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Electrical Conductivity of Thin Metal Films
(W. F. G. Swann. The electrical resistance of thin metallic films, and a theory of the mechanism of conduction in such films. Phil. Mag. 28, p. 467, 1914.)
(R. W. King. The electrical conductivity of sputtered films, Phys. Review, 10, p. 291, 1917.)
The specific ohmic resistance of metals increases sharply when the thickness of the conducting layer is reduced below a certain limit (fractions of a micron). On the other hand, the temperature coefficient of resistance of the thinnest films is negative and extremely large in absolute value. These anomalies, established by a number of investigators¹), can with difficulty be fitted within the framework of the existing electron theories of conductivity. Sir J. J. Thomson pointed out²) that the mean free path of the electron, \(\lambda\), which is proportional to the conductivity, becomes a function of the thickness of the conducting layer \(t\) when the dimensions of the latter are comparable with the free path and, consequently, special surface conditions acquire importance. For \(t > \lambda\)
\[ \lambda' = \lambda \left(1 - \frac{\lambda}{4t}\right) \tag{1} \]
where \(\lambda'\) is the free path in the thin conductor and \(\lambda\) is the normal free path at the corresponding temperature. For \(t < \lambda\):
\[ \lambda' = t \left\{\frac{3}{4} + \frac{1}{2}\log \frac{\lambda}{t}\right\}. \tag{2} \]
If Thomson’s theory fully exhausted the phenomenon, then, by finding the critical thickness of the metal at which the sharp increase in resistance begins, one could judge the magnitude of \(\lambda\). Observations show, however, that conductivity falls hundreds of times faster than would follow from Thomson’s theory. Moreover, the negative value of the temperature coefficient is not at all accounted for by the theory mentioned.
In the new works of Swann and King a fairly satisfactory interpretation of the anomalies described is given.
¹) I. Stone, Phys. Rev. 6, 1, 1898; Vincent, Ann. de Chim. et de Phys. (7), 19, 494, 1900. Longden, Phys. Rev., 11, 40, 1900. Patterson, Phil. Mag. 4, 1902. K. Baedeker, Die elektr. Erscheinungen in metall. Leitern, p. 16, 1911.
²) J. J. Thomson. Cambr. Phil. Proc. 11, 120, 1901.
5) The temperature coefficient of films of medium thickness changes sign within the given temperature interval. 6) For the thinnest films the resistance is inversely proportional to the absolute temperature, i.e.
\[ R\cdot \Theta = \mathrm{Const.}, \]
as is seen from Table (2), where the values of \(R\cdot \Theta\) are compared for film No. 21.
Table 2.
| \(\Theta\) | \(\Theta\cdot R\) |
|---|---|
| \(93^\circ\) | \(1{,}04\cdot 10^9\) |
| \(287^\circ\) | \(1{,}04\cdot 10^9\) |
| \(373^\circ\) | \(1{,}24\cdot 10^9\) |
To explain this distinctive behavior of the conductivity, Swann proposes that a film obtained by sputtering a cathode has a granular structure (which is also confirmed by direct microscopic observations)2, and he constructs an electronic theory of the conductivity of this discrete conductor. The expressions obtained by Swann do not lend themselves to quantitative verification and allow only a qualitative judgment about the course of the conductivity curves as functions of thickness and temperature. The essence of Swann’s theory reduces to the following. In a thin film of granular structure conductivity of two kinds is possible: 1) a part of the molecular complexes—grains—may be in direct metallic contact, i.e. the width of the gaps between grains does not exceed the radius of the molecular sphere of action. Such grains will always form a thin metallic conductor of enormous resistance, obeying the ordinary laws of metallic conductivity (validity of Ohm’s law, positive temperature coefficient, etc.); 2) the other group of grains may be separated by gaps larger than the radius of the molecular sphere of action, but so small that the molecular forces of one surface still do not cease acting on the neighboring forces. These gaps must be of the order of 100 Å. If a sufficiently large molecular complex bears an electric charge, then, according to the Richardson formula, the gaps separating grains of this kind may, upon application of a field, become conductors. The current emitted from a unit of surface is determined by the well-known Richardson formula
\[ i=A\Theta^{\frac12}e^{-\frac{B}{\Theta}} \]
