BROADENING OF ELECTRON ENERGY LEVELS AND NUCLEAR SPIN-LATTICE RELAXATION IN SMALL SUPERCONDUCTORS

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BROADENING OF ELECTRON ENERGY LEVELS

AND NUCLEAR SPIN-LATTICE RELAXATION IN

SMALL SUPERCONDUCTORS

E. Imánek

To cite this version:

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JOURNAL DE PHYSIQUE Colloque C2, supplément au n" 7, Tome 38, Juillet 1977, page C2-79

BROADENING OF ELECTRON ENERGY LEVELS

AND NUCLEAR SPIN-LATTICE RELAXATION

IN SMALL SUPERCONDUCTORS

E . S I M A N E K

D e p a r t m e n t of P h y s i c s , U n i v e r s i t y of California, R i v e r s i d e , California 92$02, U . S . A .

Résumé. — Pour expliquer l'absence d'effets de quantification des niveaux dans les petites particules nous proposons un mécanisme d'élargissement des niveaux mettant en jeu un passage des électrons d'un grain à l'autre par effet tunnel et thermiquement activé par interaction porteur-porteur. Nous étudions aussi l'élargissement dû aux fluctuations de supraconductivité. Ce dernier mécanisme n'agit qu'en présence d'un élargissement monoélectronique.

Abstract. — To explain the absence of level quantization effects in small particles we propose a level broadening mechanism involving intergrain electron tunneling, thermally activated by carrier-carrier interaction. We also investigate the quasiparticle broadening due to superconduc-ting fluctuations. The latter mechanism is found operative only in the presence of one-electron broadening. 1. I n t r o d u c t i o n . — A s p o i n t e d o u t b y K u b o [1], t h e n u c l e a r spin-lattice r e l a x a t i o n r a t e in fine metallic p a r t i c l e s s h o u l d b e s u p p r e s s e d b y t h e d i s c r e t n e s s of e l e c t r o n levels. S u b s e q u e n t m e a s u r e -m e n t s o n a l u -m i n u -m a n d c o p p e r p o w d e r s -m a d e b y K o b a y a s h i et al. [2], h o w e v e r , did n o t s h o w s u c h a s u p p r e s s i o n . T h i s d i s c r e p a n c y b e c a m e m o r e a p p a -r e n t afte-r t h e -r e c e n t s t u d y b y T a k a s h i et al. [3]. I n this latter w o r k a g o o d a g r e e m e n t w a s o b t a i n e d b e t w e e n t h e m e a s u r e d r e l a x a t i o n r a t e s for small a l u m i n u m particles a n d t h e t h e o r y of S o n e [4] w h i c h did n o t include a n y level q u a n t i z a t i o n at all.

T h e p r e s e n t w o r k is a n a t t e m p t t o r e s o l v e t h e a b o v e d i s c r e p a n c y . W e a r e guided b y t h e idea t h a t t h e nuclear spin-lattice r e l a x a t i o n p r o c e s s will b e e n a b l e d if t h e e l e c t r o n s of t h e small grain a r e c o u p l e d t o t h e s u r r o u n d i n g t h e r m a l r e s e r v o i r [1]. T o explain t h e spin-lattice r e l a x a t i o n r a t e s of ref. [2, 3 ] , this coupling s h o u l d p r o d u c e b r o a d e n i n g of e l e c t r o n levels w h i c h is c o m p a r a b l e w i t h t h e e l e c t r o n level spacing d u e t o finite size. T a k a h a s h i

