STUDY MATERIAL

COURSE: M.Sc Software Systems

Semester: I

SUBJECT: APPLIED PHYSICS

UNIT : IV

Staff: K. SUBRAMANIAN

SYLLABUS

UNIT I

Lasers and fibre optics: Construction and working of He-new laser - co2 Laser Ruby laser -

Semi conductor laser - Application. Types of optical fibre - singled and bundled fibres - Fbre

material - Attenuation - Dispersion - Fibre optic light sources - Detectors - Fibre optic

communication - Principles of optical recording.

UNIT II

Super conductor: Qualitative study of the phenomenon - Critical temperature and critical

field. Meissner affect - Josephson Effect - Type I and type 2 super conductors. BCS theory of

super conductivity (Qualitative) - high temperature super Conductors.

- Application: Cryotron. Magnetic leviation -Super conducting magnets.

UNIT III

Electrical properties: Free electron theory of drude and Lorentz - weidmann- Franz law -

Distinction between conductors, Semi conductors and insulators on the basic of band theory -

Factors affecting the resistivity of a conductor:

Temperature, Alloying, Pressure, Strain, Magnetic field and environment.

UNIT IV

Semi conducting materials: Intrinsic, Extrinsic semiconductors - Material preparation:

Czochralski method - Zone refining. Hall Effect in semi conductor - applications. Physics of

PN junction diode - Junction transistor. Dielectrics :

Permittivity - Dielectric constant - Dielectric polarization - - Types of polarization - Break

down mechanisms.

UNIT V

Magnetic properties : Ferro magnetism: Dornine theory - Hysteresis - Hard and soft magnetic

materials - Curie - - Weiss law - Magnetossniction. Ferrites : Preparation, Properties,

Applications - Magnetic bubble memory.

REFERENCE BOOKS:

1. Brijal and Subramanian, “Optics”, Chand and co 1995.

2. V. Raghvan, “Material science and engineering”, a first course, prentice hail of India 1991.

3. M.R. Srinivasan, ‘Physics for engineers’’, New age international pvt ltd publications,

1996.

4. Seth and Gupta, “Course in electrical engineering materi3ls”, Dhanpat Rai and Sons, 1990.

5. M. Arumugam, “Material science”, new age international pvt ltd publications, 1996.

UNIT I

LASER AND FIBER OPTICS

LASERS

Laser is an acronym for light amplification by stimulated emission of radiation. Lasers are basically optical sources that produce light radiations with high degree of intensity and coherence (monochromaticity and directionality)

Each of these characteristics that are not normally found in ordinary light makes laser a unique & the most powerful tool. Lasers find a wide variety of applications in the field of scientific research, engineering & medicine.

According to Einstein the interaction of radiation with certain matter involves three basic processes:

· · Excited atoms · ····

E₂

Incident photons ÙÙÙ> ÙÙÙ> emitted photon

ÙÙÙ> Incident photon

ÙÙÙ> emitted photon

E₁· · · · · · · · · · · ·· ·

Stimulated absorption spontaneous emission stimulated emission

Stimulated absorption: When an electromagnetic radiation of frequency n is incident on a sample of atoms, the electrons in the lower energy state (E₁) absorb the energy from the incident radiation & rise to the higher energy state (E₂). This process is called stimulated absorption.

Spontaneous emission: The atoms excited to higher energy state are unstable there. Their life time t in these states is of the order of 10^-⁸s. The electrons in these states spontaneously make transition to lower energy states emitting a photon whose energy hg is the difference between the two energy states E₂ and E₁ i.e, E₂-E₁

This type of transition of an electron from a higher to a lower energy state without any outside stimulus is called spontaneous emission. The photons so emitted are in random phases and random directions.

Stimulated emission: When a photon of energy hn = E₂-E₁ is incident on an atom which is already in an excited stateE₂, the atom being disturbed or stimulated by the incident photon, makes a transition to a lower energy state E₁emitting a photon. The emitted photon has the same frequency, phase & direction as the incident photon. This type of emission is called stimulated emission. The net effect is two identical photons in the place of one thereby increasing the intensity of the incident beam. It is this process of stimulated emission that makes possible the amplification of light in lasers.

Metastable Energy Level

Metastable states: The time interval that corresponds to 0.5 probability of transition of atoms from the excited state is called the half life of the excited state. In the visible region, the half-life of an atomic transition is 10^-8 s.

Those energy states, where the atomic transitions have a much longer half life (10^-3 s) are called metastable states. Existence of metastable state is necessary for achieving population Inversion.

(Fig. 4)

An electron excited to level A will stay much longer (say 10^-4 sec) compared to a “normal” energy level where an electron typically stays for 10^-8 sec (provided there is no external perturbation, stimulated emission can occur immediately).

IV. Population Inversion

Consider some atoms (or molecules) in space as shown in the diagram below.

(Fig. 5)

When there are more atoms with an electron in level A, we say that there is population inversion.

Essential components of a laser

The laser device basically consists of three elements.

1. A pump

2. An active medium

3. A cavity resonator or an optical cavity.

Pump: is an external energy source that pumps sufficiently high energy to excite the atoms / molecules to higher energy states & produce population inversion a necessary condition for stimulated emission. Pumps can be optical, electrical, chemical or thermal.

(i) Optical pumping: uses light energy for excitation. This is the only practical method of pumping the atoms in solid & liquid laser medium. Eg. Ruby laser uses xenon flash lamp.

(ii) Electrical discharge: a) Collision of First kind (electron-atom collision). When a current of electrons is passed through the lasing gas medium, the conduction electrons collide with a large number of atoms, ions or molecules & raise them to the excited states.e+x "x^*+e. Such a process is called a collision of First kind. This method is preferred in gaseous lasers which contain one gas species. Argon laser uses this technique.

(b) Collision of second kind (Inelastic atom-atom collision): In some lasers, the laser atoms are directly excited to the desired states by means of inelastic collisions between the atoms or molecules.

A mixture of two different gas species of atoms A & B is used in which the excited energy states of A^* & B^*coincide. Energy may be transferred from the excited state of (say) A to the other B through inelastic collision between A^* & B symbolically represented as

A^* + B ® A + B^*-DE. This method of transfer of energy is called resonant transfer or collision of second kind.. Helium-neon laser, CO₂ are notable examples using this technique of pumping.

(iii) In semiconductor lasers the electrical energy is used to pump electron from valence band to conduction band.

(iv) Chemical reaction in certain materials leaves the atoms in the excited states. Eg. dye lasers

2. Active medium: An active medium is a lasing medium which may be a solid, or a liquid or a gas with atleast 3 energy levels as follows.

a) a ground state E_g

b) a high energy pump state E_P in which the excited electrons remain for only a short period (~10 ns)

c) an intermediate level E₁ in which the excited electrons remain for a relatively longer time (~ms). Such energy states with relatively long life time are called metastable states.

_{high
energy state}E_P

pumping Metastable state E₁

_{lasing
transition}

_ground
stateE_g

The most important requirement of an active medium is to support population inversion between two energy levels involved in lasing transition.

For achieving population inversion, the energy absorption must occur for a transition different from the transition undergoing stimulated emission. This shows that, atleast a 3-level system is required for lasing action.

Generally the lasers are named after the lasing medium used. For example, ruby laser, He-Ne laser, CO₂ laser, Nd-YAG laser, argon laser and so on.

It is the active medium that decides the wavelength of the laser emitted.

Laser action has been observed in over half of the elements known, with the range of available wavelengths extending from ultraviolet to infrared region. Gases as active medium alone give rise to more than 1000 transitions. Two of the mostly used transitions in gases are 632.8nm visible radiation from neon & 10.6 mm infrared radiation from CO₂ molecule.

Though the active medium should have atleast 3 energy levels there are active media with 4-levels

In a 3-level laser, the lasing transition terminates at the ground state. At least one half of the ground state atoms will have to be pumped to the higher state in order to start the lasing action. Therefore it requires high pumping power. Generally 3-level lasers operate in pulsed mode.

In a 4-level system there is a fourth level E₂above the ground state. The laser transition does not terminate at the ground state but terminates at E₂. Since E2decays fast into the ground state, the population inversion can be maintained continuously between E₁& E₂. Moderate pumping is sufficient to maintain this. Generally A 4-level lasers operate in continuous mode.

E_{P Pump
state}E_P

non radiative transition

E₃ metastable state E₁

Lasing transition

E_{2
fast decaying level}E₂

_{non
radiative}

E₁ ground state E_g

In same lasers the amplifying medium consists of two parts, the laser host medium and the laser atoms. For example hos of the Nd: YAG laser is a crystal of yittrium, aluminium, garnet (Commonly called YAG) laser is a crystal of atoms are the trivalent neodymium ions. In gas lasers consisting of mixture of gases, the distinction between host and laser atoms is generally not made. In ruby laser Al₂O₃ is hot molecule while Cr⁺³ ions are lasing ions.

Cavity resonator: is an optical feedback device that directs the photons back & forth through the active medium so as to maintain saturation intensity within the medium.

¬· ¬· ¬·

·® ·® ·®

reflector ·· partial reflector

L=nl/2

The back and forth reflections at the two optically plane mirrors, placed parallel to each other at the two ends of the laser tube, make the emitted photons travel through the active medium several times. Those excited atoms that are not triggered first time are triggered in the subsequent passage. This ensures that all the excited atoms of the active medium participate in the stimulated emission enhancing the lasing action. Further, resonance occurs for those waves that exactly fit into the length L of the laser tube. i.e., the tube length is such that L=nl/2where n is a integral number known as longitudinal mode number. For a given length L of the tube more than one wave length will fit into the tube and laser is called multimodal. The absorption of energy being maximum under resonance condition, laser beam of maximum amplitude is produced.

Types of lasers

Based on the active medium being solid, liquid or gas, lasers are classified as solid laser, liquid lasers and gas lasers. Based on the gain of the medium lasers are classified as low density gain and high density gain lasers based on kind of output, lasers are classified as pulsed or continuous wave (CW) lasers.

Examples :

(i) Ruby laser, solid high gain pulsed laser

(ii) Neodymium laser, solid high gain operating in continuous wave mode (CW)

(iii) Semiconductor lasers, solid state, high gain.

(iv) Dye lasers: liquid high gain, operates both in pulsed and continuous wave.

[Note:In contrast to solid lasers the liquid lasers do not crack or shatter. They can be made in unlimited sizes]

(v) Helium neon laser an atomic gas laser, low gain CW mode continuous or pulsed mode is an ion laser.

(vi) Carbon dioxide laser molecular gas lowgain, CW mode power>200kW emitting at 10.6 μm is an example.

(vii) Excimer laser:- noble gas halides, (XeF, KrF, ArF) low gain, pulsed.

(viii) Argon ion – gaseous, low gain, CW.

Ruby Laser

Ruby is a crystal of aluminium oxide (Al₂ O₃) doped with chromium oxide (Cr₂O₃) in 0.05% concentration. Al & O₂ are inert atoms & chromium is the active medium

Construction

ruby rod helical flash lamp

Highly polished mirror C C C C C partially silvered mirror

· ·

pulsed high voltage

It consists of a cylindrical rod of ruby a few mm in diameter & a few cm in length with its ends perfectly flat & parallel to each other. One end is highly polished to act as mirror, while the other end is partially silvered to make it partially transmitting to get the laser out put. The crystal is surrounded by a flash lamp connected to a pulsed high voltage source. The flash lamp is a helical quartz tube with xenon enclosed in it. The rod is protected from the heat produced by the intense pulse each time, through a cold-water circulation system.

Working

3 level energy system of chromium in ruby crystal

Pumping state E_P

non radiative transition

pumping 694.3 nm

E₁metastable sate

Lasing transition

An intense flash of light from the flash lamp excites a large number of chromium ions into the energy state E_P. Some of the excited electrons make transition to the intermediate metastable state E₁, through non-radiative transition. The energy released as heat in this process will be dissipated into the lattice. A large population of electrons accumulates in this level resulting in population inversion between E_g & E₁.

