Nonmetallic behaviour in thin metal wires

NEERAJ JAIN A N D RAMJI SRIVASTAVA School of Stlrdies in Physics, Jiwrrji Universib, Gwrrlior-474011, lrldin

Received October 8 , 1982

General expressions for the temperature coefficient of resistivity (TCR) of thin metal wires have been derived by incorpo- rating the thermal expansion of the wire diameter and the wire-to-substrate expansion mismatch. A possibility of transition to nonmetallic behaviour is predicted at low temperatures.

On a obtenu des expressions gCnCrales des coefficient de variation de la rCsistivitC en fonction de la tempCrature, paur des fils mCtalliques de faible diamktre, en tenant compte de la variation du diamktre ainsi que des effets dus i I'Ccart entre la dilatation du mCtal et celle du substrat. On prCdit la possibilitC d'apparition d'un comportement non mCtallique aux basses temperatures.

[Traduit par le journal]

Can. J . Phys. 61, 979 (1983)

Introduction The size effect behaviour in thin metal wires has been

discussed by Dingle, Macdonald et al . , and Ditletsen et al. ( I ) . Electron transport has been considered in terms of the Fuchs-Sondheimer (F-S) model and it has been found that the complete expressions for wires are more complicated than the corresponding expres- sions for films. Realizing the complexity of thin wire expressions, Nordheim (2) has tried to simplify them by assuming that surface electron scattering is independent of other scattering mechanisms. His approximate re- lation gives values for the resistivity which are within 5 % of those obtained by Dingle for all the values of the parameter k (k = dl&, , the ratio of the diameter rl and bulk mean free path A,). It is also noted that the per- centage deviation becomes much lower as the k value is reduced. Normally, from such size effect theories, it is expected that the resistivity behaviour, characterized by a positive temperature coefficient of resistivity (TCR), has a tendency to vanish at T = 0 K. At extremely low temperatures (around liquid helium and lower tem- peratures), however, the k values will become very low and because the wire resistivity is highly sensitive to k, the contribution from thermal expansion of the wire material will be quite appreciable and cannot be ne- glected. Furthermore, because very thin wires are often prepared with t6e help of a supporting substrate, the effect of differential thermal stress arising from the mismatch in expansion coefficients will also contribute towards resistivity.

Theory and expression

Starting from Nordheim's expression for the wire-to- bulk resistivity ratio

and including the effect of thermal expansion in k, the TCR of thin wires can be expressed as (3, 4)

1 [21 P , = P o - 1 ( P o + "I")

where j3, is the bulk TCR and a l , is the thermal expan- sion coefficient of the wire material. Now when the effect of differential thermal stress is also included by writing

the expression for the wire TCR is modfied to (3):

where ul, is the thermal expansion coefficient of the supporting substrate and k W is Poisson's ratio of the wire material.

2.

-Results and discussion From expression [4] it is seen that the TCR values are

further reduced with the inclusion of expansion effects, and that there exists a critical value (kc) of k, below which the TCR is expected to become negative, ex- hibiting a nonmetallic behaviour. These critical k values will be obtainable only at extremely low temperatures where inelastic electron-phonon scattering is associ- ated with large mean free path values. Hence, to each value of kc there will correspond a critical value of the bulk mean free path (A,,) and a critical value of the temperature (T,), at which the metallic to nonmetallic transition is expected to occur. It may be further noted that these k values are a function of the ratios CX,,~/P, and al , /al ,v. The value of kc decreases as a lw/Po is lowered, whereas if a l , / a l , is lowered, kc is expected to increase. Thus, the observation of nonmetallic conduction should

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depend on the type of material through Po and al, and o n the supporting substrate through a,,.

We have calculated the critical values (kc) of the reduced diameter for different values of the ratio al,"/Po after (i) excluding and (ii) including the substrate-wire expansion mismatch. The typical value of a,,/a1, has been chosen to be 0.5. This is the value normallv ob- tained for noble metal wires prepared with a glass cap- illary as the substrate. p, has been assumed to be 0.3. The results have been tabulated in Table 1 .

Because most of the experimental observations with respect to the nonmetallic behaviour have been made either on disordered metals or alloys, a direct com- parison of our results with experimental ones cannot be made. Theoretical predictions as well as experimental observations in this direction have recently been reported by many workers (5- 15). Explanations sug- gested are either based on the Anderson localization theory for a one-dimensional system or electron- electron correlation effects. An alternative explanation proposed by us applies only to very pure films, whereas the experiments performed on Au-Pd (6) and W -Re a l l o y s ~ 5 ) were under extremely dirty conditions. More- over, the electron mean free paths were in the range of 0.5- 1 x lo-' cm as inferred from the film resistivity. The mean free path is thus - 100 times smaller than the wire diameter. Therefore, boundary scattering effects are of negligible importance in determining the effec- tive wire resistivity.

Recently Overcash er ~zl . ( 1 I ) have observed the per- sistance of metallic behaviour up to T = 0.4 K in single crystal Bi whiskers having a diameter as low as 140 nm and a bulk mean free path of - I00 nm. Sambles et al. (15) have given a general discussion on the resistiviy of thin wires: ~ur thermore , a general length dependence of the thin wire resistance has been discussed by Masden and Giordano (13), who have also observed localization effects in thin Pt wires (12).

Conclusion Though a direct comparison with any of the above

experiments is not possible, it is suggested that our calculations may be significant provided (i) the experi- ments are performed on very pure materials and in an extremely pure environment, and (ii) the measurements are made at very low temperatures and in a low k range. Furthermore, our calculations might suggest a more

VOL. 61, 1983

TABLE I. Calculated values of the critical reduced diameter (k,) for different values of a,,/P,, (defined in the text) assuming p, = 0.3 and a,,/a,, = 0.5 for unsupported and

glass-supported noble metal wires

~- r l , /Bo with mismatch without mismatch

appropriate explanation for the observed behaviour if the effects of thermal expansion on electron density are also taken into consideration. Moreover, the surface specularity, the nature of which is not exactly known, might also change,with temperature as well as with wire diameter. If all these effects are taken into account a more fruitful explanation may arise.

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