et al. [3] suggested t h a t coupling of e l e c t r o n s t o

s u p e r c o n d u c t i n g fluctuations m a y b e r e s p o n s i b l e for t h e b r o a d e n i n g of t h e d i s c r e t e levels. I n section 2 w e a n a l y z e this possibility a n d c o m e t o t h e c o n c l u s i o n t h a t s u p e r c o n d u c t i n g fluctuations in a n isolated grain d o not b r o a d e n t h e levels in a c c o r d with t h e reversible b e h a v i o u r of a finite c l o s e s y s t e m . S e c t i o n 3 is d e v o t e d t o a d e s c r i p t i o n of a n e w m e c h a n i s m for b r o a d e n i n g of e l e c t r o n levels in fine particle p o w d e r s . I t is b a s e d o n t h e o b s e r v a t i o n t h a t in a c t u a l s a m p l e s t h e particles h a v e a t e n d e n c y t o f o r m c l u s t e r s . T h i s e n a b l e s e l e c t r o n h o p p i n g b e t w e e n g r a i n s w h i c h is s o m e w h a t similar t o t h e impurity h o p p i n g in s e m i c o n d u c t o r s [5]. W e a l s o d i s c u s s t h e m e c h a n i s m s w h i c h c o u p l e t h e e l e c t r o n t o t h e t h e r m a l r e s e r v o i r a n d p r o p o s e o n e w h i c h involves t h e t i m e - d e p e n d e n t electric field fluctua-t i o n s p r o d u c e d b y fluctua-t h e fluctua-t h e r m a l m o fluctua-t i o n of fluctua-t h e reservoir c a r r i e r s . T h e resulting h o p p i n g r a t e is m u c h stronger t h a n t h a t d u e t o t h e p h o n o n a s s i s t e d h o p p i n g — a p r o c e s s well k n o w n in impurity c o n d u c t i o n [5]. F i n a l l y , w e c o m p a r e this b r o a d e n ing m e c h a n i s m with s o m e of t h e p r e s e n t e x p e r i m e n -tal d a t a o n small particle p o w d e r s .

2. Interaction of electrons with fluctuating C o o p e r p a i r s . — F l u c t u a t i n g C o o p e r pairs a r e d e s c r i b e d b y a (-matrix, w h i c h in a small grain h a s n o spatial d e p e n d e n c e a n d is only a function of f r e q u e n c y [6, 7]. T h e r o l e of t h e (-matrix in t h e m e c h a n i s m of level b r o a d e n i n g c a n b e u n d e r s t o o d b y referring t o figure l a w h i c h d e p i c t s t h e D y s o n e q u a t i o n for e l e c t r o n p r o p a g a t o r r e n o r m a l i z e d b y p a i r -a) b)

FIG. 1. — a) Dyson equation for electron propagator renormalized by the interaction with pair fluctuations. Light straight lines : bare electron propagator ; heavy straight lines : fluctuation renormalized electron propagator; wavy line:

t-matrix.

b) Dyson equation for the f-matrix in Hartree Approximation.

Straight lines of the box diagram: electron propagators ; light wavy line : t-matrix in the RPA approximation ; heavy wavy

line: t-matrix in Hartree approximation.

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fluctuations [7]. The inverse electron lifetime will be given by the imaginary part of the self-energy. When the poles of the t-matrix are imaginary, corresponding to a relaxation of Cooper pairs, there will be a finite lifetime-broadening of the electron levels. This is the case considered previously by Patton [6] who starts from the RPA t-matrix given by

t ;iA(o ) =

NO

f2 [1 n(T/ T,) - iww /8] (1) where

NO

is the electron density of states at the Fermi level and f2 is the volume of the particle. We note that the imaginary pole in eq. (1) is a direct consequence of the use of free-electron propagators with continuous energy spectrum [6]. Therefore it is of interest to calculate the t-matrix in the discrete electron level scheme of a small particle. Following Kadanoff and Martin [8] we have in this case

1

[t g;A(o)]-l =

-

+

C

1 tanh

-

Ek =

V k w-2Ek 2 T

where V is the strength of the electron-electron interaction and sk are the discrete energy levels. The poles of t@b~(o) are obtained graphically by plotting the function F(o, t), and looking for

1

intersections of the latter with the -

-

line. (See

v

Fig. 2). For T > T,, JF(0, T)I< 1/ V and intersections at finite o appear, corresponding to real poles. For T < T,, IF(0, T)I> 1/V, and became imaginary

indicating the onset of instability with respect to the formation of Cooper pairs [8]. In a small supercon-

FIG. 2.

-

A plot of the function

1 Ek

F(w, T,) = - - tanh -

k2Ek 2 T c - 1

Intersections of F(o, T,) with the line -

-

determine the poles

Veff

of the function tg)(w) at T = T,. The function F ( o , T,) is shown

1

to touch the - -(dashed line at

_v

o = 0 , in accord with the RPA definition of T,, The energy levels are separated by a constant

difference AE.

ductor, the inclusion of the interactions between fluctuations is known to smear the phase transition [6, 71. In the following we show, that, if these interactions are taken into account in a 'self -consistent Hartree approximation, the resulting

t-matrix t@(o) exhibits only real roots.