The first chromium ion that makes a spontaneous downward transition to from E₁to E_g radiates a photon that triggers the other excited chromium atoms in F₁ resulting in lasing action. The ruby laser emerges as a pulse, lasting as long as the excited atoms remain available in the metastable state. When the excited atoms become depleted, the lasing action ends & has to be started again by another flash.

The back & forth reflections at the end surfaces make the photon travel through the ruby rod several times. The excited chromium atoms that were not triggered first time are triggered in the subsequent passage of the photon. This ensures that almost all the excited chromium atoms participate in the stimulated emission enhancing the lasing action. The ruby rod acts as a resonant cavity for those waves that exactly fit into the length of the rod L integral number of time half a wavelength. (L=nl/2)

Since the ruby laser is a 3 level laser, the lasing transition terminates at the ground state resulting in a pulsed output.

Helium-Neon Laser

The helium–neon laser is an example of the 4-level gas laser that produces a continuous output of laser. The active medium is the neon. The helium atoms serve as a pumping medium.

Power supply Quartz tube

Partially silvered mirror

Fully silvered

mirror Electrodes polarised

laser beam

Brewster’s window

Construction: It consists of a glass or a quartz discharge tube about 1 cm in diameter & 0.5 m in length filled with helium at 1mm pressure & neon at 0.1mm pressure. The tube windows are set at Brewster’s angle to allow only the linearly polarized light parallel to the plane of incidence to participate in

stimulated emission & to reflect away from the cavity those components of light oscillating perpendicular to the plane of incidence. Two optically plane mirrors are placed at the two ends. One mirror is fully silvered to reflect 100 % light while the other is partially silvered ( 95% reflectivity) to make it slightly transmitting.

Working

A source of high voltage (~1kV) across the electrodes sends a current of electrons through the tube. These electrons collide with the helium atoms as they are present in large number & excite them to the 2s level. Since the 2s level in helium is a metastable state, the excited atoms remain there for a long time. The 5s level in neon has almost the same energy as the 2s level in helium.

Energy level diagram

2S 20.66eV 20.61eV

laser transition 632.8 nm

spontaneous transition

1S ground state 2p

Helium Neon

Under such a condition, the helium atoms in the excited state transfer their energy to the neon atoms in the ground state by a process called resonant transfer of energy. This transfer of energy will excite the neon atoms to the 5s state & de excite the helium atoms back to the ground state.

This can be symbolically represented as

Helium* + neon helium + neon*

An inverted population occurs at the 5s metastable state relative to the 3p state of neon. An occasionally emitted photon from 5s level causes stimulated emission of other atoms. The mirrors at the two ends help in building up an intense coherent beam as the light bounces back & forth repeatedly.

The stimulated transition from 5s to 3p level results in a laser beam of wavelength 632.8 nm. There are other transitions that give rise to laser beam in IR region. Eventually the atoms from the 3p level come down to the ground state 2p through an intermediate state 3s by a quick spontaneous transition. This maintains the population inversion between the 5s & 3p levels providing a continuous supply of laser. If the length of the discharge tube is adjusted to be equal to n l/2, laser beam of maximum amplitude is produced.

Though the helium atoms are present in greater number than the active atoms neon, they do not take part directly in laser transition. The fairly light helium atoms can be easily pumped up to their excited states. Without them it is not possible to directly excite neon atoms so efficiently to the required energy state.

Applications of Laser: The applications of laser are based on their special properties such as high intensity, directionality & monochromaticity .

Scientific:

1. Used in investigating the basic laws of interaction of atoms & molecules with electromagnetic wave of high intensity.

2. Used as source of monochromatic & coherent beam in the study of interference.

3. Measurement of distances & velocity of mobile objects to a high degree of accuracy.

4. Used as sources of light in optical fibers

Engineering: Due to its high intensity & directionality laser is used for

1. drilling fine holes with precision.

2. accurate welding of fine wires.

3. cutting & shaping the materials including diamonds with ease & efficiency.

4. cutting & sealing of ampules, nipples of feeding bottles. used in holography, printing.

Holography is a technique of recording the amplitude & the phase of the light reflected from objects. This helps in storing the 3-D information.

Medical applications : Laser is used in medicine as diagnostic & therapeutic instrument. Laser can cut, rupture, open, vapourise, weld, seal, cauterize, coagulate& heal human tissue. It is the most advanced surgical tool.

UNIT II

SUPERCONDUCTOR

The Meissner effect

The second defining characteristic of a superconducting material is much less obvious than its zero electrical resistance. It was over 20 years after the discovery of superconductivity that Meissner and Ochsenfeld published a paper describing this second characteristic. They discovered that when a magnetic field is applied to a sample of tin, say, in the superconducting state, the applied field is excluded, so that B = 0 throughout its interior. This property of the superconducting state is known as the Meissner effect.
The exclusion of the magnetic field from a superconductor takes place regardless of whether the sample becomes superconducting before or after the external magnetic field is applied. In the steady state, the external magnetic field is cancelled in the interior of the superconductor by opposing magnetic fields produced by a steady screening current that flows on the surface of the superconductor.
It is important to recognise that the exclusion of the magnetic field from inside a superconductor cannot be predicted by applying Maxwell's equations to a material that has zero electrical resistance. We shall refer to a material that has zero resistance but does not exhibit the Meissner effect as a perfect conductor, and we shall show that a superconductor has additional properties besides those that can be predicted from its zero resistance.

Consider first the behaviour of a perfect conductor. We showed in the previous subsection that the flux enclosed by a continuous path through zero resistance material – a perfect conductor – remains constant, and this must be true for any path within the material, whatever its size or orientation. This means that the magnetic field throughout the material must remain constant, that is, ∂B/∂t = 0. The consequences of this are shown in Figure 10 parts (a) and (b).

A comparison of the response of a perfect conductor, (a) and (b), and a superconductor, (c) and (d), to an applied magnetic field.
In part (a) of this figure, a perfect conductor is cooled in zero magnetic field to below the temperature at which its resistance becomes zero. When a magnetic field is applied, screening currents are induced in the surface to maintain the field at zero within the material, and when the field is removed, the field within the material stays at zero. In contrast, part (b) shows that cooling a perfect conductor to below its critical temperature in a uniform magnetic field leads to a situation where the uniform field is maintained within the material. If the applied field is then removed, the field within the conductor remains uniform, and continuity of magnetic field lines means there is a field in the region around the perfect conductor. Clearly, the magnetisation state of the perfect conductor depends not just on temperature and magnetic field, but also on the previous history of the material.
Contrast this with the behaviour of a superconductor, shown in Figure 10 parts (c) and (d). Whether a material is cooled below its superconducting critical temperature in zero field, (c), or in a finite field, (d), the magnetic field within a superconducting material is always zero. The magnetic field is always expelled from a superconductor. This is achieved spontaneously by producing currents on the surface of the superconductor. The direction of the currents is such as to create a magnetic field that exactly cancels the applied field in the superconductor. It is this active exclusion of magnetic field – the Meissner effect – that distinguishes a superconductor from a perfect conductor, a material that merely has zero resistance. Thus we can regard zero resistance and zero magnetic field as the two key characteristics of superconductivity.

Perfect diamagnetism

Diamagnetism is due to currents induced in atomic orbitals by an applied magnetic field. The induced currents produce a magnetisation within the diamagnetic material that opposes the applied field, and the magnetisation disappears when the applied field is removed. However, this effect is very small: the magnetisation generally reduces the applied field by less than one part in 10⁵ within the material. In diamagnetic material, B = μμ₀H, with the relative permeability μ slightly less than unity.
Superconductors take the diamagnetic effect to the extreme, since in a superconductor the field B is zero – the field is completely screened from the interior of the material. Thus the relative permeability of a superconductor is zero.

2.4 Critical magnetic field

An important characteristic of a superconductor is that its normal resistance is restored if a sufficiently large magnetic field is applied. The nature of this transition to the normal state depends on the shape of the superconductor and the orientation of the magnetic field, and it is also different for pure elements and for alloys. In this subsection we describe the behaviour in the simplest situation; we shall discuss other more complex behaviour in Section 4.
If an increasing magnetic field is applied parallel to a long thin cylinder of tin at a constant temperature below the critical temperature, then the cylinder will make a transition from the superconducting state to the normal state when the field reaches a well-defined strength. This field at which the superconductivity is destroyed is known as the critical magnetic field strength, B_c. If the field is reduced, with the temperature held constant, the tin cylinder returns to the superconducting state at the same critical field strength B_c.
Experiments indicate that the critical magnetic field strength depends on temperature, and the form of this temperature dependence is shown in Figure 11 for several elements. At very low temperatures, the critical field strength is essentially independent of temperature, but as the temperature increases, the critical field strength drops, and becomes zero at the critical temperature. At temperatures just below the critical temperature it requires only a very small magnetic field to destroy the superconductivity.

The discovery of superconductors

The phenomenon of superconductivity, in which the electrical resistance of certain materials completely vanishes at low temperatures, is one of the most interesting and sophisticated in condensed matter physics. It was first discovered by the Dutch physicist Heike Kamerlingh Onnes, who was the first to liquefy helium (which boils at 4.2 Kelvin at standard pressure). In 1911 Kamerlingh Onnes and one of his assistants discovered the phenomenon of superconductivity while studying the resistance of metals at low temperatures. They studied mercury because very pure samples could easily be prepared by distillation.

The historic measurement of superconductivity in mercury is shown in Figure 1. As in many other metals, the electrical resistance of mercury decreased steadily upon cooling, but dropped suddenly at 4.2 K, and became undetectably small. Soon after this discovery, many other elemental metals were found to exhibit zero resistance when their temperatures were lowered below a certain characteristic temperature of the material, called the critical temperature, T_c, some of which are given in Figure 2.

The discovery of superconductors

Type I and II superconductors

High magnetic fields destroy superconductivity and restore the normal conducting state. Depending on the character of this transition, we may distinguish between type I and II superconductors. The graph shown in Figure 4 illustrates the internal magnetic field strength, B_i, with increasing applied magnetic field. It is found that the internal field is zero (as expected from the Meissner effect) until a critical magnetic field, B_c, is reached where a sudden transition to the normal state occurs. This results in the penetration of the applied field into the interior. Superconductors that undergo this abrupt transition to the normal state above a critical magnetic field are known as type I superconductors. Most of the pure elements in Figure 2 tend to be type I superconductors. Type II superconductors, on the other hand, respond differently to an applied magnetic field, as shown in Figure 5. An increasing field from zero results in two critical fields, B_c1 and B_c2. At B_c1 the applied field begins to penetrate the interior of the superconductor. However, the superconductivity is maintained at this point. The superconductivity vanishes above the second critical field, B_c2. Note that there is superconductivity for an applied field in between the two critical fields despite the increasing internal magnetic field strength, and thus the loss of the Meissner effect.

Type II superconductors are the most useful because the second critical field can be quite high, enabling high field electromagnets to be made out of superconducting wire. Most compounds shown in Figure 2 are type-II superconductors. Wires made from say niobium-tin (Nb₃Sn) have a B_c2 as high as 24.5 Tesla – in practice it is lower. This makes them useful for applications requiring high magnetic fields, such as Magnetic Resonance Imaging (MRI) machines. The advantage of using superconducting electromagnets is that the current only has to be applied once to the wires, which are then formed into a closed loop and allow the current (and field) to persist indefinitely – as long as the superconductor stays below the critical temperature. That is, the external power supply can be switched off. As a comparison, the strongest permanent magnets today may be able to produce a field close to 1 Tesla. However, it is possible to obtain up to 24.5 Tesla from a niobium–tin superconductor.

levitated magnet stable

A common demonstration of the Meissner effect is to cool a high T_c superconductor (YBa₂Cu₃O₇), then place a small and strong permanent magnet on top of it to demonstrate the repulsion of the magnetic field by the superconductor as shown in Figure 6. This repulsion results in the levitation of the magnet. An explanation for this levitation is that the magnet “sees” a mirror image of itself in the superconductor, which is like a magnet floating on top of another identical magnet. This would be true if the superconductor was much larger than the magnet. In practice the superconductor may be only slightly larger than the magnet. This will result in a distorted image of the magnet, especially near the edges of the superconductor. The situation then is similar to trying to balance two magnets on top of each other. If you have ever tried to balance one magnet on top of another, you would have quickly found that it is impossible to do without physically holding it there. Left alone, the magnet will always topple over and never stay levitated. This is a well known effect in physics called Earnshaw’s theorem, which states that there can never be any stable configuration of magnetic fields that will trap another magnet.

