UNIT-IV & NANOMATERIALS 1. Free electron model 2. Kronig penny model 3. Effective mass 4. for intrinsic and extrinsic semiconductor 5. P-N junction 6. Zener diode 7. photo diode 8. solar cell 9. 10. elementary idea of nanostructures and nano materials

Free electron model: In order to explain the electrical and thermal properties of in the year 1900, H. A. Lorentz and Paul Drude propounded the free electron model. They made the following postulates: 1. The outermost electrons (or the valence electrons) of the constituent atoms of a are most weakly bound with the atoms. Hence these electrons get separated from their atoms and move freely inside the entire substance. These electrons are called the free electrons or the conduction electrons. 2. There is large no. of free electrons inside a metal and they behave like the molecules of a gas enclosed in a vessel. Hence they can also be referred as free electron gas. These free electrons are responsible for the thermal and electrical conduction of metal.

1 3. The free electrons in thermal equilibrium obey the Maxwell- Boltzmann statistics. Each electron has 3 degrees of freedom for translational motion. 4. According to Maxwell-Boltzmann statistics, mean energy 1 kT per electron at an absolute temperature T, is 2 , where k is the Boltzmann’s constant. 5. Inside the metal, the free electrons move randomly with a high speed and their speed depends on the temperature of the metal. During motion, when they collide with the positive their speed and direction of motion change such that the rate of flow of electrons in a particular direction is zero. 6. When the metal is kept in an external electric field, the free electrons get accelerated in a direction opposite to the external electric field but due to collisions with the positive ions of the metal they begin to move with a constant velocity which is called drift velocity. It is of the order of 10-4 m/s. Success: On the basis of above model, the electrical and of a metal can be explained successfully. From this model we find that electrical conductivity of metal decreases with increase in their temperature and the ratio of thermal to electrical conductivity of each metal at any temperature is constant. Failure: Although this theory explained some of the properties of the metals yet it was unable to explain following features: 1. It could not explain the variation of electron specific heat with temperature at low temperature. According to this model the electron specific heat for metal is 3R which is 2 temperature independent but experimentally it was found that specific heat of metals is temperature dependent. 2. According to free electron model the magnetic susceptibility of a paramagnetic substance is inversely proportional to temperature but experimentally it was found temperature independent. 3. The calculated on the basis of free electron model was ten times less than the experimentally calculated value. 4. It was unable to explain the behaviour of and insulators. 5. It can not explain the origin of Pauli’s paramegnetism of metals. 6. Monovalent metals (Cu, Ag) have been found to have higher electrical conductivity than divalent (Cd, Zn) and trivalent (Al, In) metals. If the conductivity is proportional to the electron concentration than monovalent metals should have lesser electrical conductivity compared to the divalent and trivalent metals. Sommerfeld’s Free Electron Theory: Sommerfeld modified the Drude Lorentz free electron model on the basis of quantum statistics. He made the following assumptions: 1. According to Sommerfeld, each free electron inside the metal experiences an electrostatic attractive force due to all the positive ions and an electrostatic repulsive force due to the other electrons. 2. The force of repulsion due to mutual interactions of electrons can be assumed to be negligible and the attractive field due to positive ions can be considered to be uniform

3 everywhere inside the crystal. Thus each free electron inside the metal is in an attractive potential field. 3. Since the crystal structure of is periodic i.e., in a solid crystal, each positive is at a definite distance from each other, therefore this potential field inside the metal must also be periodic. But for convenience Sommerfeld assumed that this potential inside the metal is constant. 4. Since no electron is emitted from the metal at an ordinary temperature, therefore it can be assumed that the electron inside the metal is more stable than outside the metal i.e., the potential energy of a stationary electron inside the metal is less than the potential energy outside the metal. 5. Thus inside the metal electron can be assumed to be inside a potential well of depth Es. Es is the difference in the energy of the electron outside and inside the well.

