Electronic Materials

# Conductors and their properties

Electric conductors can be solids, liquids and even gases. Solid conductors are metals, metal alloys and some carbon modifications. Metals are plastic compounds with characteristics that conduct electrical current and heat.

Liquid compounds are melted metals and electrolytes. The metals with the lowest melting temperatures are mercury and gallium, -39°C and +29.8°C respectively. All other metals have much higher melting temperatures.

The mechanism of electrons transported in solid and liquid metals is due to free electrons, that’s why they are called electron conduction compounds or first sort conductors.

Electrolytes, or second sort conductors, are solutes of acids, alkalis and salts and also melts of ionic compounds. Current flow through these conductors is due to the charged parts of molecules (ions). As a result electrolyte composition changes, and the electrodes have the products of electrolysis.

All gases and steams are not conducting current with low electromagnetic fields. However, if the electromagnetic field is higher than some critical level, obtaining strike and photoionisation, gas can become a conductor with an electron or ion type of conductance. High ionised gas with the equal quantity of electron and positive charges in the unit of volume is called plasma.

### Nature of conductance in metals

The classical theory about electronic metals, developed by Drude and Lorentz, is the idea about electron gas, consisting of free electrons. Electron gas has the properties of ideal gas. Considering that atoms in metal once ionised, the concentration of free electrons is:

According to the atomic-kinetic theory of ideal gases, average kinetic energy of electrons, moving chaotically, is growing linear with temperature:

When the external electric field is applied, electrons get an additional vector if the rate, oriented by the field, stimulates the electric current. Current density is:

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In this explanation we have discussed single free electrons, but it’s reasonable to discover all electrons in the electric field. Summarised impulse of electrons changes in the field and is due to collisions with lattice atoms, so the speed will be two times larger. And:

This vision about free electrons leads us to the Wideman-Franz theory, that sets up the connection between electrical conductivity and thermal conductivity. Electrons carry electrical charge and heat in conductors.

Good conductors are also good heat conductors. The process of energy absorption involves both free electrons and lattice atoms. Then for metals, heat capacity should  be higher then for insulators. However, in reality, thermal capacity of metals is nearly the same as the thermal capacity of insulators. However, there are some contradictions, which can be resolved by quantum theory. The main flaw of classical theory about electrons in conductors is application of classical statistics (statistics of Maxwell-Boltzmann), where the energy distribution of electrons is an exponential function:

Quantum statistics are based on the Pauli principle, which states that every energy state can contain only one electron. In quantum theory, the possibility to occupy the energy levels by the electrons is the following:

Where EF– Fermi energy, the energy characteristic level is where the possibility curve is symmetrical. It considers the maximum energy of an electron in metal when T=0K. Fermi energy corresponds to the term of electrochemical potential:

${\psi }_{F}=\frac{{E}_{F}}{e}$

EF relates to the compound volume, but depends on the free electrons concentration. The most frequent Fermi energy value is 3-15eV. During heating metal gets to an energy proportional to kT, however this energy excess is much less than Fermi energy. This leads us to the conclusion that metals have low heat capacity and high conductivity.

Systems of microparticles, described by Fermi-Dirac statistics, are called degenerated. The average energy of electron gas is not dependent on the temperature. Electron gas is degenerated in a metal while electrons can not exchange their energy with lattice atoms. Degeneration temperature for metals is about 104K, which is bigger than the melting temperature for metals.

The conductance for metals, basin on the Fermi-Dirac statistics, depends on the free path of the electrons:

We can conclude that free electrons concentration in metals does not differ significantly. Electrons concentration n also do not differ significantly with temperature. It means that metal conductance depends only on the free path of the electrons, the symmetry of the lattice and atom nature.

### Wiedeman-Franz theory

In conductors electron heat conductivity prevails over the other types of heat conductivity, because of the quantity of free electrons in a metal. According to kinetic theory, heat conductivity is:

${\lambda }_{\tau }=\frac{1}{2}knul\phantom{\rule{0ex}{0ex}}\frac{{\lambda }_{\tau }}{\gamma }=3{k}^{2}{e}^{–2}T=LT$

It means the ratio of heat conductivity and current conductivity is a constant at a given temperature. And good current conductors are good heat conductors.

These results more or less correspond to each other.

