Analog System & Application(3rd sem) Ranjansatya Sp2 PDF

# Unit - 1: Semiconductors * Distinction among conductors, insulators and Semiconductors with the help of Energy Band ## Energy Band A collection of a number of energy levels which are closely packed is called a band. It is of two types: 1. **Valence Band** The lower band of solids belongs...

# Unit - 1: Semiconductors * Distinction among conductors, insulators and Semiconductors with the help of Energy Band ## Energy Band A collection of a number of energy levels which are closely packed is called a band. It is of two types: 1. **Valence Band** The lower band of solids belongs to the valence electrons is known as the valence band. The band may be partially filled or completely filled with electrons. This band cannot be empty. The electrons donate energy from external electric field. Hence the electrons do not contribute to the electric current. 2. **Conduction Band** It is situated upper to the valence band in a crystalline solid. This band is empty at absolute zero temperature or partially filled. At room temperature, this band is either empty or partially filled. The electrons contribute the energy from external electric field. ## Energy Band Gap The distance between energy level of valence band and lower level of conduction band is called forbidden energy gap or band gap. The minimum amount of energy required to shift electrons from valence band to conduction band is called energy band gap. $E_g = hv = h\omega$. ## Conductors The substance having low resistivity or high conductivity is called conductor. Here, $ \rho = 10^{-2} \Omega m $ to $ 10^{-8} \Omega m $ $ \sigma = \frac{1}{\rho} = 10^{2} \Omega^{-1} m^{-1} $ to $ 10^{8} \Omega^{-1} m^{-1} $ ## Insulators The substance having high resistivity and low conductivity is called insulator. $ \rho = 10^8 \Omega m $ to $ 10^{18} \Omega m $, $ \sigma = 10^{-8} \Omega^{-1} m^{-1} $ to $ 10^{-18} \Omega^{-1} m^{-1} $. The band gap between valence band and conduction band is very large, about greater than 3eV. Example - Diamond having $E_g$ = 6eV. ## Semiconductors The substance which initially behaves like an insulator at absolute zero temperature i.e. valence band is completely filled and conduction band is completely empty. But, the band gap is less than 3eV. At room temperature, the electrons may have the capacity to jump from valence band to conduction band. The hole are created in valence band and become electrically positive like germanium, silicon. Example - Germanium (0.7eV), Silicon (0.17eV) ### Types of Semiconductor It is of two types: 1. **Intrinsic Semiconductor** These semiconductors are called pure semiconductors because they have no external or additional impurities. Through these semiconductors are insulators at 0K best at room temperature, they have capability for conductivity which is due to the charge carriers such as free electrons and holes (positively charged particles, which are due to the absence of electrons, nothing but negatively charged particles) The holes are nothing but the absence of electrons, which is due to the positively charged particles, which are negatively charged. eg: Germanium, Silicon. Ge - 1s² 2s² 2p^6 3s² 3p^6 4s² 3d^10 4p² Si - 1s² 2s² 2p^6 3s² 3p² The germanium crystal each Ge atom has four valence electrons each are bonded by covalent bond. At room temperature i.e. thermal excitation, the covalent bond breakup and a free electron comes out creating a hole at that covalent bond site. Now this free electron is responsible for conduction of current. The free electron and holes are called the intrinsic charge carriers. Because of the hole is replaced by another electron by movement of electron from side to side covalent bond. This happens due to some additional thermal energy called ionization energy. If n_e and n_h are the number of free electrons and holes then for an intrinsic semiconductor. n_e = n_h and n_e = n_h = exp(-E_g / 2kT) where, K = Boltzmann's constant T = Temperature is Kelvin 2. **Extrinsic Semiconductor** The semiconductor in which the conductivity increases