Metal Ø

E s

EF

Potential Well 6. The electrons present inside this well has so much energy that it can move inside the well, but it cannot come out of the metal surface. 7. Inside the well, all the energy states from zero to some energy EF are filled up with electrons. The energy EF is

4 called the . Thus Fermi energy is the maximum kinetic energy of the electrons inside the metal. 8. The threshold energy or the work function for an electron

inside the well is   Es  EF to cross this well. 9. He further specified the energy distribution function for an 1 f (E)  α electron viz. 1 e (E / kT ) .Substituting the value of e 1 EF kT e f (E)  (EE ) / kT as distribution function becomes 1 e F . 10. The Fermi function predicts below EF all the energy levels are completely occupied by the electrons. Merits and demerits of Sommerfeld’s model: This model successfully explained several properties of metals such as electrical conductivity, thermal conductivity, specific heat and magnetic susceptibility. But this model was unable to distinguish between behaviour of metals, semiconductors and insulators. Band Model: In order to remove the drawbacks of the Sommerfeld’s free electron model Band Model was propounded. Following features were discussed in this model: 1. According to this model the free electrons inside the metal moves in the electric field of positive ions and of other free electrons. 2. Since the crystal structure is periodic the potential energy of free electrons also changes periodically with distance hence the motion of the electron inside the metal is in the periodic potential well.

5 3. The potential energy of a free electron at a distance x in the  Ze2 potential field of an atom i is given as U (x)  and the 4 0 x graph plotted for it is a hyperbola.

U (x) x

4. Since inside the metal the atoms are arranged in a definite order, therefore we obtain the influence of the other atoms also on the potential energy curve and it’s combined effect can be viewed as:

U(x) x

i j

6 5. All the electrons from the lowest energy to Eb are bound with their atoms and can vibrate only with very small amplitude. 6. The electrons with energy between Eb and EF move anywhere within the metal, where EF is the Fermi energy level. The work function for which can be given as

  Es  EF where ES is the depth of the potential well.

U(x) x 0 Φ ES

EF Eb i j k l m n a

7. The electron is associated with the entire crystal, and not only with an atom.

Kronig-Penny Model: To explain the behaviour of electrons in the periodic potential, Kronig and Penny gave a simple one dimensional model according to which the potential energy of an electron can be represented by a periodic array of rectangular potential well as shown in the fig. below:

7 V(x)

V0

i j -b 0 a a+b 2a+b x

Here the potential peaks obtained from the hyperbolic curves have been assumed to be in form of rectangular peaks. Each potential well represents the potential near an atom. If the time period of potential is (a + b), then potential energy is zero in

0  x  a and potential energy is constant (= V0) in  b  x  0 , i.e.

In region 0  x  a,V(x)  0

And in region  b  x  0,V(x) V0 (constant) In both these regions, the Schrödinger wave equations for the th wave function ψn associated with n energy state of En electron are, d 2 2m 0  x  a, n  E   0 V  0 In region dx 2  2 n n (x) 2 d  n 2m  b  x  0,  (En V0) n  0 V  V and in region dx2 2 (x) 0 ……….(1) Here the energy of electron En is very small in comparison to the potential V0.

8 Assuming that as b tends to zero, V0 becomes infinite, Kronig and Penny obtained the following condition for the allowed wave function on solving the above equation: 2 (mV0b/  )sina  cosa  cos ka…….(2) ;  2mEn /  and k is the wave vector. 2 If P  mV0ba / , which measures the area V0b of the potential barrier, then increase in P means increase in the binding energy of electron with its potential well. Substituting the value of P in above eq. the condition for the allowed wave function is Psina  cosa  cos ka a …………..(3) 3 P  Hence for 2 the graph between the quantity on the left side of above eq and a is shown below:

 Psina    cosa  a 

d’ c’ b’ a’ +1 a b c d 3    3  4 s’ r’ 2 q’ p’ 0 p q 2 r s 4 αa h’ g’ e’ f’ -1 e f g h

The graph depicts the following facts:

