Joachim Sonntag

Bertrand Lenoir

Pawel Ziolkowski

Electronic Transport in Alloys with Phase Separation (Composites)

Keywords: Hall Effect, Giant Hall Effect, Seebeck Coefficient (Thermopower), Electron Density, Conductivity, Thermal Conductivity, Composites, Nanocomposites, Percolation Theory

A measure for the efficiency of a thermoelectric material is the figure of merit defined by 2 ZT S T ρκ = , where S, ρ and κ are the electronic transport coefficients, Seebeck coefficient, electrical resistivity and thermal conductiviy, respectively. T is the absolute temperature. Large values for ZT have been realized in nanostructured materials such as superlattices, quantum dots, nanocomposites, and nanowires. In order to achieve further progress, (1) a fundamental understanding of the carrier transport in nanocomposites is necessary, and (2) effective experimental methods for designing, producing and measuring new material compositions with nanocomposite-structures are to be applied. During the last decades, a series of formulas has been derived for calculation of the electronic transport coefficients in composites and disordered alloys. Along the way, some puzzling phenomenons have been solved as why there are simple metals with positive thermopower? and what is the reason for the phenomenon of the "Giant Hall effect"? and what is the reason for the fact that amorphous composites can exist at all? In the present review article, (1), formulas will be presented for calculation of ( )

and R in composites. R, the Hall coefficient, provides additional informations about the type of the dominant electronic carriers and their densities. It will be shown that these formulas can also be applied successfully for calculation of S, ρ , κ and R in nanocomposites if certain conditions are taken into account. Regarding point (2) we shall show that the combinatorial development of materials can provide unfeasible results if applied noncritically.

Introduction

The performance of a thermoelectric material for cooling of power generation applications, or more generally, for energy conversion, are directly related to the dimensionless figure of merit defined by coefficient, thermal conductivity, electrical conductivity, and electrical resitivty, respectively. T is the absolute temperature. The larger is ZT, the larger the thermoelectric performance of the material is. For many decades, 1 ZT ≈ was a practical upper limit realized in real materials. A further increase of ZT was limited by the fact that the transport coefficients occurring in Equation (1) are generally not independent from each other. In spite of these restrictions, during the last decade, progress was achieved by so-called nanostructured materials such as superlattices, quantum dots, nanocomposites, and nanowires.

In order to achieve further progress in the field of nanostructured materials with improved ZT, (1) a fundamental understanding of the carrier transport in these complex materials is necessary, and (2) effective experimental methods for designing, producing and measuring new material compositions with nanocomposite-structures are to be applied.

Regarding point [START_REF] Minnich | Bulk Nanostructured Thermoelectric Materials: Current Research and Future Prospects[END_REF], today it is generally accepted that application of the established classical theories for calculation of the electronic transport as the Boltzmann transport equation (BTE) can no longer be applied for the nanocomposite materials, because many of the characteristic lengths in the nanocomposite materials are smaller than the electron de Broglie wavelength (see, e.g., [START_REF] Minnich | Bulk Nanostructured Thermoelectric Materials: Current Research and Future Prospects[END_REF], Figure 6 therein). Instead, modelling transport in nanocomposites is generally assumed to require more powerful tools such as non-equilibrium Green's functions, which explicitly take account of electron wave effects [START_REF] Minnich | Bulk Nanostructured Thermoelectric Materials: Current Research and Future Prospects[END_REF].

Regarding point [START_REF] Edwards | An Investigation of the Structure of Amorphous Si 1-x Ni x through the Metal-Insulator Transition[END_REF], the combinatorial development of materials seems to be a proper method for experimental studies. From this method (applied to a certain material system) a large range of different compositions can be realized concurrently on one large substrat, for instance by deposition of thin films by simultaneous co-sputtering from two or three targets on a large substrat. Subsequently the transport coefficients can be measured with high lateral resolution applying measuring equipment such as the Potential-Seebeck Microprobe, the 3-Omegamethod and the 4-point method for S, κ , and σ , respectively. The advantage of these methods is the fact that quick and precise measurements of the transport coefficients at identical positions over the complete substrate is possible. How-DOI: 10.4236/ojcm.2019. 91002 23 Open Journal of Composite Materials ever, there is also a disadvantage of this method. Because in a continuous sample with a lateral concentration gradient the electrochemical potential µ is the same in the whole sample, both the topological structure (atomic configuration)

and electronic structure may be completelly different from those ones, if the samples with a certain composition are produced individually.

In the following we shall challenge these two points of view. We shall show that (1), the classical theories can, after all, be applied successfully for nanocomposites, particularly with respect to the electronic transport and its relation with the atomic structure and (2), the combinatorial development of materials can provide unfeasible results if applied noncritically.

On the way to this awareness, studies of amorphous alloys, as for instance a-Cr 1-x Si x alloys, have played an important role. Additional microstructure analyses of a series of amorphous transition-metal-alloys [START_REF] Edwards | An Investigation of the Structure of Amorphous Si 1-x Ni x through the Metal-Insulator Transition[END_REF] [3] [START_REF] Lorentz | Structural Studies of the Phase Separation of Amorphous FexGe 100-x Alloys[END_REF] [5] [START_REF] Van Raap | Evidence of Phase Separation in Amorphous FexSi 1-x Films[END_REF], have shown that, for large concentration ranges, they are real nanocomposites composed of different (amorphous) phases, where the phase grains are found to be of the order of 1 -2 nm [5] [6]. In correspondence to these analytical results, the a-Cr 1-x Si x thin films can be described as nanocomposites consisting of the phases 1 -Cr Si In Sections 2.1-2.4 formulas for σ , κ , S and R (the Hall coefficient) in composites will be derived and, in Section 2.5, compared with other published formulas. In Section 2.6 percolation elements will be additionally introduced in the formulas for two-phase composites. In Section 3.1 the classical formulas for the transport coefficients of the phases will be summarized applicable to large phase grains, which in Section 3.2 are extended to nanocomposites. In Sections 4-6 it will be shown that under certain conditions a discontinuity in the concentration dependence of the thermopower can occur, that the classical thermopower formula is to be supplemented by an additional term to be complete and that a noncritical application of the method of the combinatorial development of materials can provide unfeasible results. The puzzling phenomenons: why there are simple metals with positive thermopower? and what is the reason for the phenomenon of the "Giant Hall effect"? and what is the reason for the fact that amorphous composites can exist at all? have been considered in the Section 5 and Section 3.2.1.

Electronic Transport in Composites

Let us consider a two phase-composite consisting of the phases

, i A B = in a
symmetrical fashion regarding the average geometric form of the phase grains and without preferred orientations. Let us assume that each phase can be characterized by a set of transport coefficients i σ , , e i κ , i S and i R for the phase i, which are the electrical conductivity, electronic contribution to the thermal conductivity, Seebeck coefficient and the Hall coefficient, respectively, in the phase i. The corresponding transport coefficients of the composite, σ ,

Electrical Conductivity

Applying effective medium theory (EMT), let us derive the relation between i σ and σ , the electrical conductivities in the phases i and the composite, respectively. The strategy underlying the EMT is the following: a single phase grain of the phase i is considered to be completely embedded in an effective medium consisting of the two phases randomly arranged and characterized by the total transport coefficients. At the boundary face between this single phase grain and the surrounding effective medium continuity of the current densities and potentials and their gradients are to be saved.

The local electric current density J i can be written as

J E grad . i i i i i σ σ ϕ = = - (2)
In analogy to Equation (2) we write for the electric current density J in the specimen J E grad .

