21PYB102J Semiconductor Physics and Computational Methods Unit 1. PDF

21PYB102J – Semiconductor Physics and Computational Methods Unit – 1 : Session – 1 : SLO - 1 SRM Institute of Science and Technology 1 Free Electron Theory ▪The electron theory of materials is to explain the structure and properties of solids through their electronic...

21PYB102J – Semiconductor Physics and Computational Methods Unit – 1 : Session – 1 : SLO - 1 SRM Institute of Science and Technology 1 Free Electron Theory ▪The electron theory of materials is to explain the structure and properties of solids through their electronic structure. ▪It also gives information about bonding in solids, energy levels in metals and cohesive & repulsive forces in metals. Development of Free Electron Theory The Classical free electron theory [Drude and Lorentz] ▪ It is a macroscopic theory, through which free electrons in lattice and it obeys the laws of classical mechanics. Here the electrons are assumed to move in a constant potential. SRM Institute of Science and Technology The Quantum free electron theory It is a microscopic theory, according to this theory the electrons in lattice moves in a constant potential and it obeys law of quantum mechanics. Brillouin Zone Theory [Band Theory] Bloch developed this theory in which the electrons move in a periodic potential provided by periodicity of crystal lattice. It explains the mechanisms of conductivity, semiconductivity on the basis of energy bands and hence It is called as Band theory SRM Institute of Science and Technology  Classical free electron theory of metals This theory was developed by Drude and Lorentz in 1900 and hence is also known as Drude-Lorentz theory. it’s the first theory to explain the electrical conduction in conducting materials and reveals that free electrons are responsible for the electrical conduction. According to this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a container. In certain metals especially in Cu, Ag and Al valence electrons are so weakly attached to the nuclei they can be easily removed or detached such electrons are called as free electrons. But all the valence electrons in the metals are not free electrons. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the total energy of the electron is equal to its kinetic energy. SRM Institute of Science and Technology 4 Assumptions ❑ Electrons travel with constant potential and confine to the boundaries of metal ❑ All the attractive and repulsive forces are neglected ❑ The energies of free electrons are not quantized ❑ The distribution of electrons is as per the Hund’s rule and follows Pauli’s exclusion principle SRM Institute of Science and Technology Postulates : (a) In an atom electrons revolve around the nucleus and a metal is composed of such atoms. (b). The valence electrons of atoms are free to move about the whole volume of the metals like the molecules of a perfect gas in a container. The collection of valence electrons from all the atoms in a given piece of metal forms electrons gas. It is free to move throughout the volume of the metal (c) These free electrons move in random directions and collide with either positive ions fixed to the lattice or other free electrons. All the collisions are elastic i.e., there is no loss of energy. SRM Institute of Science and Technology (d). The movements of free electrons obey the laws of the classical kinetic theory of gases. (e). The electron velocities in a metal obey the classical Maxwell – Boltzmann distribution of velocities. (f). The electrons move in a completely uniform potential field due to ions fixed in the lattice. (g). When an electric field is applied to the metal, the free electrons are accelerated in the direction opposite to the direction of applied electric field. SRM Institute of Science and Technology Success of classical free electron theory: (1). It verifies Ohm’s law. (2). It explains the electrical and thermal conductivities of metals. (3). It derives Wiedemann – Franz law. (i.e., the relation between electrical conductivity and thermal conductivity) (4). It