Figure 6. Levitating permanent magnet on top of a high Tc superconductor.

So why does a levitating permanent magnet remain stable on top of a small sized superconductor? Even a little nudge causes the magnet to spring back to its original position as if somehow tied by invisible springs to that point. To explain this, we need to expose everyone’s little secret when they demonstrate this levitation experiment. If the magnet is lightly placed over a newly cooled high temperature superconductor, you should find that the magnet does not stay levitated for long. It slips off very quickly as one would expect to happen if a magnet is placed on top of another magnet, or a distorted mirror image of itself in this case. Note what everyone does in order to get the magnet to levitate stably. They hold the magnet over the superconductor and rather than letting it go, they thrust it slightly towards the superconductor. Releasing the magnet at this point causes it to remain there stably. Incredibly, if the magnet is then removed then dropped back over the superconductor, it levitates stably without the need to thrust the magnet towards the superconductors. It is as if the superconductor has “remembered” that the magnet was there. Moving the magnet back and forth parallel to the surface of the superconductor or allowing the superconductor to warm up above T_c then cooling it down again will make the levitation of the magnet unstable once more. The magnet must again be thrust towards the superconductor to achieve stability. How can this behaviour be explained? Read the next section for the answer.

Vortex states and flux pinning

Stable levitation of a permanent magnet above a small and flat superconductor only occurs for type-II superconductors. Certainly levitation occurs when using type-I superconductors but this must be in the shape of a bowl so that the permanent magnet doesn’t fall off. The answer lies in the properties of type-II superconductors for an applied magnetic field between the two critical fields, B_c1 and B_c2. In this region, a regular array of normal conducting regions is formed as shown by the dark regions in Figure 7.

Figure 7. Vortices (dark regions) in a type-II superconductor.

These normal regions allow the penetration of the magnetic field in the form of thin filaments, usually called flux lines or vortices. The vortices or vortex states are aptly named because of the swirls of electrical currents (shown on the left in Figure 7) that are associated with this state. While in the vortex state, the material can have zero resistance and has partial flux penetration. Vortex regions are essentially filaments of normal material (non-superconducting) that run through the sample when an external applied magnetic field exceeds the lower critical field, B_c1. As the strength of the external field increases, the number of filaments increases until the field reaches the upper critical value, B_c2, and the sample becomes normal.

One can view the vortex state as a cylindrical swirl of current surrounding a cylindrical normal conducting core that allows some flux to penetrate the interior of type-II superconductors. Thrusting a permanent magnet towards a type-II superconductor will cause the applied magnetic field at the superconductor to be within the region of the two critical fields. This creates the vortex states shown on the right of Figure 7. In principle, the motion of a levitating permanent magnet will cause these vortices to move. In practice, real materials (such as High T_c superconductors) have defects (missing or misplaced atoms, impurity atoms) in their crystal lattices. They are also composed of many crystals, all bound together, resulting in many crystal boundaries. The crystal defects and boundaries stop the motion of the vortices, which is known as flux pinning. This provides the stability of a levitating magnet. Pinning the motion of its magnetic field lines also means stopping the motion of the magnet. Note that flux pinning can only occur in type-II superconductors. This demonstration with high temperature superconductors indicates that they are of type-II.

BCS theory and Cooper pairs

According to classical physics, part of the resistance of a metal is due to collisions between free electrons and the crystal lattice’s vibrations, known as phonons. In addition, part of the resistance is due to scattering of electrons from impurities or defects in the conductor. As a result, the question arose as to why this does not happen in superconductors?

A microscopic theory of superconductivity was developed in 1957 by John Bardeen, Leon Cooper and J. Robert Schrieffer, which is known as the BCS theory. The central feature of the BCS theory is that two electrons in the superconductor are able to form a bound pair called a Cooper pair if they somehow experience an attractive interaction between them. This notion at first sight seems counterintuitive since electrons normally repel one another because of their like charges. This may be thought of in the following way and is illustrated in Figure 8.

Figure 8. Classical description of the coupling of a Cooper pair.

An electron passes through the lattice and the positive ions are attracted to it, causing a distortion in their nominal positions. The second electron (the Cooper pair partner) comes along and is attracted by the displaced ions. Note that this second electron can only be attracted to the lattice distortion if it comes close enough before the ions have had a chance to return to their equilibrium positions. The net effect is a weak delayed attractive force between the two electrons.

From the BCS theory, the total linear momentum of a Cooper pair must be zero. This means that they travel in opposite directions as shown in Figure 8. In addition, the nominal separation between the Cooper pair (called the coherence length) ranges from 100’s to 1000’s of ions separating them! This is quite a large distance and has been represented incorrectly in many textbooks on this subject. If electrons in a Cooper pair were too close, such as a couple of atomic spacings apart; the electrostatic (coulomb) repulsion will be much larger than the attraction from the lattice deformation and so they will repel each other. Thus there will be no superconductivity. A current flowing in the superconductor just shifts the total moment slightly from zero so that, on average, one electron in a cooper pair has a slightly larger momentum magnitude that its pair. They do, however, still travel in opposite directions.

The interaction between a Cooper pair is transient. Each electron in the pair goes on to form a Cooper pair with other electrons, and this process continues with the newly formed Cooper pair so that each electron goes on to form a Cooper pair with other electrons. The end result is that each electron in the solid is attracted to every other electron forming a large network of interactions. Causing just one of these electrons to collide and scatter from atoms in the lattice means the whole network of electrons must be made to collide into the lattice, which is energetically too costly. The collective behaviour of all the electrons in the solid prevents any further collisions with the lattice. Nature prefers situations that spend a minimum of energy. In this case, the minimum energy situation is to have no collisions with the lattice. A small but finite amount of energy is needed to destroy the superconducting state and make it normal. This energy is called the energy gap.

Although a classical description of Cooper pairs has been given here, the formal treatment from the BCS theory is quantum mechanical. The electrons have wave-like behaviour and are described by a wave function that extends throughout the solid and overlaps with other electron wave functions. As a result, the whole network of electrons behaves line one wave function so that their collective motion is coherent.

In addition to having a linear momentum, each electron behaves as if it is spinning. This property, surprisingly, is called spin. This does not mean that the electron is actually spinning, but behaves as though it is spinning. The requirement from the BCS theory is that spins of a Cooper pair be in opposite directions.

Note that the explanation and pictorial representation of a Cooper pair presented here comes directly from BCS theory. However, current HSC textbooks tend to distort this picture with unphysical situations such as the Cooper pair being within one or two atomic spacings and traveling in the same direction – each of these situations is false.

High – T_c superconductors

It has long been a dream of scientists working in the field of superconductivity to find a material that becomes a superconductor at room temperature. A discovery of this type will revolutionize every aspect of modern day technology such as power transmission and storage, communication, transport and even the type of computers we make. All of these advances will be faster, cheaper and more energy efficient. This has not been achieved to date. However, in 1986 a class of materials was discovered by Bednorz and Müller that led to superconductors that we use today on a bench-top with liquid nitrogen to cool them. Not surprisingly, Bednorz and Müller received the Nobel Prize in 1987 (the fastest-ever recognition by the Nobel committee). The material we mostly use on bench-tops is Yttrium – Barium – Copper Oxide, or YBa₂Cu₃O₇, otherwise known as the 1-2-3 superconductor, and are classified as high temperature (T_c) superconductors.

The critical temperature of some high-T_c superconductors is given in Figure 2. Critical temperatures as high as 135 K have been achieved. Whilst this is not room temperature, it has made experiments on superconductivity accessible to more people since these need only be cooled by liquid nitrogen (with a boiling point of liquid nitrogen is 77 K), which is cheap and readily available. This is in contrast to the expensive and bulky equipment that used liquid helium for cooling the traditional types of superconductors. Moreover, the superconductors have an upper critical magnetic field, B_c2, of about 100 Tesla – huge!

The crystal lattice structure of YBa₂Cu₃O₇is shown in Figure 9. Unlike traditional superconductors, conduction mostly occurs in the planes containing the copper oxide. It has been found that the critical temperature is very sensitive to the average number of oxygen atoms present. For this reason the formula for 1-2-3 superconductor is sometimes given as YBa₂Cu₃O_7-δ where δis a number between 0 and 1.

The nominal distance between cooper pairs (coherence length) in these superconductors can be as short as one or two atomic spacings. As a result, the coulomb repulsion force will generally dominate at these distances causing electrons to be repelled rather than coupled. For this reason, it is widely accepted that Cooper pairs, in these materials, are not caused by a lattice deformation, but may be associated with the type of magnetism present (known as antiferromagnetism) in the copper oxide layers. So high–T_c superconductors cannot be explained by the BCS theory since that mainly deals with a lattice deformation mediating the coupling of electron pairs. The research continues into the actual mechanism responsible for superconductivity in these materials.

Figure 9. Crystal lattice structure of High – T_c superconductors.

Applications of superconductors

The first large scale commercial application of superconductivity was in magnetic resonance imaging (MRI). This is a non-intrusive medical imaging technique that creates a two-dimensional picture of say tumors and other abnormalities within the body or brain. This requires a person to be placed inside a large and uniform electromagnet with a high magnetic field. Although normal electromagnets can be used for this purpose, they would dissipate a great deal of heat and have large power requirements. Superconducting magnets on the other hand have almost no power requirements. Once electrical current flows in the superconducting wire, the power supply can be switched off because the wires can be formed into a loop and the current will persist indefinitely as long as the temperature is kept below the transition temperature of the superconductor.

Superconductors can also be used to make a device known as a superconducting quantum interference device (SQUID). This is incredibly sensitive to small magnetic fields so that it can detect the magnetic fields from the heart (10^-10 Tesla) and even the brain (10^-13 Tesla). For comparison, the Earth’s magnetic field is about 10^-4 Tesla. As a result, SQUID’s are used in non-intrusive medical diagnostics on the brain.

The traditional use of superconductors has been in scientific research where high magnetic field electromagnets are required. The cost of keeping the superconductor cool are much smaller than the cost of operating normal electromagnets, which dissipate heat and have high power requirements. One such application of powerful electromagnets is in high energy physics where beams of protons and other particles are accelerated to almost light speeds and collided with each other so that more fundamental particles are produced. It is expected that this research will answer fundamental questions such as those about the origin of the mass of particles that make up the Universe.

Levitating trains have been built that use powerful electromagnets made from superconductors. The superconducting electromagnets are mounted on the underside of the train. Normal electromagnets on the tracks repel the superconducting magnets to levitate the train while pulling it forwards.

A future use of large and powerful superconducting electromagnets is in a possible future energy source known as nuclear fusion. When two light nuclei combine to form a heavier nucleus, the process is called nuclear fusion. This results in the release of large amounts of energy without any harmful waste. Two isotopes of hydrogen, deuterium and tritium, will fuse to release energy and helium. Deuterium is available in ordinary water and tritium can be made during the nuclear fusion reactions from another abundantly available element – lithium. For this reason it is called clean nuclear energy. For this reaction to occur, the deuterium and tritium gases must be heated to millions of degrees so that they become fully ionized. As a result, they must be confined in space so that they do not escape while being heated. Powerful and large electromagnets made from superconductors are capable of confining these energetic ions. An international fusion energy project, known as the International Thermonuclear Experimental Reactor (ITER) is currently being built in the south of France that will use large superconducting magnets and is due for completion in 2017. It is expected that this will demonstrate energy production using nuclear fusion.

