The Origin of the Giant Hall Effect in Metal-Insulator Composites *

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       B A dn n d − = ⋅ ⋅ υ β υ [1] [2], provided that long-range diffusion does not happen during film production (n is the electron density in the phase A. A υ and B υ are the volume fractions of the phase A (metallic phase) and phase B (insulator phase). 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 Cu1-y(SiO2)y and Ni1-y(SiO2)y films.


Introduction
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 [3]- [5].A prominent example for both scientific challenge and practical application is the Giant Hall effect (GHE) in metal-insulator where ζ is the quotient of the volume or atomic fractions 2 of the two coexisting phases.( ) n ζ is the electron density in the phase A with ( ) . β is a constant for a given alloy, which is determined by the average potential difference between the two 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 [2] (conductivity) and [18] [19] (Seebeck coefficient).
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 α ′ .For Cu 1-y (SiO 2 ) y and Ni 1-y (SiO 2 ) y it follows from ( ) − , where y is the volume fraction of SiO 2 .This finding is illustrated in Figure 1(a) and Figure 1(b), where the absolute R values measured by Zhang et al. [12], Saviddes et al. [20] and Pakhomov et al. [10] are drawn versus η .The signs of the R values are negative.For Ni 1-y (SiO 2 ) y , Figure 1(b), the extraordinary R val- ues (taken from Fig. 3 in [10]) are drawn.(nearly free electrons -NFE).For a more precise discussion, we have to separate the contribution of the metallic phase to R, which can be done applying effective medium theory (EMT, [2], Sec.IVA therein).
The known EMT-formula for the Hall coefficient derived by Cohen and Jortner [21] is where σ and R are the electrical conductivity and Hall coefficient of a composite, respectively.i σ and i R are the corresponding transport parameters of the phase i.3 i υ is the volume fraction of the phase i (i stands for the phase A or B).
As will be argued in Sec.3.1, Equation (3) seems to be a good approximation for two-phase composites if  , as typical for M-I composites.Therefore, in Sec. 2 a R formula will be derived which holds for A B σ σ  as well.In Sec.3.1 this R formula and Equation (3) will be compared and its applicability to M-I composites will be checked.In Sec.3.2 it will be applied to a quantitative discussion of the GHE in M-I composites.In Sec.3.3 the effect of the grain size on the GHE will be discussed.In Sec. 4 the results will be summarized.

Derivation of the R Formula
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 i E and i σ ↔ are the electric field and the magnetoconductivity tensor [22] in this grain.For the electric current density outside of this grain we write analogously where E and σ ↔ are the electric field and the magnetoconductivity tensor outside of this grain (effective medium).For the determination of the coefficients in i σ ↔ we start with the equation for i J under the influence of an electrical and magnetic field, [23]- [25] ( ) ( ) ) is always perpendicular to B .In a composite, however, B and E (or i E ), are generally not perpendicular to each other because of the spherical boundary between a phase grain and its surroundings.Without loss of generality, the external fields applied to the sample, ext E and B , have the directions of the X and Z axes, respectively.Then Equation (6)  and Equation (4) lead to where 1 0 with , where i γ and γ are the angle between i E and B , respectively between E and B .µ and i µ are the Hall mobility in the composite and phase i, respective- ly.
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 0 = B , this demand is fulfilled by following from the EMT-formula for σ , [27] [28]   0. 2 For the case 0 ≠ B , the tensor properties of i σ ↔ and σ ↔ , Equation (7) and Equation (8), are to be taken into account.Equation (9) expressed in tensor form reads ( ) ( ) where the identities = have been used.Equation ( 11) determines the coeffi-J.Sonntag cients of Equation ( 8) as a function of the coefficients of Equation (7).Inserting Equation (7) and Equation ( 8) into Equation (11) and comparing coefficients for the tensor elements, we get following from the tensor elements xy σ or yx σ , where quadratic and higher powers of ξ , i ξ are neglected, i.e., Equation ( 12) and the following Equations ( 13), ( 14) are low-field approximations.Within this approximation the parameters i ν and ν do not have an influence on the result.From the tensor elements xx σ , yy σ , or zz σ , Equation ( 10) follows.
Substituting ξ and i ξ in Equation ( 12) by R and i R and considering Equation ( 9) we get the R formula for two-phase composites: 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 , , , 0

Comparison between Equation (3) and Equation (13)
For three examples of two-phase composites, in ) is shown, where σ 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 ("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, i D .However, the C & J formula [21] [26] does not contain i D .The differences between Equation ( 13) and the curves "C & J" are the larger the larger the difference between and respectively, and for σ , Equation ( 9) gives ( ) Starting at 1 A υ = , with decreasing A υ both σ and ( ) .This result corresponds to the fact that for 1 3 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 C&J 1 0 R > even for 1 3

A υ <
, where all the metallic granules are separated by adjacent insulating phase regions, that is, electron transport through the sample does not happen, if additional J. Sonntag tunneling is excluded.These two (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.