9 1. Since the maximum and minimum possible values of the term cos a  are +1 and -1 hence two horizontal lines are drawn on the Y-axis at y = +1and at y = -1. 2. Solution of the eq.(3) can be given by the intersection points a, b, c,….and a’, b’, c’…….which means that the solution of the eq.(3) can be possible only in some specific regions. 3. The ranges π to q, 2π to r,……represents the forbidden energy gap. Thus the Kronig –Penny model, we get the following conclusions: 1. In the energy spectrum of metals there exist several bands separated by the forbidden energy region. The energy band completely filled with electrons are called the valence band and the energy band which is either completely empty or is partially filled is called the conduction band.

n Conduction band i

y g r l e a t n Forbidden energy gap s e

y r n c o r t c

e Valence band l E

2. As the value of αa increases, the width of the allowed energy bands increases. 3. With increase in the binding energy V0 of electrons or with increase in the value of P, the width of a particular allowed energy band decreases and when the binding energy becomes infinite, the allowed energy becomes very narrow i.e. the energy spectrum becomes the line spectra. In other

10 words at P = ∞ energy levels become discrete while at P = 0 the energy levels become continuous. E

u

t E2

Forbidden s energy gap r E1

q p 0 0.4 1 0 P/4π 4π/P

4. At the wave vector k  n / a the energy is discontinuous and these values of k correspond to the boundaries of Brillouin Zones. For n  1we get the first Brillouin zone. 5. The energy in an energy band is a periodic function of k. 6. The no. of total possible wave functions in an energy band is equal to the no. of unit cells. 7. The velocity of free electron is zero at the top and bottom of an energy band and it is maximum at the point of inflexion of energy band. 8. At T  0k , the effective no. of electrons in a completely filled band is zero while the effective no. of electron is max. in a band filled upto the point of inflexion. At absolute

11 zero the energy level completely filled by the electron is called the Fermi level

En

Es

F P EF P

   -k1 0 +k1  a a

Effective mass: 1. In solid state physics, a particle's effective mass is the mass it seems to carry in the semi classical model of transport in a crystal. 2. It can be shown that, under most conditions, electrons and holes in a crystal respond to electric and magnetic fields almost as if they were free particles in a vacuum, but with a different mass. 3. This mass is usually stated in units of the ordinary mass of -31 an electron me (9.11×10 kg).This experimentally determined electron mass is called the effective mass m* 4. The cause for deviation of the effective mass from the free electron mass is due to the interactions between the drifting electrons and the atoms in a solid.

12 Expression for effective mass: According to wave mechanics, the velocity of electron corresponding to the wave vector k is equal to the group velocity of waves representing it i.e. d   dk , where ω is the angular frequency of the de-broglie waves. dE E   d  If E is the energy of electron, then or  hence, 1 dE    dk ……….(1) Let there be only one electron initially in the k state in the first Brillouin zone. Now if an external electric field ε is applied on the electron for a very short duration dt, then displacement of electron in time dt will be dt and force on electron due to the electric field will be e . Hence the gain in the energy of electron dE  force x displacement e dE dE  edt  dt or  dk  dE  e dE  dk  dt or  dk   dk dk e  or rate of change in wave vector dt  ……………(2) d 1 d 2 E dk a   . from (1),acceleration of electron dt  dk 2 dt Substituting the value of dk/dt from (2) we get ,

13 e d 2E a   dk 2 ……….(3) comparing the above equation with Newton’s second law we conclude that the proportionality factor may be regarded as mass and is known as effective mass m* .Thus the effective mass of  m*  2 the electron m* is d E ………..(4) dk 2 The effective mass of the electron can be determined with the help of graph plotted between the energy E and wave vector k. We find that up to E < EF, i.e. in the lower half part of the energy band the value of m* is positive and in the upper half of part of the band E > EF the value of m* is negative. At the points of inflexion i.e. at E=EF the value of m* becomes infinite. An electron with negative effective mass is called an and electron-hole pair is called an exiton.