σ σ ϕ = = - (3)
i ϕ and ϕ are the local and total electrostatic potential and

grad i i E ϕ = - and grad E ϕ = -
are the local and total electric field, respectively. Now we demand that the total current density is equal to the average of the local current density,

J J , i = (4)
and the same for the potential grad grad .

i ϕ ϕ = (5)
Let us assume a spherical inclusion of the phase i with the radius 0 r , embedded in an uniform medium with the average conductivity σ and that for this enclosed phase i the local transport equation Equation ( 2) holds. With Equations ( 3)-( 5) we get J grad grad .

i i i σ ϕ σ ϕ = - = - (6)
The local potential i ϕ obeys the Laplace equation 0,

i ϕ ∆ = (7)
where the boundary conditions

, i ϕ ϕ = (8) J J n n i = (9
following from Equation [START_REF] Sonntag | Disordered Electronic Systems. III. Thermoelectric Power in Alloys with Phase Separation[END_REF], Equation ( 6) can be written as

. i i i a a σ σ = ( 14
)
With the boundary condition Equation ( 8) it follows that

, i a b c = + (15)
and with Equations ( 10)-( 12) for 0 r r = , ( )

2 . i i b c a σ σ - = (16)
Equations ( 15), ( 16) resolved for i a providing

3 2 i i b a σ σ σ = + (17)
and introduced in Equation ( 14) provide

1 . 2 2 i i i σ σ σ σ σ σ = + + (18)
Replacing the averages … by i i υ Σ … we get 0, 2

i i i i σ σ υ σ σ -= + ∑ ( 19
)
where i υ is the volume fraction of the phase i.

Equation ( 19) is the EMT formula for σ . It holds for composites with more than two phases as well. The first authors who derived Equation [START_REF] Halpern | The Thermopower of Binary Mixtures[END_REF] independently from each other, were Odelevskii [START_REF] Odelevskii | Raschet obobshchennoy provodimosti geterogennykh system[END_REF] and Landauer [START_REF] Landauer | The Electrical Resistance of Binary Metallic Mixtures[END_REF].

Thermal Conductivity

The schema shown in Section 2.1 can be applied to all the other transport parameters provided that a current density can be defined which is a function of a gradient of a potential, analoguously to Equation [START_REF] Edwards | An Investigation of the Structure of Amorphous Si 1-x Ni x through the Metal-Insulator Transition[END_REF]. For the thermal conductivity the corresponding relations read

Q J grad e T κ = - (20)
and

Q , J grad , i e i i T κ = - (21)
where Q J and Q J i are the total and local electronic thermal current density, respectively. Applying the same formalism as described in Section 2.1, however

κ κ υ κ κ - = + ∑ (22)
For the total thermal conductivity composed of both an electronic contribution e κ and an nonelectronic one ne κ , it follows an analogous formula 0, 2

i i i i κ κ υ κ κ -= + ∑ ( 23
)
where i κ is given by

, , i e i ne i κ κ κ = + (24)
provided that interactions between the various modes of heat transport can be neglected.

, ne i κ is the non-electronic contribution to i κ .

Thermopower

The following derivation will be based on J and S J , the electric and entropy-flux density, respectively. The corresponding local flux densitities, J i and , J S i , can be written as [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF] [10] ( )

J grad grad , i i i i e S T σ µ   = -   (25) ( ) ( ) 2 , , J grad grad ,
S i i i i i i e i S e S T T σ µ σ κ = - + ( 26
)
where e is the elementary charge. ( e is the charge of the electron.) i µ is the electrochemical potential in the phase i.

According to the strategy underlying the EMT, we demand continuity of the entropy-flux density and the electrochemical potential and their gradients, at the boundary face between a single phase grain and its surrounding (effective medium), where additionally J J 0 i = = is to be fulfilled. Setting J 0 i = in Equation (25), and inserting into Equation [START_REF] Xia | The Effective-Medium Approximation for the Thermoelectric Power of Polycrystals: Application to a Model for La 2-x M x CuO 4[END_REF], one obtains for the local entropy-flux density, ( ) , ,

J

grad .

e i S i i i e S T κ µ = - (27)
In analogy to Equation [START_REF] Cohen | Effective Medium Theory for the Hall Effect in Disordered Materials[END_REF] we write for the total entropy-flux density J S in the specimen ( )

J grad , e S e ST κ µ = - ( 28
)
where µ is the electrochemical potential of the composite.

The Equations ( 27) and ( 28) have the same structure as Equations ( 2) and (3);

that is why, we can apply the same formalism as described in Section 2. (Sonntag, [START_REF] Sonntag | Disordered Electronic Systems. III. Thermoelectric Power in Alloys with Phase Separation[END_REF] and [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF], respectively). For the one-band models, it makes little difference whether one chooses to interpret thermoelectric phenomenons on the basis of J & Q J or J & S J . However, for two-band or multiband models characteristic for semiconductors, the use of J & Q J leads to complications, which are not to be expected, if J & S J is used as a basis (Harman and Honig [9], p. 28).

Hall Coefficient

Let us consider a non-magnetic two-phase composite. Under the same conditions as assumed in Section 2.1 the local electric current density in a single grain of the phase i (i = A or B) can be written as J ,

i i i σ = E (33)
where i E and i σ are the electric field and the magnetoconductivity tensor [START_REF] Czycholl | Theoretische Festkörperphysik[END_REF] in this grain. For the electric current density outside of this grain we write analogously J , σ = E [START_REF] Ziman | Prinzipien der Festkorpertheorie[END_REF] 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 J i under the influence of an electric and magnetic field, [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF] [14] [START_REF] Kirejew | Semiconductor Physics[END_REF] ( ) ( )

3 4 2 11, 12 , 13, 2 J
K K K . i i i i i i i i i i i i e e e m m = + × + ⋅ E E B B E B (35)
, K rs i are the transport integrals defined by

( ) ( ) 0 , 2 2 , 4 K d 3 1 r s i i rs i i i i f E T E N E E m E τ µ ∂ = - ∂ + ∫ B ( 36
)
with the Fermi-Dirac distribution function, ( )

0 1 , . 1 e i B i E k T f E T µ - = + (37)
( )

i N E , i m and i τ are the density of states, the effective mass and the relaxation time, respectively, of the carriers in the phase i.

i i i i i i i ξ σ σ ξ ξ ν ξ     = -   +   +   (38)
where

i i i i B R B ξ µ σ ≡ = .
Analogously we write for σ ,

2 2 1 0 1 0 1 0 0 1 ξ σ σ ξ ξ νξ     = -   +   +   (39)
( ) ( ) ( ) 2 , 3 1 3 1 2 0 i A B A A B B f σ σ σ σ σσ υ σσ υ σ ≡ + -+ -- = (40)
following from the EMT formula for σ , Equation (19).

For the case 0 ≠ B , the tensor properties of i σ and σ , Equation (38) and Equation (39), are to be taken into account. Equation [START_REF] Zhang | Observation of Giant Hall Effect in Non-Magnetic Cermets[END_REF] expressed in tensor form reads ( ) ( )

3 1 3 1 2 0, A B A A B B σ σ σσ υ σσ υ σσ + -+ -- = (41)
where the identities A B B A σ σ σ σ = and i i σσ σ σ = have been used. Equation [START_REF] Savvides | Localization in the Metallica Regime of Granular Cu-SiO 2 Films[END_REF] determines the coefficients of Equation (39) as a function of the coefficients of Equation [START_REF] Sonntag | The Effect of the Band Edges on the Seebeck Coefficient[END_REF]. Inserting Equation [START_REF] Sonntag | The Effect of the Band Edges on the Seebeck Coefficient[END_REF] and Equation (39) into Equation [START_REF] Savvides | Localization in the Metallica Regime of Granular Cu-SiO 2 Films[END_REF] and comparing coefficients for the tensor elements, we get

( ) ( ) ( ) ( ) ( ) 2 3 1 3 1 , 4 3 1 3 1 A B A B A A A B B B A A B B σ σ ξ ξ σσ ξ υ σσ ξ υ ξ σ σσ υ σσ υ + + -+ - = - -- - (42)
following from the tensor elements xy σ or yx σ , where quadratic and higher powers of ξ , i ξ are neglected, i.e., Equation (42) and the following Equations ( 43), [START_REF] Mael | Specific Heat of Amorphous Mo x Ge 1-x through the Metal-Insulator Transition[END_REF]

) ( ) ( ) 2 2 2 3 1 3 1 2 A A B A B B A B A B R R R σ σ σ υ σ σ σ υ σ σ σ σ + -+ + -         = + (43)
The same formalism can also be applied to composites with more than two phases leading to relatively complex formulas for R. A self-contained and more manageable description of these R formulas is given by

( ) 2 2
, , , 0

i i i i A B i R R f σ σ σ σ σ σ =   ∂ ∂ + =   ∂ ∂   ∑ (44) with ( ) ( ) , , , , , 2 . 2
i i i i i A B i A B i f σ σ σ σ σ σ υ σ σ = =     - = +     +     ∑ ∏ (45)
Equations ( 43)- [START_REF] Rogatchev | Comparison of the Specific Heat and the Conductivity of Amorphous Ti x Si 100-x Alloys across the Metal-Insulator Transition[END_REF] were firstly published by Sonntag [START_REF] Sonntag | The Origin of the Giant Hall Effect in Metal-Insulator Composites[END_REF].