by addition of a suitable impurity atom to an intrinsic semiconductor by suitable amount is called extrinsic semiconductor. The process of adding impurity atom to an intrinsic semiconductor is called doping. The atoms are called dopants. The concentration of dopant atom does/ should not change the crystal lattice not more than 1% of the crystal atom. According to valence electrons, the dopant atom is classified into two types i.e. (a) Pentavalent Impurity: The impurity atom having five valence electrons are called pentavalent impurity. Example - Phosphorous, Antimony, Arsenic, Bi, Sb etc. (b) Trivalent Impurity: The impurity atom having three valence electrons are called trivalent impurity. Example - Boron, Aluminium, Indium, Ga, Th etc. ### Classification 1. **n-type semiconductor** The semiconductor which is doped with a pentavalent dopant atoms. Then this semiconductor is called n-type semiconductor. Example : Consider for Ge atom, It has four valence electrons and As’ has five valence electrons. The four electrons of Ge atom are replaced by four electrons of As’ atom during doping and free attaching with the parent As atom. As atom donates free electron, hence the n-type semiconductor is called donor type semiconductor. The n-type semiconductor is electrically neutral. Here, the majority charge carriers are free electrons, which conducts electricity and holes are minority charge carriers. Hence Fermi level in n-type semiconductor is very close to conduction band. 2. **p-type semiconductor** When a trivalent impurity atom is added to the semiconductor like Ge atom, in Si, then p-type semiconductor is formed. Eg: Let us consider for Ge and In, both Ge and In have four and three valence electrons respectively. After combination the In atom is replaced by In atom in place of a Ge atom. Due to breaking four covalent bonds, one of the covalent bonds remain absent with one free electron i.e. hole. This hole is replaced by one free electron coming out from next covalent bond. This hole is responsible for conduction of current. This hole provides p-type semiconductor. The p-type semiconductor is also called acceptor type semiconductor. At room temperature, the electrons in the valence band are easily transferred from the valence band to acceptor level or Fermi level. When an external field is applied then the holes take the valence band to the conduction band producing large hole, which carries out current. The p-type semiconductor is electrically neutral. The p-type semiconductor is also called acceptor type semiconductor. ### Electron and Hole Density Let n_i is the electron concentration and p_i is the hole concentration. Then for intrinsic semiconductor : $n_i \times p_i = n_i^2 \to e^{n_i}$. When N_D is the donor atom density of n-type semiconductor. N_A is the acceptor atom density of p-type semiconductor, But in p-type semiconductor N_D = 0. So, $p_p = n_i + N_A$ $ \implies p_p = N_A$ $ But, n_i^2 = n_n \times p_n $ $ \implies n_n = \frac{n_i^2}{p_n} \implies n_n = \frac{n_i^2}{N_A}$ ie density of minority carrier is equal to the impurity atom density ### Drift Velocity The average velocity of free electrons and holes inside a semiconductor when an electric field is applied is known as drift velocity. Free electrons and holes inside a semiconductor consume random motion. They move with all possible velocities in all possible directions. So, the average charge velocity of charge carriers due to thermal energy is zero. Let v_1, v_2, ... v_n are the velocities of 1st, 2nd, ... nth free electrons inside a semiconductor due to thermal energy. $ \implies v_1 + v_2 + ... + v_n = 0 $ $ \implies \frac{v_1+v_2+...