### Temperature dependence of metal conductivity

Elementary particles are characterised by dual-wave duality. It means electron movements in a crystal can be described as a flat waves, with wavelength described by de Broglie’s equation:

$\lambda =\frac{h}{{m}_{0}u}=\frac{h}{\sqrt{2{m}_{0}E}}$

This flat wave distributes through the potential field of ideal crystalline  lattice without energy losses. Then for an ideal crystal, free path is infinite and resistance is equal to zero. Thus resistance of ‘technically pure’ metals,  like Au, Pb, Cu, Ag and some others, is striving to zero, when temperature is striving to zero.  When there are some imperfections in an ideal crystal structure, electrons are dissipating on the crystal lattice. Dissipation appears when the defect’s size is bigger than a fourth of the electron wavelength. In metals energy of electrons in the conductance band is about 3-15, what corresponds to the wavelength 3-7A. It means any impurities in the crystal structure hinders the electron wave distribution. It means a crystal’s resistance is increasing.

For pure metals, the only reason that can prevent electron waves distribution is heating oscillations of the lattice atoms. ρT is thermal electrical resistance of a metal. When the temperature is increasing, atom oscillation amplitudes and potential field fluctuations are increasing too. It is heightening electron dissipation and metal resistance too.

Free path for electrons in these conditions is inversely proportional to the temperature:

$l=\frac{{k}_{e}}{2\mathrm{\pi NkT}}$

Where kis a factor of elasticity (for the atom, tending to return it to the initial position). We have to remember that this formula does not work for low temperatures, because heating oscillations become insignificant in this case (oscillation amplitude and frequency are decreasing). Interaction of an electron and oscillating atom changes electron impulse insignificantly. In the oscillation theory of atoms, lattice temperature is considered around some characteristic temperature ϴD, called Debye temperature. Debye temperature considers the maximum frequency of atomic heating oscillations in the lattice:

${\theta }_{D}=\frac{h{v}_{max}}{k}$

Debye temperature depends on the connection bond between atoms in a lattice. When T > ϴD, resistivity for conductors is shown as the following:

${\rho }_{T}=\frac{1}{\gamma }=\frac{2\mathrm{\pi m}\stackrel{*}{\mathrm{n}}{\mathrm{ku}}_{\mathrm{F}}}{{\epsilon }^{2}{k}_{e}}T=aT$

This linear temperature function for resistivity works for temperatures higher than:

$\frac{2}{3}{\theta }_{D}$

For the most common metals Debye temperature is about 400K-450K, then the formula above is valid for room temperature and above. For lower temperatures ρT~T5. Figure 11 describes temperature dependence of resistance of most materials.

A relative change of resistance when the temperature changes for 1 Kelvin, is called the temperature coefficient of resistance, which is shown as the following:

Positive sign of αρ states that ρ is growing. αρ for pure metals usually has the 1/T temperature dependence. And for most metals at room temperature αρ~0.004K-1.

### The influence of impurities and other structural defects on metal resistance

Electron waves are dissipating on the lattice atoms and on the solid structure impurities that disrupt the crystal potential field. Electron wave dissipation on the static defects does not depend on the temperature, which is why, when the temperature is striving to zero, the resistance is striving to some considered value, called residual resistance.

The Matissen rule about resistance of a metal compound: the impedance of the conductor is a composed of the resistance due to scattering by thermal vibrations of the lattice atoms, and on the defects of the structure of the impedance of the compound which is composed of the resistance due to scattering by thermal oscillations of the lattice atoms and on the defects of the compound structure.

$\rho ={\rho }_{T}+{\rho }_{res}$

An exception to this rule are superconductors, in which resistance disappears after a certain critical temperature.

The biggest impact into the metal’s resistance brings electron dissipation on the defects of compound, that can be an impurity, or alloying. Any impurity addition is increasing the compound resistance, even if it has higher conductivity than solvents.

It was established by an experiment, that resistance changes proportionally to the defects concentration, when defects concentration is small.

Different impurities impact the residual resistance in different ways. The effect of the dissipation on the impurities depends on the exciting potential in the lattice. This potential is higher when there are bigger differences between the impurity valence and the solvent atom valence.

With the exception of impurities, residual resistance is affected by the internal compound defects – vacancies, dislocations, grains boundaries etc. Concentration of point defects grows exponentially with temperature. Vacancies and interstitial appears during the affecting of the metal with high energy radiation. Resistance of the compound can tell you about the rate of radiation irradiation of the lattice.

Stress affects compound resistance highly. The rate of this affect depends on the stress character.