UNIT III

ELECTRICAL PROPERTY

Electron theory Drude and Lorenz
The Drude model of electrical conduction was proposed in 1900 by Paul Drude to explain the transport properties of electrons in materials (especially metals). The model, which is an application of kinetic theory, assumes that the microscopic behavior of electrons in a solid may be treated classically and looks much like a table football machine, with a sea of constantly jittering electrons bouncing and re-bouncing off heavier, relatively immobile positive ions.
The two most significant results of the Drude model are an electronic equation of motion,
and a linear relationship between current density

J

and electric field

E

,
Here

t

is the time and

p

q

n

and

m

, and

τ

are respectively an electron's momentum, charge, number density, mass, and mean free time between ionic collisions. The latter expression is particularly important because it explains in semi-quantitative terms why Ohm's Law, one of the most ubiquitous relationships in all of electromagnetism, should be true.

Lorentz force

F is the force (in newtons)

E is the electric field (in volts per metre)

B is the magnetic field (in teslas)

q is the electric charge of the particle (in coulombs)

v is the instantaneous velocity of the particle (in metres per second)

× is the vector cross product

All the quantities written in boldface are vectors (in particular, F, E, v, B).
The Lorentz force law has a close relationship with Faraday's law of induction.
A positively charged particle will be accelerated in the same linear orientation as the E field, but will curve perpendicularly to both the instantaneous velocity vector v and the B field according to the right-hand rule (in detail, if the thumb of the right hand points along v and the index finger along B, then the middle finger points along F).
The term qE is called the electric force, while the term qv × B is called the magnetic force.^[2] According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force,^[3] with the total electromagnetic force (including the electric force) given some other (nonstandard) name. This article will not follow this nomenclature: In what follows, the term "Lorentz force" will refer only to the expression for the total force.
The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the Laplace force.
Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by Johann Tobias Mayer and others in 1760, and electrically charged objects, by Henry Cavendish in 1762, obeyed an inverse-square law. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when Charles-Augustin de Coulomb, using a torsion balance, was able to definitively show through experiment that this was true.^[4] Soon after the discovery in 1820 by H. C. Ørsted that a magnetic needle is acted on by a voltaic current, André-Marie Ampère that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements.^[5][6] In all these descriptions, the force was always given in terms of the properties of the objects involved and the distances between them rather than in terms of electric and magnetic fields.^[7]
The modern concept of electric and magnetic fields first arose in the theories of Michael Faraday, particularly his idea of lines of force, later to be given full mathematical description by Lord Kelvin and James Clerk Maxwell.^[8] From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents,^[9] however, in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects. J. J. Thomson was the first to attempt to derive from Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in cathode rays, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as $\tfrac{1}{2}q\mathbf{v} \times \mathbf{B}$ . Thomson was able to arrive at the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the displacement current, included an incorrect scale-factor of a half in front of the formula. It was Oliver Heaviside, who had invented the modern vector notation and applied them to Maxwell's field equations, that in 1885 and 1889 fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object.^[10] Finally, in 1892, Hendrik Lorentz derived the modern day form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between matter and the luminiferous aether and sought to apply the Maxwell equations at a microscopic scale. Using the Heaviside's version of the Maxwell equations for a stationary ether and applying Lagrangian mechanics, Lorentz arrived at the correct and complete form of the force law that now bears his name.^[11][12]

Charged particle drifts in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomgeneous magnetic field, grad H

In many cases of practical interest, the motion in a magnetic field of an electrically charged particle (such as an electron or ion in a plasma) can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

Wiedemann–Franz law

In physics, the Wiedemann–Franz law states that the ratio of the electronic contribution to the thermal conductivity (κ) and the electrical conductivity (σ) of a metal is proportional to the temperature (T). Theoretically, the proportionality constant L, known as the Lorenz number, is equal to
This empirical law is named after Gustav Wiedemann and Rudolph Franz, who in 1853 reported that κ/σ has approximately the same value for different metals at the same temperature. The proportionality of κ/σ with temperature was discovered by Ludvig Lorenz in 1872.
Qualitatively, this relationship is based upon the fact that the heat and electrical transport both involve the free electrons in the metal.

The mathematical expression of the law can be derived as following. Electrical conduction of metals is a well known phenomenon and is attributed to the rather free conduction electrons. It is measured as sketched in the figure. The current density j is observed to be proportional to the applied electric field and follows Ohm's law where the prefactor is the specific electrical conductivity. Since the electric field and the current density are vectors we have expressed Ohm's law here in bold face. The conductivity can in general be expressed as a tensor of the second rank (3×3 matrix). Here we restrict the discussion to isotropic, i.e. scalar conductivity. The specific resistivity is the inverse of the conductivity. Both parameters will be used in the following.
Drude (c. 1900) realized that the phenomenological description of conductivity can be formulated quite generally (electron-, ion-, heat- etc. conductivity). Although the phenomenological description is incorrect for conduction electrons, it can serve as a preliminary treatment.
The assumption is that the electrons move freely in the solid like in an ideal gas. The force applied to the electron by the electric field leads to an acceleration according to
This would lead, however, to an infinite velocity. The further assumption therefore is that the electrons bump into obstacles (like defects or phonons) once in a while which limits their free flight. This establishes an average or drift velocity V_d. The drift velocity is related to the average scattering time as becomes evident from the following relations.

Preparation of semiconductor materials

Semiconductors with predictable, reliable electronic properties are necessary for mass production. The level of chemical purity needed is extremely high because the presence of impurities even in very small proportions can have large effects on the properties of the material. A high degree of crystalline perfection is also required, since faults in crystal structure (such as dislocations, twins, and stacking faults) interfere with the semiconducting properties of the material. Crystalline faults are a major cause of defective semiconductor devices. The larger the crystal, the more difficult it is to achieve the necessary perfection. Current mass production processes use crystal ingots between 100 mm and 300 mm (4–12 inches) in diameter which are grown as cylinders and sliced into wafers.
Because of the required level of chemical purity and the perfection of the crystal structure which are needed to make semiconductor devices, special methods have been developed to produce the initial semiconductor material. A technique for achieving high purity includes growing the crystal using the Czochralski process. An additional step that can be used to further increase purity is known as zone refining. In zone refining, part of a solid crystal is melted. The impurities tend to concentrate in the melted region, while the desired material recrystalizes leaving the solid material more pure and with fewer crystalline faults.
In manufacturing semiconductor devices involving heterojunctions between different semiconductor materials, the lattice constant, which is the length of the repeating element of the crystal structure, is important for determining the compatibility of materials.

UNIT IV

SEMI-CONDUCTORS

Semiconductors:

A Semiconductor is a substance which has resistivity in between conductors and insulators, e.g. Germanium, Silicon, Selenium, Carbon etc. it is not just the resistivity alone that decides whether a substance is semiconductor or not. for e.g. it is just possible to prepare an alloy whose resistivity falls within the range of Semiconductor but the alloy can not be regarded as a Semiconductor.

In fact, Semiconductor has a number of peculiar properties which distinguish them from conductors, insulators and resistance materials.

Properties of Semiconductors

The resistively of a semiconductors is les than an insulator but more than a conductors.

Semi conductors have negative temperature coefficient of resistance i.e. the resistance of a semiconductors decrease with the increase in temperature and vice-versa.

When a suitable metallic impurity leg arsenic, gallium, etc is added to a semiconductor, its current conducting properties change appreciably.

Classification of semiconductors

Semiconductors are classified as:

Intrinsic (Pure) Semiconductors
Extrinsic (Impure) Semiconductors

Intrinsic Semiconductors

A pure semiconductor is called an intrinsic semiconductor. Two of the most important semiconductors are silicon and germanium and both belong to the IV column of the periodic table. This means that these have 4-valence elements in the outer orbit (Tetravalent). The crystalline structure of both silicon and germanium is the tetrahedral pattern as shown below:

Thus, each of the 4values electrons of an atom are shared by 4 neighboring atoms to form 4 covalent as shown above.

At absolute zero temperature i.e. at ok, all the covalent bonds are complete. Hence no free electron is available for the conduction of current.

At room temperature the valence electron acquires energy greater than the energy binding it to the nucleus. So the valence electron leaves the atom and as a result, a covalent bond is broken.

The electron which leaves the atoms is called free electron and the vacancy created in the covalent bond due to release of electron is called a hole (i.e. deficiency of an electron). the hole is equivalent to a the charge i.e.

This hole can be filled by an electron from the neighboring covalent bond where another hole is created. This process of movement of valence electron or free electron from one bond to another bond continues and hence holes also move throughout the semiconductor as shown below:

Silicon crystal with one covalent bond broken »

Thus, in a semiconductor, these are 2 kinds of charge carries i.e. free electron (-e) and hole (te) and the semiconductor in which no. of free electron per (ne) unit volume is equal to the no of holes per unit volume (N_h) is called an Intrinsic Semiconductor.
i.e.

In case of intrinsic semiconductor,

And
In case of extrinsic semiconductors,
n_e ≠ n_h

Extrinsic Semiconductors (Impure) (Doped Semiconductors)

Intrinsic (pure) semiconductor has thermally generated current carries (holes and electron). These current carries are small in no. at room temperature and hence the conductivity of an intrinsic semiconductor is low. If temperature of intrinsic conductor is increased considerably to have more and more change carries to increase its conductivity, it may get damaged. So to increase the conductivity of an intrinsic semiconductor, some suitable impurities can be added in it which increases the conductivity of the semiconductor. This process of adding impurities in the intrinsic semiconductor is called Doping and the semiconductor obtained after adding impurities is called Doped / Extrinsic / Impure semiconductor.

Type of Extrinsic Semiconductors

Depending upon the type of impurity added in the Intrinsic Semiconductor, Extrinsic Semiconductor is classified into two categories:

P-type Semiconductor

N-type Semiconductor

P-type Semiconductor

When a small amount of trivalent impurity having 3 valence electron like gallium (Ga), aluminum (Al), Boom (B), etc is added to a pure / intrinsic semiconductor.
The addition o trivalent impurity provides a large no. of holes in the semiconductor. Such impurity which produce p-type semiconductor is known as accepts impurities because the holes created can accept the electron.

Explanation of formation of p-type semiconductor considers a pure germanium crystal. When a small amount of trivalent impurity like gallium is added to a germanium crystal, there exists a large no. of holes in the crystal.

This is because gallium is trivalent i.e. its atom has 3 valence electrons. Each atom of gallium fits into germanium crystal but now only 3 covalent bonds can formed.

In the 4th covalent covalent bond, only germanium atom contributes one valence electrons which gallium has no valence electron to contribute as all its there valence electron bond are already engaged in the covalent bonds with the neighboring germanium atoms.

In other words, fourth bond is incomplete; being short of are electron, this missing electron is called a hole.

For each gallium atom added, one hole is created. A small amount gallium provides millions of hole.

This can be further understood by the following fig:

Germanium crystal forming a p-type semiconductor by adding Gallium (Ga) a trivalent impurity.

The crystal of this type has an excess of holes or positive charge carries and hence known as p-type semiconductor or positive type semiconductor.

Thus, majority charge carries in p-type semiconductors are holes and the minority charge carriers are electrons which are thermally generated.

Semiconductor Devices

The devices formed with the suitable combination of n-type and p-type Semiconductor are known as semiconductor devices e.g. Semiconductor Diodes, Transistors.

In a piece of semiconductor material, if one half is doped by p-type impurity (Trivalent type) and the other half is doped by n-type impurity (Pentavalent type), a pn junction is formed. The place dividing the 2 halves or zones is called pn junction.

In other words, when a p-type semiconductor is suitable joined to n-type semiconductor, the contact surface called pn junction.

Formation of pn junction

One common method of making PN junction is called Alloying. In this method, a small block of indium (trivalent impurity) is placed on an n-type germanium slab as shown in fig below:-

The system is then heated to a temperature of about 500^oC.The indium and some of the germanium melt to from a small puddle of molten Germanium Indium mixture as a shown.