explains optical properties of metals. SRM Institute of Science and Technology Drawbacks of classical free electron theory: The phenomena such a photoelectric effect, Compton effect and the black body radiation couldn’t be explained by classical free electron theory. According to the classical free electron theory the value of specific heat of gas at constant volume is given by 3/2 R, where R is universal gas constant, but experimentally it was observed that the specific heat of a metal by its conduction electron is given by 10-4 RT. Thus, the experimental value of Cv is very much lesser than the expected value of Cv. According to classical free electron theory Cv is independent of temperature, but the experimental value of Cv is directly proportional to temperature. Hence classical free electron theory fails to explain Cv. SRM Institute of Science and Technology Electrical conductivity of semiconductor or insulators couldn’t be explained using this model. Though K/σT is a constant (Wiedemann – Franz Law) according to the Classical free electron theory, it is not a constant at low temperature. Ferromagnetism couldn’t be explained by this theory. The theoretical value of paramagnetic susceptibility is greater than the experimental value. SRM Institute of Science and Technology  21PYB102J – Semiconductor Physics and Computational Methods Unit – 1 : Session – 1 : SLO - 2 SRM Institute of Science and Technology 11 Quantum free electron theory Classical free electron theory could not explain many physical properties. To overcome the drawbacks of Classical free electron theory, In 1928, Sommerfield developed a new theory applying quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metal. This theory is called quantum free electron theory. SRM Institute of Science and Technology 12 The following are the assumptions of quantum free electron theory: ❑ The free electrons in a metal can have only discrete energy values. Thus the energies are quantized. ❑ The electrons obey Pauli’s Exclusion Principle, which states that there cannot be more than two electrons in any energy level. ❑ The distribution of electrons in various energy levels obey the Fermi-Dirac quantum statistics. ❑ Free electrons have the same potential energy everywhere within the metal, because the potential due to ionic cores is uniform throughout the metal. ❑ The force of attraction between electrons & lattice ions and the force of repulsion between electrons can be neglected. ❑ Electrons are treated as wave-like particles Merits of quantum free electron theory 1. It successfully explains the electrical and thermal conductivity of metals. 2. We can explain the Thermionic phenomenon. 3. Temperature dependence of conductivity of metals can be explained by this theory. 4. It can explain the specific heat of metals. 5. It explains magnetic susceptibility of metals. Demerits of quantum free electron theory 1. It is unable to explain the metallic properties of exhibited by only certain crystals. 2. Failed to give difference of metals/semiconductors/ insulators 3. It is unable to explain why the atomic arrays in metallic crystals should prefer certain structures only.  21PYB102J – Semiconductor Physics and Computational Methods Unit – I: Session – 2: SLO - 1 SRM Institute of Science and Technology 1 Quantum Free Electron Theory – Density of States in 3D  DENSITY OF STATES Definition: Density of States Z (E) dE is defined as the number of available electron states per unit volume in an energy interval (dE). To find the number of energy levels in a cubical metal piece and to find number of electrons that can be filled in a given energy level, let us construct a sphere of radius ‘n’ in the space. 3 21PYB102J Module-I Lecture-2 Expression for Density of States in 3D (1) Expression for Density of States in 3D (2) Expression for Density of States in 3D (3) Expression for Density of States in 3D (4) Expression for Density of States in 3D (5)  21PYB102J – Semiconductor Physics and Computational Methods Unit – I: Session – 2: SLO - 2 SRM Institute of Science and Technology 9  Band Theory of Solids - Lattice, Reciprocal Lattice, Concept of Energy Bands in Solids, Bloch Theorem  1D Lattice Atoms (or ions) are arranged with equal spacing a called lattice parameter.  