Comparison with Other Transport Formulas for Composites

Thermopower

As we are interested in a direct comparison between the different thermopower formulas for composites or heterogeneous materials, the transport coefficients in the phase, i κ , i S , and i σ , are set to be constant, although this is not realized in real composites caused by the condition of a common electrochemical potential as well as because of the specific features at

1 3 i υ < as discussed in Section 3.2.2. Additionally, , i e i κ κ = is set.
The thermopower formula derived by Airapetiants ([17], Equation ( 12) therein), called Air S , is 1 2

i i i i Air i i S g S g σ σ = - ∑ ∑ (46)
with ( )( )

3 . 2 2 i i i i g υ κ σ σ κ κ = + + (47)
In Figure 1, Air S vs B υ is shown for a composite consisting of two phases, A and B, with the coefficients, 18.5 V K

A S = - µ , 370 V K B S = - µ , 1 1 7450 cm A σ - - = Ω ⋅ , 1 1 7 cm B σ - - = Ω ⋅ , 1 1 0.022 W K cm A κ - - = ⋅ ⋅ , 1 1 0.070 W K cm B κ - - = ⋅ ⋅
which were applied by Airapetiants [START_REF] Airapetiants | Thermal Electromotive Force and Additional Thermal Conductivity of Statistical Mixtures[END_REF], already, where κ and σ are calculated by Equation ( 23) and Equation ( 19), res- pectively. For comparison the thermopower is calculated with our formula, Equation [START_REF] Mclachlan | Electrical Resistivity of Composites[END_REF] and Equation ( 23) for κ applying the same numbers for i κ and i S [ i σ does not occur in Equation ( 31)]. The result S is drawn in Figure 1 as well.

( )

Air B
S υ , Equation ( 46), and our ( )

B S υ , Equation (31
i i i i Web i i i i S D S D κ υ υ κ = - ∑ ∑ (48) with 1 ( )( ) , 2 2
i i i i D σ κ κ σ σ = + + (49)
provides the same concentration dependence as those of Airapetiants [START_REF] Airapetiants | Thermal Electromotive Force and Additional Thermal Conductivity of Statistical Mixtures[END_REF],

( )

Air B
S υ , shown in Figure 1, if κ and σ in Equations ( 48), ( 49 23) and Equation [START_REF] Halpern | The Thermopower of Binary Mixtures[END_REF], respectively. This result seems to be surprizing, because at the end of the derivation of theirs formulas Webman et al. have written: "Our effective-medium result… differs from that previously derived by Airapetiants [START_REF] Airapetiants | Thermal Electromotive Force and Additional Thermal Conductivity of Statistical Mixtures[END_REF], as the latter treatment did not involve a completely self-consistent averaging procedure."

Also the thermopower formula derived by Halpern [START_REF] Halpern | The Thermopower of Binary Mixtures[END_REF] (Equation [START_REF] Barnard | Thermoelectricity in Metals and Alloys[END_REF] therein),

( ) ( ) , A B B A A A B B B A Halp A B B A A B B A S S S S S σ σ κ σ κ σ κ σ κ σ κ σ σ κ σ κ - - = + - - (50)
provides the same concentration dependence as those by Airapetiants [START_REF] Airapetiants | Thermal Electromotive Force and Additional Thermal Conductivity of Statistical Mixtures[END_REF], Air S ;

the same refers also to the thermopower formula derived by Balagurov ([20], Equation ( 7) therein), ( )

A B A B A Bal A A B B A A S S S S σ σ κ κ σ κ σ κ σ σ -   = + -   -   (51)
as well as to those by Bergman and Levy ( [START_REF] Bergman | Thermoelectric Properties of a Composite Medium[END_REF], Equation ( 50) and Equation ( 9) therein), ( )

, B B Berg B A B A A B B S S S S γ σ γ σ γ σ γ σ - = + - - (52)
with

2 , T S γ κ σ = + ( 53
)
2 i i i i T S γ κ σ = + ( 54
)
if κ and σ in Equations ( 50)-( 53) are again calculated by Equation [START_REF] Hurvits | A Quantitative Analysis of the Thermoelectric Power Measurements on Composite Al-Ge Films[END_REF] and Equation [START_REF] Halpern | The Thermopower of Binary Mixtures[END_REF], and i γ and γ in Equation ( 52) are replaced by i κ and κ , respectively, i.e., the second term in Equation ( 53) and Equation ( 54) is neglected [START_REF] Hurvits | A Quantitative Analysis of the Thermoelectric Power Measurements on Composite Al-Ge Films[END_REF] [23] (quantity of higher order).

Additionally, in Figure 1 it is drawn the concentration dependence of Her S calculated by the formula by Herring ([24], second equation of Equation [START_REF] Sonntag | The Effect of the Band Edges on the Seebeck Coefficient[END_REF] therein]), ( )

1 2 3 3 Her S S S S κ ρ κ ρ   = - - +       (55)
with

1 ρ σ - =
, derived for a randomly inhomogeneous medium. If S , κ and ρ are interpreted as

i i i S υ ∑ , i i i υ κ ∑ and i i i υ ρ ∑ , respectively, then Equation (55) leads to 1 2 3
3

Her i i j j i i i j i i i i i i i S S S S κ ρ υ υ υ υ κ υ ρ     = - - +           ∑ ∑ ∑ ∑ ∑ (56)
with

1 i i ρ σ - = .
It is noteworthy that Bergman and Levy [START_REF] Bergman | Thermoelectric Properties of a Composite Medium[END_REF] have concluded that for a two-component isotropic composite their result, Equation ( 52 The reason for the differences is the neglection of the quadratic term of the thermopower-coefficient in the heat current density ("quantity of higher order") before J 0 i = is set. Neglection of this term, "

2 i i S
σ ", leads to an additional term in the resulting , J S i , Equation ( 27), after setting J 0 i = ; this additional term depends on σ as well. Such a term does not occur, if all the terms are maintained, before J 0 i = is set. This statement is independent of the question whether the EMT formula is derived on the basis of J & Q J or J & S J .

The starting equations applied by Webman et al. [START_REF] Webman | Thermoelectric Power in Inhomogeneous Materials[END_REF] (Equations (2.2b) and

(2.2a) therein) read J grad grad , P T

σ ϕ = - + (57)
J grad grad , Q T PT κ ϕ = - + (58)
where P Sσ = . Setting J 0 = , it follows that grad grad , T S ϕ = [START_REF] Ricker | Thermoelektrische Eigenschaften reiner Metalle in der Umgebung der Schmelztemperatur[END_REF] and replacing gradT in Equation ( 58) by Equation ( 59) we get J grad ,

Q ST S κ σ ϕ   = -+     (60)
Equation ( 60) contains σ , κ and S. This does not occur if the complete formula, ( )

2 J grad grad , Q S T T PT κ σ ϕ = -+ + ( 61
)
is applied as the starting formula instead of Equation [START_REF] Joffe | Physik der Halbleiter[END_REF]. In Equation ( 28 J is more universal as Q J . That is why, in [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] J & S J has been applied as a basis. There are differences between Equation (31) derived in [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] on basis J & S J and the formula derived on basis J & Q J (Equation ( 30) in [START_REF] Sonntag | Disordered Electronic Systems. III. Thermoelectric Power in Alloys with Phase Separation[END_REF]). While in the last one the chemical potentials of the phases, 0 i µ , occur, in the former one only the common electrochemical potential, µ , occurs mediated through the i S described by Equation (73).

