Band Theory of Solid

Objectives • Effective Mass of electron• Concept of Holes• Energy Band Structure of Solids:

Conductors, Insulators and Semiconductors• Semiconductors

Intrinsic and Extrinsic Semiconductors• Type of diodes

Simple Diode Zener Diode

Effective Mass of electronAn electron moving in the solid under the

influence of the crystal potential is subjected to an electric field.

We expect an external field to accelerate the electron, increasing E and k and change the electron’s state.

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Concept of HolesConsider a semiconductor with a small number of

electrons excited from the valence band into the conduction band.

If an electric field is applied,

• the conduction band electrons will participate in the electrical current• the valence band electrons can “move into”

the empty states, and thus can also contribute to the current.

Concept of Holes

If we describe such changes via “movement” of the “empty” states – the picture will be significantly simplified. This “empty space” is called a Hole.

“Deficiency” of negative charge can be treated as a positive charge.

Holes act as charge carriers in the sense that electrons from nearby sites can “move” into the hole.

Holes are usually heavier than electrons since they depict collective behavior of many electrons.

Electrical current for holes and electrons in the same direction

• To understand hole motion, one requires another view of the holes, which represent them as electrons with negative effective mass m*.

• For example the movement of the hole think of a row of chairs occupied by people with one chair empty, and to move all people rise all together and move in one direction, so the empty spot moves in the same direction

Energy Band Structure of Solids Conductor, Semiconductor and Insulator

In isolated atoms the electrons are arranged in energy levels.

Energy Band in SolidThe following are the important energy

band in solids: Valence band Conduction band Forbidden energy gap or Forbidden band

Valance band

The band of energy occupied by the valance

electrons is called valence band. The electrons in the

outermost orbit of an atom are known as valance

electrons. This band may be completely or partial filled.

Electron can be move from one valance band to

the conduction band by the application of external

energy.

Conduction band

The band of energy occupied by the conduction

electrons is called conduction band. This is the

uppermost band and all electrons in the conduction

band are free electrons.

The conduction band is empty for insulator and

partially filled for conductors.

Forbidden Energy Gap or Forbidden band

The gap between the valance band and conduction

band on energy level diagram known as forbidden band

or energy gap.

Electron are never found in the gap. Electrons may

jump from back and forth from the bottom of valance

band to the top of the conduction band. But they never

come to rest in the forbidden band.

According to the classical free electron theory, materials are classified in to three types:

Conductors

Semiconductors

Insulators

Conductors

There is no forbidden gap and the conduction band and valence band are overlapping each other between and hence electrons are free to move about. Examples are Ag, Cu, Fe, Al, Pb …. Conductor are highly electrical conductivity. So, in general electrical resistivity of conductor is very low and

it is of the order of 10-6 Ω cm. Due to the absence of the forbidden gap, there is no structure

for holes. The total current in conductor is simply a flow of electrons. For conductors, the energy gap is of the order of 0.01 eV.

Semiconductors

Semiconductors are materials whose electrical resistivity lies between insulator and conductor. Examples are silicon (Si), germanium (Ge) ….

The resistivity of semiconductors lie between 10-4 Ω cm to 103 Ω cm at room temperature.

At low temperature, the valence band is all most full and conduction band is almost empty. The forbidden gap is very small equal to 1 eV.

Semiconductor behaves like an insulator at low temperature. The most commonly used semiconductor is silicon and its band gap is 1.21 eV and germanium band gap is 0.785 eV.

When a conductor is heated its resistance increases; The atoms vibrate more and the electrons find it more difficult to move through the conductor but, in a semiconductor the resistance decreases with an increase in temperature. Electrons can be excited up to the conduction band and Conductivity increases.

InsulatorsIn insulator, the valence band is full but the

conduction band is totally empty. So, free electrons from conduction band is not available. In insulator the energy gap between the valence and

conduction band is very large and its approximately equal to 5 eV or more.

Hence electrons cannot jump from valence band to the conduction band. So, a very high energy is required to push the electrons to the conduction band.

The electrical conductivity is extremely small. The resistivity of insulator lie between 103 to 1017 Ωm, at

the room temperature Examples are plastics, paper …..

Types of semiconductorsSemiconductors

Intrinsic Semiconductor Extrinsic Semiconductor

p - type n - type

Intrinsic Semiconductor The intrinsic semiconductor are pure semiconductor materials. These semiconductors posses poor conductivity. The elemental and compound semiconductor can be intrinsic

type. The energy gap in semiconductor is very small. So even at the room temperature, some of electrons from

valance band can jump to the conduction band by thermal energy.

The jump of electron in conduction band adds one conduction electron in conduction band and creates a hole in the valence band. The process is called as “generation of an electron–hole pair”.

In pure semiconductor the no. of electrons in conduction band and holes in holes in valence bands are equal.

Extrinsic Semiconductor

Extrinsic semiconductor is an impure semiconductor formed from an intrinsic semiconductor by adding a small quantity of impurity atoms called dopants.