E

E

  kx a a

Allowed Bands

14

dE m dk

  k    x  a a a a

Semiconductors: Semiconductors are the materials whose conductivity lies between conductors and insulators. According to Band Theory they are characterized by a narrow energy band gap(Eg~1eV)

15 1. The Fermi energy EF is midway between the valence band and the conduction band. 2. At T=0, the valence band is filled and the conduction band is empty 3. However for semiconductors the band gap energy is relatively small (1-2eV) so appreciable numbers of electrons can be thermally excited into the conduction band 4. Hence the electrical conductivity of semiconductors is poor at low T but increases rapidly with temperature. Semiconductors can be classified into two categories: 1. Elemental Semiconductors 2. Compound Semiconductors Elemental Semiconductors: Chemically pure semiconductors are known as elemental or intrinsic semiconductors. Pure Ge and Si are well known examples of elemental semiconductors. They are tetravalent and have four electrons in the outermost orbit of the atom.

16 There are two types of charge carriers in semiconductors: electrons in conduction band and holes and valence band. All charge carriers, electrons and holes are thermally generated. Electrons and holes are equal in numbers because they are always formed as electron hole pairs and they are evenly distributed throughout the crystal. The behaviour of intrinsic semiconductors with temperature can be studied: 1. at 0 k:  At 0 k, all valence electrons are strongly bounded to their atoms and they spend most of the time between neighbouring atoms  It takes large energy to force an electron out of the bond.  Therefore there are no free electrons drifting about within the material at a temperature of absolute zero.  Because of this semiconductors at 0 k cannot conduct electricity.

E

Si Si Si Conduction Band

Ec

Si Si Si EF Eg

Ev

Valence Band Si Si Si

Distance

17 2. at room temperature:  Thermal energy can dislodge some electrons from their bonds.  Whenever a covalent bond is ruptured by thermal energy, a valence electron becomes free.  These electrons make transitions from valence band to conduction band after acquiring thermal energy.  Simultaneous to the generation a free electron, an empty space known as hole arises in the valence band.  These thermally generated electron-hole pair causes electrical conduction in intrinsic semiconductors.

T>0 k E

Si Si Si free Broken Ec Covalent electron Bond EF Eg Si Si Si Ev Vacancy

Si Si Si Distance

Compound Semiconductors: Two group IV elements, III-V group elements or II-VI group elements have average of four; hence they also show the semiconductor properties and are better known as the compound semiconductors, e.g. IV-IV elements: SiC III-V elements: GaP, GaAs, InAs

18 II-VI elements: ZnS, CdS, CdSe, CdTe Extrinsic Semiconductors: If a small amount of pentavalent or trivalent impurity is added into a pure semiconductor crystal, then the conductivity of the crystal increases appreciably and the crystal is known as extrinsic semiconductor. The process of adding impurity is known as doping and the impurity element is known as dopant. They can be classified into two categories: 1. P- type semiconductors 2. N-type semiconductors P- type semiconductors: 1. when a trivalent ( boron, , gallium or indium) atom replaces a Ge (or Si) atom in a crystal lattice only three valence electrons are available to form covalent bonds with the neighbouring Ge (or Si) atoms. 2. This results into an empty space known as hole. 3. When a voltage is applied this vacancy is filled by the electron bound to the neighbouring Ge (or Si) atom thereby creating new vacancy there. 4. This process continues and hole moves in the crystal lattice. 5. The conduction mechanism in these semiconductors with acceptor impurities is predominated by positive carriers which are introduced into valence band. This type of semiconductors is known as p-type semiconductors.

19

Conduction Band

Ec

Acceptor E Level A Eg Ev Valence Band

6. In p-type semiconductor the holes are the ‘majority carriers’ and the few electrons thermally excited from the valence band into the conduction band are ‘minority carriers’.

N- Type semiconductors: 1. When a small amount of pentavalent (antimony, phosphorous or arsenic) atom is added to Ge (or Si) four of these valence electrons form bonds with the neighbouring Ge(or Si) atoms. 2. The fifth electron is loosely bound. 3. At room temperature this extra electron becomes disassociated from its atom and move through the crystal as a conduction electron when a voltage is applied to the crystal. 4. This extra electron is called ‘donor’ and the crystal is known as n-type semiconductor. 5. The impurity atoms introduce discrete energy levels for the electrons just below the conduction band, called the donor levels. 20 6. In n-type semiconductor majority carriers are electrons while the minority carriers are holes formed due to thermally ruptured covalent bonds.

