PHYSICAL ELECTRONICS(ECE3540) CHAPTER 3 – INTRODUCTION TO THE QUANTUM THEORY OF SOLIDS Brook Abegaz, Tennessee Technological University, Fall 2013 1 Tennessee Technological University Friday, September 13, 2013
Chapter 3 – Introduction to the Quantum Theory of Solids Chapter 2: application of quantum mechanics and Schrodinger’s wave equation to determine the behavior of electrons in the presence of various potential functions. an electron bound to an atom or bound within a finite space can take on only discrete values of energy; Energies are quantized! Pauli Exclusion Principle: only one electron is allowed to occupy any given quantum state. 2 Tennessee Technological University Friday, September 13, 2013
Chapter 3 – Introduction to the Quantum Theory of Solids Chapter 3: generalization of these concepts to the electron in a crystal lattice. Determine the properties of electrons in a crystal lattice, and to determine the statistical characteristics of the very large number of electrons in a crystal. Since current in a solid is due to the net flow of charge, it is important to determine the response of an electron in the crystal to an applied external force, such as an electric field. 3 Tennessee Technological University Friday, September 13, 2013
Allowed and Forbidden Energy Bands The energy of the bound electron is quantized: Only discrete values of electron energy are allowed. It is possible to extrapolate the single ‐ atom results to a crystal and qualitatively derive the concepts of allowed and forbidden energy bands. 4 Tennessee Technological University Friday, September 13, 2013
Allowed and Forbidden Energy Bands 5 Tennessee Technological University Friday, September 13, 2013
Allowed and Forbidden Energy Bands 6 Tennessee Technological University Friday, September 13, 2013
Formation of Energy Bands The wave functions of the two atom electrons overlap, which means that the two electrons will interact. This interaction or perturbation results in the discrete quantized energy level splitting into two discrete energy levels. 7 Tennessee Technological University Friday, September 13, 2013
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Kronig ‐ Penney Model The concept of allowed and forbidden energy bands can be developed more rigorously by considering quantum mechanics and Schrodinger’s wave equation. The result forms the basis for the energy ‐ band theory of semiconductors. The solution to Schrodinger’s wave equation, for a one ‐ dimensional single crystal lattice, is made more tractable by considering a simpler potential function in the Kronig–Penney model, which is used to represent a one ‐ dimensional single ‐ crystal lattice. 9 Tennessee Technological University Friday, September 13, 2013
Kronig ‐ Penney Model For a single crystalline lattice, the Kronig Penney model gives the relation between the wave number parameter k=2 π / λ , total energy E (through the parameter α 2 =2mE/ ħ 2 ), and the potential barrier bV 0 . It is not a solution of Schrodinger’s wave equation but gives the conditions for which Schrodinger’s wave equation will have a solution. where a = width of the region, b = width of the barrier, and V o = amplitude of the potential barrier. 10 Tennessee Technological University Friday, September 13, 2013
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Electrical Condition in Solids Covalent bonding of Silicon determines how the Silicon crystal is formed. As the temperature increases some valence electrons of the Si atom can break the covalent bond structure and jump into the conduction band. 12 Tennessee Technological University Friday, September 13, 2013
Electrical Condition in Solids In terms of the k ‐ space diagram: 13 Tennessee Technological University Friday, September 13, 2013
Electrical Condition in Solids Drift Current: electric current due to applied electric field. for a collection of positively charged ions having: a) Volume density N(cm ‐ 3 ) b) Average drift velocityV d (cm/s) Drift Current Density a collection of positively charged ions with a volume density N (cm − 3 ) and an average drift velocity v d (cm/s), then the drift current density would be: 14 Tennessee Technological University Friday, September 13, 2013
Electron Effective Mass Movement of electrons in a lattice affects the mass of electrons, which results in a different movement of electrons than in a free space. Effective mass is a parameter that relates the quantum mechanical results to classical force equations. The parameter m , called the effective mass, takes into account the particle mass and also takes into account the effect of the internal forces. If E is the energy of the electron at the conduction band, E is the applied electric field, e is the charge of the electron, and a its acceleration, then: 15 Tennessee Technological University Friday, September 13, 2013
Quantum Theory of Solids in 3D Particular characteristics of three dimensional crystals in terms of E versus k plots, band gap energy and effective mass are studied. The distance between atoms varies as the direction through the crystal changes, for e.g. in [100] planes and in [110] plane directions. Different directions encounter different potential patterns and thus different k space boundaries. For crystal lattices, the E versus k diagram is plotted such as [100] direction is along the +k axis and [111] direction is along the –k axis. 16 Tennessee Technological University Friday, September 13, 2013
Quantum Theory of Solids in 3D Direct Band Gap Semiconductors = semiconductor lattice whose minimum conduction band energy and maximum valence band energy occurs at the same k. Example is GaAs. Transition between a valence band state and conduction band state occurs without a change in Crystal Momentum. These materials are better suited for semiconductor lasers and optical devices. Indirect Band Gap Semiconductors = semiconductor lattice whose minimum conduction band energy and maximum valence band energy occurs at different k. Example are Si, Ge, GaP, AlAs. 17 Tennessee Technological University Friday, September 13, 2013
Quantum Theory of Solids in 3D 18 Tennessee Technological University Friday, September 13, 2013
Density of States Function Aim We want to find density of carriers in a semiconductor 1 st find the number of available states at each energy level. 2 nd find the number of electrons by multiplying number of states with the probability of occupancy. 19 Tennessee Technological University Friday, September 13, 2013
Density of States Function It involves determining the density of allowed energy states as a function of energy in order to calculate the electron and hole concentrations. It is important to find out the available number of electrons and holes available for conduction and to describe the V ‐ I characteristics in a semiconductor. Density of states in a semiconductor equals density of number of solutions of Schrödinger’s wave equation to unit volume and energy. 20 Tennessee Technological University Friday, September 13, 2013
Density of States Function In a crystal lattice, if a potential function V(x, y, z) exists as a potential well such as: V(x, y, z) = 0 for 0 < x < a, 0 < y < a, 0 < z < a and V(x, y, z) = ∞ otherwise, (a free electron confined to three-dimensional infinite potential well), Using wave number k = n π /a, and therefore n = n x + n y + n z , 21 Tennessee Technological University Friday, September 13, 2013
Density of States Function Now, distance between two quantum states: Volume V k of a single quantum state: Differential volume is (4 π k 2 )dk because total volume = 4/3 π k 3 . Differential density of quantum states in space which is also where 2 is for two spin states allowed for each quantum state, 1/8 is for positive regions of each quantum state k x , k y , k z , 4 π k 2 dk is the differential volume, and ( π /a) 3 = volume of one quantum state. 22 Tennessee Technological University Friday, September 13, 2013
Density of States Function Substitute k 2 , k and dk/dE as To find: This gives the total number of Quantum States between E and dE. Then dividing by the volume a 3 gives the density of quantum states as a function of energy. 23 Tennessee Technological University Friday, September 13, 2013
Density of States Function This equation gives the density of allowed electron quantum states using the model of a free electron with mass m, bounded in a three dimensional infinite potential well. In general, for semi ‐ conductors, density of allowed energy states equals in conduction band: ‐ In valence band: ‐ 24 Tennessee Technological University Friday, September 13, 2013
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