The Giant Hall Effect in M-I Composites
For A R 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), ln , where β ′ is a constant for a given M-I composite: For Cu 1-y (SiO 2 ) y and Ni 1-y (SiO 2 ) y it follows from Figure 1(c) and Figure 1(d where β β′ ≈ .n is the electron density in the metallic phase and υ and A υ are the volume fractions of the insulator phase (B) and metallic phase (A), respectively.B υ and A υ are identical with y and 1 y − , respectively, if the insulating phase consists only of SiO 2 and the metallic phase only of Cu or Ni.In this case, β β′ = .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 β ′ is only an approximation for β .
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. 5There the parameter β was in- terpreted to be a constant for a given composite, which is determined by the average potential difference between the phases, V ∆ . 6Phase 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 R with decreasing metal content is essentially caused by a decrease of n due to electron transfer to the insulator phase (SiO 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., [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] A σ and A µ are the conductivity and Hall mobility, respectively, of the phase A. e 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 i H H  .H is the external field applied to the specimen and i H is the internal field produced by the quantum mechanical exchange forces ( [30], p. 341).An electron does not distinguish between H and i H .It moves according to the Lorentz force determined by w H and the electrical field E. One can as-

J. Sonntag
sume that w H is nearly proportional to H as long as i H 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 xy ρ is linearly proportional to H.If so, the measured R val- ues 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-SiO 2 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 A R due to electron transfer should be essen- tially smaller as in M-I composites containing a noble metal as metallic phase.For instance, in Mo 1-y (SnO 2 ) y ( [7], Fig. 2 therein), we do not find an exponential change of A R with increasing ( ) ( ) ), the experimental R values [7] of Mo-SnO 2 fluctuate slightly where the average of A R calculated by Equation ( 16) remains nearly independent of y.Only approaching the M-I transition (

y >
), A R increases drastically. 7ow the question arizes: why do we find an exponential dependence of ( ) 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. [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 (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 A R (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 ( ) n ζ dependence be- cause the electron transfer is made up entirely of the s electron density.For M-I composites containing a transition-metal, an exponential ( ) n ζ 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 σ with y (respectively ( ) 1 y y − ) 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, V ∆ , 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, σ and A σ decrease exponentially with decreasing metal content in correspon- dence with the exponential increase of R. For some M-I composites, in where A µ 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 ( ) ( ) The only exception in Figure 3, where such an exponential concentration dependence of σ , respectively A σ , does not occur, is represented by the annealed W 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.[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.

The Effect of the Grain Size on the GHE
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. [36] [37]).This decrease of A D with decreasing metal content even continues in the dielectric regime, as found for Ni 1-y (SiO 2 ) y , Pt 1-y (SiO 2 ) y and Au 1-y (Al 2 O 3 ) y thin films ( [29], Fig. 17 therein), where A D decreases from 4 nm at 0.5 y ≈ to 1 nm at 0.9 y ≈ .For co-sputtered granular Ni 1-y (SiO 2 ) y films, Abeles et al. found that the average particel size, A D , decreases with Ni content: A D = 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 A D , which de- creases 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 A D .With increasing A D , 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 σ and A σ which can be essentially smaller than before annealing.One typical example is W 1-y (Al 2 O 3 ) y , [29], Figure 3: Before annealing, ( ) ( )  is 8 EHE is applied in [11] for the extraordinary Hall effect in magnetic M-I composites.

Figure 1 .
Figure 1.Experimental Hall coefficient data at 5 K versus

Figure 2 (
b), Figure 2(d), and Figure 2(f), the concentration dependence of R related to its values at 1 A υ = is shown, calculated by Equation (13), and compared with Equ- ation (3), denoted as "C & J".In Figure 2(a), Figure 2(c), and Figure 2(e), the corresponding concentration dependence of the Hall mobility µ ( R σ = (a) and Figure 2(c): The "C & J" curves decrease dramatically with increasing A υ and pass through a pronounced minimum at 1 In contrast, the µ curves calculated by Equation (13) agree with the expectation: Fig- ure 2(a): µ agrees with A µ for all A υ ; Figure 2(c) and Figure 2(e): µ increases and decreases with in- creasing A υ , respectively.A possible interpretation for such dramatic decrease of µ at 1 3 A υ =

A
σ and B σ .On the other hand, for the limiting case, A B σ σ = , Equation (3) and Equation (13) agree.(2) Another striking difference between Equation (13) and Equation (3) is represented by the boundary case " 0 B σ = and 0 A σ ≠ ", for which one obtains

Figure 3 ,
the NFE approximation the connection between A σ and n is given by

in Figure 3
if the concentration dependences of L or A µ can ei- ther be neglected or change exponentially with For W 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.
This finding suggests that the colossal increase of R is caused by one (!) effect acting in the complete metallic regime.Inserting 1