_{1}

Near the metal-insulator transition, the Hall coefficient R of metal-insulator composites (M-I composite) can be up to 104 times larger than that in the pure metal called Giant Hall effect. Applying the physical model for alloys with phase separation developed in [1] [2], we conclude that the Giant Hall effect is caused by an electron transfer away from the metallic phase to the insulating phase occupying surface states. These surface states are the reason for the granular structure typical for M-I composites. This electron transfer can be described by
[1] [2], provided that long-range diffusion does not happen during film production (n is the electron density in the phase A.
*u*
* _{A }*and

*u*are the volume fractions of the phase A (metallic phase) and phase B (insulator phase).

_{B}*β*is a measure for the average potential difference between the phases A and B). A formula for calculation of R of composites is derived and applied to experimental data of granular Cu

_{1-y}(SiO

_{2})

_{y}and Ni

_{1-y}(SiO

_{2})

_{y}films.

Nanocomposites play a growing role in both scientific research and practical applications because of the possibility of combination of special properties which cannot be reached in classical materials [^{4} times larger than that in the pure metal [

Applications of the GHE we find in magnetic field sensing elements, in read heads of magnetic recording devices and magnetic switching devices. Other examples for practical applications of nanocomposites are biomedical ones, materials with improved corrosion resistance, and thermoelectric materials with higher efficiency for energy harvesting, environmentally friendly refrigeration, direct energy transformation from heat into electricity, and temperature sensors.

As reasons for the GHE, quantum size effects and quantum interference effects on the mesoscopic scale have been discussed [^{1}:

For large ranges of concentration there is

(1) Phase separation between two phases called phase A and phase B, where each phase has its “own” short-range order (SRO),

(2) The phase separation leads to band separation in the conduction band (CB) and valence band (VB) connected with the phases A and B, respectively, and the electrons are freely propagating and the corresponding wave functions are extended over connected regions of one phase as long as the phase forms an infinite (macroscopic) cluster through the alloy.

(3) Between the two coexisting phases there is electron redistribution (electron transfer) which can be described by

where ^{2} of the two coexisting phases.

The points (1) and (2) imply the fact that each phase can be characterized by its own transport coefficients which can be calculated, in principle, by classical transport theory as done in [

Since M-I composites also consist of two separate phases with phase grains at the nanoscale, it is obvious to ask whether Equation (1) is reflected in the concentration dependence of the Hall coefficient R of M-I composites as well. Indeed, we have found that in the metallic regime of Cu_{1-y}(SiO_{2})_{y} and Ni_{1-y} (SiO_{2})_{y} thin films, the concentration dependence of R can be approximated by linear relations

with constant slope_{1-y}(SiO_{2})_{y} and Ni_{1-y}(SiO_{2})_{y} it follows from _{2}. This finding is illustrated in _{1-y}(SiO_{2})_{y},

The known EMT-formula for the Hall coefficient derived by Cohen and Jortner [

where ^{3}

As will be argued in Sec. 3.1, Equation (3) seems to be a good approximation for two-phase composites if

Let us consider a non-magnetic two-phase composite, where the phase grains are spherical without preferred orientations and arranged in a symmetrical fashion and each phase i can be characterized by a set of transport coefficients. The local electric current density in a single grain of the phase i (i = A or B) can be written as

where

where

where

with

At the interface between a single phase grain and its surroundings continuity of the normal components of the current density and the tangential components of the potential gradient are to be fulfilled. For the limiting case

following from the EMT-formula for

For the case

where the identities

following from the tensor elements

Substituting

The same formalism can also be applied to composites with more than two phases leading to relatively complex formulae for R. A self-contained and more manageable description of these R formulae is given by

with

For three examples of two-phase composites, in

(1) The most striking difference appears in

A possible interpretation for such dramatic decrease of

The differences between Equation (13) and the curves “C & J” are the larger the larger the difference between

(2) Another striking difference between Equation (13) and Equation (3) is represented by the boundary case “

and

respectively, and for

Starting at

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

where _{1-y}(SiO_{2})_{y} and Ni_{1-y}(SiO_{2})_{y} it follows from ^{4} This finding suggests that the colossal increase of

where_{2} and the metallic phase only of Cu or Ni. In this case,^{5} There the parameter ^{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 _{2}) 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 SiO_{2} phase, the transferred electrons occupy surface states on the SiO_{2} phase. This is the reason for the granular structure: spherical metal grains are embedded in the amorphous SiO_{2} phase (see, e.g., [

(NFE approximation) is of the order of one, depending slightly on the magnetic field. [

This assumption is supported by the experimental finding by Xiong et al. [

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-SiO_{2} the NFE approximation is surely also a good one, because Ni has only 0.55 4s valence electrons per Ni atom ( [

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 _{1-y}(SnO_{2})_{y} ( [_{2} fluctuate slightly where the average of ^{7}

Now the question arizes: why do we find an exponential dependence of _{1-y}(SiO_{2})_{y} although Ni is a transition-metal? X-ray emission spectra of amorphous and crystalline Ni_{1-y}Si_{y} and Pd_{1-y}Si_{y} alloys by Tanaka et al. [_{1-y}(SiO_{2})_{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

In summary, for M-I composites containing a noble metal, we expect an exponential

Comparing granular M-I composites with amorphous transition-metal―metalloid alloys ( [

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 [

2) In M-I composites,

where _{1-y}(Al_{2}O_{3})_{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}(SiO_{2})_{y} descussed earlier.

The only exception in _{1-y}(Al_{2}O_{3})_{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. [

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. [

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-y}Ge_{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. [_{1-y}(SiO_{2})_{y}, Pt_{1-y}(SiO_{2})_{y} and Au_{1-y}(Al_{2}O_{3})_{y} thin films ( [_{1-y}(SiO_{2})_{y} films, Abeles et al. found that the average particel size,

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

As mentioned earlier ( [

On the other hand, at sufficiently high temperatures, appreciable diffusion can take place leading to additional growth of

Therefore, the GHE decreases or disappears by annealing at sufficiently high temperatures [_{1-y}(Al_{2}O_{3})_{y}, [_{1-y}(Al_{2}O_{3})_{y} samples [

This can also explain the experimental finding [_{1-y}(SiO_{2})_{y} is about 60, but 700 in Cu_{1-y}(SiO_{2})_{y}: the size of the granules in Zn_{1-y}(SiO_{2})_{y} is much larger (_{1-y}(SiO_{2})_{y}, for which _{1-y}(SiO_{2})_{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.

A formula is derived for the Hall coefficient R of composites and applied to a discussion of the concentration dependence of R in M-I composites. From the empirical relation _{1-y}(SiO_{2})_{y} and Ni_{1-y}(SiO_{2})_{y} thin films, it is concluded that both the GHE and the granular structure typical for M-I composites are caused by electron transfer from the metallic phase to the

insulating phase which obeys

atomic diffusion does practically not play a role during the film deposition process. It 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.

In M-I composites, the decrease of electron density n in the metallic phase occurs as interface charges occupying surface states on the insulating phase which is responsible for the granular structure.

The author is appreciative to MEAS Deutschland GmbH a TE Connectivity LTD company for supporting this work. He also would like to thank Professor Stolze from the University of Dortmund for a critical reading of the manuscript and Stefan Lange for technical support.

Joachim Sonntag, (2016) The Origin of the Giant Hall Effect in Metal-Insulator Composites. Open Journal of Composite Materials,06,78-90. doi: 10.4236/ojcm.2016.63008