Electric conductivity

Electric conductivity is the reverse of the resistivity. It corresponds to the conductance of a material portion of 1 m length and 1 m² of section.
Among the best conductors, it there a:
Metals like the money, copper, gold or the aluminum for which the charge carriers are the free electrons.
Solutions of electrolytes having ions in solution. For these last, the value of conductivity depends on the nature of the ions present in the solution and their concentrations. The conductivity of a solution can be measured using a conductimeter.
Certain materials, like the semiconductors, have a conductivity which depends on other physical conditions, like the temperature or the exposure to the light. These properties are more and more made profitable to produce sensors.
In IF conductivity is measured in S.m-1 (siemens per meter), but generally measurement with a conductimeter gives the result in mS.cm-1 (millisiemens per centimetre).
Current use
Largely used in chemistry, its unit in the international System of units (SI) is the siemens per meter (1 S⁄m = A2.s3.m-3.kg-1). It is the report/ratio of the density of current by the intensity of the electric field. It is the reverse of that of the resistivity. The symbol generally used to indicate conductivity is the Greek letter sigma : σ, which varies according to materials of 108 S.m-1 à 10-22 S.m-1.
Other uses of conductivity
In the field of electrostatics and the magnetostatic one, one more generally uses electric conductivity γ expressed in (Ω.m)-1. The unit of σ is homogeneous with that of γ insofar as the siemens is homogeneous in Ω-1.
The conductivity of an aqueous solution makes it possible to estimate its load in ions, it is generally expressed in μS⁄cm.
The law of Nernst-Einstein makes it possible to calculate conductivity according to other fundamental parameters of material:
σ = DZ²e²C ⁄ kBT
D is the coefficient of diffusion of the species charged considered
Z is the number of loads carried by the species
E is the elementary charge, that is to say 1,602×10-19 C
C is the concentration of the species
kB is the Boltzmann constant, that is to say approximately 1,3806*10-23 J.K-1
T is the absolute temperature, expressed in Kelvins.
In chemistry, the law of Kohlrausch makes it possible to calculate conductivity according to the concentration [Xi] of n ions Xi present in solution. σ = zi * λi * Ci, with Zi the number of loads of the ion. For example, Zi = 2, for the ion sulfates SO2-4.
Ionic molar conductivity λi is a size characteristic of an ion, it is the contribution of the ion to the electric conductivity of the solution. It depends in particular on the concentration, the temperature, the load and the size of the ion. For a solution, conductivities are added : σ = Σi σi and the law of Kohlrausch takes the following general form then : σ = Σi zi * λi * Ci.
Electric conductance
Subjected to a potential difference, the bodies let pass a certain quantity of current. The conductance is a representation of this capacity to let pass the current. It is thus the reverse of resistance. G = 1 ⁄ R, one from of deduced another electric formula resulting from the law of Ohm : G = I ⁄ U.
The conductance is expressed in the international system in mho (symbol : S).
In an ionic solution : G = σ * S ⁄ L.
Electric conduction in crystalline oxides
The crystalline oxides, when they are stoechiometric, are electrical insulators : they can be described like quasi-ionic crystals, close to salts, the loads are related to the atoms and are not mobile. The electric insulators are besides frequently ceramics or glasses, let us note however that ceramics and glasses all are not of oxides and that glasses are amorphous solids but.
However, the variations with stoichiometry give rise to specific defects which allow an electric conduction.
Ionic and electronic conductivity
The electric current can result from the movement of two types of loads.
The ions (anions and cations): the migration of the ions involves the displacement of the associated load.
Electronic loads: free electrons and holes of electron.
The displacement of the ions can be done in two manners.
That is to say the ions slip between the fixed ions of the network, one speaks about interstitial movement.
Either there exists a gap in network, an ion of the network can then jump in the empty position, one speaks about lacunar movement.
The transported load is not the load of the ion itself, but the difference between the load of the ion and the load which one would have if the network were perfect at this place, which one calls the effective load.
For example : in alumina Al2O3, the ion aluminum in the network has a load 3+, the natural load of an aluminum site is thus 3+. So now the site is occupied by an ion of iron Fe2+ in substitution, then the site is in deficit of positive load; its effective load is thus -1. In the notation of Kröger and Vink, one notes this FeAl’. Thus, a displacement of the positive ion Fe2+ corresponds in fact to the displacement of a negative charge in the network.
An interstitial position is empty in a perfect crystal, its natural load is thus null. In this case, the effective load of the site is the real load of the species which occupies it.
A free electron or a hole of electron is considered in interstitial position. Their displacement follows a traditional law of Ohm. They can however be captured by an ion and modify the local load, for example : MM + e’ → MM’.
they move then with the ion.
The displacement of the ions can be the only fact of thermal agitation, one speaks then about diffusion, the generated electric current being a consequence of this migration. But displacement can also be created by a chemical gradient of potential, an electrostatic gradient of potential.