From the Dyson equation shown in figure l b we obtain

The parameter g ( T ) is obtained, by calculation of the box diagram in figure lb, in the form

where om = w(2 m

+

1) T. Solving eq. (3) for t("(w-= 0) we get

The proper branch of this solution is the one which involves the upper sign on the r.h.s. of eq. (5). In fact, if one takes the limit g--t 0 of the latter one recovers the RPA result t$bA(w = 0). From eqs. (4) and (5) i t follows that t $)(o = 0) > 0. Eq. (3) then implies

Consequently there are intersections of the F(o, T )

1

curves and the - - line, corresponding to real V&

poles of t @(o).

This result in in accord with the above assertion that there should be no superconducting instability in a small superconductor. Furthermore, if the self-energy in figure 1 is calculated with the use of tg(w) there will be no lifetime broadening due to superconducting fluctuations ; since the self-energy has no imaginary part. Hence, we conclude that in an isolated grain, the renormalization by fluctuations in not a sufficient mechanism for lifetime broadening of the discrete electron levels.

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MECHANISM FOR ENERGY LEVEL WIDTH I N SMALL PARTICLES C2-81

where E = ( T

-

Tc)/ T, and AE is the level separa- tion due to finite size of the grain [I]. This result indicates that the fluctuation mechanism [3] is quite effective for T

----

T,, provided there is a coupling of electrons to the thermal reservoir.

3. Thermally activated intergrain hopping.

-

We describe now a mechanism for the above coupling which should be operative in samples where the fine particles are in a good mechanical contact with each other [lo]. Electron micrographs of the powder samples show frequently a tendency of the particles to form clusters. This can be understood by taking into account the attraction between particles due to the Van der Waals force. For an orientational estimate of the latter we calculate the energy of attraction U of two metallic spheres of

radius R = 100

A,

separated by a distance of closest points s = 5

A.

Following ref. [ll], we have

where kg is the Boltzmann constant and hw is the plasmon energy, which, for metals, is of the order of 10 eV. Hence it is practically impossible to dissociate the particle-pair at room temperatures.

As a result of this cluster formation, most of the particles in the sample experience strong perturbations due to thermal excitations of the neighbouring grains. These perturbations will produce transitions between electron energy levels. We consider transitions which involve thermally activated hopping of electrons between grains [lo]. The latter mechanism is known to be responsible also for the electrical conduction in granular films in the dielectric regime [12]. We note that the proposed intergrain hopping does not necessarily imply a high electrical conduc- tivity of the powder as a whole. Imperfect packing will inhibit the percolation of electrons across the sample:-ih addition, the fluctuations of the size of the particles produce energy level mismatch between neighboring grains thus contributing to an Anderson-type of localization of the electrons 1131. To calculate the hopping rate between the neighbouring grains a and b we first consider the quantum mechanical mixing of the corresponding electron orbitals

4,

and

41,

yielding orbitals of the pair-complex [15]

4i =

4 a

+

~ 4 b

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'h

= 4 b -

~ 4 a

where y

--

T'I2, Tbeing the transmission coefficient of the potential barrier due to oxide coating of the particles. The electron is initially in orbital

4i

with energy Ei t Ej (we assume Ea

<

Eb). A real transition to orbital c$~ takes place provided the energy difference Ej - Ei is supplied by the thermal reservoir. A conventional mechanism for the electron-reservoir interaction is the electron-phonon

coupling familiar from impurity conduction in semiconductors 1-53. For small particles the relevant deformation potential arises from vibrationally induced changes of their volume. Numerical esti- mates of the hopping rate due to the latter mechanism were made [14]. For aluminum particles of radius 100 and at T = 1 K we find a rate which is one to two orders of magnitude too small to account for the required width of levels.

A more effective electron-reservoir coupling results from interaction between the electron and the electrostatic potential fluctuations V(r, t) due to the thermal motion of the carriers surrounding the grain a. The corresponding interaction Hamilto- nian is

where I/J = ci +i

+

cj

4j.

The operator c f ci acts to

transfer the electron from

4i

to

cpj.