### Electrical properties of metal alloys

Metal alloys with disordered structure are used widely. These metal alloys are characterised with the solvent crystalline lattice, but the lattice period is different. Many metals, having a similar type of crystalline lattice, mixes in any proportions and creates a continuous series of the solids. Statistical distribution of different kinds of atoms in the crystalline lattice provokes fluctuations in the periodical field of the lattice, which leads to strong electrons dissipation.

ρal = ρh + ρres, where ρh is a dissipation of the electron on the atom’s heating oscillations, ρres is a residual resistance due to the impurities in the crystalline structure.

A feature of the metal alloys is that ρres can be higher than ρh. For many two-component alloys, residual resistance can be described with the following expression – ρres = Cab = Cb(1 – b), where a and b are atomic fractions of the components in the alloy, C is a characterisation constant. This expression is a Nordheim Law.

The alloy resistance is bigger the farther it is from the pure metals by its consistence. Where the residual resistance has its maximum value, with equal amount of both components in the alloy, i.e. a=b. Nordheim Law works precisely if there is no phase transitions in the alloy, or it does not contain transient or rare metals. Examples are Au-Ag, Au-Cu and Cu-Ag.

### High frequency metals resistance

Electric current distributes unevenly along the metal section in high frequency conditions. Current density is at its maximum on the surface of the conductor and decreases deep into the conductor. This phenomena is called surface effect (or skin effect). Uneven current distribution is explained by the conductor field. Magnet flux for the conductor is ϕ = Li, where  is a conductor inductance. Magnet flux changes provoke the EMF of self-inductance:

${e}_{L}=–L\frac{di}{dt}$

If the current changes by sinus function , then EMF is the following:

EMF of self-inductance is opposite to the direction of the current flow and slows it down. When the conductor carries alternating current, alternating magnet flux appears inside and out of the conductor, and different parts of the conductor section are in different conditions. EMF has the maximum value for central parts of the conductor and is minimal for its surface part. And current density is weakened the most at the centre of the conductor, and weakened the least at the surface. Current extrusion to the conductor surface is stronger with frequency growth. If the current direction is considered as the following expression is valid:

${J}_{x}={J}_{0}exp\left(–\frac{z}{∆}\right)$

Where J0 is a current density on the surface, Δ– is a depth of magnet field penetration into the conductor. Connection of the field penetration with physical characteristics of the compound is the following:

$∆=\sqrt{\frac{2}{\omega \gamma {\mu }_{0}\mu }}$

Where µ0=4π10-7, Hn/m – magnet constant. With high frequencies, current is equal to zero for all conductor cross-section except thin surface layer.

### Thin metal film resistance – surface effects

Metal thin films are used in microelectronics as intercomponent connections, connection pads, capacitor plates, magnetic and resistive elements of the integral circuits. Electric properties of metal thin film can significantly differ from the properties of bulk metals and alloys. The first reason is the variety of structural characteristics of the thin films, obtained with the method of molecular beams condensation in the high vacuum atmosphere. In varying condensation conditions, thin film structure can change from an absolutely disordered finely dispersed state (amorphous condensate), to the perfect monocrystalline layers (epitaxial films).

The second reason properties change in metal thin films are surface effects (surface affects’ role is prevailing above volume effects). The surface effect in our case appears when thin film width is commensurate with a free path of the electrons. In this case film properties depend on the geometrical characteristics of a conductor. This film resistance is described by the formula:

${\rho }_{\delta }=\rho \left(1+\frac{l}{\delta }\right)$

Where ρ is bulk metal resistance, l is a free path of the electrons for bulk metal, δ is a thin film width.

Surface dissipation of the electrons in the room temperature for thin films becomes significant when the thin film depth 200Ȧ-300Ȧ. However, with low temperatures, when an electron’s free path is increasing, the surface affect becomes significant with a bigger width of the thin film.

### Contact phenomena and thermo-EMF

During the contact of two different metals there are contact potential differences that appear between them. This phenomena has been discovered by A. Volta in 1797. In accordance to quantum theory, contact potential differences appear between two materials proportional to the difference of their Fermi energies.

$eU=E\frac{1}{F}–E\frac{2}{F}$

The system made by two different conductors, make a closed loop, which is called thermocouple. (Figure 11). When contact has a different temperature in the closed loop, current is flowing through the circuit. This current is called thermoelectric current. If the loop is broken, then we have a potential difference on the contacts of the circuit. This potential difference is called thermo-EMF.

