For three examples of two-phase composites, in Figure 2(b), Figure 2(d), and Figure 2(f), the concentration dependence of R related to its values at is shown, calculated by Equation (13), and compared with Equation (3), denoted as “C & J”. In Figure 2(a), Figure 2(c), and Figure 2(e), the corresponding concentration dependence of the Hall mobility () is shown, where is calculated by Equation (9). There are two essential differences between the two solutions Equation (3) and Equation (13):

(1) The most striking difference appears in Figure 2(a) and Figure 2(c): The “C & J” curves decrease dramatically with increasing and pass through a pronounced minimum at, although and, respectively. In contrast, the curves calculated by Equation (13) agree with the expectation: Figure 2(a): agrees with for all; Figure 2(c) and Figure 2(e): increases and decreases with increasing, respectively.

A possible interpretation for such dramatic decrease of at (“C & J” curves) could be additional scattering centres in the added phase boundaries. Such an effect by the phase boundaries is expected to be the more pronounced the smaller the sizes of the phase grains,. However, the C & J formula [21] [26] does not contain.

The differences between Equation (13) and the curves “C & J” are the larger the larger the difference between and. On the other hand, for the limiting case, , Equation (3) and Equation (13) agree.

(2) Another striking difference between Equation (13) and Equation (3) is represented by the boundary case “and”, for which one obtains

and

respectively, and for, Equation (9) gives

Starting at, with decreasing both and decrease continuously until they vanish at

. This result corresponds to the fact that for there is no longer a connected metal cluster through the composite (in correspondence with the assumption made earlier that the phase grains are spherical without preferred orientations and arranged in a symmetrical fashion). This result is, however, not reflected by Equation (17) which gives even for, where all the metallic granules are separated by adjacent insulating phase regions, that is, electron transport through the sample does not happen, if additional tunneling is excluded.

These two differences, (1) and (2), suggest the fact that Equation (16) represents the physical situation better than Equation (17). Therefore, in the following, Equation (13), respectively Equation (16), will be applied in a discussion of the Hall coefficient in M-I composites.

For calculated by Equation (16) applied to the R data of Figure 1(a), Figure 1(b), we find that they can be approximated by a relation similar to Equation (2),

where is a constant for a given M-I composite: For Cu_1-y(SiO₂)_y and Ni_1-y(SiO₂)_y it follows from Figure 1(c) and Figure 1(d), and with the coefficient of determination and, respectively.⁴ This finding suggests that the colossal increase of is caused by one (!) effect acting in the complete metallic regime. Inserting (NFE approximation) in Equation (19) leads to Equation (1), or in differential form,

where. n is the electron density in the metallic phase and. and are the volume fractions of the insulator phase (B) and metallic phase (A), respectively. and are identical with y and, respectively, if the insulating phase consists only of SiO₂ and the metallic phase only of Cu or Ni. In this case,. If, however, a certain portion of the metalloid atoms is dissolved in the metallic phase and/or a certain portion of the metal atoms is solved in the insulating phase, then is only an approximation for. Equations (1) and (20) agree with the equations (15a) and (15b) in [1] , respectively, which describe electron transfer between the phases in amorphous transition-metal―metalloid alloys.⁵ There the parameter was interpreted to be a constant for a given composite, which is determined by the average potential difference between the phases,.⁶ Phase B is the phase with the deeper potential. Because of this analogy, Equation (19) suggests the following interpretation of the GHE: The colossal increase of with decreasing metal content is essentially caused by a decrease of n due to electron transfer to the insulator phase (SiO₂) which can be described by Equation (1), respectively Equation (20).

Because the Fermi level lies in the energy gap between the valence band and conduction band of the insulator SiO₂ phase, the transferred electrons occupy surface states on the SiO₂ phase. This is the reason for the granular structure: spherical metal grains are embedded in the amorphous SiO₂ phase (see, e.g., [29] , Figs. 13-16 therein). A minimum energy is realized if, firstly, the transferred (pinned) electrons are arranged on spherical surfaces and, secondly, the insulating phase forms very thin layers around the metal grains providing the largest possible surface to accommodate the large number of transferred electrons. This electron transfer from the metallic phase to the phase boundaries provides the logical explanation for the granular structure in M-I composites. Such a granular structure has been found in many M-I films [7] [13] [15] [29] . This proposal applies to magnetic M-I composites as well. For nonmagnetic M-I composites the parameter C in

(NFE approximation) is of the order of one, depending slightly on the magnetic field. [23] [24] and are the conductivity and Hall mobility, respectively, of the phase A. is the elementary charge. For magnetic M-I composites Equation (21) holds approximately if “=” is replaced by “” considering the effect of the additional internal magnetic field due to the magnetization: An electron sees the effective magnet field, where. H is the external field applied to the specimen and is the internal field produced by the quantum mechanical exchange forces ( [30] , p. 341). An electron does not distinguish between H and. It moves according to the Lorentz force determined by and the electrical field E. One can assume that is nearly proportional to H as long as is nearly proportional to the magnetization produced by H.

