Size Effects in the Thermal Variations of the Hall Coefficient

A number of experimental investigations of the temperature T dependence of the Hall coefficient RHF of thin metallic films have been reported1-9 in the past few years, ttowever some data have only been interpreted in terms of the temperature dependence of the electron density r ,9, even if the transport properties of these metallic films agree3,6,7,9,1 0 with the well known FuchsSondheimer theory. 11 In a previous paper we have derived an expression for the correction in the temperature coefficient, 3RH of the Hall coefficient R/4F of thin film arising from the temperature dependence of the bulk mean free path lo. Unfortunately in the case of partially specular scattering of electrons on the external surfaces the expression of/3R/4 is somewhat complicated and its numerical evaluation requires the use of a digital computer. In order to allow an easier calculation this letter proposes an alternative approach to derive an analytical expression for nH: it is taken into account that for nearly specular scattering on external surfaces the surface scattering can be treated with good approximation by the Cottey method 3 which states that the film resistivity PF and its temperature coefficient/F are respectively given by: 3,1 4 PO/PF F(p) t.t{p -1/2 + (1 -u2)ln(1 + p-l)} (1)

In a previous paper 12 we have derived an expression for the correction in the temperature coefficient, 3RH of the Hall coefficient R/4F of thin film arising from the temperature dependence of the bulk mean free path lo.Unfortunately in the case of partially specular scattering of electrons on the external surfaces the expression of/3R/4 is some- what complicated and its numerical evaluation requires the use of a digital computer.
In order to allow an easier calculation this letter proposes an alternative approach to derive an analytical expression for nH: it is taken into account that for nearly specular scattering on external surfaces the surface scattering can be treated with good approximation by the Cottey method 3 which states that the film resistivity PF and its temperature coefficient/F are respectively given by: 3,1 4 PO/PF F(p) t.t{p -1/2 + (1 -u2)ln(1 + p-l)} (1) where" 14 G(p)=p dp = p 3p--+(1-3p 2)ln(l+p-1) (3) Here/o is the temperature coefficient (t.c.r.) of the bulk resistivity Po which can be written in the form" o d In po/dT -d In lo/dT (4) The physical parameter p is related to the specularity parameter p and to the reduced thickness k by: 13 p=k-In k" [I-p]-1", p>0.5 (5) with" k a/lo.
We have previously shown 16 that in the case of transverse magnetic field the Hall coefficient of thin metallic R,F can be expressed in terms of film resistivity PF and its t.c.r.F" RHF/RHo (PF'F)/(RO "flO) (6) where RHO is the Hall coefficient in bulk material.By combining Eqs.(1), ( 2) and (6) we get" D(U)] -' (7) with" D(U)= C(u)/F(u) (8) Taking into account the value of the bulk Hall coefficient RHO : assuming that thermal variations in thickness a and specutarity parameter can be neglected; the validity of these assumptions has been recently established.

D(/1)
where" 1--3/12 } +/1 (15) 3n is the temperature coefficient of the electronic density r/previously defined as' 3n d In ?/dT (16) Eq. ( 13) predicts that RH depends on the reduced thickness k; this behaviour can be easily explained since it has been found that RHF markedly depends on film thickness ,1 7 for relatively thin films (k < 0.5).13) that 3RH + 3n gives the expression of the change in temperature of the Hall coefficient due to the temperature dependence of the reduced thickness; hence, as expected, the ratio I(RH + (3n)/[3o decreases (for a given thickness) with increasing values of the specularity parameter p; moreover it appears that the magnitude of (/3RH + (3n)/{3o rapidly decreases with increasing values of k and finally vanishes for thick films (for example when p 0.75 for k > 0.1).
More generally marked size effects could be observed only if the magnitude of/3n is considerably smaller than the generally positive o value.For example for p 0.5 and k 0.5 we obtain a deviation less than 20% until the ratio /0 keeps values greater than 0.2.
As a consequence of this study we estimate, as suggested in a previous paper, 12 that the temperature dependence of the Hall coefficient of thin metallic films may be (in the relatively high temperature range (T > 150 K)) interpreted in terms of thermal variations in both the electronic density 7(T) and the mean free path po(T); nevertheless, this last effect is significant only if the magnitude of/n is con- siderably less than/30.
Further insight is thus given for the interpretation of thermal variations in Hall coefficient of thin metallic films. FIGURE

Figure shows theoretical plots
Figure shows theoretical plots of the ratio (R + n)/o for different values of the specularity parameter p.It is clear from Eq. (13) that 3RH + 3n