Conduction Band

Ec Eg ED Donor Level

Ev

Valence Band

Fermi Level: 1. "Fermi level" is the term used to describe the top of the collection of electron energy levels at absolute zero temperature. 2. This concept comes from Fermi-Dirac statistics. Electrons are and by the Pauli exclusion principle cannot exist in identical energy states. 3. So at absolute zero they pack into the lowest available energy states and build up a "Fermi sea" of electron energy states. 4. The Fermi level is the surface of that sea at absolute zero where no electrons will have enough energy to rise above the surface.

21 5. The concept of the Fermi energy is a crucially important concept for the understanding of the electrical and thermal properties of . 6. Both ordinary electrical and thermal processes involve energies of a small fraction of an electron volt. But the Fermi energies of metals are of the order of electron volts. 7. This implies that the vast majority of the electrons cannot receive energy from those processes because there are no available energy states for them to go to within a fraction of an electron volt of their present energy. 8. Limited to a tiny depth of energy, these interactions are limited to "ripples on the Fermi Sea. At higher temperatures a certain fraction, characterized by the Fermi function, will exist above the Fermi level. 9. The Fermi level plays an important role in the band theory of solids. In doped semiconductors, p-type and n-type, the Fermi level is shifted by the impurities, illustrated by their band gaps. The Fermi level is referred to as the electron in other contexts.

22

Fermi Dirac Distribution Function: The Fermi function f(E) gives the probability that a given available electron energy state will be occupied at a given temperature. The Fermi function comes from Fermi-Dirac statistics and has the form

The basic nature of this function dictates that at ordinary temperatures, most of the levels up to the Fermi level EF are filled, and relatively few electrons have energies above the Fermi level. The illustration below shows the implications of the Fermi function for the electrical conductivity of a semiconductor.

The band theory of solids gives the picture that there is a sizable gap between the Fermi level and the conduction band of the semiconductor. At higher temperatures, a larger fraction of the

23 electrons can bridge this gap and participate in electrical conduction. Although the Fermi function has a finite value in the gap, there is no electron population at those energies. The population depends upon the product of the Fermi function and the electron . So in the gap there are no electrons because the density of states is zero. In the conduction band at 0K, there are no electrons even though there are plenty of available states, but the Fermi function is zero. At high temperatures, both the density of states and the Fermi function have finite values in the conduction band, so there is a finite conducting population.

Fermi Level in Intrinsic semiconductors:

E Conduction Band

Ec

EF Eg

Ev Valence Band

Distance

Concentration of electron in Conduction- Band: 1. In conduction band the electrons are free to move anywhere as a free particle of effective mass me*.

24 2. The electrons in conduction band per unit volume lying between energy E and E+dE can be calculated as:  n   Z(E)F(E)dE EC Where EC is the energy at the bottom of the conduction band and Z(E) is the density of states at the bottom of the conduction band and have the value, 4 3 1 Z(E)  (2m*) 2 (E  E ) 2 h3 e C , for E > EC F(E) is the Fermi-Dirac function and is given by, 1 F(E)  1 e(EEF ) / kT  3 1 4 * 2 2 1 n  (2me ) (E  EC ) dE  h3 1 e(EEF )/ kT EC

if E  EF   kT then 1 in the denominator of F(E) is negligible, i.e.  3 1 4 * 2 2 1 n  (2me ) (E  EC ) dE  h3 e(EEF )/ kT EC  4 3 1 n  (2m* ) 2 (E  E ) 2 e(EEF )/ kT dE h3 e  C EC  4 3 1 n  (2m* ) 2 (E  E ) 2 e(EEC EC EF )/ kT dE h3 e  C EC  4 3 1 n  (2m* ) 2 e(EC EF )/ kT (E  E ) 2 e(EEC )/ kT dE h3 e  C EC The integral in above eq. is of the standard form which has a solution of the following form,