+ v_n}{N} = 0$ $ \implies \overline{v} = 0 $ When an electric field is applied to the semiconductor, then an electric force will act on each electron, So, the force acting on each electron is given by, $ \implies \overrightarrow{F}= -e\overrightarrow{E} $ and the acceleration of the electron is given by, $ \implies \overrightarrow{a} = \frac{\overrightarrow{F}}{m} = \frac{-e\overrightarrow{E}}{m} $ Drift motion of free electrons, they collide with each other and between two consecutive collisions, they move more in straight line and produce drift velocity. Let v_1, v_2, ... v_n are the velocities of 1st, 2nd, ... nth free electrons inside the semiconductor after the application of electric field and t_1, t_2, t_3, ... t_n are the time interval between two successive collisions of 1st, 2nd, ... nth free electrons then - $v_1' = v_1 + at_1 $ - $ v_2' = v_2 + at_2 $ - $ v_n' = v_n + at_n$ $ \implies \overline{v_d} = v_1' + v_2' + ... + v_n' $ $ \implies \overline{v_d} = \frac{v_1+v_2+...+ v_n}{N} + a\frac{t_1+t_2+...+t_n}{N} $ $ \implies \overrightarrow{v_d} = \overline{v}t + a\overline{t} $ where, $ \overline{t} $ = Relaxation time or Average time between two consecutive collisions $ \implies\overrightarrow{v_d} = a \overline{t} $ $ \implies\overrightarrow{v_d} = \frac{-e\overrightarrow{E}}{m} \overline{t} $ ### Conductivity and Mobility of Semiconductors The relation between current density and drift velocity is given by $ I = - n e a \overrightarrow{v_d } $ So, $ I \propto \overrightarrow{v_d} $ For electron, $ \overrightarrow{v_d} = \frac{-e \overrightarrow{E}}{m} \overline{t} $ and for hole, $ \overrightarrow{v_d} = \frac{+e \overrightarrow{E}}{m} \overline{t}$ ## Relation between Drift Velocity and Current Let us consider a semiconductor of length l and cross sectional area A having lots of free electrons with a constant drift velocity $v_d$. If all the electrons move with constant drift velocity $v_d$ and the time taken by extreme electron to reach the cross section is t. Then, all the electron will cross the cross-sectional area with t seconds. $ \implies $ volume of conductor = Al $ \implies $ In = nAl = enA($\overrightarrow{v_d}$)l = enA( $\overrightarrow{v_d}$) $ \implies $ In = e h A ($\overrightarrow{v_d}$) [n = hole density] But, the free electrons and hole of semiconductor are in the state of random motion due to variation in thermal energy. When they carry drift under the influence of electric field, the current density due to drift velocity of free electron is given by $ \implies J_e = \frac{I_e}{A} = \frac{-neAl(\overrightarrow{v_d})}{A} = -ne(\overrightarrow{v_d})$ where, e = the charge of electron $ (\overrightarrow{v_d}) $ = the drift velocity of electron n = the free electron density Again the current density due to drift velocity of hole is given by, $ \implies J_h = \frac{I_h}{A} = \frac{e h A(\overrightarrow{v_d})}{A} = eh(\overrightarrow{v_d})$ where, h = the hole density $ (\overrightarrow{v_d}) $ = the drift velocity of hole $ \implies $ Total free electrons current density due to both free electrons and hole is given by, $ \implies J = I_n +(J_e) $ $ \implies J = -ne\overrightarrow{v_d} $ $ \implies $ J indicates that it is oppositely directed to $I_n$ ### Conductivity and Mobility of Semiconductors The relation between current and drift velocity is given by $ I = -nea \overrightarrow{v_d} $ $ \implies I \propto \overrightarrow{v_d} $ For electron, $\overrightarrow{v_d}= \frac{-e\overrightarrow{E}}{m} \overline{t} $ So, $ I = - nea (\frac{-e\overrightarrow{E}}{m}\overline{t}) $ $ \implies I = \frac{ne^2 a \overline{t}}{m} \overrightarrow{E}$ and for hole, $\overrightarrow{v_d} = \frac{+e\overrightarrow{E}}{m}\overline{t}$ $ \implies I = \frac{ne^2 a \overline{t}}{m} \overrightarrow{E}$ Now substituting the values of I and $ \overrightarrow{v_d} $ in $ J = ne \overrightarrow{v_d} $, we get: - $J= \frac{ne^2 a \overline{t}}{m} \overrightarrow{E}$ But, the conductivity of semiconductor is defined as - $\sigma = \frac{J}{E} = \frac{\frac{ne^2 a \overline{t}}{m} \overrightarrow{E}}{\overrightarrow{E}} $ $ \implies \sigma = \frac{ne^2 \overline{t}}{m} $ This is the expression for conductivity. The mobility is defined as the ratio of the drift velocity $ \overrightarrow{v_d} $ to the electric field E. $ \implies \mu_e = \frac{\overrightarrow{v_d}}{E} $ $ \implies \mu_e = \frac{\frac{-e\overrightarrow{E}}{m} \overline{t}}{\overrightarrow{E}}$ $ \implies \mu_e = \frac{-e \overline{t}}{m} $ $ \implies \mu_h = \frac{+e \overline{t}}{m} $