ON THE CHARGE NEUTRALITY OF METALLIC FINE PARTICLES

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ON THE CHARGE NEUTRALITY OF METALLIC

FINE PARTICLES

A. Kawabata

To cite this version:

JOURNAL DE PHYSIQUE Colloque C2, supplément au n° 7, Tome 38, Juillet 1977, page C2-83

ON THE CHARGE NEUTRALITY OF METALLIC

FINE PARTICLES

A. KAWABATA (*)

Research Institute for Fundamental Physics, Kyoto University, Kyoto 606 Japan

Résumé. — On étudie l'effet des fluctuations de charge dans les petites particules en RMN et RPE. On montre que le temps de relaxation des spins nucléaires ne décroît pas simplement avec la taille de la particule.

Abstract. — The effects of charge fluctuations of metallic fine particles on NMR and ESR are investigated. It is shown that the relaxation rate of nuclear spins does not simply decrease when the particle size becomes smaller.

1. Introduction. — The charge neutrality is an essential assumption in discussing the electronic properties of metallic fine particles [1]. Kubo has argued that the large electrostatic energy prohibits the thermal fluctuation of the charge. In this paper we will discuss the effects of quantum fluctuations due to tunneling of electrons between the particles. The effects of tunneling is very important, for in many cases the experiments are done on aggregates of fine particles separated by layers of insulator, e.g. the oxide of the metal. Among the various phenomena which are caused by the discreteness of the energy levels, absence of the relaxation of nuclear spin and the electronic spin [2] are the most direct evidences of the discrete energy level. So far there have been several reports on the absence of the relaxation of electron spins [3], while it has not been observed for nuclear spins on particles of semi-macroscopic size [4] even when the level discreteness seems to manifest itself in other facts, e.g., Knight shift. Thus it is meaningful to in vestigate the effects of the tunneling on ESR and NMR, and in the following we will confine ourselves in these problems.

Consider the case of particles with odd number of electrons. Then the situation is analogous to that of magnetic insulators, for magnetic moment is produced because of electron correlation i.e. because of coulomb repulsion between electrons. Generally, the relaxation time of nuclear spins in magnetic insulators are very short, and we can expect that it can be short also in fine particles, rather than be long as is expected from a simple consideration. Below we will be mainly concerned on this point.

(*) Present address : Department of Physics, Gakushuin University, Toshima-ku, Tokyo 171 Japan.

2. Model. — In the absence of tunneling the energy levels in each particles are quantized, and with eia and aia<T we denote the energy eigen-value and the anihilation operator of the eigen-state a in the ith particle, respectively, a being the spin suffix. The electrostatic energy due to excess charge q is given by [1] q2/2 a, a being the radius of the particle, and is equal to UAn2, where

U= e2/2 a with e the electronic charge and Aw, is the number of the excess electrons in the ith particle. Thus the total Hamiltonian of the electrons in the aggregates of fine particles is of the form,

I = I & , a L at o+ S [ / A n ? + I , (1)

where we have neglected the dependence of U on the particle, and §ft is the tunneling Hamiltonian

U aP<r

It is hard to justify this form of Hamiltonian from the first principle. Moreover, there is an ambiguity in the choice of tiatp for the states a and j8 with different energy eigen values, for generally they are determined in such a way that they should give a correct rate of tunneling as a real process. Here we assume that tiajp is practically non vanishing for adjacent pairs of ij and for such a and j3 that \sia— eip\ :£ t2/8, t being an appropriate average of |t,„^|. This assumption is rather crucial and the condition for the charge neutrality is much depen-dent on it (for instance, the assumption that t-m\p is constant for any a. and /3 leads to very large charge fluctuation as is written in the abstract). However, it does not affect the problem of the spin reso-nance.

