Electrostatic Fields in and around Conductors, 

Semiconductors and Superconductors

Introduction

Every material in the world can be defined in terms of how well it conducts electricity. Certain things, such as cold glass, never conduct electricity. They are known as insulators. Materials which do conduct electricity, like copper, are called conductors.  In the middle are materials known as semiconductors, which do not conduct as well as conductors, but can carry current. Last, are materials called superconductors, which when brought down to very low temperatures  turn into superhighways of current, they conduct electricity without any resistance. We are going to give you  information about conductors, semiconductors and superconductors and their behavior under fields.

Conductors in Static Electric Field

Introduction:

By the simplest definition, a conductor material is a material that can conduct electricity. The electrons in the outermost shells of a conductor material atom are very loosely held and migrate easily from one atom to another. Conductors stay in between semiconductors and superconductors in conductivity.

Under Static Electric Field

If  one is to inject some positive (or negative) charges inside a conductor material, he will set up an electric field inside the conductor. After that the electric field will cause the charges away from each other until they reach the surface of the material. After all the charges reach the surface and redistribute themselves according to the shape of the material, the electric field and the charge in the conductor will vanish.

        Table 1-1

In the same manner, if we put a conductor in a static electric field in the beginning the electric field will penetrate inside the material. As time passes the free electrons in the conductor will move in the opposite direction of the electric field and the holes will move in the same direction with the electric field which causes the positive charges and negative charges being collected on opposite sides of the material creating a field opposing the static electric field and a current density in the same direction with the electrostatic field. We can investigate the current density formula and the below graph for a better understanding of these situations.

 Ј (current density) = σ (conductivity).E (electric field)

    Figure 1-1

In part (1) all the electric field is penetrating through the material causing a current density in the same direction.

In part (2) as a result of the current in the material there happens to be an electric field opposing the external electric field so now only part of the external electric field is penetrating and the magnitude of the current density got smaller (can be seen by looking at the formula). The electric field inside the material is still in the + x direction with a magnitude equal to E - Ei.

In part (3) the same action continues as in part (2) and J gets smaller and Ei gets bigger. As a result the electric field penetrating through the material got even more smaller than part (2).

In part (4) E = Ei which means there is no more electric field in the material and the current density is zero. We can see that the negative charges are collected on the left side and positive charges on the right that creates a potential difference and the electric field Ei. We have to remember that and electric field is from positive to negative, the external electric field attracted the electrons to its source and repelled the protons away from its source.

The charge distribution on the material depends on the shape of the material and there will newer be tangential component of electric field on the surface because the it would cause the charges on the surface to move. That means under static conditions the E field on a conductor surface is everywhere normal to the surface which makes its surface an equipotential surface.

    Figure 1-2

In the above picture there is another conductor and in static condition E equals E_in so that the net electric field inside the material is zero. 

E_in is the electric field created by the charges inside the material and if we take the external electric field (E) away the charges will redistribute inside the material making the internal electric field zero again. The redistribution process takes a certain amount of finite time depending on the conductivity of the material.

Electric Field Components on the Surface of the Conductor (Boundary conditions)

    Figure 1-3

As the scalar line integral of the static electric field intensity around any closed path vanishes, if we consider the abcda counter in the above picture and let cb=ad=0 the below equality holds

    (1-1)

As the loop we are talking about is a to b and then c to d (letting ad and cb equal to zero), the vector ab must equal the negative of the vector cd. That means the tangential component of the electric field on the surface of the conductor is zero. It is logical, as if there was a tangential component on the surface it would cause the charges on the surface to move.

As we know that there is electric field on the surface and it does not have a component parallel to the surface, it has to be normal to the surface. In order to find the normal component we are going to use Gauss's law which states that the total outward flux of the electric field intensity over any closed surface in free space is equal to the to the total charge enclosed in the surface divided by . To start with we choose a Gaussian surface in the form of a thin pillbox, having the top face in free space and the bottom face in the conductor. We should remember that the electric field inside a conductor is zero under static conditions, so the bottom part of the conductor is in zero electric field. Using an integral form of Gauss's law we get the below equality

    (1-2)

which asserts that, the normal component of the E field at a conductor/free space boundary is equal to the surface charge density on the conductor divided by the permittivity of space.

    Table (1-2)

The equations on the two tables holds only under static conditions.

    Figure 1-4

Electric Field Lines

    Figure 1-5

In the above figures we can see the electric field lines of a positive and a negative point charge. 

The figure below shows the electric field lines between two point charges. Equal charges repel, opposite charges repel.