Hall Coefficient

The EMT formula for the Hall coefficient of two-phase composites derived by

Cohen and Jortner [START_REF] Cohen | Effective Medium Theory for the Hall Effect in Disordered Materials[END_REF] is

( ) 2 2 2 , 0. 2 i i i i A B i R R σ σ υ σ σ = - = + ∑ (62)
As will be argued in the following, Equation ( 62) seems to be a good

approximation for two-phase composites if A B σ σ ≈ , but not if A B σ σ , as
typical for metal-insulator composites.

For three examples of two-phase composites, in Figure 2 A υ = is shown, where σ is calculated by Equation (19). There are two essential differences between the the two solutions Equation [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF] and Equation ( 43):

(1) The most striking difference appears in Figure 2 A possible interpretation for such dramatic decrease of µ at

1 3 A υ =
("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 [27] [28] does not contain i D .

The differences between Equation ( 43) and the curves "C&J" increase with increasing difference between A σ and B σ . On the other hand, for the limiting case, A B σ σ = , Equation (62) and Equation (43) agree.

(2) Another striking difference between Equation ( 43) and Equation ( 62 ( )

(43) 3 1 1 1 , 2 A A R R υ - = (63)
and

( ) C&J 3 1 1 1 , 4 A A R R υ + = (64)
respectively, and for σ , Equation (19) gives ( ) . This result corresponds to the fact that for 1 3

3 1 . 2 A A υ σ σ - = (65)
A υ <
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 ( 64)

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 tunneling is excluded.

These two differences, ( 1) and ( 2), suggest the fact that Equation (63) represents the physical situation better than Equation ( 64).

Composites with Non-Spherical Phase Grains

An essential assumption for the derivation of the EMT formulas, Equations ( 19), ( 22), ( 31), ( 32), ( 43) and ( 44) was the fact that the phase grains are spherical. In real composites this assumption is often not fulfilled, especially not for manmade composites with large phase grains. A typical feature for these composites is the fact that the transition from an infinite phase i cluster through the composite does not occur at

1 3 i υ =
but at larger values called the percolation edge. For such cases McLachlan and co-workers [START_REF] Mclachlan | An Equation for the Conductivity of Binary Mixtures with Anisotropic Grain Structures[END_REF] [START_REF] Mclachlan | Electrical Resistivity of Composites[END_REF] have modifyed the EMT formulas by introduction of percolation elements in the formulas for two-phase composites leading to the generalized form (GEMT)

1 1 1 1 1 1 1 1 0 t t t t A B A B t t t t A B A A σ σ σ σ υ υ σ σ σ σ - - + = + ⋅ + ⋅ (66)
and

1 1 1 1 1 1 1 1 0 t t t t A B A B t t t t A B A A κ κ κ κ υ υ κ κ κ κ - - + = + ⋅ + ⋅ ( 67
)
where A is given by ( )

1 c c A υ υ = - . c
υ is the volume fraction of the phase A, where the actual percolation threshold is assumed to occur. t represents the asymmetry of the microstructure. Vaney et al. [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF] have applied this idea to the EMT formula for the thermopower. With this modification Equation (32) reads now

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
1 1 1 1 1 1 1 1 0, t t t t A A B B A B t t t t A A B B S S S S S A S S A S σ σ σ σ υ υ σ σ σ σ - - + ≈ + ⋅ + ⋅ (68)
Vaney et al. [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF] have studied real composites composed of a crystalline phase "Bi 0.4 Sb 1.6 Te 3 " (≡phase A) and a glassy phase "Si 10 As 15 Te 75 " (≡phase B) produced by spark plasma sintering (SPS) of powder mixtures with different volume fractions 0.1, , 0.5

A υ =
. Inserting the i σ and i S data (given by Vaney et al. [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF]) in Equation ( 32), it follows the concentration dependence of S on A υ as drawn in Figure 3(a). phase grains in the man-made composites as studied in [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF] are arranged randomly (accidentally), that is, the transition from a continuous A phase cluster through the composite to separated A phase grains does not occur at 1 3

A υ =
, but it is shifted to smaller values of A υ . These percolation effects are considered by the generalized form (GEMT), Equation (68).

In Figure 3 there is an excellent agreement with the experimental data.

Both in [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF] and in Figure 3, the values for i S and i σ were assumed to be independent of i υ . This is, however, only an approximation, because the Open Journal of Composite Materials electrochemical potential, µ , in the two phases is generally different, as long as the phases are separated from each other. In the composite, however, µ must be uniform. This is ensured by an electron transfer n δ between the phases.

The temperature dependent part of n δ leads to a change of d dT µ

. This provides an additional contribution to the thermopower,

1 d d S
e T µ ∆ = as will be discussed in Section 3.1. For a calculation of d dT µ

, knowledge of the band structure data of the phases is necessary. For the composite considered by Vaney et al. [START_REF] Vaney | Effective Medium Theory Based Modeling of the Thermoelectric Properties of Composites: Comparison between Predictions and Experiments in the Glass-Crystal Composite System Si 10 As 15 Te 75 -Bi 0.4 Sb 1.6 Te 3[END_REF], the contribution of d dT µ to S is surely small, because A S and B S have the same sign. However, for composites with different signs of A S and B S , the effect of d dT µ can be essentially larger. That is, a really sensitive hardness test of Equation [START_REF] Sonntag | Disordered Electronic Systems: Concentration Dependence of the DC Conductivity in Amorphous Transition-Metal-Metalloid Alloys (Metallic Regime)[END_REF], respectively Equation (68), would be a composite with 0

A S > and 0 B S < or vice versa.

Formulas for the Transport Coefficients of the Phases

Composites with Large Phase Grains

"Large phase grains" means that the classical transport theory can be applied separately to each of the phase grains. This fact is immediarely plausible for crystalline phases, but also for amorphous phases as argued in [START_REF] Sonntag | Disordered Electronic Systems. III. Thermoelectric Power in Alloys with Phase Separation[END_REF] [32] [START_REF] Sonntag | Disordered Electronic Systems. II. Phase Separation and the Metal-Insulator Transition in Metal-Metalloid Alloys[END_REF].

First, we focus our attention on σ , e κ and S. In the framework of the kinetic transport equations, for a zero-magnetic field situation the transport coefficients of the phase i can be written as [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF]

i i i i i S e T µ - = (71)
where 0 i µ is the chemical potential for the phase i.

0 i
S is the Seebeck coefficient of the phase i for the (hypothetic) case that electron transfer does not happen between the different phases and that the band edge does not depend on T. Therefore an additional term, S ∆ , is to be introduced considering these effects realizing a common electrochemical potential in the composite,

. i µ µ = (72)
In the composite the Seebeck coefficient of the phase i is given by [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] 0 0 1 d , d

C A V B A B C A A B B A B A A B E E E T T T T T T T e p n e n n µ µ µ µ µ ϕ υ µ ϕ υ ∂ ∂ ∂ ∂ + + - ∂ ∂ ∂ ∂ ∂ ∂ = + - ∂ ∂   ∂ ∂ -   ∂ ∂   +   ∂ ∂ -   ∂ ∂   (74)
where

, C A E T ∂ ∂
E E F i B B i F i i F i F i k T k T E N E E E E µ = = - = -     (77)
(lowest order in the powers of

, B F i k T E ) with 2 3 2 2 3 , 3 , 8 π F i i i h E n m   =     (78)
where

, F i E
is the Fermi energy and i m the effective mass in the phase i, respectively. h is Planck's constant. The second equation in Equation (77) corresponds to the NFE-approximation.

For metallic phases, Equations (69)-( 71) provides (NFE-approximation, [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF] [10] [START_REF] Kirejew | Semiconductor Physics[END_REF]) For metallic phases, the non-electronic contribution to i κ , , ne i κ , can often be neclected compared to , e i κ , if the carrier densities are not too small. If not, κ is to be calculated by Equation ( 23) under consideration of Equation ( 24), where the , ne i κ are to be determined separately; that becomes especially important for semiconducting phases and if the phase i does not form a macroscopic cluster.