The System is then lowered and puddle begins to solidify. Under proper conditions, the atoms of the indium impurity will be suitably adjusted in the germanium slap to form a single crystal. The addition indium overcomes the excess of electron in the n-type germanium to such an extent that it creates a p-type region as shown in figure:

Hence, in this way a p-n junction is formed.

Characteristics of p-n junction Diode

A p-n junction is also known as crystal or semiconductor

Symbolic representation of p-n junction diode

The arrow head represents p-type semiconductor

The bar represents n-type semiconductor

The arrow further represents the direction of conventional current through the diode

Characteristics

The variation of current with the applied voltage to the junction diode known as the characteristic of p-n type junction diode.

Forward Bias Characteristic
Reverse Bias Characteristic

Forward Bias Characteristic

Forward Bias: When a battery of e.m.f greater than the barrier potential (V_b) is connected to a p-n junction diode in such a way that the terminal of the battery is connected to the p-region and the negative terminal of the battery is connected to the n-region of the junction diode, then the p-n junction diode is said to be Forward Biased.

To understand the meaning of forward biasing clearly. Let us first get acquainted with the following terms:

Depletion Layer: The thin region around the function containing immobile positive and negative charges is known as depletion layer.

Reason for the formation of Depletion layer

As we know, there is a high concentration of holes in the p-region and high concentration of electrons in the n-region of p-n junction. The holes from p-region and electrons from n-region diffuse through the junction .the electrons which diffuses through the junction to p-region recombine with holes. As a result of this recombination, holes disappear and an excess negative charge appears in the p-side of the junction. Also, when the holes diffuse through the junction, an excess positive charge appears in n-side of the junction. This can be easily seen from the fig given above

Thus, these positive and negative ions on both sides of the junction form a depletion layer. This layer is known as depletion layer because it is depleted of free and mobile charges carriers. It’s thickness is about 10^-3 m o 10 ^-6 m.

Dielectric Behavior

A dielectric is an electrical insulator that can be made to exhibit an electric dipole structure (displace the negative and positive charge so that their center of gravity is different).

Capacitance

When two parallel plates of area A, separated by a small distance l, are charged by +Q, –Q, an electric field develops between the plates

E = D/ee₀

where D = Q/A. e₀ is called the vacuum permittivity and e the relative permittivity, or dielectric constant (e = 1 for vacuum). In terms of the voltage between the plates, V = E l,

V = Dl/ee₀= Q l/Aee₀= Q / C

The constant C= Aee₀/l is called the capacitance of the plates.

Field Vectors and Polarization

The dipole moment of a pair of positive and negative charges (+q and –q) separated at a distance d is p = qd. If an electric field is applied, the dipole tends to align so that the positive charge points in the field direction. Dipoles between the plates of a capacitor will produce an electric field that opposes the applied field. For a given applied voltage V, there will be an increase in the charge in the plates by an amount Q' so that the total charge becomes Q = Q' + Q₀, where Q₀is the charge of a vacuum capacitor with the same V. With Q' = PA, the charge density becomes D = D₀ E + P, where the polarization P = e₀(e–1) E.

Types of Polarization

Three types of polarization can be caused by an electric field:

Electronic polarization: the electrons in atoms are displaced relative to the nucleus.
Ionic polarization: cations and anions in an ionic crystal are displaced with respect to each other.
Orientation polarization: permanent dipoles (like H₂O) are aligned.

Frequency Dependence of the Dielectric Constant

Electrons have much smaller mass than ions, so they respond more rapidly to a changing electric field. For electric field that oscillates at very high frequencies (such as light) only electronic polarization can occur. At smaller frequencies, the relative displacement of positive and negative ions can occur. Orientation of permanent dipoles, which require the rotation of a molecule can occur only if the oscillation is relatively slow (MHz range or slower). The time needed by the specific polarization to occur is called the relaxation time.

Dielectric Strength

Very high electric fields (>10⁸ V/m) can free electrons from atoms, and accelerate them to such high energies that they can, in turn, free other electrons, in an avalanche process (or electrical discharge). This is called dielectric breakdown, and the field necessary to start the is called the dielectric strength or breakdown strength.

Dielectric Materials

Capacitors require dielectrics of high e that can function at high frequencies (small relaxation times). Many of the ceramics have these properties, like mica, glass, and porcelain). Polymers usually have lower e.

Ferroelectricity

Ferroelectric materials are ceramics that exhibit permanent polarization in the absence of an electric field. This is due to the asymmetric location of positive and negative charges within the unit cell. Two possible arrangements of this asymmetry results in two distinct polarizations, which can be used to code "0" and "1" in ferroelectric memories. A typical ferroelectric is barium titanate, BaTiO₃, where the Ti⁴⁺ is in the center of the unit cell and four O^2- in the central plane can be displaced to one side or the other of this central ion (Fig. 19.33).

Hall Effect

If a piece of conductor (metal or semiconductor) carrying a current is placed in a transverse magnetic field, an electric field is produced inside the conductor in a direction normal to both the current and magnetic field. this phenomenon is known as the Hall Effect and the generated electric field is called the Hall Field.

If a rectangular piece of conductor carrying a current I in the positive X-direction and subjected to a magnetic flux density B in the positive Z-direction. The current carriers will experience a Lorentz force in the negative Y-direction. As a result the carriers are deflected towards the bottom surface of the sample and are accumulated there. If the current carriers are electrons as in the case of an n-type semiconductor this accumulation will make the bottom surface negatively charged with respect to the top surface. Therefore an electric field called the Hall field will be developed along the negative Y-direction. The force on the current carrying electrons due this Hall field will oppose the Lorentz force. An equilibrium condition is reached when these two forces balance each other. At this stage no further accumulation of electrons takes place on the bottom surface and the Hall field reaches a steady value.

If the current carriers are holes then the accumulation of carriers on the bottom surface will make this surface positively charged relative to the top surface. In this hall field is produced along the positive Y direction. The force on the holes due to the hall field opposes the Lorentz force and balances it under equilibrium conditions preventing further accumulation of holes. The hall field then attains its steady value.

If E_His the Hall field in the Y direction the force due to this field on the carrier of charge “e” is “eE_H” . The average Lorentz force on a carrier is evB where v is the drift velocity in the x-direction. In equilibrium these two forces balance, i.e.,

eE_H= evB

J= n_cev

E_H= BJ/n_ce

E_H= R_HBJ where R_H= 1/n_ce=Hall Co-efficient

If the current carriers are electrons the charge on the carrier is negative and hence

R_H= -1/ne

If the current carriers are holes the charge on the carrier is positive and hence

R_H= 1/pe

With the sign of the Hall coefficient, we can determine whether the sample is of n-type or p-type semiconductor.

Determination of Hall Coefficient

The hall coeffiencient is determined by measuring the hall voltage that generates hall fiend. If V_H is the hall voltage across the sample of thickness “d” then

V_H = E_Hd

V_H=R_HJBd

If “w” is the width of the sample then it’s cross sectional area is dw and the current density is given by J=I/dw where I is the current flowing through the sample.

Therefore V_H=R_HIB/dw

R_H=V_Hw/IB

Use of Hall Effect:

i) Determination of semi-conductor type

For an n-type semiconductor the hall coefficient is negative whereas for a p-type the hall coefficient is positive.

ii) Determination of Carrier Concentration:

by measuring the hall –co-efficient the carrier concentration of a semi-conductor can be determined.

n=1/R_H

iii) Determination of mobility:

If the conduction is due to one type of carrier then

σ = neμ where μ is the mobility

μ = σ R_Hthus the mobility can be determined

iv) Measurement of magnetic flux density

Since the hall voltage V_H is proportional to the magnetic flux density B for a given current I through the sample, the Hall Effect can be used as the basis for the design of a magnetic flux density meter.

v) Hall Effect Multiplier:

If the magnetic flux density B is produced by passing a current “i” through an air-core coil, B will be proportional to “i”. The hall voltage is thus proportional to the product of I and i. This forms the basis of a multiplier.

Advantages of Semi-conductor Devices

The advantages of semi-conductor devices over vacuum tubes are the following.

i) As the name implies vacuum tube require vacuum, but the semi-conductor devices do not require vacuum.

ii) Vacuum tubes have filaments and so require filament power. Semi-conductor devices do not have filaments and hence no filament power is required.

iii) Semi-conductor devices are smaller, lighter in weight and mechanically very rugged.

iv) Operation of semi-conductor devices requires low voltage power supplies. The power consumption is also smaller than that of the corresponding vacuum tubes.

v) The semi-conductor devices require very small warm-up time and therefore operate immediately after the supply voltages are switched on

vi) The performance of semiconductor devices is more reliable than that of vacuum tubes.

vii) The operating life of semi-conductor device is appreciably longer than that of its equivalent tube.

JUNCTION TRANSISTORS

INTRODUCTION:

When a pn junction is biased, in the forward direction it offers a very low resistance and when biased in the reverse direction, it offers a very high resistance. At a given value of current, the power developed across a high resistance or a reverse biased junction is always greater than the power developed across a low resistance or a forward biased junction. A crystal having two p-n junctions can be manufactured: a one junction giving low resistance (forward biased) and the other giving a high resistance (reverse-biased).

If a small signal voltage is introduced into the low resistance region and taken out of the high resistance region there will be a large power gain provided the current in the two regions is approximately of the same value. Consequently, this system can transfer a signal current from a low resistance to a high resistance. Such a device is given the name

Trans-(fer) (Res)-istor.

Working of a transistor:

Since the transistor has two PN junctions there are four possible ways in which we can bias this device. These conditions are:

a) Both junctions reverse biased: when both junctions are reverse biased the device will have very large input and output resistances. The current through the device is practically zero and the device is said to be cut-off and acts as a open switch.

b) Both junctions forward biased: in this condition both the junctions offer an easy flow of current. Inputs as well as output resistance is very low. The device is said to be in saturation and acts as a closed switch.

c) Emitter- Base junction forward biased and collector base junction reverse biased: in this condition the emitter base junction resistance is small whereas collector-base junction acts as the output junction, the device will have a small input resistance and a large output resistance. Under this condition if a signal is given across the input junction, we shall get an amplified signal across the output. The transistor is said to work in the active region with this “ biasing arrangement”

d) Emitter-Base junction forward biased and collector-Base junction forward biased:

This is termed as the inverted condition. The transistor action ceases in this condition. The condition is not practically used.

In condition (a) and (b) transistor acts as an open switch. These conditions are therefore used where a transistor is used as a switch. Condition (c) is used where the purpose of the device is signal amplification.

Biasing the transistor in the active region

Here the diagram shows the biasing of an NPN transistor. Emitter-base junction is forward biased and the collector –base junction is reverse biased using Vee and Vcc respectively. The biasing circuits for the two junctions are completed through the switches S1 and S2. For a PNP transistor, the biasing batteries are reversed as compared to the NPN biasing.

Suppose when both the switches are in OFF condition, C-B junction as well as E-B junctions are unbiased and there exist space charge regions at both the junctions .

When we close the switch S1, negative terminal of the battery Vee is now connected to the emitter. Since the battery positive is connected to the base, the emitter-base junction is forward biased and it’s potential barrier is reduced. The electrons are repelled by the negative potential at the emitter, diffuse through the junction, and reach the base region where they are attracted by the positive of the battery. The holes in the base region diffuse through the junction, reach the N-region.some of the electrons meet the holes, and recombine with them. With the continuous flow of electrons and holes, there results a flow of current through the junction from the base to the emitter.

The total current in the circuit results due to the flow of electrons and holes. The emitter is heavily doped and has a large no: of free electrons. The base region is lightly doped and contains a small no: of holes. Therefore, the current component flowing due to electrons is large and the current due to holes is small. The ratio of electron current component to the hole current is generally in excess of 100:1.