    Figure 1-6

As we know that the electric field lines on the surface of a conductor is perpendicular to the surface we can draw the electric field lines between a point charge and a conductor. This will have a similarity to the drawing of the electric field lines between two point charges which makes us think that there happens to be a mirror charge in the conductor.

    Figure 1-7

As opposite charges repel each other the mirror charge has to have the opposite charge with the point charge.

Same theory holds for objects.

    Figure 1-8

Electric Field and Semiconductors

Introduction:

Semiconductors are a group of materials having electrical conductivities  intermediate between metals and insulators. All the properties of conductors under static electric field hold for semiconductors, only they have less carriers inside them. A perfect semiconductor crystal with no impurities or lattice defects is called an intrinsic semiconductor. I such materials there are no charge carriers at 0 K, so we will raise the temperature in order to create electron-hole pairs. In addition to that it is possible to create carriers in semiconductors by purposely introducing impurities into the crystal which is called doping. There are two types of doped semiconductors, n-type (mostly electrons) and p-type (mostly holes). If we combine an n-type material and p-type material we will build a p-n junction.

P-N Junction and the Electric Field Inside

    Figure 2-1

 In the above picture as we combine the two materials electrons start diffusing from the n-type (mostly electrons) leaving positively charged ions to p-type (mostly holes) and the opposite from p-type to n-type. But this situation does not last long as the increasing potential difference in the depletion region causes an electric field from the n-type material to the p-type material shown below

                                           Figure 2-2

.                                  

The current due to diffusion of the charges is called the diffusion current which is from the p-type material to the n-type material and the current due to the electric field (because the electric field pushes the positive charges to the p-type material and the negative charges to the n-type material) is called the drift current. After a certain time these currents equal to each other in magnitude in the opposite directions, but in the depletion region there is always carriers passing from one side to another.

We have to ask ourselves why there is electric field in the depletion region but no electric field outside this area in the p and n material. The answer is easy if we refer to the current density equation again

Ј (current density) = σ (conductivity).E (electric field)

(2-1)

There is current in the depletion region because of the carriers moving from one material to another, but there is no current outside this are so there is no electric field.

Superconductors and Magnetic Fields

Introduction:

In 1908 Heike Kammerlingh Onne succeeded in liquefying helium, which provided and ideal cold bath for experiments at temperatures close to absolute zero. In 1911 he made the discovery that when certain substances were cooled they become superconductors. In 1933 Walter Meissner and Robert Ochsenfeld discovered that a superconducting material will repel a magnetic field. The reason behind this is that in a superconductor the induced currents exactly mirror the field that would have otherwise penetrated the superconducting material - causing the magnet to be repulsed. This phenomenon is known as diamagnetism and is today often referred to as the "Meissner effect". The Meissner effect is so strong that a magnet can actually be levitated over a superconductive material. In 1962 scientists at Westinghouse developed the first commercial superconducting wire, an alloy of Niobium and Titanium. The first use of this wire in high-energy, particle-accelerator electromagnets, however, did not come until 1987 when it was used at the Fermilab Tevatron.

Superconductors are substances (element, inter-metallic alloy, or compound) that conduct electricity without any resistance, when they are sufficiently cooled. Also, because there is no resistance in a super conductor, there is no heat dissipated by the super conductor. To add since there is no heat dissipated, there must not be any energy loss. You can see this from the fact that V/I is proportional to the volume integral of J•є (energy), where J and E are the conduction current density and the electromotive intensity. The conductance in a super conductor goes to infinity, because of the fact that G = 1/R where G is the conductance and R is the resistance.

A superconductor will not allow any magnetic field to freely enter it. This is because microscopic magnetic dipoles are induced in the superconductor that opposes the applied field.

                                                    Figure 3-1

    Figure 3-2

    Figure 3-3

This induced field then repels the source of the applied field, and will consequently repel the magnet associated with that field. This implies that if a magnet was placed on top of the superconductor when the superconductor was above its Critical Temperature (T_c), and then it was cooled down to below T_c, the superconductor would then exclude the magnetic field of the magnet. This can be seen quite clearly since a magnet itself is repelled, and thus is levitated above the superconductor.

Movie 1: Levitation in action

A superconductor is immersed in liquid nitrogen to provide cooling below the critical temperature. A magnet is placed in the air above the superconductor and left there levitating. Nothing but magnetic interaction keeps the magnet from falling down.

Movie 2: Finding a better levitating position

The levitating magnet has a preferential position above the superconductor and returns there after a small perturbation by a human finger. When the magnet is pushed hard towards the superconductor, it changes the magnetic field distribution in the superconductor, and a new position becomes preferential.