3. Condition for the charge neutrality. — We consider two adjacent particles 1, 2 with even

C2-84 A. KAWABATA number of electrons, and neglect other particles for

the time being. We assume that a 5 0 and

p

5 0 are

occupied in the particles 1 and 2, respectively, at 0 K in the absence of tunneling. This state will be

denoted by !Po. Then, with the use of perturbation theory, the ground state with tunneling is given by

('1

and the amplitude of the non-neutral state are given by

+ C

C

2

1

ta,

I2

a > o p r o ( ~ p - &a - 2 U)' ' (3')

According to the assumption on tap, the states which contribute to A,, are those with energy less than t measured from the Fermi level, and hence we have

A,,

-

t 6 / U Z S 4 , (4)

where S is the average level spacing. The factor

t 4 / S 4 comes from the number of state in the inteval

t 2 / S of energy (we assume t > 6 ) . If we consider

z

adjacent particles instead of one, the right hand side of (4) is to be multiplied by

z.

Thus the condition for charge neutrality is

It is easy to see that this condition is valid for a particle with odd number of electrons. Such a particle has one unpaired spin under the condition ( 9 , and those spins are free in the absence of tunneling (we neglect the effects of spin-orbit coupling). As in the case of magnetic insulator, the tunneling of electrons between particles gives rise t o exchange coupling (Anderson's kinetic exchange [5]).

4. Coupling between spins.

-

Consider two

particles with odd number of electrons. We assume that at 0 K in the absence of tunneling the states

a, ,G

<

0 are occupied by two electrons with diffe- rent spin directions and that each of the states

a,

p =

0 are occupied by one electron. When the spins of two unpaired electrons are parallel, the correction to the ground state energy due to tunneling is given by

(I) Hereafter the suffixes 1 and 2 will be suppressed.

+

C

p c o Ep - E o - 2 U

On the other hand, when the spins are anti-parallel we have

AE, = AE,

+

2

1

t,I2

e o - E o - 2 U ' (7)

and hence the coupling of spins are antiferromagne- tic with coupling energy

Thus, if the values of t and U are not much

different for different particles, there is a possibility of the ordering of spin orientation with Tc of order

zt2/ U. However, as will be shown below, Tc is

generally very small and the ordering of spins is of no simple matter because of the complexity of the system. Therefore we will not go further on this problem and assume that the spins are parama- gnetic.

The coupling energy J can be written in the form

where use has been made of (5), and we observe that J < S, if the condition for the charge neutrality is fulfilled.

5. Relaxation of nuclear spin.

-

Here we discuss

the longitudinal relaxation time TI of nuclear spin. We write the Hamiltonian of the hyperfine coupling in the form

X,,

= A s-(@'(R) u@(R)) (10) where A is a constant, s the spin of the nucleus located at R, a the spin matrix, and +(R) is the field operator of the electrons, In the case when the energy levels are quasi-continuous, TI is determin- ed by the Korringa mechanism ;

h / TI = h / TK = 2 ~ r r n ( ~ D ' ( A

/

V)' , (1 1)

where D is the density of states without spin degeneracy, T the temperature and V is the volume of the particle.

ON THE CHARGE NEUTRALITY OF METALLIC FINE PARTICLES C2-85

theory of NMR in magnetic insulators [ 6 ] , we separated by potential barriers. Generally t is

obtain determined in such way that it gives the correct

transmission coefficient P of the barrier. If we

$=

(-$yI-:

dt exp neglect the electrostatic energy, the characteristic

(12) time T of the escape of an electron from the particle

-

A l w is phenomenologically given by

=-(vyeXP{-Z(<~}, ws 117 = P v F / a , (17)

where v , / a is the frequency of the collision of the where w , is the Zeeman energy of an electron, and electron with the surface. On the other hand,

w , is the frequency of spin fluctuation given by quantum theory gives

W E =

3 z T 2 / S . (13) h - 1 / ~ = 2 n-zt2/S, (18)

If the effect of the discrete levels on T 1 is absent, since 116 is the density of states in the limit of

T , is given by ( 1 1 ) with D = 116. To compare ( 1 1 ) quasi-continuous levels. Equating the right hand and (12), we investigate the ratio sides of (17) and ( I S ) , we obtain

T , / T , = d % w , k , TS-2 exp { ( W , / W , ) ~ / ~ } . (14) P = 2 n z t 2 a h l v , 6 . (19)

As can be seen from (9) and (13), we have

~ , = - d 3 ~ / 8 S ~ / t , (15)

and from (5) we have w ,

<

S. Thus, provided that

k , T 5 6 , the effect of the energy level quantization

is to make T I small rather than large unless w, < w,. 6 . Effects on ESR spectrum.