The process of adding impurities to the semiconductor crystal is known as doping.

This added impurity is very small of the order of one atom per million atoms of pure semiconductor.

Depending upon the type of impurity added the extrinsic semiconductors are classified as:

(1) p – type semiconductor(2) n – type semiconductor

The application of band theory to n-type and p-type semiconductors shows that extra levels have been added by the impurities.

In n-type material there are electron energy levels near the top of the band gap so that they can be easily excited into the conduction band.

In p-type material, extra holes in the band gap allow excitation of valence band electrons, leaving mobile holes in the valence band.

p – type semiconductor

The addition of trivalent impurities such as boron, aluminum or gallium to an intrinsic semiconductor creates deficiencies of valence electrons,called "holes". It is typical to use B2H6 diborane gas to diffuse boron into the silicon material.

n – type semiconductor

The addition of pentavalent impurities such as antimony, arsenic or phosphorous contributes free electrons, greatly increasing the conductivity of the intrinsic semiconductor. Phosphorous may be added by diffusion of phosphine gas (PH3).

Simple Diode (p n- junction Diode) The two terminals are called Anode and Cathode. At the instant the two materials are “joined”, electrons

and holes near the junction cross over and combine with each other.

Holes cross from P-side to N-side and Free electrons cross from N-side to P-side.

At P-side of junction, negative ions are formed. At N-side of junction, positive ions are formed.

Depletion region is the region having no free carriers.

Further movement of electrons and holes across the junction stops due to formation of depletion region.

Depletion region acts as barrier opposing further diffusion of charge carriers. So diffusion stops within no time.

Current through the diode under no-bias condition is zero.

Reverse bias

Positive of battery connected to n-type material (cathode).

Negative of battery connected to p-type material (anode).

Reverse bias…..

Free electrons in n-region are drawn towards positive of battery, Holes in p-region are drawn towards negative of battery.

Depletion region widens, barrier increases for the flow of majority carriers.

Majority charge carrier flow reduces to zero. Minority charge carriers generated thermally can cross

the junction – results in a current called “reverse saturation current” Is , Is is in micro or nano amperes or less. Is does not increase “significantly” with increase in the reverse bias voltage

Forward bias Positive of battery connected to p-type (anode)

Negative of battery connected to n-type (cathode)

Forward bias…

Electrons in n-type are forced to recombine with positive ions near the boundary, similarly holes in p-type are forced to recombine with negative ions.

Depletion region width reduces. An electron in n-region “sees” a reduced barrier at the

junction and strong attraction for positive potential. As forward bias is increased, depletion region narrows

down and finally disappears – leads to exponential rise in current.

Forward current is measured in milli amperes

Zener Diode

A diode which is heavily doped and which operates in the reverse breakdown region with a sharp breakdown voltage is called a Zener diode.

This is similar to the normal diode except that the line (bar) representing the cathode is bent at both side ends like the letter Z for Zener diode.

In simple diode the doping is light; as a result, the breakdown voltage is high and not sharp. But if doping is made heavy, then the depletion layers becomes very narrow and even the breakdown voltage gets reduced to a sharp value.

Working Principle

The reverse breakdown of a Zener diode may occur either due to Zener effect or avalanche effect. But the Zener diode is primarily depends on Zener effect for its working.

When the electrical field across the junction is high due to the applied voltage, the Zener breakdown occurs because of breaking of covalent bonds and produces a large number of electrons and holes which constitute a steep rise in the reverse saturation current (Zener current IZ). This effect is called as Zener effect.

Zener current IZ is independent of the applied voltage and depends only on the external resistance.

I-V characteristic of a Zener diode

The forward characteristic is simply that of an ordinary forward biased junction diode. Under the reverse bias condition, the breakdown of a junction occurs.

Its depends upon amount of doping. It can be seen from above figure as the reverse voltage is increased the reverse current remains negligibly small up to the knee point (K) of the curve.

At point K, the effect of breakdown process beings. The voltage corresponding to the point K in figure is called the Zener breakdown voltage or simply Zener voltage (VZ), which is very sharp compared to a simple p-n junction diode. Beyond this voltage the reverse current (IZ) increases sharply to a high value.

The Zener diode is not immediately burnt just because it has entered the breakdown region.

The Zener voltage VZ remains constant even when Zener current IZ increases greatly.

The maximum value of current is denoted by IZ

max and the minimum current to sustain breakdown is denoted by IZ min. By two points A and B on the reverse VI characteristic, the Zener resistance is given by the relation, rz = ( Δ VZ / Δ IZ) -----(1)

Zener diode Applications:

I. Zener diodes are used as a voltage regulator.II. They are used in shaping circuits as peak

limiters or clippers.III. They are used as a fixed reference voltage in

transistor biasing and for comparison purpose.IV. They are used for meter protection against

damage from accidental application of excessive voltage.