Variation with stoichiometry
Let us consider an element M, and this MnO2 element oxidizes it. One can describe it like a salt, (Mz+n, O2-2).
The variation with stoichiometry can come from two factors: thermodynamic balance with the atmosphere and doping.
Thermodynamic balance with the atmosphere
The oxide and the reduced element are in balance following the reaction of oxidation, n M + O2 ↔ MnO2.
According to the pressure partial of dioxygene and the temperature, balance moves on a side or other. Under the conditions where the oxide is stable, one will have variations with stoichiometry, the formula of becoming oxide:
Mn-xO2 : the oxide is known as overdrawn in cation.
Mn+xO2 : the oxide is known as surplus in cation.
MnO2-y : the oxide is known as overdrawn in anion.
MnO2+y : the oxide is known as surplus in anion.
The oxide can contain foreign elements. These elements can be, of the impurities, introduced involuntarily into the present or manufactoring process in the natural product, of the voluntary additions to modify the oxide reaction of.
These impurities can slip between the ions of the network, they are then known as interstitial, or can replace atoms of the network, they are then known as in substitution.
The doping elements can introduce a nonnull effective load. This creation of load will allow an electric conductivity, either in ionic form, or in electronic form, by collecting electrons of other sites thus creating holes of electron, or by emitting free electrons.
Laws phenomenologic of conduction
If one subjects oxide to an electric tension, the nonnull relative loads are put moving. By doing this, that creates a gradient of concentration, that the diffusion tends to level. If there is a stationary mode, one can describe this movement in a statistical total way by the law of Nernst-Einstein:
vi = DiFi ⁄ kT.
vi is the mean velocity of species I considered
Di is the coefficient of diffusion of this species I in the crystal.
Fi is the electrostatic force to which species I is subjected.
k is the Boltzmann constant.
T is the absolute temperature.
This law is similar to a fluid friction: speed, in stationary mode, is proportional to the force.
One can thus connect local electric conductivity σi due to species I with the coefficient of diffusion:
σi = Diie²ci ⁄ kT
zi is the effective load of species I many loads.
e is the elementary charge.
ci is the concentration of species I at the place considered.
Total electric conductivity σ is the sum of electric conductivities for each species : σ = Σiσi.
Variable arranges hopping
The Variable arranges hopping is a theory in physics of the condensed matter which describes the electric mechanism of conduction that one observes in the noncrystalline systems called according to the disordered situations, systems, or amorphous systems. According to Nevill Mott, the author of the variable theory arranges hopping (VRH) established in 1968-1969, this mechanism dominates electric conduction at low temperature in the disordered materials, for which the states of energy are generally localized within the meaning of Anderson close to the Level of Fermi. When the VRH is the mode of conduction dominating in a system, then the electric conductivity of this one follows the law of Mott : σ = σoe-(T0 ⁄ T)1 ⁄ n + 1 where T0 is called the temperature of Mott and depends on mainly the length of localization on the charge carriers as well as theirs density on the level of Fermi and N represents the dimensionnality of the system considered.
The model of Mott
To arrive at the identification of this mode of conduction, Mott analyzes initially the mechanisms intervening in the transfer of an electron of a site towards another and built its model while postulating that conductivity is primarily controlled by the factors which intervene in this transfer. By comparing this one to a probability P, Mott considers that the latter is only the resultant of two other probabilities : first is proportional to the factor of Boltzmann : exp [-W ⁄ kT] and second is proportional to the rate of covering of the functions of wave associated with the sites intervening in the transfer with the loads : exp [-2αR].
In these expressions W = 3 ⁄ [4πR³ N (EF)] is the energy spent by the electron so that it can carry out its jump, K is the Boltzmann constant and 1 ⁄ α is the radius of localization of the functions of wave. N (EF) is known as being the density of states on the level of Fermi.
Thus the probability of transfer is thus the resultant of these two probabilities and is proportional to : P ~ exp [-2αR - (W ⁄ kT)].
A jump of electron will not be possible which if this probability is maximum and that occurs only when the argument of the exponential function presents minima, thay is to say when ∂ ⁄ ∂R [ -2αR - (W ⁄ kT)] = 0, which gives : R = (T0 ⁄ T)¼, the carryforward of this expression of R in the resulting probability P gives the conductivity of Mott.
The VRH : an intelligent consumption of energy
The examination of the preceding expressions shows that energy concerned during the electronic transfer of a site towards another, is all the more small as the distance is long. Indeed, according to the VRH, the jumps carried out by a load within a material are optimized not in a space with 3 dimensions (3D) but in a space with four dimensions divided into three dimensions for space (3D) and for energy (1D). A jump between two very distant sites but of very close energy levels could be more probable than a jump between two very close sites but of very different energies. The cost of a jump between two sites does not depend only any more on the geometrical distance which separates these two sites but also from the energy difference between these two sites. This law derogates from our usual perception of the energy expense associated with any displacement. This type of conduction contrasts with traditional conduction known as by activation, which makes that the charge carrier moves only one site towards its close close relation.
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