2D Lattice Atoms (or ions) are periodically arranged in 2D space. Two lattice parameters a and b uniquely define the 2D lattice.  Reciprocal Lattice and Brillouin Zone in 1D Direct lattice. Reciprocal lattice.  Brillouin Zone in 2D Also called Wigner-Seitz primitive cell in the reciprocal lattice.  Concept of Energy Bands in Solids (1) When two identical atoms are far apart, the allowed energy levels for a given principal quantum number (for example n = 1) consist of one double degenerate level. That is, both atoms have exactly the same energy. When they are brought closer, the doubly degenerate energy levels will split into two levels by the interaction between atoms. This splitting occurs due to the Pauli’s exclusion principle.  Concept of Energy Bands in Solids (2) As N isolated atoms are brought together to form a solid, it causes a shift in the energy levels of all N atoms, as in the case of two atoms. However, instead of two levels, N separate but closely spaced levels are formed. When N is very large, the result is essentially a continuous band of energy. This band of N levels can extend over a few eV at the inter-atomic distance of the crystal.  An Example: Energy Bands in Silicon When many Si atoms are brought together, the energy bands are formed in such a way it divides into two and in between there are no energy levels present. The gap between these bands are called band gap. The band lying below is called valence band owing to the presence of valence electrons which are tightly bound to the parent atoms. The band lying higher than the valence band is called conduction band which is responsible for conducting current in the material.  Band Theory of Solids Developed by Felix Bloch in 1928. (PhD thesis) Free-electron approximation is abandoned. Coulombic interaction between valence electrons and positively charged metal ions is included. Independent-electron approximation is retained. Explains band structure in solids. Felix Bloch (1905 - 1983)Prize in Nobel Physics, 1952  Bloch’s Theorem A fundamental theorem in the quantum theory of crystalline solids. Statement: The wave function of an electron moving in a periodic 1D lattice is of the form u(x) has the same periodicity as the lattice. Alternate statement of Bloch’s theorem: Band Theory of Solids - Kronig-Penney Model l=h/p 2p/k=h/p p=(h/ 2p)k 2 Derivation of Time-Independent Schrodinger Equation 3 18PYB101J Module-III Lecture-8 4 5  6 18PYB101J Module-III Lecture-8  1D Lattice Atoms (or ions) are arranged with equal spacing a called lattice parameter.  2D Lattice Atoms (or ions) are periodically arranged in 2D space. Two lattice parameters a and b uniquely define the 2D lattice.  Band Theory of Solids Developed by Felix Bloch in 1928. (PhD thesis) Free-electron approximation is abandoned. Coulombic interaction between valence electrons and positively charged metal ions is included. Independent-electron approximation is retained. Explains band structure in solids. Felix Bloch (1905 -1983) Nobel Prize in Physics, 1952  Bloch’s Theorem A fundamental theorem in the quantum theory of crystalline solids. Statement: The wave function of an electron moving in a periodic 1D lattice is of the form u(x) has the same periodicity as the lattice. Alternate statement of Bloch’s theorem: Periodic Ionic Potential in a 1D crystal  Kronig-Penney Model of Ionic Potential in a 1D crystal K-P model is an idealized periodic potential representing 1D crystalline solid. Schrodinger’s equation needs to be solved in region I and II. Solution of Schrodinger’s equation yields allowed values of energies of the electron in the 1D crystal.  