25   x1/ 2eaxdx  1  3/ 2 where a  0 2a kT   4 * 3/ 2 (EC EF )/ kT  3/ 2 n  3 (2me ) e  (kT)  h  2  3/ 2 2m*kT n  2 e e(EC EF ) / kT or,  2  ,  h  3/ 2 2m*kT N  2 e putting C  2  we get  h  n  N e(EC EF )/ kT C , where NC is known as effective density in conduction band. Concentration of holes in Valence-Band: If F(E) is the probability for the occupancy of an energy state E by an electron, then the probability that the energy state is vacant is given by [1- F(E)]. Since a hole represents a vacant state in valence band, the probability for occupancy of a state at E by a hole is equal to the probability of absence of electron at that state. The hole density in the valence band is therefore given by,

EV p   Z(E)[1 F(E)]dE  1 1 [1 F(E)] 1 (EE ) / kT  (E E) / kT E  E  kT 1 e F 1 e F if F then the exponential factor in the denominator of above eq. will be much grater than unity hence for all values of E in the valence band, [1 F(E)]  e(EF E)/ kT

26 EV (E E)/ kT hence, p   Z(E)e F dE  4 3 1 Z(E)  (2m* ) 2 (E  E) 2 h3 h V E V 4 3 1 p  (2m* ) 2 (E  E) 2 e(EF E)/ kT dE  3 h V  h E V 4 3 1 p  (2m* ) 2 (E  E) 2 e(EF EV EV E)/ kT dE  3 h V  h E 4 3 V 1 p  (2m* ) 2 e(EF EV )/ kT (E  E) 2 e(EV E)/ kT dE 3 h  V h  3/ 2 2m*kT p  2 h e(EV EF ) / kT  2   h  * 3/ 2 2mhkT (EV EF ) / kT N  2 or, p  NV e where V  2  is known as  h  effective density of states in the valence band. Fermi level in intrinsic semiconductor: In an intrinsic semiconductor, the free electron and hole concentrations are equal, i.e. n  p

(EC EF ) / kT (EV EF ) / kT NCe  NV e taking log on both sides and rearranging the terms, we get  (E  E ) N (E  E ) C F  ln V  F V kT NC kT N  (E  E )  kTln V  (E  E ) C F F V NC N 2E  kTln V  E  E F C V NC 27 1 N (E  E ) E  kTln V  C V F 2 NC 2 substituting the values of NV and NC we get, * 3 mh (EC  EV ) (EC  EV ) EF  kTln  * * E  * if mh  me then F 4 me 2 2 i.e. Fermi level is mid-way between the valence band and conduction band ,in intrinsic semiconductor. Fermi level in extrinsic semiconductor: The Fermi level of an extrinsic semiconductor is determined by the intrinsic properties of the material and the dopant levels. This includes a function with three terms.

EF = EF (Intrinsic)+EF (Impurities) E  E 3 m*  N  N   C V  kTln p  kTsinh 1 D A  *   2 4 mn  2ni  Term (1) is the centre of the band gap. Term (2) is the effect of unequal hole and electron effective masses. Term (3) is the effect of the donor and acceptor atoms. At low, but not too low, temperature the extrinsic term dominates. At higher temperature the concentration of intrinsic carrier’s increases and the intrinsic terms dominate. That is, the -1 sinh term tends to zero because ni gets large.

28

With this approximations for n type semiconductors we have  N  N  E  E (Intrinsic)  kTln D A  F F    ni  and for p type semiconductors  N  N  E  E (Intrinsic)  kTln A D  F F    ni  P-N junctions: A semiconductor device can be defined as a unit which consists, partly or wholly, of semiconductor materials and which can perform useful functions in electronic apparatus, e.g. p-n junction diode, transistor etc. P-N junction is a system of two semiconductors in physical contact, one with excess of electrons (n- type) and the other with excess of holes (p- type). There are three methods for preparing p-n junction:

