# Electronic Transport in Alloys

Key words:
Giant Hall effect, Seebeck coefficient (Thermopower), conductivity, Electron density, Ioffe-Regel criterion, Composites, Nanocomposites, Minimum metallic conductivity

During the last decades I have derived a series of formulas for calculation of the electronic transport coefficients in composites and disordered alloys. Along the way, some puzzling phenomenons have been solved:

1) Why there are simple metals with positive thermopower?
2) What is the reason for the phenomenon of the ”Giant Hall effect” in metal-insulator composites?
3) What is the reason for the fact that amorphous metallic composites can exist?
4) Is there a lowest Minimum metallic conductivity?

The answer to 1) is given by the formula (4). The answer to 2) is given by the formula (5) in connection with (6). The answer to 3) is given by the formula (5). The answer to 4) is given by the formulas (10) and (11) following from the formula (9).

# List of publications

 formula number published in (1) Phys. Rev. B 73, 045126 (2006) (2) J. Phys.: Condens. Matter 21 (2009) 175703 and J. Mater. Chem. C, 4, 10973 (2016) (4),(3) J. Phys.: Condens. Matter 22 (2010) 235501 (5) Phys. Rev. B 40, 3661 (1989) (6)-(8) Open J. of Composite Materials 6 (2016) 78 (9) Phys. Rev. B 71, 115114 (2005) (Appendix) (10),(11) Phys.Rev.B73, 045126 (2006) (Appendix B)

#### Thermopower (Seebeck coefficient) for alloys with phase Separation

\label{formel00}
\sum_i\upsilon_i\frac{\sigma_i/S_i-\sigma/S}{\sigma_i/S_i + 2\sigma/S}\approx 0

Formula (1)

\label{formel01}
\sum_i\upsilon_i\frac{\kappa_{e,i}/S_i-\kappa_e/S}{\kappa_{e,i}/S_i + 2\kappa_e/S}=0

Formula (2)

where $$S_i$$ is given by
\label{formel01b}
S_{i} = S_{i,0} + \frac{1}{|e|}\frac{d\mu}{dT}.

Formula (3)

#### Thermopower in homogeneous alloys

\label{formel05}
S = S_0 + \frac{1}{|e|}\frac{dE_c}{dT}

Formula (4)

$$S_0$$ is the classical thermopower formula for a homogeneous alloy. ($$S_{i,0}$$ is the classical thermopower formula for the phase $$i$$.) The additional term in (4), „$$\frac{1}{|e|}\frac{dE_c}{dT}$$“, follows as limiting case of (2) for an one-phase alloy. It is the reason for positive Seebeck coefficient of many metals, for instance: Cu, Ag, Au, Li.

#### Electron density in alloys with amorphous phase separation (electron transfer between the phases)

\label{formel02}
dn = -\beta \cdot n\cdot d\zeta

Formula (5)

#### Hall coefficient formula for two-phase composites

\label{formel07}
R = \frac{\sigma_A^2 R_A \left[\sigma_B+\sigma (3 \upsilon_A-1)\right] + \sigma_B^2 R_B \left[\sigma_A+\sigma (3 \upsilon_B-1)\right]}{\sigma(\sigma_A \sigma_B + 2\sigma^2)}

Formula (6)

#### General Hall coefficient formula for composites with two or more phases

\label{formel08}
\left(R \sigma^2 \frac{\partial}{\partial \sigma} + \sum_{i} R_i \sigma_i^2 \frac{\partial}{\partial \sigma_i} \right)  f(\sigma,\sigma_i) = 0,

Formula (7)

where
\label{formel09}
f(\sigma,\sigma_i) = \left(\prod_{i} \left(\sigma_{i}+2\sigma\right)\right) \left(\sum_{i}\upsilon_i\frac{\sigma_{i}-\sigma}{\sigma_{i}+2\sigma}\right)

Formula (8)

#### Ioffe-Regel criterion (Alternative interpretation)

\label{formel03}
k_{F} L \geq c^* = \frac{1}{4}

Formula (9)

#### Minimum metallic conductivity; strong scattering

\label{formel06a}
\sigma_{min} = \frac{c^{*2}}{6}\Bigm(\frac{\mathrm{e}^2}{h}\Bigm)\frac{1}{d} = \frac{1}{96}\Bigm(\frac{\mathrm{e}^2}{h}\Bigm)\frac{1}{d}

Formula (10)

#### Minimum metallic conductivity; general case

\label{formel06}
\sigma_{min} = \frac{2 c^{*2}}{3 \pi}\Bigm(\frac{\mathrm{e}^2}{h}\Bigm)\frac{1}{L} = \frac{1}{24 \pi}\Bigm(\frac{\mathrm{e}^2}{h}\Bigm)\frac{1}{L}

Formula (11)

### Symbols

$$S$$ – Seebeck coefficient

$$\sigma$$ – electrical conductivity

$$\kappa_{\mathrm{e}}$$ – electronic contribution to the thermal conductivity

$$\upsilon_i$$ – volume fraction of the phase $$i$$

$$i$$ characterizes the phase in a composite, $$i = A, B, \ldots$$

$$R$$ – Hall coefficient

$$n$$ – electron densitiy [in a two-phase composite $$n$$ is the electron densitiy in the phase with the higher potential ($$\equiv$$ phase $$A$$)]

$$\zeta = \upsilon_B/\upsilon_A$$

$$\beta$$ – a constant for a given alloy, which is determined by the average potential difference between the two phases.

$$E_c$$ – band edge of the conduction band

$$T$$ – temperature

$$\mu$$ – electrochemical potential

$$k_F$$ – wave number at the Fermi surface

$$L$$ – mean free path of the electronic carriers

$$d$$ – average atomic distance