1 3 2 2 3 π e 2 , 3 i i i L n h σ   =     (79)
For the Hall coefficient of the phase i in a nonmagnetic composite, i R in NFE-approximation is given by

, i i i i C R e n µ σ = - = ⋅ (82)
where i µ is the Hall mobility of the phase i and C is a parameter of the order of one depending slightly on the magnetic field. [START_REF] Kirejew | Physik der Halbleiter[END_REF] [START_REF] Kirejew | Semiconductor Physics[END_REF].

The volume fractions of the phases, i υ , can be calculated from the atomic concentrations of the composite and the phases, x and i x , respectively. For a two-phase composite, the i υ can be determined by ( ) ( )

1 1 1 , A A A B B B x x x x υ υ -   - =-= +   -       (83)
where A  and B  are the atomic densities in the phases A and B, respectively.

Nanocomposites

Electron Density in Nanocomposites

Let us consider the class of amorphous transition-metal-metalloid alloys in more detail: it was one of the most important results that amorphous transition-metalmetalloid alloys do generally not consist of randomly mixed atoms, but they form composites consisting of amourphous phases which differ regarding their short range order (SRO). Interpreting a series of transport data of amorphous transitionmetal-metalloid alloys, Sonntag [START_REF] Sonntag | Disordered Electronic Systems: Concentration Dependence of the DC Conductivity in Amorphous Transition-Metal-Metalloid Alloys (Metallic Regime)[END_REF] has drawn the following conclusions: For large ranges of concentration there is (i) amorphous phase separation between two different amorphous phases called phase A and phase B, where each phase has its "own" short-range order (iii) Between the two coexisting phases there is electron redistribution (electron transfer) which can be described by

( ) ( ) exp , A n n ζ βζ = - ( 84
)
where ζ is the quotient of the volume or atomic fractions of the two coexisting phases. ( ) n ζ is the electron density in the phase A with ( )

0 A n n =
. β is a constant for a given alloy, which is determined by the average potential difference between the two phases.

For a series of amorphous transition-metal-metalloid alloys, conclusion (i) and conclusion (ii) are now confirmed experimentally or supported by inde- [START_REF] Abkemeier | Hopping Conduction in Hydrogenated Amorphous Si 1-y Ni[END_REF]. For details see [START_REF] Sonntag | Disordered Electronic Systems. II. Phase Separation and the Metal-Insulator Transition in Metal-Metalloid Alloys[END_REF] (Section 1 therein). Although the conclusion (iii) is not yet confirmed or supported by independent authors, there are a series of experimental findings supporting this conclusion (iii), as demonstrated in [START_REF] Sonntag | Disordered Electronic Systems: Concentration Dependence of the DC Conductivity in Amorphous Transition-Metal-Metalloid Alloys (Metallic Regime)[END_REF], where for some amorphous transition-metal-metalloid alloys it is shown that ρ versus ( )

pendent authors [2] [3] [4] [5] [6] [43] [44] [45] [46]
1 x x -
follows an exponential concentration dependence (see Figures 123456in [START_REF] Sonntag | Disordered Electronic Systems: Concentration Dependence of the DC Conductivity in Amorphous Transition-Metal-Metalloid Alloys (Metallic Regime)[END_REF]). For instance, a-Cr 1-x Si x thin films measured by Helms et al. [39] and shown in Figure 4 follow an exponential dependence of ρ versus ( )

1 x x -.
As argued in [START_REF] Sonntag | Disordered Electronic Systems: Concentration Dependence of the DC Conductivity in Amorphous Transition-Metal-Metalloid Alloys (Metallic Regime)[END_REF], one can assume that for 0.25 x > a-Cr 1-x Si x thin films consist of the two amorphous phases a-Cr 3 Si (=phase A) and a-Si (=phase B), i.e., the silicon concentrations in the two phases are characterized by 0.25

A x = and 1 B
x = , respectively, whereas for 0 0.25 x ≤ <

, the films are one-phase. In the two-phase range, ζ ( B A

X X =

) can be calculated by

B A A B X x x X x x ζ - = = - ( 85
)
As can be seen in Figure 4(b) (lower grafic), ρ follows an exponential dependence on ζ ( B A

X X =

). This result is in correspondence with Equation (84) if the effect of B σ (about Equation ( 19)) as well as the effect of a variation of the electronic mean free path L can be neglected. Because the scattering in a-Cr 1-x Si x is very strong, L is very small. However, L cannot be smaller than the average atomic distance, d; therefore one can assume that it is nearly independent of x, that is, d L

≈ and A σ ( 1 A ρ =
) is expected to be directly proportional to the electron density, n. The effect of the second phase (a-Si = phase B) on ρ ( 1 σ = ) (about Equation ( 19)) can be neglected, because in a-Cr 1-x Si x the concentration dependence of B σ is similar to A σ . This con- clusion follows from calculations done in [START_REF] Sonntag | Disordered Electronic Systems. III. Thermoelectric Power in Alloys with Phase Separation[END_REF] (see Figure 4(c) therein).

Support for Equation (84) comes also from Hall coefficient measurements of metal-insulator composites as shown in Figure 5. The sizes of the phase grains are of the order of some atomic distances as well. In Figure 5 discrepance. For magnetic metal-insulator composites Equation (82) 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 82), only by a factor which is nearly constant. Therefore, we assume that the EMT formula for R, Equation [START_REF] Yoshizumi | The Metal-Insulator Transition and Superconductivity in Amorphous Molybdenum-Germanium Alloys[END_REF], 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 ( [START_REF] Schulze | Metallphysik[END_REF], p. 271).

Additional support for Equation (84) comes from experimental resistivity data of amorphous (Cr 1-x Si x ) 1-z N z and (Cr 1-x Si x ) 1-y O y thin films (in preparation).

Amorphous Metals

The electron transfer n δ between the phases described by Equation (84) leads to a lowering of the total energy of the composite compared with a situation, where the phases exist alone. This is the reason for the fact that metallic composites with an amorphous structure can exist at all. Almost invariably, amorphous metals contain a metalloid as one of the constituents. e.g., Au-Si, Pd-Si, Fe-P-C, … [START_REF] Masumoto | Structural Stability and Mechanical Properties of Amorphous Metals[END_REF]. That is, one-phase amorphous metals practically do not exist. If yet (for instance after an evaporation process at extremely low temperatures), then there is the strong suspicion that during the preparation process impurities as oxygen and nitrogen are incorporated in the metal film.

Of course, for the crystalline state the energy gain is surely larger compared with that of the amorphous state. However, the transition from the amorphous state to the crystalline one realized by atomic diffusions processes requires additional energy to overcome energy barriers.

The Giant Hall Effect (GHE)

As reasons for the very large values of the Hall coefficient in metal-insulator composites as shown in Figure 5 for the examples Cu 1-y (SiO 2 ) y and Ni 1-y (SiO 2 ) y (known as the "Giant Hall effect"), quantum size effects and quantum interference effects on the mesoscopic scale have been discussed. [START_REF] Zhang | Observation of Giant Hall Effect in Non-Magnetic Cermets[END_REF] [51] [52] [START_REF] Zhang | Giant Hall Effect in Nonmagnetic Granular Metal Films[END_REF] [54] [55] However, to our knowledge, until now, there is no explanatory model which can interpret the phenomenon of GHE. Now, we believe that Equation (84) gives a simple and plausible explanation for the GHE.