29 1. The grown junction method 2. The alloying method 3. The method. In the grown junction method, a crystal of semiconductor is grown, for example from an initially n-type melt, which while the crystal is being formed, is counter doped by adding enough acceptor impurities so that the subsequently grown portion of the crystal is p- type. But this technique involves the difficulty of locating the junction in the grown crystal and the difficulty of attaching leads to the narrow grown junction regions and thus it is not adopted frequently. This process generally produces graded p-n junctions. In the alloying method an alloy pellet or foil is melted upon a semiconductor base crystal, which contains impurities that give rise to the conductivity type opposite to that of the original crystal. After melting it the temperature is raised to high value such that the molten alloy dissolves away the underlying semiconductor to some extent. The liquid phase which we get then is cooled slowly. Then some of the dissolved semiconductor atoms re-crystallize at the liquid solid interface. Since the liquid phase contains impurities associated with the conductivity type opposite to that of the original crystal, the re- grown material will be of the opposite conductivity and there will be a p-n junction at the interface between original and re- grown material. But this process generally produces abrupt p-n junctions. But because of ease and simplicity this method is quite popular. On the other hand in diffusion method, an impurity atom is diffused at an elevated temperature into a base crystal whose

30 conductivity type is opposite to that which is produced by the presence of the diffusing impurity in the crystal lattice. Diffused junctions can be either graded or abrupt depending upon the diffusion time. This method is also popular because of its simplicity. p-n junction characteristics: Following features can be observed in a p-n junction:

p-type n-type + - - + - - + + - + - + - - - + - + + + + - + - - + - + -+ Concentration of Concentration of acceptors holes

y t

i s

n +

e

d

e Distance

g _

r a

h Depletion region

C

Negative space Positive space charge charge

e

g

a t

l o

v V2 VB -X X 1 0 2 V 1

31

1. On each side of the junction there exist free carriers which were shown in the fig. by un-circled charges. These free charges, holes on the p-side and electrons on the n-side, are present because of the thermal ionisation of the donors and acceptors. 2. Along with the free charges there exist ionised impurities, which can not move shown encircled in the fig. The acceptors impurities on the p-side are negatively charged. On the other hand the donor impurities on the n-side are positively charged as they have supplied electrons to the conduction band. 3. Due to thermal agitation the holes from the p-side diffuse over to the n-side and electrons from the n-side diffuse into the p- region. 4. After the formation of junction the charges on both side penetrates the boundary region to get recombined. 5. Away from the boundary charged impurity atoms get neutralised by the space charge of the free carriers present in both the regions by recombination. 6. Near the junction region, the impurities have no neutralizing space charge and thus near the junction on the n-side there exist ionised donors and on the p-side there exist ionised acceptors. 7. Because of such uncovered ions a potential barrier or junction barrier is developed. 8. The region across the p-n junction in which the potential changes from positive to negative is called depletion region which consist immobile charges and also known as space charge region.

32 V-I characteristic of P-N junction: When a dc voltage is applied to a device it is said to be biased. A p-n junction can be biased in two ways: 1. Forward Biased 2. Reverse Biased P-N junction under forward biased: To apply forward bias positive terminal of battery is connected to the p-side and negative terminal of battery is connected to the n-side.

Depletion region

- - + + - - + + - - + + - - + + p n

VB

33 p n - + - + - + - + free electron flow

VB-V

The applied forward potential establishes an electric field which acts against the potential barrier field and reduces the barrier height. It increases the probability of majority carriers crossing the junction and thus diffusion Jdiff increases. Since it does not affect the influence of minority carriers hence drift current density Jdrift remains unaffected. Here the negative terminal of external causes an increase in electron energy and an upward shift of all energy levels on the n- side. Similarly, the positive terminal causes an increase in hole energy and hence a lowering of all levels on p-side. As the displacement of energy levels occur in opposite direction, the Fermi levels EFn and EFp get separated by an amount of energy eV added by the voltage source and thus the height of the potential barrier is reduced by an amount of energy e(VB-V).