Movie 3: Lifting superconductor without touching it

At room temperature magnetic field lines from the magnet penetrate the superconductor without restraint. After cooling by liquid nitrogen they get trapped by microscopic inhomogeneities in the superconductor. The trapped magnetic lines then serve as invisible threads holding the two objects together at a certain distance

Movie 4: Smooth landing during warming up

When the superconductor is taken out of the liquid nitrogen, its temperature slowly starts increasing. As a result, the superconducting properties weaken, and the levitation force gradually gives way to the gravity.

For this experiment to be successful, the force of repulsion must exceed the magnet's weight. One must keep in mind that this phenomenon will occur only if the strength of the applied magnetic field does not exceed the value of the Critical Magnetic Field, H_c for that superconductor material. That is that the Force of repulsion > Mw*g, where Mw is the weight of the magnet applied to the superconductor and g is acceleration due to gravity. There is also a force of attraction that will allow the superconductor to be lifted up when the magnet is lifted.

    Figure 3-4

    Figure 3-5

This magnetic repulsion phenomenon is called the Meissner Effect and is named after the person who first discovered it in 1933. It remains today as the most unique and dramatic demonstration of the phenomena of superconductivity.

Type I Superconductors

There are two types of Super conductors Type I & Type II. The Type I category of superconductors is mainly comprised of metals and metalloids that show some conductivity at room temperature. Type 1 superconductors are characterized as the "soft" superconductors and exhibit a very sharp transition to a superconducting state.

    Figure 3-6

  They require the coldest temperatures to become superconductive and to slow down molecular vibrations sufficiently to facilitate unimpeded electron flow in accordance with what is known as BCS theory. BCS theory suggests that electrons team up in Cooper pairs in order to help each other overcome molecular obstacles.

 Scientists call this process phonon-mediated coupling. Type I super conductors have one critical magnetic field for any given temperature. If they are in a magnetic field that is weaker than the critical magnetic field, they have zero resistance and show perfect diamagnetism. If the magnetic field is stronger than the critical magnetic field, resistance is greater than zero, and there is flux penetration. Here is a list of Type I super conductors and their critical temperature.

 Table 3-1

*Note: Tc's given are for bulk (alpha form), except for Platinum, which is a compacted powder.

**Note 2: Normally bulk carbon (amorphous, diamond, graphite, white) will not superconduct at any temperature. However, a Tc of 15K has been reported for elemental carbon when the atoms are configured as a single-walled nanotube. An analog may exist for nanotubes made from silicon, boron-nitride or tungsten-disulfide as well.

***Note 3: Sources disagree as to whether lutetium requires high compression pressures to enter a superconductive state. However, they do agree that it has a Tc of 0.1 K.

Below is a Graph showing a Type I superconductor and the measured µoM (M = Magnetization) versus an applied magnetic field. Figure taken from Tipler, Paul A. Physics: For Scientists and Engineers. New York : Worth Publishers, 1991.

    Figure 3-7                                                                           

Type II Superconductors

Except for the elements vanadium, technetium and niobium, the Type 2 category of superconductors is comprised of metallic compounds and alloys. The recently-discovered superconducting "perovskites" (metal-oxide ceramics that normally have a ratio of 2 metal atoms to every 3 oxygen atoms) belong to this Type 2 group. They achieve higher Tc's than Type 1 superconductors by a mechanism that is still not completely understood. Conventional wisdom holds that it relates to the planar layering within the crystalline structure. Although, other recent research suggests the holes of hypo charged oxygen in the charge reservoirs are responsible. (Holes are positively-charged vacancies within the lattice.) The superconducting cuprates (copper-oxides) have achieved astonishingly high Tc's when you consider that by 1985 known Tc's had only reached 23 K. To date, the highest Tc attained at ambient pressure has been 138K. One theory predicts an upper limit of about 200 K for the layered cuprates (Vladimir Kresin, Phys. Reports 288, 347 - 1997). Others assert there is no limit. Either way, it is almost certain that other, more-synergistic compounds still await discovery among the high-temperature superconductors. Type 2 superconductors - also known as the "hard" superconductors - differ from Type 1 in that their transition from a normal to a superconducting state is gradual across a region of "mixed state" behavior. A Type 2 will also allow some penetration by an external magnetic field into its surface. Type II superconductors have two critical magnetic fields, Bc1<Bc2. For a magnetic field B less than Bc1, the superconductor acts like a type I, and for a magnetic field B greater than Bc2, the substance behaves as a normal material. A unique phenomenon occurs when the magnetic field is between Bc1 and Bc2. In this case, the superconductor has zero resistance but allows partial flux penetration. This is said to be the vortex state. In the vortex state there are cores of normal material, surrounded by material in the superconducting state. As the magnetic field increases, the number of normal cores increases until eventually, the material becomes non-superconducting. Here is a list of some Type II super conductors.