Kronig-Penney Model in a 1D crystal Properties of well-behaved wave functions y is continuous First derivative of y is also continuous  Kronig-Penney Model in a 1D crystal Boundary conditions at x = 0: (y is continuous) (1) (derivative of y is continuous) (2)  Kronig-Penney Model in a 1D crystal Periodic boundary conditions at x = +a and x = -b: (y is continuous) (3) (derivative of y is continuous) (4)  Four equations we got with four unknown A, B, C, D (1) (2) (3) (4) The determinant of coefficients of A, B, C and D must be equal to zero for nontrivial solutions of A, B, C, and D. After equating determinant to zero, we will get  Kronig-Penney Model in a 1D crystal Value of the determinant: The above equation is a transcendental equation. No analytical solutions are possible. Only graphical solutions can be obtained.  Case 2: Electron in a Periodic Potential Only values of 𝛼𝑎 for which f lies between +1 and -1 are allowed.  Case 2: Electron in a Periodic Potential Only values of 𝛼𝑎 for which f lies between +1 and -1 are allowed.  E-k Diagram for an Electron in a Periodic Potential 3rd band 2nd band 1st band Discontinuities in E indicate forbidden energies of the electron. Case 1: Free Electron Kronig-Penney Model: Conclusion  Concept of Energy Bands in Solids (1) When two identical atoms are far apart, the allowed energy levels for a given principal quantum number (for example n = 1) consist of one double degenerate level. That is, both atoms have exactly the same energy. When they are brought closer, the doubly degenerate energy levels will split into two levels by the interaction between atoms. This splitting occurs due to the Pauli’s exclusion principle.  Concept of Energy Bands in Solids (2) As N isolated atoms are brought together to form a solid, it causes a shift in the energy levels of all N atoms, as in the case of two atoms. However, instead of two levels, N separate but closely spaced levels are formed. When N is very large, the result is essentially a continuous band of energy. This band of N levels can extend over a few eV at the inter-atomic distance of the crystal.  An Example: Energy Bands in Silicon When many Si atoms are brought together, the energy bands are formed in such a way it divides into two and in between there are no energy levels present. The gap between these bands are called band gap. The band lying below is called valence band owing to the presence of valence electrons which are tightly bound to the parent atoms. The band lying higher than the valence band is called conduction band which is responsible for conducting current in the material.  21PYB102J – Semiconductor Physics and Computational Methods Unit – I : Session – 4: SLO - 1 SRM Institute of Science and Technology 1  Problem Solving Calculate the density of states per unit volume with energies between 0 and 1 eV. Solution 1𝑒𝑉 3 1𝑒𝑉 4𝜋(2𝑚)2 න 𝑔 𝐸 𝑑𝐸 = න 𝐸𝑑𝐸 ℎ3 0 0 1 4π(2𝑚)3/2 2 3 = 3 𝐸2 ℎ 3 0 = 4.5 x 1021 states/cm3  Problem Solving sin∝𝑎 Find the Variation of 𝑃 ′ + cos ∝ 𝑎 = cos 𝑘𝑎 with ∝ 𝑎 for P’ = ∞ ∝𝑎 For P’ → ∞ , spectrum becomes a line. sin ∝ 𝑎 = 0 ∝ 𝑎 = ±𝑛𝜋 ∝𝑎 2 𝑛 2 𝜋2 8𝜋2 𝑚𝐸 ∝ = 2 = 𝑎 ℎ2 𝑛2 ℎ 2 𝐸= 8𝑚𝑎 2 This expression shows that the energy of the particle is discrete. Width of the allowed band Decreases with increase in P’, i.e., with more binding energy of electrons  𝑖𝑓 𝑃 = 0, 𝑡ℎ𝑒𝑛 Cos ∝ 𝑎 = Cos ka 8𝜋2 𝑚𝐸 ∝=k→ 𝑘2 = 2 ∝ = ℎ2 2𝜋 ℎ2 𝑘 2ℎ2 ( )2 ℎ2 𝐸= = λ = 8𝜋 2 𝑚 8𝜋 2 𝑚 2𝑚λ2 Using de-Broglie’s formula for wave-particle duality ℎ2 𝑝 2 𝑝2 E= ( ) = 2𝑚 ℎ 2𝑚 This is just equivalent to the case of free particle. For P= 0 the free electron model and energy spectrum is (quasi) continuous  21PYB102J – Semiconductor Physics and Computational Methods Unit – I : Session – 4: SLO - 2 SRM Institute of Science and Technology 1  Problem Solving Introduce Time Independent Schroedinger Equation in 1D For 3D 2m   2 2 ( E  V )  0  Problem Solving Further simplification and re-arrangement lead to Hamiltonian form Time-Dependent Schroedinger Equation  2      V   i 2  2m  t  2 where H = 2  V 2m  Problem Solving To solve the problem for a particle in a 1-dimensional box Step 1. Define the Potential Energy, V Step 2. Solve the Schrödinger Equation Step 3. Define the wavefunction Step 4. Define the allowed energies Step 1. Define the Potential Energy, V The potential energy is 0 inside the box (V=0 for 0