34 R + - mA + V -

t n e r r u c

d r a w r o F Forward voltage

E Depletion re gion p-Type n-Type

Je n

J e(VB -V) ep

EFn eVF EFp Forward bias J hn

J hp

Energy band diagram of a p-n junction under forward bias P-N junction under reverse biased: To apply reverse bias positive terminal of battery is connected to the n-side and 35 negative terminal of battery is connected to the p-side. The applied reverse potential establishes an electric field which acts in the same direction as that of the potential barrier field and thus increases the barrier height by an amount of energy of e(VB+V). In this case the energy levels of n-side are displaced downwards and energy levels of p-side are displaced upwards. Due to the large height of the barrier diffusion of majority carriers totally stops while drift of minority carriers is not affected and thus we get only drift current in this case. R + - μA + V -

Reverse voltage VBR R e v e r s e

c u r r e n t

36 p n - - + + - - + + - - + + - - + +

VB+V

Depletion region E p-Type n-Type

E lec tro Je n p dr ift e(VB +V)

EFp eVR EFn

J Reverse bias hn ho le dr ift

Energy band diagram of a p-n junction under reverse bias

37 PHPTODIODE  Its working is based on photo conduction from light.  A photo diode is a semiconductor made of photosensitive semiconductor material.  In such a diode, a provision is made to allow the light of suitable frequency to fall on it.  The conductivity of p-n photodiode increases with the increase in intensity of light falling on it.  Symbolically, a photodiode is shown in the fig. below:  Fig shows an experimental arrangement in which the photodiode is reverse biased but the voltage applied is less than the break down voltage.  When visible light of energy greater than forbidden energy

gap (i.e. h>Eg)is incident on a reverse biased p-n junction photodiode additional electron-hole pairs are created in the depletion layer (or near the junction).  These charge carriers will be separated by the junction field and made to flow across the junction.  The value of reverse saturation current increases with the increase in the intensity of incident light as shown in the fig. below:  It is found that the reverse saturation current through the photodiode varies almost linearly with the light flux.  The photodiodes are used preferably in reverse bias condition because the change in reverse current through the photodiode due to change in light flux can be measured directly proportional to the light flux. But it is not so when the photodiode is forward biased.

38  When the photodiode is reverse biased, then a certain current exists in the circuit even when no light is incident on the p-n junction of photodiode.  This current is called dark current. A photodiode can turn its current ON and OFF in nanoseconds. Hence photodiode is one of the fastest photodetector. Photodiodes are used for following purposes: 1. In photodetection for optical signals. 2. In demodulation for optical signals. 3. In switching the light on and off. 4. In optical communication equipments. 5. In logic circuits that require stability and high speed. 6. In reading of computers, punched cards and tapes etc.

Solar cell:  Solar cell is basically a solar energy converter.  It is a p-n junction device which converts solar energy into electric energy.  A solar cell is symbolically shown in fig. (a) and in construction along with circuit in fig. (b)

+

-

39a LIGHT

METAL FINGER ELECTRODES GLASS +

P R N

METAL CONTACT - b

 A solar cell consists of silicon or gallium-arsenide p-n junction diode packed in a can with glass window on top.  The upper layer is of p-type semiconductor.  It is very thin so that the incident light photons may easily reach the p-n junction.  On the top face of p-layer, the metal finger electrodes are prepared in order to have enough spacing between the fingers for the light to reach the p-n junction through p-layer.

 When photons of light (of energy h>Eg) fall at the junction, electron-hole pairs are generated in the depletion layer (or near the junction) which move in opposite directions due to junction field.  The photo generated electrons move towards n-side of p-n junction.  The photo generated holes move towards p-side of p-n junction.

40  They will be collected at the tow sides of the junction, giving rise to a photo voltage between the top and bottom metal electrons.  The top metal contact acts as positive electrode and bottom metal contact acts as negative electrode.  When an external load is connected across metal electrodes a photo current flows. Applications: 1. Solar cells are used for charging storage batteries in say time, which can supply the power during night times. 2. The solar cells are also used in artificial satellite to operate the various electrical instruments kept inside the satellite. 3. They are used for generating electrical energy in cooking food. 4. Solar cells are used in calculators, wrist watches and light meters (in photography).

41