Electronic Transport in Nanocomposites

For many alloys with phase separation the phase grains are very small so that an application of the classical transport equations to the phase grains (Section 3.1) does not seem to be appropriate. In spite of this reservation, both the BTE and the approximation of free electrons (NFE-approximation) may be good descriptions for a phase i of a composite, as long as it forms an infinite cluster in the composite and the scattering processes are elastic. For amorphous transition-DOI: 10.4236/ojcm.2019.91002 44 Open Journal of Composite Materials metal-metalloid alloys this condition is fulfilled for

1 3 i υ >
. This point of view is justified in [START_REF] Sonntag | Disordered Electronic Systems. II. Phase Separation and the Metal-Insulator Transition in Metal-Metalloid Alloys[END_REF] (Section 4.2 therein) and [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] (Section 5.2. therein). That is, formulas for the transport coefficients of the phases, Equations ( 79)-(82), may be a good description if the phase i forms a macroscopic cluster through the composite corresponding to 1 3 i υ > . For this case the phase i can form a quasi-continuous energy band and the scattering is elastic. With decreasing i υ , the macroscopic i phase cluster decomposes into separate phase grains; for amorphous composites, this transition occurs very precisely at 1 3 i υ =

. The reasons are the following [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] [33]:

(1) As the grain diameters are very small (of the order of ~1 -2 nm [START_REF] Regan | Anisotropic Phase Separation through the Metal-Insulator Transition in Amorphous Alloys[END_REF] [6], the number of grains (of the same sort i) must be very large to form a cluster for which a quasi-continuous energy band is realized. Accidentially formed agglomerates of some (for instance 10 or 100 or 1000) neighbouring grains (with ~1 -2 nm diameter) are not sufficient for forming a quasi-continuous energy band.

(2) During the film deposition of a composite, the atoms of the different atom sorts arrive at the substrate equally distributed; therefore the different phase grains (A and B) can also be assumed to be locally equally distributed in the amorphous composite, because the diffusion paths during solidification are very short, which is a prerequisite for forming amorphous composites.

A phase is an "electronic phase" determined by a solution of the Schrödinger equation; after hitting at the substrate, the atoms move locally only so long until they can form a phase which corresponds to a solution of the Schrödinger equation. That is why, the phase grains of the same sort i are also locally equally distributed, as the compositions of the different phases are very different, i.e., the local distribution of the i phase grains is not completely random, as, e.g., assumed within the framework of the classical percolation theory. For such a locally equally distributed arrangement of the i phase grains in an amorphous matrix (formed by the rest of the composite), it follows that this merging to a macroscopic cluster through the sample occurs very precisely at a specific concentration i υ ; and this specific concentration is

1 3 i υ =
, as follows, for instance, from Equation ( 19) setting

0 j σ = , but 0 i σ ≠ ( j i ≠ ). (For a
comparison with the classical percolation theory concerning this critical value of i υ , see also [33] (Section 4.1 therein).

(3) A macroscopic i phase cluster is only realized, if all the atoms in this i phase cluster are directly connected to atoms belonging to the same phase sort i. When, for instance, two grains of the same phase sort i are separated by a monoatomic layer of a different(!) phase (j), these two i phase grains cannot be considered as (nearly) one i phase cluster, because the overlap of the electron wave functions is interrupted by this monoatomic layer. Within an i phase grain or i phase cluster the wave functions, δ outside this macroscopic cluster. This "falling off" is comparable with the decrease of the molecular orbitals of (large) molecules at their molecule boundaries (see, e.g., [START_REF] Atkins | Physikalische Chemie[END_REF], pp. 409 and pp. 435, or [START_REF] Atkins | Physical Chemistry[END_REF]). Because of this "non-overlapping" of the ( )

r ψ
belonging to different phases, they do not "penetrate" each other, i.e., there are sharp boundaries between the different (electronic) phases.

Considering the fact that the phase i does not form a quasi-continuous energy spectrum for

1 3 i υ <
, but there is a discrete energy spectrum typical for separate grains, then there are no electronic states immediately above and below µ (within the energy range B k T ), i.e., the electrons cannot be activated to higher energies (at the hot end of the sample) and cannot deliver energy (at the colder end), if 1 3 i υ < . Under this condition it follows that (at the temperature T)

, 0 e i κ = for 1 3 i υ <

. For i S the situation is analogous: for ) are expected to be essentially smaller than calculated by Equations ( 80) and (81), because the d states are localized.

Discontinuity in the Concentration Dependence of the Thermopower

A specific feature of composites with 0 A S > and 0 B S < or vice versa is the fact that a discontinuity (step) in the calculated S vs. i υ can occur, which is an additional possibility to check experimentally the thermopower formula. One example for a composite with different signs of A S and B S is a-Cr 1-x Si x consisting of the two amophous phases [START_REF] Sonntag | Disordered Electronic Systems. II. Phase Separation and the Metal-Insulator Transition in Metal-Metalloid Alloys[END_REF] 1 -Cr Si Now, the calculations of [START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] we have done once more, but for 0.25 As this discontinuity occurs at 0 S = , this phenomenon opens the possibility

( ) ( ) ( ) ( ) ( ) 2 , , , , , 4 3 1 3

Extension of the Classical Thermopower Formula

In the limit 0 B υ = , the composite degenerates to a homogeneous alloy con- sisting exclusively of the phase A. On the opposite side, for 1 B υ = we get a homogeneous alloy consisting exclusively of the phase B. For these two limiting cases the formulae must hold as well. Setting 0 B υ = , it follows from Equation (74) that

0 , d , d C A A E T T T µ µ ∂ ∂ = + ∂ ∂ (87)
and taking into account Equations ( 73), ( 81), (68) and Equation ( 22), ( ) ( )

2 2 0 , , π 1 1 . 3 C A B A A A A F A E k T r S S e E e T T µ ∂ +   ∂ + = = + +   ∂ ∂   (88)
Analogously it follows for 1

B υ = that 0 , d , d V B B E T T T µ µ ∂ ∂ = - ∂ ∂ (89) ( ) ( ) 2 2 0 , , π 1 1 . 3
V B B B B B B F B E k T r S S e E e T T µ ∂ +   ∂ + = = + -   ∂ ∂   (90)
Inserting Equation (75) in Equations ( 88), (90) we get ( ) p. 316). For more details see [START_REF] Sonntag | The Effect of the Band Edges on the Seebeck Coefficient[END_REF].

Constraints of the Combinatorial Development of Materials

The combinatorial development of materials is a very effective method to get experimental data about a certain material system, because a large range of different compositions can be realized concurrently on one large substrat, for instance by deposition of thin films by simultaneous co-sputtering from two or three targets on a large substrat. On the other hand, this method is to be used with great caution because the results can be completely different from a situation where the different compositions are produced separated in single manufacturing processes. The reasons are the following ones.

In a composite a common electrochemical potential is realized by electron transfer between the different phases. This can lead to the fact that in one of the phases another topological structure (atomic configuration) is more favorable than for the case that this phase exists alone (as homogeneous material). And in nanocomposites the electron transfer has an essential larger effect on the carrier densities and therefore on the transport coefficients in the phases compared with composites with large phase grains.

Figure 7 shows two series of resistivity measurements of Cr . Details of film deposition, analysis and resistivity measurements are described in [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. Additionally the resistivity data of co-sputtered Cr 1-x Si x thin films by Helms et al. [39] are shown (circles; already shown as a logarithmic representation in Figure 4), where the single samples were separated from each other during the deposition process. cosputtered from a chromium target and a silicon target and deposited on 4 inch glass wafers (the deposition conditions are specified in [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF] [63]). Three parallely lying measuring lines have been measured with a distance of 4.5 mm to each other. The first series extend continuously about the concentration range 0 0.30 x < <

, the second one about 0.22 0.80 x < <

. The corresponding film thickness were 200 nm > d > 80 nm and 165 nm > d > 131 nm, respectively. Additionally resistivity data received from Helms et al. [39] are drawn. These data were produced by co-sputtering on glass as well, but with the difference that each measuring point represent a single sample; these single samples were separated from each other during the deposition process, i.e., there were no electrical connections between the single samples. The structure of all the films were detected to be X-ray amorphous, where for the chromium rich samples microcristalline precipitations has been found; number and average size of these DOI: 10.4236/ojcm.2019.91002 50 Open Journal of Composite Materials precipitations decrease with the silicon content.