This assumption is supported by the experimental finding by Xiong et al. [31] that (for not too small fields H), in the granular Co-Ag system, the Hall resistivity is linearly proportional to H. If so, the measured R values differ from the calculated R values, Equation (21), only by a factor which is nearly constant. Therefore, we assume that the EMT-formula for R, Equation (13), can be applied to magnetic composites as well.

If the metallic phase of a M-I composite is a noble metal, the NFE approximation is a good one for the metallic phase, above all as the Fermi surface moves away from the Brillouin zone boundary as n decreases. For the metallic phase in Ni-SiO₂ the NFE approximation is surely also a good one, because Ni has only 0.55 4s valence electrons per Ni atom ( [30] , p. 271).

If the metallic phase of a M-I composite is a transition-metal, the electron transfer is expected to be composed of both the d and s electrons. As the d density of states at the Fermi level is essentially larger than the s density of states, the principal share of electrons transferred to the insulating phase, is made up of d electrons, that is, the s electron density in the metallic phase remains relatively large. Because the electronic transport is determined by the s valence electrons in the A phase, the effect of the electron transfer on the electronic transport in the metallic phase is expected to be relatively small, and the increase of due to electron transfer should be essentially smaller as in M-I composites containing a noble metal as metallic phase. For instance, in Mo_1-y(SnO₂)_y ( [7] , Fig. 2 therein), we do not find an exponential change of with increasing: for (i.e.), the experimental R values [7] of Mo-SnO₂ fluctuate slightly where the average of calculated by Equation (16) remains nearly independent of y. Only approaching the M-I transition (),increases drastically.⁷

Now the question arizes: why do we find an exponential dependence of in Ni_1-y(SiO₂)_y although Ni is a transition-metal? X-ray emission spectra of amorphous and crystalline Ni_1-ySi_y and Pd_1-ySi_y alloys by Tanaka et al. [32] have shown that there are strong bonds between d orbitals (of Ni and Pd) and Si p orbitals leading to a stronger splitting of the d band into a bonding and antibonding fraction, where the former is lifted, whereas the latter lies below the Fermi level. Analogously, for Ni_1-y(SiO₂)_y one can also expect strong bonds between Ni d orbitals and Si (and O) p orbitals which leads to a strong reduction or disappearance of the d density of states at the Fermi level. Therefore, we find an experimental increase of (Figure 1(d)). Moreover, there is strong evidence for the assumption that the metallic phase does not consist of Ni alone, but that there is a certain fraction of Si (and O atoms) dissolved in the metallic phase.

In summary, for M-I composites containing a noble metal, we expect an exponential dependence because the electron transfer is made up entirely of the s electron density. For M-I composites containing a transition-metal, an exponential dependence can be expected if the d density of states at the Fermi level is strongly reduced, for instance caused by a hybridization of the d states with the p states of the metalloid.

Comparing granular M-I composites with amorphous transition-metal―metalloid alloys ( [1] ), we state that the exponential increase of R and the exponential decrease of with y (respectively) is essentially caused by the same phenomenon: decrease of the electron density in the metallic phase due to electron transfer to the metalloid or insulator phase. The essential difference between these two material classes is the fact that in the metalloid phase of the amorphous transition-metal―metalloid alloys an incompletely occupied sp band can exist ( [2] , Sec. IIA therein) for accepting the transferred electrons. In contrast, in the insulator phase of M-I composites only localized states on the surface of it are available for acceptance of the transferred electrons. This difference is also the reason for the different microscopic structures of M-I composites and amorphous transition-metal―metalloid alloys. Another, rather quantitative difference is the fact that the decrease of n is essentially larger than in amorphous transition-metal―metalloid alloys, as the average potential difference between the phases, , is essentially larger.

Our electron transfer model is compatible with a series of other experimental findings:

1) The GHE occurs both in magnetic M-I composites and non-magnetic ones suggesting a mechanism independent from magnetism [13] .

2) In M-I composites, and decrease exponentially with decreasing metal content in correspondence with the exponential increase of R. For some M-I composites, in Figure 3, and are drawn versus. In the NFE approximation the connection between and n is given by

where is the mobiliy of the carriers which is assumed to be equal to the Hall mobility introduced in Sec. 2. h is Plancks constant. L is the (elastic) mean free path of the electronic carriers in the (metallic) phase A. Because of Equation (22) the exponential concentration dependence of n, Equation (1), is also reflected by the concentration dependence of in Figure 3 if the concentration dependences of L or can either be neglected or change exponentially with as well. For W_1-y(Al₂O₃)_y we assume that there are strong bonds between W d orbitals and Si (and O) p orbitals, comparable with the situation in Ni_1-y(SiO₂)_y descussed earlier.