Table 3-2

Below is a Graph showing a Type II superconductor and the measured µoM (M = Magnetization) versus an applied magnetic field. Figure taken from Tipler, Paul A. Physics: For Scientists and Engineers. New York : Worth Publishers, 1991.

    Figure 3-8

Applications

Electric Power

High Temperature superconductors (HTS) can be used in the production of more cost effective motors and generators. Also High temperature superconductor power cables can carry 2 to 10 times more power in equally or smaller sized cables, because of the fact that with zero resistance there is no heat loss during the transmission of current over the transmission lines. A large scale shift to superconductivity technology depends on whether wires can be prepared from the brittle ceramics that retain their superconductivity at 77 K while supporting large current densities.

Cutaway of HTS Power Transmission Cable

Shows how Liquid nitrogen circulates through the hollow core of an High temperature Superconductor assembly, which is surrounded by layers of thermal and electrical insulation encapsulated in an outer steel jacket.

    Figure 3-7

Cable cross-section

 Figure 3-8

Transportation

The use of superconductors for transportation has already been established using liquid helium as a refrigerant. Prototype levitated trains have been constructed in Japan by using superconducting magnets. High temperature superconductor coils create strong magnetic fields that produce the effect of levitation by repulsion or attraction. This is the principle underlying magnetically levitated (maglev) trains. Maglev trains hover above a magnetic field without any physical contact with a track during operation. As a result, high speeds of up to 500 miles per hour are possible with only a small consumption of energy. For short trips, Maglev trains are competitive with short airplane flights of up to 500 miles!

     Figure 3-9

    Figure 3-10

Note: The video stream in the beginning of the page shows a water drop, a frog, a strawberry, a tomato and bug in levitation. The ring is a strong magnet which makes the objects diamagnets. An object does not need to be superconducting to levitate. Normal things, even humans, can do it as well, if placed in a strong magnetic field. Although the majority of ordinary materials, such as wood or plastic, seem to be non-magnetic, they, too, expel a very small portion (0.00001) of an applied magnetic field, i.e. exhibit very weak diamagnetism. Such materials can be levitated using magnetic fields of about 10 Tesla.

Medical Industry

Magnetic resonance imaging (MRI) is playing an ever increasing role in diagnostic medicine. Superconductive magnetic coils are an important portion of this whole-body scanner. Since these coils are capable of producing very stable, large magnetic field strengths, they generate high quality images. Currently, low temperature superconductors are employed in these coils, but the eventual use of High Temperature Superconductor materials will greatly enhance the cost-benefit aspect of the application.

Figure 3-11

Questions 

      1)   Under static conditions, what 1will be the electric field and charge in a conductor ?

      2)      Under static conditions, what will be the normal and the tangential component of the electric 

            field be equal to on the surface of a conductor ?

      3)   Why there is electric field in the depletion region but no field outside it ?

4)      What are the differences between Type I and Type II superconductors ?

5)      What causes a magnet to be levitated by a superconductor ?

6)      Why do superconductors conduct electricity so well ?

7)      What must the force of repulsion exceed in order for the Meissner effect to be successive ?

Answers 

       1)      Under static conditions, both the electric current and the charge in a conductor will be zero.

     2) Under static conditions, the tangential component of the electric field is always zero on the surface and the 

           normal component is equal to surface charge density on the conductor divided by the permittivity of space.

       3)     In order to have electric field we need to have current, there is current in the depletion region but no current 

            outside it.

 4)    Type I superconductors have a lower critical temperature than type II.

Type I superconductors do not allow partial flux penetration, while type II superconductors do allow partial flux penetration.

Type I super conductors have one critical magnetic field for any given temperature, while type II superconductors have two critical magnetic fields.

Type I superconductors exhibit a very sharp transition to a superconducting state, while type II superconductors transition from a normal to a superconducting state is gradual across a region of "mixed state" behavior.

 5)      Levitation is caused by diamagnetism or the Meissner effect.

 6)      The conductance in a super conductor goes to infinity, because the resistance goes to zero.

 7)      The force of repulsion must exceed the weight of the magnet placed over the superconductor.  

Bibliography

Tipler, Paul A.,  Physics: For Scientists and Engineers, New York, Worth Publishers, 1991.  

Cheng, K.C., Field and Wave Electromagnetics, Addison-Wesley Publication Company, USA, 1992.

Streetman B.G. and Banerjee S., Solid State Electronic Devices, Prentice Hall, New Jersey, 2000.

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