The current flows into the base terminal and is called the base current. The current completes its path through the emitter terminal to the battery negative. The current flowing out of the emitter terminal is termed the emitter current and an arrow is used to show the direction of this current. In NPN transistors, this arrow points outwards while the arrow points inwards for PNP transistors.

In the diagram the collector is open circuited and collector current is zero. And Ib=Ie

Now if we switch on the S2 and switch off S1, then the collector base junction is reverse biased and thus the majority carriers moves away from the junction which increases the depletion width of the collector base junction. During this condition there will be a minority charge carrier flow which is denoted as Icbo(reverse saturation current)

Icbo stands for the collector base current with emitter open.

And finally if we close both the switches then the transistor is said to be operating in active region, with emitter-base forward biased and the collector-base reverse biased.

During this condition the electrons in the emitter and the holes in the base moves towards the junction. At this junction some holes and electrons combine with each other and are lost. However because of the extreme thinness of the base layer and because of the attraction of the relatively high positive voltage at the collector, almost all the electrons will diffuse through the base and to the collector and produce an electron current in the collector. This current is called the collector current.(Ic). Which is of the order of few mill amperes? To make the collection of electrons efficient the base collector junction has got greater area than the emitter base junction.

The base current is due to the small fraction of recombinations in this thin region. Since the base region is very small, the base current is a small fraction of the collector current. This means that the collector is slightly less than the emitter current.

Ie=Ib + Ic

Amplifying action of a transistor:

Since the base emitter region is forward biased the depletion region around this region is much smaller that that around base collector region which is reverse biased. Consequently the internal resistance (Reb) of the emitter base junction is much smaller than the resistance (Rbc) of the base-collector region. Now the power dissipation in the base collector junction is Pbc and that at the emitter base junction is Peb.

Pbc=(Ic)²Rbc Peb=(Ie)²Reb

Here we know Ic is almost equal to the Ie itself, and Rbc is greater than the Reb, and thus the Pbc>>Peb, in other words the output power is always greater than the input power.

This is the amplifying action of the transistor.

Current gain in a Transistor:

There are two different current amplification factors in a transistor α, β.

Ie=Ic+Ib

α = Ic/Ie

β =Ic/Ib

Since Ie=Ic+Ib,

Ic/Ie+Ib/Ie=1

α + 1/(Ie)/Ib =1

α + 1/(Ic +Ib)/Ib =1

α + 1/ {(Ic/Ib) + 1}=1

α + 1/ {β +1}=1

1/{ β +1} =1- α

1+ β =1/{1- α}

β =1/{1- α} –1

β = α /{1- α}

And β x {1- α} = α

β- α β = α

β = α + α β = α (1 + β)

α = β/(1 + β)

Transistor connections:

There are 3 ways in which we can connect a transistor in a circuit.

They are Common Emitter configuration, Common Base Configuration,

Common Collector Configuration.

Common Base Configuration:

The npn transistor is biased as shown. The emitter junction is forward biased and the

Collector junction is reverse biased. The loads resistance R is connected in the output circuit. An ac signal source Vs is connected in series with the input circuit which is the emitter circuit. The emitter voltage Veb varies with time in accordance with the signal. As a result the emitter current will also vary with time. The collector current is a function of the emitter current. Hence Ic will also vary with time in accordance with the in put signal this varying collector current passes through the load resistance R. thus a varying voltage appears across the load. This is the output voltage Vo. As the emitter is forward biased the resistance of the emitter is circuit is small ( r ). If the input voltage is Vs, the variation in the emitter circuit is

Ie =Vs/r

The collector current is almost equal to the emitter current. Hence the variation in the collector current is also Vs/r. As this current flow through a load resistance R, the voltage across the load resistance which is equal to output voltage is

Vo= Ic x R =Ie x r = {Vs x R}/r

The ratio of the output voltage to input voltage is called the voltage gain. It is denoted by

Av, thus

Av = Vo/Vs = IcR/Ie x r = α R/r

Common Emitter Configuration:

The grounded emitter configuration is the most popular of the three types.

Current gain = Ic/Ib

Voltage Gain = {Ic x R}/{Ib x r}

Where “r” corresponds to the resistance of the base circuit.

Thus voltage gain = current gain x resistance gain

Power gain = {Ic²R}/{Ib²r}

Common Collector Configuration:

In the grounded collector the common terminal is the collector. The input Impedance is quite high because of the reverse bias that exists between the base and collector. The current gain for this configuration is slightly greater than β.

Current gain =Ie/Ib={Ic+Ib}/Ib={Ic/Ib} + 1 = β +1

The voltage gain in this configuration is always less than 1. power gain is achieved mainly because of the current gain.

Characteristic	CE	CB	CC
Current gain	Large	1	Large
Voltage Gain	Large	Large	1
Power Gain	Largest	Large	Lowest
Input Resistance	Low	Lowest	Highest
Output Resistance	High	Highest	Lowest
Phase shift wrt input	180	Zero	Zero

Transistor Characteristics:

These are helpful in studying the operation of a transistor when connected in a circuit. The three important characteristics of a transistor are,

i) input characteristics

ii) output characteristics

iii) constant transfer characteristics

Common Base Circuit:

The static characteristics of an NPN transistor connected in the common base configuration can be determined by the circuit as shown.

Milli ammeters included in series with the emitter and collector circuits measure Ie and Ic. Similarly voltmeters are connected across E and B to measure voltage Vbe and across C and B to measure Vcb. The two potentiometer resistors R1 and R2 supply variable voltages from the emitter and collected dc supplies.

Common Base Characteristics:

i) Input characteristics: it shows how Ie varies with Vbe while voltage Vcb is kept constant. The input characteristics for Ge and Si are shown in figure.

Both curves are exactly similar to the forward characteristic of a pn diode which in essence is what the emitter junction is. The characteristic may be used to find the input resistance of the transistor. Its value is given by the reciprocal of its slope.

Rin = {ΔVbe/ Δ Ie}, Vcb kept constant.

Since the characteristic is initially non-linear, Rin will vary with the point of measurements. Its value over linear part of the characteristic is about 50 ohms.

ii) Output characteristics: this shows how Ic varies with Vcb when Ie is kept constant.

The reciprocal of the slope of the near horizontal part of the characteristic gives the output resistance of the transistor which it would offer to an Input signal. Since the characteristic is linear over most of its length, Rout is very high, a typical value being 500 K ohms.

Rout = ΔVcb/ Δic

iii) Current transfer characteristics:

This shows the how the Ic varies with changes in Ie when Vcb is kept constant.

α = ΔIc/ Δie

Common Emitter Circuit:

The static characteristic of a transistor connected in the CE configuration is as shown. A micrometer is connected in series with the nase to measure Ib. similarly a milliammeter is included in the collector circuit to measure Ic. A voltmeter is connected across base and emitter terminals for measuring Vbe. Potentiometer R2 connected across dc power supply Vbb is used to vary Ib and Vbe. A second voltmeter is connected across collector emitter terminals to measure the output collector –emitter voltage Vce.

Common Emitter Characteristics:

i)Input characteristics : this shows how Ib varies with change in Vbe when Vce is held constant at a particular value.

Rin = ΔVbe/Δib

ii) Output Characteristics:

It indicates the way in which Ic varies with changes in Vce when Ib is held constant.

Rout = ΔVce/ΔIc. The value will be varying from 10Kohms to 50K ohms

iii) Current transfer characteristics:

It indicates how Ic varies with Ib. β =ΔIc/Δib

It is seen that small collector current flows even when Ib =0. It is the common emitter leakage current. It is due to the flow of minority carriers across the reverse CB junction.

Common Collector Circuit:

The static characteristics of a transistor connected in the CC configuration. a micro ammeter is connected in series with the base to measure Ib. similarly milliammeter is included in the emitter circuit to measure Ie. A voltmeter is connected across base and collector terminals for measuring Vcb. Potentiometer R2 connected across the dc supply Vbb is used to vary Ib and Vbe. A second voltmeter is connected across emitter collector terminals to measure the Vce.

Common Collector Characteristics:

The common collector output characteristics are Ie plotted vs. Vce for several fixed values of Ib. the common collector current gain characteristics are Ie plotted versus Ib for several fixed values of Vce.

UNIT V

MAGNETIC PROPERTIES

Magnetism

· Magnetism is a phenomenon by which a material exerts either attractive or repulsive force on another.

· Basic source of magnetic force is movement of electrically charged particles. Thus magnetic behavior of a material can be traced to the structure of atoms.

· Electrons in atoms have a planetary motion in that they go around the nucleus. This orbital motion and its own spin cause separate magnetic moments, which contribute to the magnetic behavior of materials. Thus every material can respond to a magnetic field.

· However, the manner in which a material responds depend much on its atomic structure, and determines whether a material will be strongly or weakly magnetic.

Bohr magnetrons

· Magnetic moment due to spin of an electron is known as Bohr magnetrons, M_B.

· where q is the charge on the electron, h – Planck’s constant, m_e– mass of electron.

· Bohr magnetrons is the most fundamental magnetic moment.

Why not all materials are magnets?

· As every material consists spinning electrons, each of them could be a magnet. Fortunately, not so

There are two reasons for it.

· First: according to Pauli exclusion rule, two electrons with same energy level must have opposite spins – thus so are their magnetic moments, which cancel out each other.

· Second: orbital moments of electrons also cancel out each other – thus no net magnetic moments if there is no unpaired electron(s).

· 224.10274.94mAXmqhMeB−==π

· Some elements such as transition elements, lanthanides, and actinides have a net magnetic moment since some of their energy levels have an unpaired electron.

Magnetic dipoles

· Magnetic dipoles are found to exist in magnetic materials, analogous to electric dipoles.

· A magnetic dipole is a small magnet composed of north and south poles instead of positive and negative charges.

· Within a magnetic field, the force of field exerts a torque that tends to orient the dipoles with the filed.

· Magnetic forces are generated by moving electrically charged particles. These forces are in addition to any electrostatic forces that may already exist.

· It is convenient to think magnetic forces in terms of distributed field, which is represented by imaginary lines. These lines also indicate the direction of the force.

Magnetic field

· If a magnetic field is generated by passing current I through a coil of length l and number of turns n, then the magnetic field strength, H (units A/m), is given by

· Magnetic flux density (induction) is the measure of lines within a medium. It has units as Weber (Wb) /m²or tesla and is

· where μ – permeability. It is a specific property of the medium, and has units as Wb/A.m or henry (H) /m.

· Relative magnetic permeability, is defined as

· μ_ris a measure of the degree material can be magnetized. where μ₀– m

· HMmχ=HMHBrμμμμ000=+= lnIH= HBμ=0μμμ=r

· χ_mis called the magnetic susceptibility and is given as

Types of Magnetism

· A material is magnetically characterized based on the way it can be magnetized.

· This depends on the material’s magnetic susceptibility – its magnitude and sign.

· Three basic magnetisms are:

o Dia-magnetism

o Para-magnetism

o Ferro-magnetism. Anti-Ferro-magnetism and ferri-magnetisms are considered as subclasses of ferro-magnetism.

Dia-magnetism

· Very weak; exists ONLY in presence of an external field, non-permanent.

· Applied external field acts on atoms of a material, slightly unbalancing their orbiting electrons, and creates small magnetic dipoles within atoms which oppose the applied field. This action produces a negative magnetic effect known as diamagnetism.

· The induced magnetic moment is small, and the magnetization (M) direction is opposite to the direction of applied field (H).

· Thus the relative permeability is less than unity i.e. magnetic susceptibility is negative, and is in order of -10^-5.

· Materials such as Cu, Ag, Si, Ag and alumina are diamagnetic at room temperature.

Para-magnetism

· Slightly stronger; when an external field is applied dipoles line-up with the field, resulting in a positive magnetization. However, the dipoles do not interact.

· Materials which exhibit a small positive magnetic susceptibility in the presence of a magnetic field are called para-magnetic, and the effect is termed as para-magnetism.