As can be seen in Figure 7, in the overlapping range between the two series " 0 0.30 x < < " and " 0.22 0.80 x < < ", there are very large ρ differences, nearly a factor of 1.7 … 1.8. The factor between the " 0 0.30 x < < " series and the Helms samples is even 3.0 … 3.1.

Figure 8 shows experimental thermopower data of the same samples as shown in Figure 7 supplemented by a third series for the concentration range 0.49 0.96 x < < with the corresponding film thickness of 70 nm < d < 260 nm [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. The structures were detected to be X-ray amorphous. S has been measured applying the Potential-Seebeck Microprobe (PSM) from the company LOT-Oriel Group Europe [START_REF] Ziolkowski | Application Overview of the Potential Seebeck Microscope[END_REF] [65].

The S curves for the concentration ranges 0 0.30 x < < and 0.22 0.80 x < < as well as for the single samples (triangles) correspond relatively well. However, between the second series and the third one there are large differences. The discontinuity at 0 S = occurring in the S curve of the single samples does not occur in the co-sputtered samples. . Details of film deposition, analysis and thermopower measurements are described in [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. Additionally it is drawn the experimental S data of the Cr 1-x Si x thin films shown in Figure 6 already. DOI: 10.4236/ojcm.2019.91002 51 Open Journal of Composite Materials

Figure 9 shows thermopower data of a-(Cr 1-x Si x ) 1-y O y thin films co-sputtered as described in [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF] for Cr 1-x Si x thin films, where additionally an oxygen gradient was realized, lateral and perpendicular to the wafer. S has been measured at the complete wafer and after that the wafer was sawed producing single 8 × 2 mm 2 pieces. Now S was measured once more at these pieces. The S vs.

( )

1 x
x curves for the "wafer" and the "pieces" are shifted nearly parallely to each other, where the zero-crossing of S is shifted from The experimenal results shown in Figures 789demonstrate strikingly that the combinatorial development of materials can provide results which can be different from samples produced by single procedures with a given composition.

Conclusions

Formulas have been presented for calculation of σ , κ , S, and R in composites.

We have shown that these formulas can also be applied successfully to nanocomposites if certain conditions are taken into account, especially the ( )

1 x
x of a a-Cr 1-x Si x -O thin film with an additional O gradient vertical to the wafer surface as well as in the same direction as those of silicon concentration gradient (measured for the complete wafer and the pieces after sawing to 8 × 2 mm 2 pieces). The measuring positions with respect to the wafer, before and after sawing, were the same ones. This figure is taken from [START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. DOI: 10.4236/ojcm.2019.91002 52 Open Journal of Composite Materials phenomenon of electron transfer between the different phases in the nanocomposite. It is argued that, under certain conditions, an noncritically appli- cation of the combinatorial development of materials can provide unfeasible results.

The formulas and the theory described in the present article give answers to some mysterious puzzles, for which the scientific research had no final answers:

1) Why there are simple metals with positive thermopower? 2) What is the reason for the phenomenon of the "Giant Hall effect"? 3) What is the reason for the fact that amorphous metals can exist at all? 4) Until to the end of the twentieth century amorphous metallic alloys were assumed to be a random and homogeneous distribution of the metal atoms in the amorphous matrix. On this basis, a quantitative calculation applying classic theories was not successful.

The answers given in the present article are the following:

1) The reason for positive thermopower of some metals comes from the temperature dependance of the band edge: If d d 0 C E T > and if this effect overcompensates for the influence of the scattering term in the thermopower formula, Equation (91), then the thermopower is positive.

2) The reason for the phenomenon of the "Giant Hall effect" in metalinsulator composites is the exponential reduction of the electron density in the metallic phase due to electron transfer from the metallic phase in direction to the insulating phase, described by Equation (84). The transferred electrons are pinned at the phase boundaries between the phases.

3) The reason for the fact that amorphous metals can exist is an electron transfer between the phases described by Equation (84). This electron transfer leads to a lowering of the total energy of the composite compared with a situation, where the phases exist alone. (Amorphous metals are generally composites.) 4) Experimental and theoretical studies at amorphous transition-metal-metalloid alloys have shown that in these amorphous alloys there exists amorphous nano-scaled phase separation between two different amorphous phases, where each phase has its "own" short-range order (SRO) and each phase may be described by its "own" band structure. Within this framework the formulas for σ , κ , S, and R described in this paper can be applyed.

  κ , σ and ρ are the Seebeck (or thermopower)

  = . grad r is the gradient into the radial direction. J n i and J n are the normal components of the current density in the sphere i and the surrounding DOI: 10.4236/ojcm.2019.91002 25 Open Journal of Composite Materials effective medium, respectively. Equation (7) has the solution medium. ϑ is the angle between the direction of E and the position vector r with

(

Figure 1 .

 1 Figure 1. Comparison of Air S , Her S , and S vs. B υ calculated by Equation (46), Equation (56), and Equation (31), respectively, where 18.5 V K A S = -µ ,

) are again 1 We

 1 have replaced the averages … in the original equations by Webman et al. [webman] and Herring [herring] by i i υ ∑ … , Equations (48) and (55), respectively. DOI: 10.4236/ojcm.2019.91002 31 Open Journal of Composite Materials calculated by Equation (

  Figure 2(f), the concentration dependence of R related to its value at 1 A υ = is shown, calculated by Equation (43), and compared with Equation (62), denoted as "C&J". In Figure 2(a), Figure 2(c), and Figure 2(e), the corresponding concentration dependence of the Hall mobility µ (

>

  Figure 2(e): µ increases and decreases with increasing A υ , respectively.

  ) is represented by the boundary case " 0 B σ = and 0 A σ ≠ ", for which one obtains DOI: 10.4236/ojcm.2019.91002 34 Open Journal of Composite Materials

Figure 2 .

 2 Figure 2.

For 1 3

 3 A υ >the calculated S on A υ curve agree with the experimental data. For 1 3 A υ < the EMT formulas are no longer a good description, because the DOI: 10.4236/ojcm.2019.91002 36 Open Journal of Composite Materials

Figure 3

 3 Figure 3. S versus A υ for [ ] [ ] 0.4 1.6 3 10 15 75 1 Bi Sb Te Si As Te A A υ υ composites produced by spark plasma sintering (SPS) [29] and comparison with the experimental data (full points, 300 K T = ). (a), calculated by Equation (32), (b), calculated by the GEMT equation, Equation (68), for 0.16 c υ =

  (b), S versus A υ is drawn calculated by Equation (

µ

  is an additional term taking into account the change of µ with temperature. One consequence of Equation (73) is the fact that the thermopowers of the phases, i S , dependent on i υ . For a Open Journal of Composite Materials calculation of d dT µ , knowledge of the band structure data of the phases is necessary. Assuming a two-phase composite consisting of the phase A with electron conductivity and the phase B with hole conductivity, d dT µ

(

  SRO), (ii) the amorphous phase separation leads to band separation in the con-DOI: 10.4236/ojcm.2019.91002 40 Open Journal of Composite Materials duction 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 with respect to connected phase ranges.

  (a) and Figure 5(b) the experimental original data of Cu 1-y (SiO 2 ) y and Ni 1-y (SiO 2 ) y taken from [40] DOI: 10.4236/ojcm.2019.91002 42 Open Journal of Composite Materials

Figure 5 .