where \(A\) is a constant depending on the number of free electrons per unit volume of the metal, and \(B\) is proportional to the work necessary for an electron to overcome the surface molecular forces; for platinum \(A=3{,}7\cdot10^{?}\), \(B=5{,}22\cdot10^4\). It is easy to see that \(A\) cannot depend on the width of the gap; on the contrary, \(B\) may decrease when the gap is sufficiently small, when the molecular forces of neighboring surfaces are mutually weakened to a certain degree. For constant \(B\), i.e. comparatively wide
gaps, the thermo-ionic conductivity is immeasurably small, and only for gaps satisfying the indicated conditions can it attain a considerable value. The resistance of such gaps must, obviously, vary depending on the applied voltage and possess a negative temperature coefficient. Grains with such gaps may link up into chains and form conductors of the second kind. The combination of two types of conductors in a thin Swan film explains all the anomalies of conductivity. In the thinnest films conductors of the second type predominate; in comparatively thick films the conductivity is mainly “metallic”; and, finally, in films of intermediate thicknesses the conductivity is effected now by one method, now by the other.
R. Wood1 wrote that thin metallic films, cut across with a diamond, nevertheless retain a considerable conductivity, and supposed the existence of an “electron atmosphere” explaining, in the given case, the mechanism of conductivity. Subsequent investigations showed that the conductivity is explained by metallic chains extending from one edge of the slit to the other; in the absence of such chains the gap becomes conducting only when the width is \(<500\,\mu\mu\). Wood’s cut films are a complete analogue of Swan’s films: they possess a large negative temperature coefficient, etc. In any case, the question of the conductivity of films of granular structure is, apparently, closely connected with the question of the conductivity of thin, gas-like films.
Swan’s theory is of a purely qualitative character; in any case, to explain relation (3) it is necessary to assume that \(B\) in formula (4) depends not only on the width of the gap between the grains, but also on the temperature. It is possible that this dependence is explained by compression of the gaps during the thermal expansion of the grains.
King, in his work, also proceeds from the assumption of the granular structure of thin films, considering, however, that the conductivity is explained exclusively by metallic chains of grains connecting one edge of the film with the other. Let \(N\) be the number of grains per unit area; \(n\) of them are connected into chains and determine the conductivity \(c\). One may suppose that
\[ n = c \cdot f(N), \]
where \(f(N)\) decreases slightly in magnitude with increasing \(n\). For the region of small changes of \(N\), King arrives at the following simple expression:
\[ \log c = (\rho + 1)\log N + \mathrm{Const}, \tag{5} \]
where \(\rho\) is a constant. In contrast to Swan, King operated only with a single film, upon which new quantities of metal were successively deposited. The lead wires from the film were soldered into the cathode tube, and the resistance could be measured at any moment of deposition. In order to satisfy the conditions of the theory (small changes of \(N\)), the deposition intervals were taken very small: 1 and 2 seconds. The accuracy of measuring the deposition time was guaranteed by an automatic falling switch of the inductor. The work was carried out with platinum, gold, and silver. The results are fully
confirmed relation (3), i.e., the logarithm of the sputtering time is, within sufficiently broad limits, linearly related to $\log c$. King estimates the thickness of his films from optical data as $1$–$6\,\mu\mu$; the temperature effect was not studied by him.
King’s theory only supplements Steam’s theory by establishing the functional dependence itself; it does not, however, explain the negative value of the temperature coefficient and does not permit one to conclude, as the author does, that the conductivity of thin films is of a purely metallic character. Equation (5) indicates only that $C$ is a certain statistical function of $N$, while the statistical elements may be either King’s metallic chains or Steam’s “gaps”; in other words, the validity of equation (5) merely confirms the granular structure of the film.
The condensation of metal molecules into molecular complexes occurs, according to King’s supposition, in the space between the cathode and the glass plate. It is possible, however, that the surface of the plate itself is the determining factor in the formation of grains.
S. Vavilov.