The electrostatic p o t e n t i a l d i f f e r e n c e between t h e grains A V(t) =

Va

- Vt, is a random function of time. The fluctuations of A V(t) induce energy conserving transitions i + j with the probability rate

The contribution to AV due to distant carriers is strongly diminished by the presence of Debye- Hiickel screening [15] and one can calculate

( A V(t) A V(0) ) by including only the carriers hopping on and off the nearest-neighbour grains [lo]. Denoting the average time of residence of a carrier on a given grain by T we then have from (11)

Where A Vtr) is the potential due to r-th neighbours,

z

is the coordination number of grain a, and Ec is the charging energy [12]. We note that the factor exp(- E,/2 T ) gives the probability of a grain to be

singly ionized. To estimate the lifetime broadening

6E = h / r , we assume an octahedral coordination

and calculate 1/r from (12) self-consistently, with use of relation 1 / ~ = 6 Wij. For (wi

-

wj)2 T~ 4 1, we get

6 E = 0 . 1 y e x p C - E c / 4 T ) ( e 2 / R ) . (13)

The charging energy E, can be estimated using the

model of a concentric capacitor [12]. For particles of radius R = 100

A

separated by an oxide layer of dielectric constant E = 5 and of thickness s = 5

A

we obtain E, = e2 S/ &R2 = 1.4 x 10--3 eV. Inserting this value into (13) we get, at T

-

1 K, a complete smearing of levels (SE

-

AE = eV) for

y = 5 x low2. This is a smaller value than that required by the phonon assisted hopping [14] and

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leads to an acceptable barrier transmission coeffi- cient [12].

The R-2 dependence of

Ei

implies, via eq. (13), a pronounced dependence of 6E on the particle size.

Recent far-infrared studies [16] show a distinct energy level structure f o r smaller particles (2 R = 150A) and a smeared spectrum for larger

particles (2 R = 400

A),

which is in accord with eq.

(13) (see the note added in proofs). Since the

intergrain hopping makes the electron propagatbr similar to that of a dirty superconductor [17], one can understand why the theory of Sone [4], based on

such propagators, gives a good agreement with experiment [3].

I should like to thank Professor D. MacLaughlin for many helpful discussions and critical reading of the manuscript.

Note added in proofs : A more recent study by Granqvist et al. (Phys. Rev. Lett. 37 (1976) 625) did

not reveal this level structure for ultrafine A1 particles (2 R = 50

A).

The far-infrared absorption

is probably dominated by the dielectric losses in the amorphous surface oxide layer (see E. Sim~nek,

Phys. Rev. Lett. 38 (1977) 1161).

References [I] KUBO, R., J. Phys. Soc. Japan 17 (1%2) 975.

[2] KOBAYASHI, S., TAKAHASHI, T., and SASAKI, W., J. Phys.

Soc. Japan 31 (1971) 1442.

[3] TAKAHASHI, T., KOBAYASHI, S., and SASAKI, W., Solid

State Commun. 11 (1975) 681.

[4] SONE, J., J. LOW Temp. Phys. 23 (1976) 699.

[5] Mom, N. F., and DAVIS, E. A., in Electronic Processes in

Non-crystalline Materials (London : Clarendon Press) 1971, p. 104.

[6] PAT~ON, B., Phys. Rev. Lett. 27 (1971) 1273.

[7] SIMANEK, E., in Proceedings of the International School of Physics Enrico Fermi, on Local Properties at Phase Transitions, Course LIX, July 1973 (Academic Press :

New York and London) 1975, p. 838.

[8] KADANOFF, L. P., and MARTIN, P. C., Phys. Rev. 124 (1%1) 6.

[9] DAVIES, P. C. W., in The Physics of Time Asymetry (University of California Press : Berkeley and Los Angeles) 1974, p. 154.

[lo] SIMANEK, E., Phys. Lett. 59A (1976) 231.

[11] MITCHELL, D. J., and NINHAM, B. W., J. Chem. Phys. 56 (1972) 1117.

[I21 ABELES, B., SHENG, P., COWS, M. D., and ARIE, Y., Adv.

Phys. 24 (1974) 407.

[I31 ANDERSON, P. W., Phys. Rev. 109 (1958) 1492. [I41 SIMANEK, unpublished results.

[IS] DEBYE, P., and HUCKEL, W., Phys. 2 . 2 4 (1923), 185, 305. [16] TANNER, B. D., SIEVERS, A. J., and BUHRMAN, R. A., Phys.

Rev. B 11 (1975) 1330.