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

  but they do not overlap between two i phase grains or i phase clusters separated by a monoatomic layer of a different(!) phase (j). DOI: 10.4236/ojcm.2019.91002 45 Open Journal of Composite MaterialsIf a sufficiently large number of i phase grains form a macroscopic cluster, -continuous energy band, while the wave functions fall off exponentially in a very short distance r

  and i S change discontinuously. Such discontinuities are especially to be expected in composites with metallic phases.For amorphous transition-metal-metalloid alloys, " approximation, because, at the boundary faces between the different phases, there are p-d bonds, i.e., d orbitals of the Cr atoms (of the A phase grains) overlap with p orbitals of the boundary faces atoms on the B phase grains resulting to a p-d band, which is incompletely occupied. (For a detailed discussion see Section 2.1 of[START_REF] Sonntag | Disordered Electronic Systems. II. Phase Separation and the Metal-Insulator Transition in Metal-Metalloid Alloys[END_REF]). Because of this p-d overlapping, this (incompletely occupied) p-d band exists also if levels of the s states in the separate A phase grains are discret, the electrons can be activated to higher energies (at the hot end of the sample) and deliver energy (at the colder end) by a transition to the p-d band. That is why,

and 1 Bx

 1 ≈ . (Amorphous phase separation is typical for many amor- phous transition-metal-metalloid alloys[START_REF] Edwards | An Investigation of the Structure of Amorphous Si 1-x Ni x through the Metal-Insulator Transition[END_REF] [3] [4][5] [6]). In[START_REF] Sonntag | Thermoelectric Power in Alloys with Phase Separation (Composites)[END_REF] S versus x had been calculated for a-Cr 1-x Si x applying Equation (31) for 0.25 A x = and 0.90 B x = (bold line in Figure 1 of [12]). DOI: 10.4236/ojcm.2019.91002 46 Open Journal of Composite Materials

  of the calculated curves agree with the experimental data. The discontinuity of S versus i υ for composites with vice versa has its origin in the mathematic structure of the formula

S

  + , which both show a discontinuity (step) at the same concentration, when ( ) Sand ( ) S + passes the value "0" coming from negative values crossing to positive values or vice versa. The physics follows only one of them, ( ) S -, as suggested by the results of [11] (compare Figure 4(g) and Figure 4(h) with Figure 5 therein).

Figure 6 .

 6 Figure 6. Thermopower versus x for a-Cr 1-x Si x at 300 K T = calculated by Equation (32) with 0.25 A x = and 0.90 B x =

Figure 7 .

 7 Figure 7. Experimental resistivity data of Cr 1-x Si x thin films cosputtered from a chromium target and a silicon target. By variation of the sputtering rates two different concentration ranges about 4 inch glass wafers are realized, 1) 0 0.30 x < < , and 2) 0.22 0.80 x < <. Details of film deposition, analysis and resistivity measurements are described in[START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. Additionally the resistivity data of

Figure 8 .

 8 Figure 8. Experimental thermopower data of Cr 1-x Si x thin films for the same samples as shown in Figure 7 supplemented by a third series 0.49 0.96 x < <. Details of film deposition, analysis and thermopower measurements are described in[START_REF] Sonntag | High-Throughput Characterization of the Seebeck Coefficient of a-(Cr 1-x Si x ) 1-y O y Thin Film Materials Libraries as Verification of the Extended Thermopower Formula[END_REF]. Additionally it is drawn the experi-

Figure 9 .

 9 Figure 9. S vs.

  and

							, V B E ∂	T ∂ are the band edge shifts with temperature. n
	(	A ≡ ) is the electron density in the phase A. p ( B n n ≡ ) is the hole density in the
	phase B.	, C A E	and	, V B E	are the energies of the band edges of the conduction
	band (CB) and valence band (VB) in the phase A and phase B, respectively. In
	Equation (74),	0 ∂ ∂ =, i T ϕ	, C A E ∂	0 ∂ =, and n	, V B E ∂	0 ∂ = is assumed. The p
	first assumption corresponds to the fact that the electrostatic potential does not
	depend on T, the second and third ones are equivalent to the assumption that
	, C A E	and		, V B E	do not depend on occupation of the CB and VB. From Equation
	(73) and Equation (74) it follows immediately the fact that i S depends on i υ .
		Now, d dT µ	can be calculated by Equation (74) if
									0 i T µ ∂ = -∂	2 2 6 π B F i , k T E	,	(75)
									0 i i n µ ∂ = ∂	, F i i n 3 2 E	,	(76)
	are taken into account and if the band edge shifts for the CB and VB,	, C A E ∂	T ∂
	and	, V B E ∂	T ∂ , are known. Equations (75) and (76) follow from the Fermi-
	Dirac-statistics, where 0 i µ is given by
					0	,	π	2 2 2 6 d d ln	( )	,	,	2 2 2 , 12 π

  assumption is supported by the experimental finding by Xiong et al.[START_REF] Xiong | Extraordinary Hall Effect and Giant Magnetoresistance in the Granular Co-Ag System[END_REF] 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 values differ from the calculated R values, Equation (

	H	w =	i + , where i H H 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 (

[START_REF] Schulze | Metallphysik[END_REF]

, 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 electric field E. One can assume that w H is nearly proportional to H as long as i H is DOI: 10.4236/ojcm.2019.91002 43 Open Journal of Composite Materials nearly proportional to the magnetization produced by H. This

  , this puzzle is now solved. Positive sign of thermopower can also be expected for crystalline metals: As a trend, positive sign of thermopower will be measured, if the conduction band edge shifts to higher values with increasing T and if this effect overcompensates for the influence the scatterring term. Thus, for the crystalline Cu, Ag, Au, and Li the experimental thermopowers also have

	second term in Equation (91) represents the effect of the temperature depen-
	dance of the band edge on the thermopower. Therefore, we can say that for
	normal metals positive sign of thermopower will be measured if d E T > d C	0
	and if this effect overcompensates for the influence of the first term in Equation
	(91). This conclusion holds exactly if	1 z = . If	z =	3 2	, this fact is to be
	considered as a tendency.
	As mentioned earlier, it is not yet completely clear whether	z =	3 2	or	1 z = .
	We believe that for metals	z =	3 2	is correct. However, the final answer
	depends on the question whether or not Equation (81) contains exclusively the
	scattering contribution [corresponding to 0 S , Equation (81)]. This question is a
	matter of future studies.
	Simple Metals with Positive Thermopower
	It is a widely forgotten mystery, why the thermopower of a metal as simple as
	lithium is positive [10] [58] [59] [60] [61]. Considering only the scatterring term
	of Equation (91) (as it was classical standard theory according to Equation (81)),
	positive sign was incomprehensible. However, considering the second term of
	2 2 3 B k T z r F e E = -π + Equation (91)positive sign; at 0˚C they are +1.7, +1.4, +1.1, and +11.5 μV/K, respectively ([48], d 1 , d C E S e T + (91)
					S =	π	( ) F k T z r 2 2 3 B e E +	+	d d E V e T 1	,	(92)
	with	z =	3 2	, where the index i is omitted and	, C A E ∂	T ∂ and	, V B E ∂	T ∂ are
	replaced by d	d E T and d C	d E T , respectively, because a homogeneous metal V
	consists only of one phase. Each of the two limiting cases, Equation (91) and
	Equation (92), represents a homogeneous metal with electron conductivity and
	hole conductivity, respectively, and constant carrier density, i.e., interband
	transfer of electrons (in dependence on temperature) are not considered.
	The term " 0 i µ ∂	T ∂ " reflects the lowering of the chemical potential with T
	described by the Fermi-Dirac-statistics. In Equations (87)-(92) it is assumed that
	Equation (81) does exclusively represent the scattering part of S. It is, however,
	not completely clear, whether Equation (81) does indirectly contain this term
	" 0 i µ ∂	T ∂ ", already. If so, then in Equations (88), (90)-(92) the term " 0 i µ ∂	T ∂ "
	(respectively "	-	2 2 6 π B F i , k T E	") is to be deleted, and the value for z is to be replaced by
	1 z = .
	Equation (81) agrees with the first term in Equation (91) if	1 z = is set. It
	represents the contribution of the scatterring on the thermopower, whereas the
	DOI: 10.4236/ojcm.2019.91002						48	Open Journal of Composite Materials

Acknowledgements

The authors thank Mr. Alan Savan from the Institute of Materials, Faculty of Mechanical Engineering of the Ruhr-University Bochum, and Mr. Michael Kieschnik from RUBION, Ruhr-University Bochum, for co-sputtering the Cr-Si thin films and the RBS analysis, respectively, shown in Figures 789.

R and A R versus

( )

follow an exponential dependence also for Ni 1-y (SiO 2 ) y , although we had assumed non-magnetic materials for the derivation of Equation ( 43), wherease Ni is a magnetic material. This is however not a

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.