be calculated when the magnetic field is parallel to the chain axis, this is related to the lifting of the degeneracy mentioned above. AgG2)(N) is, however, the exact counterpart of Agy)(N) and both terms can be broken up into four parts corresponding to four different surface contri- butions. For Agy) the four terms can be expressed

in the following forms :

and

where the functions of R and N must be evaluated numerically for arbitrary R and N. AgY,O)(N) comes from the contribution to the g shift of the second term in eq. (7). It is the term which breaks the selection rule in I . Physically it thus comes from the

first order contribution of the surface energy to the wave function. A similar analysis shows that Agy,b)(N) results from the normalization of the wave function, while Agpc)(N) and Agy*d)(N) result from the first and second order corrections to the energy.

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C2-136 C.W. MYLES AND J. BU?TET

eq. (10). This is accomplished by multiplying by - A/2 G ( z ) and replacing the functions f1,")(R, N), etc., by functions which differ from those in eq. (10) by the fact that an energy denominator squared enters the latter functions while only entering in the first power in eq. (10).

We have calculated AgY)(N) and AgL2)(N) and the individual terms numerically as a function of N

( N < 150) for various values of R. We present our

results for the case with R = - 0 . 9 in figure 3 . The

f

FIG. 3. - Ag values in an open linear chain with surface effects.

The open circles represent r g y ) ( x , N) and the triangles

F g ~ ( x , N), where

rg

= ( G ( Z ) / ~ ) ~ Ag. The smooth curve corres-

ponds to a l / N decrease. The numerical values are given for the case R = - 0.9, A F ( x ) / G ( x ) = 0.1 and A F ( z ) / G ( z ) = - 0.09.

large oscillations in the values of AgY) and A g p are due to the fact that the denominators of Agy,*)(N) and Ag$2.a)(N) are small if there is a state in the z band (see Fig. 1) which is close in energy to the unpaired level in the x, y band. A detailed analysis shows that A g ! 2 . " ) ( ~ ) is the sum of two different contributions which are respectively proportional t o ( A F / N 6 ) 2

-

const. and

( A F

/

N G ( z ) ) ~ ( G ( z ) / 6 *)

-

1

/

N , where the average

energy difference 6" between two states in the x, y and

z

band is proportional to 1 / N . The numerical results show indeed that the average &jfsn)(N) approaches a constant at large N . The total average

&y)(N) contribution falls off nevertheless to zero

as 1 / N since -'the constant term in ggS*")(N) is compensated by a constant term in rgjf.b)(N) which is opposite in sign. The situation is different for

Tgi2.")(N) (where

rgg=

( G ( Z ) / A ) ~ A ~ ) , which is the sum of two contributions proportional to ( A F / N S ) '

-

const. and

(AF/ NG(z))' ( G ( z ) / G * ) =

-

const. While the first contribution is compensated by the corresponding term in zgp,b)(N), the second is not and leads to the approximately N independent contribution of the average rg$)(IV) at large N shown in figure 3. It is clear that the average value of AgL2)(N) is constant as long as the quantum limit conditions are satisfied. In our model these conditions are equivalent to fiw,

<

2 d 3 ( z ) / N since the conditions,

eq. ( 2 ) , on the spin-orbit matrix elements are

always fulfilled. At X band frequency we obtain

N-e lo5.

It should be noted that the term Agf.")(N) is essentially the contribution Kawabata [3] consider- ed and that he found it to go like a2 in the quantum limit, where a is the particle diameter. Thus, while our one dimensional model does not conform exactly to his results, it suggests that there is something special about this term, since it does not fall to zero with increasing N . The exact N

dependence of this term depends on the estimate of the squared spin-orbit matrix element

)( k q

-1

Hsol k m

+

)I2. In Kawabata's work this quan-

tity is proportional to 6 A g ; / a

-

l / a 4 where Agb is

the bulk metal Ag, while in this model it is proportional to A 2 ( A F / N G ( ~ ) ) 2

-

1/W. Note also from figure 3 that for our model at R = - 0.9 the

contribution Agi2)(N) is larger than A g y ) ( N ) if

N > 400. However, except for some N values, the

surface corrections to Ag are smaller in magnitude than the average deviation of AgY1(N) from the

N = value, for N < 4 000.

5. Conclusion.

-

We have shown in a very simple model that the change in Ag as a function of

N is the sum of three terms : AgY1(N), A g y ) ( N ) and Ag',2'(N). A g P ( N ) arises from the change in the

spacing of the particle energy levels as a function of

N ; it goes to the N = a value as 1 / N and it is the

largest contribution if N < 4 000. A g y ) ( N ) and

Ag(,"(N) arise from surface effect contributions

which break the wave vector selection rule for the spin-orbit coupling matrix elements. Although for very large N values AgL2)(N), which is similar to the

g shift calculated by Kawabata, is the largest

contribution, our analysis shows that for smaller

N

the first order term in spin-orbit coupling can not be neglected. As a consequence, the prediction that the g shift tends to the bulk g shift for small particles and that the line widths are very small may not be valid.

One of' the basic limitations of our inclusion of

surface effects in the model is the assumption that

the surface perturbation will not mix bands. As a result, only spin-orbit transitions between levels close in energy but situated in different bands are taken into account. A careful further study should show if band mixing in a small paiticle is an important contribution to the Ag. Some further work is also needed to extend this model to two and three dimensions. In particular, the N dependence of the squared spin-orbit matrix element appearing in eq. (1) is crucial to predict the behaviour of Ag as a function of N .

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g SHIFT IN SMALL METAL PARTICLES

References

[I] KUBO, R., J. Phys. Soc. Japan 17 (1962) 975. [7] MOORE, R. A., J. Phys. F 5 (1975) 459.

C21 B. W-, CO1lOque XIV (North-Holland,

_[a]

_STONE,_{A. J.,}_{Proc. R. Soc. London A 271 (1963)}_424._See

Amsterdam), 1967 468. also SLICHTER, C. P., Principles of Magnetic Resonance [3] W A B A T A , A., J. Phys. Soc. Japan 29 (1970) 902. _{(Harper and Row), 1963.}

[4] GORDON, D. A., Phys. Rev. B 13 (1976) 3738.

[5] CHATELAIN, A., MILLET, J.-I.. and MONOT, R., J. A P P ~ . [91 FROST, A. A. and MUSULIN, B., J. Chem. phys. 21 (1953)

Phys. 47 (1976) 3670. 572.

[6] DE GRAAF, A. M. and OVERHAUSER, A. W., Phys. Rev. 180 [lo] YAFET, Y., Solid State Phys. 14 (Academic Press) 1963.