UNIT-IV SEMICONDUCTOR & NANOMATERIALS 1. Free electron model 2. Kronig penny model 3. Effective mass 4. Fermi level for intrinsic and extrinsic semiconductor 5. P-N junction 6. Zener diode 7. photo diode 8. solar cell 9. Hall effect 10. elementary idea of nanostructures and nano materials Free electron model: In order to explain the electrical and thermal properties of metals 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 metal 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 ions 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 thermal conductivity 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 mean free path 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 semiconductors 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 solid is periodic i.e., in a solid crystal, each positive ion 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 Es 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 Fermi energy. 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) (EE ) / 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 E and E move Es EF b F anywhere within the metal, where EF is the Fermi energy level. The work function for which can be given as 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/ )sina cosa 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 Psina cosa 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: Psina cosa 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.