$U~{\alpha }_{T}\left({T}_{2}–{T}_{1}\right)$

αT is a relative thermo-EMF. αT depends on the metals nature and their temperature. Thermo-EMF is combined of three components. The first one is related to temperature dependence of contact potential difference. When the metal temperature is increasing, its Fermi energy is lowering down too. It means that hot contact of the thermocouple should have higher Fermi energy than a cold one. Hence, contact components of thermo-EMF appears in the thermocouple. The second component of the thermal-EMF is related to the charge carrier diffusion from the hot contact to cold one.

The average energy of electrons in a metal changes with temperature. Electrons on the hot contact have bigger kinetic energy, bigger rate and diffuse faster. This diffusion flow in the temperature gradient direction creates thermo-EMF components. The third component of the thermo-EMF appears in a loop due to electron movement by phonons (quants of thermal energy). Their flow also moves to the cold contact. All components of the thermo-EMF is considered by a small concentration of electrons, located close to the Fermi level. Thermo-EMF for metals is relatively small and is considered by the expression:

${\alpha }_{T}\approx {\mathrm{\pi }}^{2}\frac{\mathrm{k}}{\mathrm{e}}\frac{\mathrm{kT}}{{\mathrm{E}}_{\mathrm{F}}}$

For the thermocouple loop relative thermo-EMF is the difference between thermo-EMF of both contacts:

${\alpha }_{T}\approx \alpha \frac{1}{T}–\alpha \frac{2}{T}$

α1 and α2are total relative thermo-EMF of both contacts.

To consider a total, relative thermo-EMF Plumbum is used as a standard, because its thermoelectric properties are very weak. Superconductors are very good standards for thermo-EMF consideration at low temperatures, because their thermo-EMF is always zero. Thermo-EMF has a negative sign if the hot contact of the thermocouple is a positive sign, its natural for most simple metals. Absolute thermo-EMF strongly depends on the temperature and can change its sign during heating.

### Classification of conductor materials – Materials with high conductance

These materials are metals with electrical conductance less then 0.1mOhm*m.

Cuprum. Cuprum advantages are the following:

1. small resistance;
2. high mechanical strength;
3. normal resistance to corrosion;
4. good processability – its rolling to lists, stripes and wires;
5. It’s easy to solder.

Its resistance is sensitive to impurities. With 0.5% of Zn, Cd or Ag it’s resistance decreases by five percent. With 0.5% of Ni, Sn or Al, it’s resistance decreases by 40%. Be, As, Fe, and Si decreases resistance to 55% or more. Cuprum disadvantage is its atmospheric corrosion when creating oxides and sulfur films. Oxidation rate grows with temperature. Cuprum is not suitable for low current contacts, because of oxidation.

Cuprum is used in electronics for wires, cables, switchgear buses, transformer coils, current-carrying elements and others. Solid cuprum is used when it’s important to ensure mechanical strength, rigidity and stability to abrasion. Soft cuprum is in use when flexibility and plasticity is important. Special electrovacuum types of cuprum is used for production of clystrons, magnetrons, RF components, waveguides and resonators.

Aluminium. Of secondary importance, after cuprum, and metal with high resistance is aluminum. Its resistance is 1.5 times higher than cuprum. And aluminium is 3.5 times lighter than cuprum. Aluminium provides bigger conductivity per mass unit, due to its low density. The disadvantage of Aluminium is low mechanical strength. Aluminium oxidises and covers with a thin film of oxide on its surface with high electric resistance. This film prevents aluminium from corrosion, but creates a big transition resistance in contact points of aluminium wires. It makes standard soldering difficult. Aluminium thin films are used in integral circuits as contacts and interconnections. Aluminium creates very good contact to silicот. A disadvantage of Aluminium is its susceptibility to electromigration, which leads to resistance growth and breaks in interconnection.

### Superconductor metals and alloys

Many metals and alloys have strong resistance, which degrades when temperature is striving to absolute zero. This phenomena is called superconductivity, and the temperature of transition to superconductivity is critical temperature Tsup. The first phenomena of superconductivity has been discovered at mercury Hg (Tsup=4.2K) by Netherlands physicist H. Cammerling-Onnes.

If we induce electric current in a superconductive loop, it won’t exist for long. Resistance was considered by magnet flux degradation, induced by current in a loop. It is 10-25m0hm*m.

The phenomena of superconductivity appears when electrons are attracted towards each other. It is possible in the field of positively charged ions, which is lowering Columb forces of electrons attraction. Only electrons that take part in the process of conductivity, placed close to the Fermi level, can attract to each other. These electrons with opposite directed impulse vectors connects to couples, called Couper couples. Electrons interaction with heating oscillations of the lattice, phonons, and plays a crucial role in the creation of Couper couples. Electrons can create and absorb phonons in a solid compound.