The only exception in Figure 3, where such an exponential concentration dependence of, respectively, does not occur, is represented by the annealed W_1-y(Al₂O₃)_y samples. This phenomenon will be discussed in Sec. 3.3.

3) With increasing y the temperature coefficient of resistivity, TCR, decreases and changes sign from positive to negative. [6] [11] [12] [14] [15] [34] The reason is an activation of localized electrons to the conduction band of the metallic phase. This conductivity contribution by activation is in competition with the positive contribution to the TCR due to scattering. The activation contribution is the larger the larger the amount of transferred electrons, i.e., the larger y, in correspondence to Equation (1).

In earlier papers it was suggested “that the GHE is a result of the drastic reduction of both the effective electron density and (in case of EHE) the effective carrier mobility”⁸ (Pakhomov et al. [11] ) or a drastic reduction of carrier density (Jing et al. [35] ). These two suggestions [11] [35] correspond to our physical model summarized in Sec. I. We emphasise, however, that it is not any effective electron density or carrier density (electrons or holes), but it is the real electron density which is reduced in the M-I composites.

Approaching the M-I transition, the charging energy arising from the positively charged metal ions grows more and more and one could assume that such ‘metal’ phase cannot exist, because the electrostatic contribution by the positive ions increases more and more as n decreases. However, the growth of the electrostatic energy is not unbounded; decrease of n is accompanied with a decrease of the sizes of the metal grains. For granular Al_1-yGe_y films, with increasing y the sizes of the metal grains decrease from 10 - 20 nm (on the metallic-rich side) to sizes <2 nm beyond the MIT (Rosenbaum et al. [36] [37] ). This decrease of with decreasing metal content even continues in the dielectric regime, as found for Ni_1-y(SiO₂)_y, Pt_1-y(SiO₂)_y and Au_1-y(Al₂O₃)_y thin films ( [29] , Fig. 17 therein), where decreases from 4 nm at to 1 nm at. For co-sputtered granular Ni_1-y(SiO₂)_y films, Abeles et al. found that the average particel size, , decreases with Ni content: = 14 nm, 9.4 nm, 5.7 nm, and 3.7 nm for 87, 67, 56 and 37 vol % Ni, respectively ( [29] , Fig. 11 therein).

We suppose that the electron transfer described by Equation (1), respectively Equation (20), holds also beyond the M-I transition. This assumption correlates with the concentration dependence of, which decreases continuously through the M-I transition as cited.

As mentioned earlier ( [18] , Sec. IVA therein), Equation (1), is part and result of a complex energy balance realized during solidification of the alloy, where the sizes of the phase grains are part of this balance. Equation (1) holds for situations, where atomic diffusion does practically not play a role because of the high cooling rate during the film deposition process. Because of this suppression of the long-range diffusion, the EMT provides a more realistic description of the electrical properties of disordered alloys with phase separation than any percolation description. This is justified in [2] (Sec. IVA therein).

On the other hand, at sufficiently high temperatures, appreciable diffusion can take place leading to additional growth of. With increasing, for instance due to annealing, the electron transfer to the phase boundaries can no longer be expected to follow Equation (1). Otherwise, the growth of the electrostatic energy could be shoreless.

Therefore, the GHE decreases or disappears by annealing at sufficiently high temperatures [14] . This phenomenon is also reflected by the concentration dependences of and which can be essentially smaller than before annealing. One typical example is W_1-y(Al₂O₃)_y, [29] , Figure 3: Before annealing, is approximately linear in, but after annealing at 1200˚C it is not. Reason is the fact that after annealing the metallic phase grains are essentially larger than before, for instance for (Abeles et al. [38] , Fig. 2 therein), whereas for the unannealed samples (Abeles et al. [38] , Fig. 1 therein). Because of the large phase grains in the annealed W_1-y(Al₂O₃)_y samples [38] , the electron transfer (related to) is essentially smaller than in the unannealed samples. Elsewise, the electrostatic energy would be too large.

This can also explain the experimental finding [14] that the maximum of the enhancement of R in Zn_1-y(SiO₂)_y is about 60, but 700 in Cu_1-y(SiO₂)_y: the size of the granules in Zn_1-y(SiO₂)_y is much larger (, [14] , p.608) than in Cu_1-y(SiO₂)_y, for which is given as the minimum value ( [14] , p. 606). Apparently, in Zn_1-y(SiO₂)_y a certain measure of atomic diffusion has been happen during film deposition, so that this balance was shifted to smaller electron transfer, i.e., Equation (1) does no longer apply.