-

Quenching of the relaxation due to level quantization is expected also for electron spin. In this case, the exchange coupling between electrons with itself gives no effect on ESR spectrum. Observations of the effects of level discreteness on resonance lines have been reported by several people [3]. Monot, Narbel and Bore1 have observed a spectrum with inhomogeneous broadening, which is ascribed to the variation of effective g shift in different particles [2]. On the other hand in the experiment by Gordon the broadening is attributed to the fluctuation of hyperfine field caused by the small number of nuclear spins. In both cases the width of the broadening is of order 10 gauss, and it will be narrowed if the exchange coupling between spins on adjacent particles are strong enough. The condition for the narrowing is given by w ,

>

WB,

where W B is the width (in energy) of the broadening

without narrowing. In (8) and (13) a typical value of Uis of order 0.1 eV and W B

-

lop7 eV and thus we obtain

t > 1 0 - ~ e v

-

1 K k B (16)

as the condition of the narrowing.

7 . Estimation of t and discussions. - In this section we estimate the value of t on a simplified model i.e. electrons confined in small spaces

For a = 50

A,

and v , = los cm, the condition (16)

of the narrowing of ESR line gives

Thus value of P i s much larger than the characteris- tic one in tunneling experiments. In fact the potential barrier of 5 eV in height and 3

A

in width gives P - Therefore in general we can expect the condition (16) does not hold and that the broadening of the ESR line can be observed.

As regards the relaxation of nuclear spin, from P < it follows that w , / p ~ t 10 gauss, and

generally the condition w , 4 w , of the absence of

relaxation is fulfilled. Thus within the investigation on the present simplified model we can not explain the presence of the relaxation of nuclear spin. It is possible, however, to choose such a value of t that

7 1 and w ,

-

w,, if it is regarded as a parameter.

In this case the condition for the charge neutrality is fulfilled, while the quenching of the relaxation can not be observed. For instance, f o r

T = S / k , = 5 K , U = 0.1 eV, t / k , = 15 K, z = 6 ,

we find that 7

-

0.1, and w , / p ,

-

3 kG. For

such particles the inhomogeneous broadening of ESR absorption line will not be observed. It is of great interest to observe ESR and NMR on the same specimen.

As for the particles with even number of electrons, at T = 0 the ground state in a particle is of zero total spin. However, the probability of finding the system in the excited state with non zero total spin will not be very small unless T + 6, and such states can contribute to the relaxation nuclear spins.

References

[I] Kueo, R., J. Phys. Soc. Japan 17 (1962) 976. GORDON, D. A., Phys. Rev. 13 (1976) 3738 and the papers

[2] KAWABATA, A., J. Phys. Soc. Japan 29 (1970) 902. cited therein.

131 TAUPIN, C., J . Phys. Chem. Solids 28 (1967) 41. [4] KOBAYASHI, S., TAKAHASHI, T. and SASAKI, W., J. Phys. SAIKI, K., FUJITA, T., SHIMIZU, Y., SAKOH, S. and WADA, Soc. Japan 36 (1974) 714.

N . , J. Phys. Soc. Japan 32 (1972) 447. [S] ANDERSON, P. W., Phys. Rev. 115 (1959) 2.

MONOT, R., NARBEL, C. and BOREL, J. P., NUOVO Cimento [6] JACCARINO, V., Magnetism Vol. HA. G. T. Rado and H. Suhl