Transition of the metal to a superconductivity state happens in a very small range of temperatures. Vacancies, impurities, structure diversity, grains boundaries and other defects, expand the temperature range for superconductivity. Electrons, responsible for superconductivity transition does not make an energy exchange with the lattice. That’s why superconductors are characterised with low thermal conductivity.

An important feature of the superconductors is that the magnetic field does not penetrate to the superconductor, it’s attenuating on the surface of the superconductor. This phenomena is called the Meisener effect and happens because the magnetic field creates a circular current on the superconductor surface, that creates a magnetic field compensating for external magnetic fields. The depth of the magnetic field penetration is about 10-6 – 10-7m. It means superconductors are ideal diamagnets with μ = 0 permeability. Superconductors are pushed out of the magnetic field as any other diamagnet.

Superconductivity state can be broken if a magnetic field is more than some critical point Hsup. There are two types of superconductors that exist: Type I and Type II, and they can be considered by way of transition into the superconductivity state. Type I superconductors make transition abruptly, as soon as the magnetic field reaches the critical point Hsup. Type II superconductors make transition into the superconductivity state gradually, and are characterised by low Hlow and high Hhigh magnetic field critical points. Between these two points material is in a transitional heterogenous state with a normal and superconductive phase. The rate of their volumes depends on the magnetic field, and penetrates to the compound gradually.

${H}_{sup}\left(T\right)={H}_{sup}\left(0\right)\left(1–\frac{{T}^{2}}{{T}_{sup}}\right)$

Hsup  (0) is a magnetic field at 0K.

Superconductivity can also be broken by the current through superconductor Isup. For Type I superconductors, current density stops at the level of the critical magnetic field point on the superconductor surface. In the case of a long wire with a round cross-section, critical current is Isup = 2πrHsup(T). Twenty-six metals are characterised with superconductivity with transition temperatures less than 4.2K. Thirteen elements can make a transition into a superconduction state with high pressure conditions, like Si, Ge,Te, Su and some others. Some intermetallic compounds and alloys can also make a transition into superconductors. The total quantity of superconductors are around 2,000.

The main superconductor application is obtaining extremely high magnetic fields. Superconductive solenoids allow obtaining of super-high magnetic fields higher than 107A/m. For normal solenoids with an iron coil, maximum is a 106A/m magnetic field.

### High resistance alloys

High resistance alloys, are conductive alloys with resistivity not less then 0.3 mOhm*m. These alloys are used for production of electric equipment, standard resistors, electric heating equipment and others. These alloys have to be characterised not only with high resistance, but also small thermo-EMF αT. Conductors in electric heating equipment  should work in the air with temperatures up to 1,000°C.

Manganin – the main alloy, based on Cu, is used for production of electrical measurement equipment and standard resistors.

Сonstantan – Cu-Ni alloy, consistence of ni corresponds to the highest ρ value and the lowest αρ value. Constantan is good with processing. αρ usually negative, and close to zero.

Chromonikel alloy is used for production of heating electric equipment.

### Precious metals

Precious metals are the most stable metals – gold, silver, platinum and palladium. They can be found in the form of nuggets or different ores. Very pure metals can be obtained with chemical, electrolyte, metallurgical and other treatment.

Gold.  This metal is used as a material for contact pads, anti-corrosion surfacing for RF resonators, internal surfaces of waveguides. The main advantage of gold is it’s resistance to oxidising and sulfurs creation at room and higher temperatures.

Stress limit of golden wire is 150MPa, relative extension is about 40%.

Silver. This metal is resistant to oxidising and has the lowest resistance among the other precious metals. Stress limit of the silver wire is 200MPa – relative extension is about 50%. Silver is used in a variety of contacts for electronics. High heat conductivity and electric conductivity sustain the lowest contact heating and quick heat removal from contact pads.

Platinum. This metal does not react with oxygen and is resistive to chemical reagents. Platinum is good for processing and stretching to the wires and stripes. Platinum does not create sulfur oxides in the atmosphere, but creates stable transition resistance.

Palladium.  This metal is close to platinum with its properties, and sometimes replace it. In electronics, palladium is used due to its ability to absorb hydrogen. Its diffusing into palladium with temperatures 150-300°C and pressures 0.015-0.1MPa; releases back as a solid from palladium with the temperatures 350-500°C.