One always thinks of electrons as the carrier for electricity. This is true in metals. But in other media, it is not always the case. Think of an electrolyte. In this media, carriers are ions. In semiconductors, carriers are electrons or holes depending on the material doping that defines polarity.

*How can we say that?*

The fundamental experiment showing this, that can also determine whether a piece of semiconductor is P type or N type is the Hall effect.

If a current flows in a material and a magnetic field is applied perpendicular to the current, the carriers trajectory is bent in a direction that depends on the carriers polarity. Then charges accumulate in a direction that is perpendicular to both the current and the magnetic field until the resulting electric field compensates for the magnetic field so that the carriers trajectory is restored. In that equilibrium state, the transverse voltage is proportional to the current and to the magnetic field.

The voltage polarity depends on the material polarity. It can be checked that both voltage polarities exist depending on the material doping.

This demonstrates that in N type material, carriers are negative and in P type material, they are positive. These positive carriers are called “holes” as they look like a missing electron. They “weigh” about three times as much as electrons. This is normal since in a semiconductor material that has four peripheral electrons per atom with doping atoms with only three peripheral electrons, moving one hole in a given direction is somewhat equivalent to moving three electrons in the opposite direction.

*What happens when current flows from one media to another one?*

The current carrier changes in transition regions.For instance when a metal connects to a P type semiconductor, carrier types change in a very short region at the contact.

*What happens in different media such as ionic liquids *?

In ionic liquids, carriers are ions and both positive and negative ions move and contribute to the current. A significant difference with electronic conductors is that ions mobility is much lower and depends on ions types. As a result, conductivity depends on frequency. In addition ions can bend, rotate or vibrate and this affects the conductivity vs. frequency curve. Because of causality principle, permittivity also depends on frequency.

If the input of a linear system is the sum of two elementary inputs, the output is the sum of the elementary outputs resulting each from one of the elementary inputs.

Out = A ⋅ In

Now:

- If In = 0 , Out = Out0 = 0
- If In = x1 , Out = Out1 = A ⋅ x1
- If In = x2 , Out = Out2 = A ⋅ x2
- If In = (x1+x2) , Out = A ⋅ ( x1+x2 )=Out1+Out2

This property is often used to simplify circuit calculations as it allows contributions to be calculated separately and summed up later:

It must be noted that this property applies only to linear systems. It does not apply to non linear systems and even not to affine systems.

Practically, if an affine system exhibits a small offset term, this property can be considered granted and it is often used:

Out = A ⋅ In + B

Now:

- If In = 0 , Out = Out0 = B
- If In = x1 , Out = Out1 = A ⋅ x1 + B
- If In = x2 , Out = Out2 = A ⋅ x2 + B
- If In = (x1+x2) , Out = A ⋅ ( x1+x2 ) + B= Out1 + Out2
**- B**

If B ≃ 0 , then Out ≃ Out1+Out2

However, results should always be checked in such cases as they are potentially incorrect

In any electric circuit at equilibrium, the two Kirchhoff laws apply:

- The Kirchhoff Current Law (KCL)
- The Kirchhoff Voltage Law (KVL)

This law states that the sum of currents at a node is zero.

This law states that the sum of voltages around a loop is zero.

Circuit theory makes extensive use of ideal voltage and current sources.

- An ideal voltage source generates a voltage that does not depend on current.
- An ideal current source generates a current whatever the voltage.

Actual sources exhibit some internal resistance that make them non ideal. A voltage source has its internal resistance connected in series while a current source has its internal resistance connected in parallel. Internal resistance is usually low for a voltage source and high for a current source. However, “low” and “high” are relative values and one can wonder about “medium” internal resistances.*Is a source with a medium internal resistance a voltage source or a current source?*

Well, it depends... It depends on the load. We can say that if a source internal resistance is lower than the load resistance, we are more on the voltage source side. On the contrary, if a source internal resistance is higher than the load resistance, we are more on the current source side. But we can go further: If we hide a source and its internal impedance in a box and give access only to the external connections, there is no means of making the difference between a voltage source or a current source only by measuring the voltage versus current characteristics. They are equivalent. This is the basis for the Norton-Thévenin equivalence or Norton-Thévenin transform.

To go even further, we can say that we are always allowed to use that transform if it makes calculations or even just understanding easier.

P-N junctions are extremely common in ICs. They are the base of the bipolar transistor operation. They exist in MOS transistors and in many other components as parasitic or isolation diodes. It is then important to understand the operation of the P-N junctions.

As the name states, a PN junction is made from two different semiconductor materials in contact.

Let's make an experiment again. A pulse generator, 3 meters of coax cable, some resistors and an oscilloscope are sufficient to demonstrate propagation.

The calculation is based on integration. Let's consider two concentric cylinders with length L and radius r and r+dr, where dr is a very small distance. The cylinders areas are very similar and equal to:

A = 2 ⋅ π ⋅ r ⋅ L

Resistance is proportional to resistivity ρ , proportional to current flow length and inversely proportional to area. In this case length is equal to dr then:

dR= ρ ⋅ dr 2 ⋅ π ⋅ r ⋅ L

Now, we can sum up these elementary resistances from inner radius R1 to outer radius R2:

R = ∫ R1 . R2 . ρ . 2 . π . r . L . dr = ρ . 2 . π . L . ( Ln R2 - Ln R1 ) = ρ . 2 . π . L . Ln (R2 / R1)

As a log is involved, the resistance does not change much. And the resistance does not depend on R1 or R2 absolute value but it depends on the ratio.

In integrated circuits, all the components are located at the surface of the silicon chip, so usually only cylinder halves have to be considered. The resistance is then doubled:

R = ρ . π . L . Ln (R2 / R1)

The calculation is based on integration. Let's consider two concentric spheres with radius r and r+dr, where dr is a very small distance. The spheres areas are very similar and equal to:

A=4 ⋅ π ⋅ r 2

Resistance is proportional to resistivity ρ , proportional to current flow length and inversely proportional to area. In this case length is equal to dr then:

dR= ρ ⋅ dr 4 ⋅ π ⋅ r 2

Now, we can sum up these elementary resistances from inner radius R1 to outer radius R2:

R= ∫ R1 R2 ρ 4 ⋅ π ⋅ r 2 ⋅ ⅆ r = ρ 4 ⋅ π ⋅ [ -1 r ] R1 R2 = ρ 4 ⋅ π ⋅ ( 1 R1 - 1 R2 )

If R2 → ∞ : R= ρ 4 ⋅ π ⋅ R1

This resistance characterizes the inner sphere and does not depend on the location of the outer sphere. The outer surface can differ from a sphere, the result does not change.

An intermediate case is R1 ≪ R2 : R ≃ ρ 4 ⋅ π ⋅ R1

In integrated circuits, all the components are located at the surface of the silicon chip, so usually only sphere halves have to be considered. The resistance is then doubled: R ≃ ρ 2 ⋅ π ⋅ R1

This calculation takes place when it comes to evaluate the electrical resistance of a substrate or well tie or the thermal resistance of a small component.