· In the absence of an external field, the orientations of atomic magnetic moments are random leading to no net magnetization.

· When an external field is applied dipoles line-up with the field, resulting in a positive magnetization.

· However, because the dipoles do not interact, extremely large magnetic fields are required to align all of the dipoles.

· In addition, the effect is lost as soon as the magnetic field is removed.

· Since thermal agitation randomizes the directions of the magnetic dipoles, an increase in temperature decreases the paramagnetic effect.

· Para-magnetism is produced in many materials like aluminium, calcium, titanium, alloys of copper.

1−=rmμχ

· Magnetic susceptibility of these materials is slightly positive, and lies in the range +10^-5to +10^-2.

Ferro-magnetism

· Both dia- and para- magnetic materials are considered as non-magnetic because they exhibit magnetization only in presence of an external field.

· Certain materials possess permanent magnetic moments even in the absence of an external field.

· This is result of permanent unpaired dipoles formed from unfilled energy levels.

· These dipoles can easily line-up with the imposed magnetic field due to the exchange interaction or mutual reinforcement of the dipoles. These are chrematistics of ferro-magnetism.

· Materials with ferro-magnetism (Examples: Fe, Co, Ni, Gd) possess magnetic susceptibilities approaching 10⁶.

· Above the Curie temperature, ferro-magnetic materials behave as para-magnetic materials and their susceptibility is given by the Curie-Weiss law, defined as

cmTTC−=χ

where C – material constant, T – temperature, T_c– Curie temperature.

· Ferro Magnets are very strong; dipoles line-up permanently upon application of external field. Has two sub-classes:-

o Anti-ferro-magnetism:

o Ferri-magnetism:

Anti-ferro-magnetism

· Dipoles line-up, but in opposite directions, resulting in zero magnetization.

· Eg: Mn, Cr, MnO, NiO, CoO, MnCl₂

· Exchange interaction which is responsible for parallel alignment of spins is extremely sensitive to inter-atomic spacing and to the atomic positions. This sensitivity causes anti-parallel alignment of spins.

· When the strength of anti-parallel spin magnetic moments is equal, no net spin moment exists, and resulting susceptibilities are quite small.

· One noticeable characteristic of anti-ferro-magnets is they attain maximum susceptibility at a critical temperature called Neel temperature. At temperatures above this, anti-ferro-magnets become para-magnetic.

Domain theory of Ferromagnetism

In order to explain the fact that ferromagnetic materials with spontaneous magnetization could exist in the demagnetized state Weiss proposed the concept of magnetic domains. Weiss built on earlier work carried out by Ampere, Weber and Ewing suggesting their existence. The findings of this work revealed that within a domain large numbers of atomic moments are aligned typically 10¹²-10¹⁸, over a much larger volume than was previously suspected. The magnetization within the domain is saturated and will always lie in the easy direction of magnetization when there is no externally applied field. The direction of the domain alignment across a large volume of material is more or less random and hence the magnetization of a specimen can be zero.

Magnetic domains exist in order to reduce the energy of the system. A uniformly magnetized specimen as shown in figure 5(a) has a large magneto static energy associated with it. This is the result of the presence of magnetic free poles at the surface of the specimen generating a demagnetizing field, H_d. From the convention adopted for the definition of the magnetic moment for a magnetic dipole the magnetization within the specimen points from the south pole to the north pole, while the direction of the magnetic field points from north to south. Therefore, the demagnetizing field is in opposition to the magnetization of the specimen. The magnitude of H_d is dependent on the geometry and magnetisation of the specimen. In general if the sample has a high length to diameter ratio (and is magnetized in the long axis) then the demagnetising field and the magnetostatic energy will be low.

The break up of the magnetisation into two domains as illustrated in figure 5(b) reduces the magnetostatic energy by half. In fact if the magnet breaks down into N domains then the magnetostatic energy is reduced by a factor of 1/N, hence figure 5(c) has a quarter of the magnetostatic energy of figure 5(a). Figure 5(d) shows a closure domain structure where the magnetostatic energy is zero, however, this is only possible for materials that do not have a strong uniaxial anisotropy, and the neighboring domains do not have to be at 180º to each other.


(a)	(b)	(c)	(d)

Figure 5: Schematic illustration of the break up of magnetisation into domains (a) single domain, (b) two domains,
(c) four domains and (d) closure domains.

The introduction of a domain raises the overall energy of the system, therefore the division into domains only continues while the reduction in magnetostatic energy is greater than the energy required to form the domain wall. The energy associated a domain wall is proportional to its area. The schematic representation of the domain wall, shown in figure 6, illustrates that the dipole moments of the atoms within the wall are not pointing in the easy direction of magnetisation and hence are in a higher energy state. In addition, the atomic dipoles within the wall are not at 180º to each other and so the exchange energy is also raised within the wall. Therefore, the domain wall energy is an intrinsic property of a material depending on the degree of magnetocrystalline anisotropy and the strength of the exchange interaction between neighboring atoms. The thickness of the wall will also vary in relation to these parameters, as strong magnetocrystalline anisotropy will favor a narrow wall, whereas a strong exchange interaction will favour a wider wall.

Figure 6: Schematic representation of a 180º domain wall.

A minimum energy can therefore be achieved with a specific number of domains within a specimen. This number of domains will depend on the size and shape of the sample (which will affect the magnetostatic energy) and the intrinsic magnetic properties of the material (which will affect the magnetostatic energy and the domain wall energy).

The microscopic ordering of electron spins characteristic of ferromagnetic materials leads to the formation of regions of magnetic alignment called domains.

The main implication of the domains is that there is already a high degree of magnetization in ferromagnetic materials within individual domains, but that in the absence of external magnetic fields those domains are randomly oriented. A modest applied magnetic field can cause a larger degree of alignment of the magnetic moments with the external field, giving a large multiplication of the applied field.

These illustrations of domains are conceptual only and not meant to give an accurate scale of the size or shape of domains. The microscopic evidence about magnetization indicates that the net magnetization of ferromagnetic materials in response to an external magnetic field may actually occur more by the growth of the domains parallel to the applied field at the expense of other domains rather than the reorientation of the domains themselves as implied in the sketch.

Some of the more direct evidence we have about domains comes from imaging of domains in single crystals of ferromagnetic materials. The sketches above are after Young and are adapted from magnified images of domain boundaries in single crystals of nickel. They suggest that the effect of external magnetic fields is to cause the domain boundaries to shift in favor of those domains which are parallel to the applied field. It is not clear how this applied to bulk magnetic materials which are polycrystalline. Keep in mind the fact that the internal magnetic fields which come from the long range ordering of the electron spins are much stronger, sometimes hundreds of times stronger, than the external magnetic fields required to produce these changes in domain alignment. The effective multiplication of the external field which can be achieved by the alignment of the domains is often expressed in terms of the relative permeability.

Domains may be made visible with the use of magnetic colloidal suspensions which concentrate along the domain boundaries. The domain boundaries can be imaged by polarized light, and also with the use of electron diffraction. Observation of domain boundary movement under the influence of applied magnetic fields has aided in the development of theoretical treatments. It has been demonstrated that the formation of domains minimizes the magnetic contribution to the free energy.

Hysteresis

When a ferromagnetic material is magnetized in one direction, it will not relax back to zero magnetization when the imposed magnetizing field is removed. It must be driven back to zero by a field in the opposite direction. If an alternating magnetic field is applied to the material, its magnetization will trace out a loop called a hysteresis loop. The lack of retraceability of the magnetization curve is the property called hysteresis and it is related to the existence of magnetic domains in the material. Once the magnetic domains are reoriented, it takes some energy to turn them back again. This property of ferromagnetic materials is useful as a magnetic "memory". Some compositions of ferromagnetic materials will retain an imposed magnetization indefinitely and are useful as "permanent magnets". The magnetic memory aspects of iron and chromium oxides make them useful in audio tape recording and for the magnetic storage of data on computer disks.

Hysteresis Loop

It is customary to plot the magnetization M of the sample as a function of the magnetic field strength H, since H is a measure of the externally applied field which drives the magnetization .

Ferrites

Ferrites are chemical compounds, ceramic with iron(III) oxide Fe₂O₃ as their principal components ^[1]. Many of them are magnetic materials and they are used to make permanent magnets, ferrite cores for transformers, and in various other applications.

Many ferrites are spinels with the formula AB₂O₄, where A and B represent various metal cations, usually including iron. Spinel ferrites usually adopt a crystal motif consisting of cubic close-packed (fcc) oxides (O²⁻) with A cations occupying one eighth of the tetrahedral holes and B cations occupying half of the octahedral holes—that is, the inverse spinel structure.

The magnetic material known as "ZnFe" has the formula ZnFe₂O₄, with Fe³⁺ occupying the octahedral sites and half of the tetrahedral sites. The remaining tetrahedral sites in this spinel are occupied by Zn²⁺.^[2]

Some ferrites have hexagonal crystal structure, e.g. barium ferrite BaO:6Fe₂O₃ or BaFe₁₂O₁₉.

Properties

Ferrites are usually non-conductive ferromagnetic ceramic compounds derived from iron oxides such as hematite (Fe₂O₃) or magnetite (Fe₃O₄) as well as oxides of other metals. Ferrites are, like most other ceramics, hard and brittle. In terms of the magnetic properties, ferrites are often classified as "soft" and "hard" which refers to their low or high coactivity of their magnetism, respectively.

Soft ferrites

Ferrites that are used in transformer or electromagnetic cores contain nickel, zinc, and/or manganese compounds. They have a low coercivity and are called

Soft ferrites. The low coercivity means the material's magnetization can easily reverse direction without dissipating much energy (hysteresis losses), while the material's high resistivity prevents eddy currents in the core, another source of energy loss. Because of their comparatively low losses at high frequencies, they are extensively used in the cores of RF transformers and inductors in applications such as switched-mode power supplies (SMPS).

The most common soft ferrites are manganese-zinc (MnZn, with the formula Mn_aZn_(1-a)Fe₂O₄) and nickel-zinc (NiZn, with the formula Ni_aZn_(1-a)Fe₂O₄). NiZn ferrites exhibit higher resistivity than MnZn, and are therefore more suitable for frequencies above 1 MHz. MnZn have in comparison higher permeability and saturation induction.

Hard ferrites

In contrast, permanent ferrite magnets are made of hard ferrites, which have a high coercivity and high remanence after magnetization. These are composed of iron and barium or strontium oxides. In a magnetically saturated state they conduct magnetic flux well and have a high magnetic permeability. This enables these so-called ceramic magnets to store stronger magnetic fields than iron itself. They are cheap, and are widely used in household products such as refrigerator magnets. The maximum magnetic field B is about 0.35 tesla and the magnetic field strength H is about 30 to 160 kilo ampere turns per meter (400 to 2000 oersteds). The density of ferrite magnets is about 5g/cm³.

Production

Ferrites are produced by heating an intimate mixture of powdered precursors pressed into a mold. During the heating process, calcinations of carbonates occurs:

MCO₃ → MO + CO₂

The oxides of barium and strontium are typically supplied as their carbonates, BaCO₃ or SrCO₃. The resulting mixture of oxides undergoes sintering. Sintering is a high temperature process similar to the firing of ceramic ware.

Afterwards, the cooled product is milled to particles smaller than 2 µm, small enough that each particle consists of a single magnetic domain. Next the powder is pressed into a shape, dried, and re-sintered. The shaping may be performed in an external magnetic field, in order to achieve a preferred orientation of the particles (anisotropy).

Small and geometrically easy shapes may be produced with dry pressing. However, in such a process small particles may agglomerate and lead to poorer magnetic properties compared to the wet pressing process. Direct calcination and sintering without re-milling is possible as well but leads to poor magnetic properties.

Electromagnets are pre-sintered as well (pre-reaction), milled and pressed. However, the sintering takes place in a specific atmosphere, for instance one with an oxygen shortage. The chemical composition and especially the structure vary strongly between the precursor and the sintered product.

To allow efficient stacking of product in the furnace during sintering and prevent parts sticking together, many manufacturers separate ware using ceramic powder separator sheets. These sheets are available in various materials such as alumina, zirconia and magnesia. They are also available in fine medium and coarse particle sizes. By matching the material and particle size to the ware being sintered, surface damage and contamination can be reduced while maximizing furnace loading.

Uses

Ferrite cores are used in electronic inductors, transformers, and electromagnets where the high electrical resistance of the ferrite leads to very low eddy current losses. They are commonly seen as a lump in a computer cable, called a ferrite bead, which helps to prevent high frequency electrical noise (radio frequency interference) from exiting or entering the equipment.

Early computer memories stored data in the residual magnetic fields of hard ferrite cores, which were assembled into arrays of core memory. Ferrite powders are used in the coatings of magnetic recording tapes. One such type of material is iron (III) oxide.

Ferrite particles are also used as a component of radar-absorbing materials or coatings used in stealth aircraft and in the absorption tiles lining the rooms used for electromagnetic compatibility measurements.

Most common radio magnets, including those used in loudspeakers, are ferrite magnets. Ferrite magnets have largely displaced Alnico magnets in these applications.

It is a common magnetic material for electromagnetic instrument pickups, because of price and relatively high output. However, such pickups lack certain sonic qualities found in other pickups, such as those that use Alnico alloys or more sophisticated magnets.

magnetic bubble

magnetic bubble memory description

Magnetic bubble memory technology has advanced considerably since the concept was introduced by Bell Telephone Laboratories in 1967. Research indicated that small cylindrical magnetic domains, which are called magnetic bubbles, can be formed in single-crystal thin films of synthetic ferrites or garnets when an external magnetic field is applied perpendicularly to the surface of the film. These bubbles can be moved laterally through the film by using a varying magnetic field. These characteristics of magnetic bubbles make them ideally suited for serial storage of data bits; the presence or absence of a bubble in a bit position is used to define the logic state. Since the diameter of a bubble is so small (as little as a tenth of a micrometer), many thousands of data bits can be stored in a single bubble-memory chip. In the spring of 1977 Texas Instruments was the first to market a 92,304-bit bubble memory. This bubble memory is much like magnetic tape or magnetic disc memory storage in that it is nonvolatile meaning that the data is retained even when power is no longer applied to the chip. Since bubble memories are a product of solid-state technology (there are no moving parts), they have higher reliability than tape or disc storage and do not require any preventive maintenance. In addition, the bubble memory is small and lightweight and is, therefore, an excellent choice for compact designs and portable applications.

functional operation of bubble memories

The basic bubble-memory package contains the bubble-memory chip, magnetic field coils, and permanent magnets as shown in Figure 1. A rotating magnetic field created by two mutually perpendicular coils causes the data in the form of magnetic bubbles to move serially through the magnetic field in a manner similar to data in a semiconductor shift register. Two permanent magnets provide non volatility and allow for the stable existence of magnetic-bubble domains. Interfacing circuits that are compatible with standard TTL devices complete the memory module to allow a convenient building-block concept for the nonvolatile memory system.

The chip is composed of a nonmagnetic crystalline substrate upon which a thin crystalline magnetic epitaxial film is grown. Only certain materials exhibit the properties necessary to form magnetic bubbles and these include orthoferrites, hexagonal ferrites, synthetic garnets, and amorphous metal films. Among these, the synthetic garnets have the best combination of the desired properties. Synthetic garnets support the formation of small magnetic bubbles that allow high-density data storage. The bubbles are highly mobile and are stable over a fairly wide range of temperatures.

The material chosen for the substrate depends on several factors. The crystalline structure should be compatible with that of the magnetic film, it should have nearly the same coefficient of expansion, and it should be nonmagnetic. The most-used garnet substrate with these properties is gadolinium gallium garnet (GGG). The magnetic film grown on this substrate has a crystalline structure that will allow the formation of magnetic domains (bubbles) in a plane perpendicular to the substrate.

Without the influence of an external magnetic field, these magnetic domains form random serpentine patterns of equal area, minimizing the total magnetic energy of the magnetic film (see Figure 2). The magnetic field of the serpentine domains tends to line up primarily along a single axis (the "easy" axis) that is perpendicular to the plane of the film. If an external magnetic field is applied, its energy tends to expand domains polarized in the direction of the field and to shrink those polarized opposite to the field until they become small cylinders embedded in a background of opposite magnetization. Viewed on end, these cylinders have the appearance of small circles or bubbles with diameters from 2 to 30 micrometers. Increasing the field further causes the bubble to collapse or to be "annihilated". The external field provides a bias that makes the bubbles stable. This bias, being a static field, can be readily provided by permanent magnets with no expenditure of power.

Before bubbles can be shifted through the magnetic film, they must be generated in accordance with input data. Bubbles are generated by locally altering the bias field with a magnetic field produced by a pulse of current through a microscopic one-turn metallized loop. This loop is located on a secondary layer immediately above the magnetic film on the surface of the chip. Given a current of the correct amplitude and polarity through the one-turn loop, a localized vertical magnetic field opposite to that of the permanent magnets is produced. This localized field establishes a domain wall inversion in the magnetic film resulting in bubble creation.

Once a bubble has been created, a method is then required to move the bubble domain along a predetermined path. This is accomplished by the deposition of chevron-shaped patterns of a soft magnetic material on the chip surface above the magnetic epitaxial film. When magnetized sequentially by a magnetic field rotating in the same plane, these chevron propagation patterns set up magnetic polarities that attract the bubble domain and establish motion. Figure 3 shows the various polarities at different positions of the rotating magnetic field. In actual practice the rotating in-plane magnetic field is implemented by applying a two-phase alternating current to the two coils shown in Figure 1.

One possible implementation for the magnetic bubble memory is a long shift register. As shown in Figure 4 the bubbles would shift under the influence of the rotating magnetic field following the path determined by the placement of chevron patterns. Even though this approach offers the simplest design and interface control, it suffers a major disadvantage of having the slowest access time. The reason for this is that after a data bit is entered or written it must circulate through

the entire shift register before it can be retrieved or read. Another problem with this single loop design is that a single fault in the shift register structure produces a defective bubble memory chip. This results in a low processing yield and a high cost to the consumer.

For these reasons TI has chosen the major-minor loop architecture, which offers a dramatic improvement in access time. As shown in Figure 5, during a write operation (data entry), data is generated one bit at a time in the major loop. The data is then transferred in parallel to the minor loops where it circulates until the next time data is to be read out of the memory.

During a write operation data are introduced into the major loop by pulses of current through the hairpin loop of the generator. The major loop is essentially a unidirectional circular shift register from which data can be transferred in parallel to the minor loops. Thus a block of data is entered in the major loop and shifted until the first data bit is aligned with the most remote minor loop. At that time, each parallel transfer element receives a current pulse that produces a localized magnetic field causing the transfer of all the bubbles in the major loop to the top bit position of the corresponding minor loop. Once data is written into the magnetic bubble memory, new data may be written only by first removing the old data by doing a destructive read. In this operation bubbles are transferred from the minor loops and annihilated by running them into the Permalloy guard rail that usually surrounds bubble devices.

During a read operation the data block to be accessed in the minor loops is rotated until it is adjacent to the major loop. At this time the data block is transferred in parallel to the major loop. The block of data is them serially shifted to the replicator where the data stream is duplicated. The duplicated data takes the path to the magneto-resistive detector element. The presence of a bubble in the detector lowers the resistance resulting in a corresponding increase in detector current, which can be detected via a sense amplifier. The original data stream remaining in the major loop is rotated and transferred back into the minor loops thus saving the data for further operations.

The magnetic-bubble-memory devices are fabricated using fine geometries that make the manufacture of perfect devices a difficult task. In order to increase production yields and achieve correspondingly lower costs, redundant minor loops on the bubble-memory chip allow some loops to be defective. Defective loops are determined at final test and a map of these loops is supplied to the end user so that the defective loops can be avoided in the final memory system. This redundancy of minor loops can be handled in several ways. The map could be written into a software program that would direct data to be stored only to the perfect minor loops, but this would require a unique software package for each memory system. Alternatively, the map could be stored in the MBM (magnetic-bubble memory) itself with some risk of being written over with new data. The recommended approach is to store the map in a programmable read-only memory (PROM). Each bit in a page of data would then be written to the MBM or read from it in accordance with the contents of the PROM, thus preventing data bits from the defective minor loops from mingling with valid data. Of course all this requires control circuitry in addition to that necessary for the timing and control of the alternating current in the field coils, the transfer of data to and from the minor loops, and the replication and detection of the magnetic bubbles.

interfacing with bubble memories

Since the magnetic-bubble memory requires accurate current pulses for the generate, replicate, and transfer operations, an interface circuit called a function driver is needed to convert the digital input control signals to the required current pulses. Also, the two field coils each require a triangular current drive 90 degrees out of phase with each other. This requirement is satisfied with another set of interface circuits (coil drivers and diode array) that is driven with digital input signals. The output signal amplitude of the MBM is relatively small, about 3 millivolts. For this to be useful in a system, the output is converted to standard TTL levels with the use of a set of interface circuits (RC networks and sense amplifiers). The block diagram in Figure 6 shows the connection of all these interface circuits as a memory module. This modular building block promotes efficient construction of mass memories.

The control and timing signals for the memory module are derived from the function-timing generator. This integrated circuit provides input timing control to the function driver, coil drivers, and sense amplifier on a per-cycle basis. The function-timing generator provides control signals to the memory module as shown in Figure 7. These signals provide control for five basic operations: generate, replicate, annihilate, transfer-in, and transfer-out. The function-timing generator also initiates the rotating magnetic field and precisely synchronizes the timing of other control signals with this field.

The time at which a particular data bit is detected in the MBM may not exactly match the time at which it is needed in the system. The sense amplifier not only increases the voltage level of the detected data, but also provides temporary storage of the data bits in a circuit called a D-type flip-flop. The sense amplifier receives a control input from the function timing generator to transfer the detected data into the internal flip-flop. In addition, the function-timing generator provides the control signals necessary to put the existing data in a known position during a power shut down. When the system is turned on again, the stored data can then be accurately located and retrieved.

In a typical system the major computing and data processing is done by a microprocessor. To provide a convenient interface from the microprocessor to the MBM system, a custom controller is needed for the read, write, and memory-addressing operations. The TMS5502/TMS9916 MBM controller responds to commands from the microprocessor system and sends control signals to the function timing generator necessary to access a page (or pages) of data. The controller maintains page-position information, handles serial-parallel data conversion between the bubble memory and the microprocessor, and generates the control signals to the function-timing generator to perform read and write operations while handling the redundancy of the minor loops.

Advantages of bubble memories

The future growth of distributed process systems will be greatly impacted by magnetic-bubble memories. These microprocessor-based systems demand high-density mass storage at low cost. Magnetic-bubble memories satisfy all of these requirements with definite advantages over the existing magnetic storage technologies. MBM's advantages over moving-head disks or floppy disks are low access time (the time ncessary to retrieve the desired data), small physical size, low user entry cost, no maintenance, and higher reliability.

The advantages of MBM's over random-access memories (RAM's) are nonvolatility, potentially lower price per bit, and more bits per chip. The RAM has the advantage of much better access time, higher transfer rate, and simpler interfacing.

In summary, the main MBM advantages are the low wnetry price versus disks for the low-end user, nonvolatility versus semiconductor memories, and high-density storage in a small physical space. Because magnetic bubble memories are a solid-state, nonvolatile technology, they are ideally suited for portable applications as well as providing memory for traditional processing systems. Industrial applications include memory for numerical control machines and various types of process control. Solid-state bubble memories are more reliable in harsh environments; they are affected much less by shock, vibration, dirt, and dust than electromechanical magnetic memories. Innovative new products include data terminals, calculators, word processing, voice storage, and measurement equipment.

A typical bubble memory circuit, from a datasheet later in the publication.

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Friday, 27 April 2012