Physical Electronics Slides of Chapter 3blogs.cae.tntech.edu/.../Physical-Electronics-Slides-of-Chapter-3.pdf · and Schrodinger’s wave equation to ... Tennessee Technological University

PHYSICAL ELECTRONICS(ECE3540)

Brook Abegaz, Tennessee Technological University, Fall 2013

Friday, September 13, 2013Tennessee Technological University 1

CHAPTER 3 – INTRODUCTION TO THE QUANTUM THEORY OF SOLIDS

Chapter 3 – Introduction to the Quantum Theory of Solids Chapter 2: application of quantum mechanicsand Schrodinger’s wave equation todetermine the behavior of electrons in thepresence of various potential functions.

an electron bound to an atom or boundwithin a finite space can take on only discretevalues of energy; Energies are quantized!

Pauli Exclusion Principle: only one electron is allowed to occupy any given quantum state.


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 statisticalcharacteristics of the very large number ofelectrons in a crystal.

Since current in a solid is due to the net flow ofcharge, it is important to determine the responseof an electron in the crystal to an appliedexternal force, such as an electric field.


Allowed and Forbidden Energy Bands

The energy of the bound electron isquantized: Only discrete values of electronenergy are allowed.

It is possible to extrapolate the single‐atomresults to a crystal and qualitatively derive theconcepts of allowed and forbidden energybands.






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.



Kronig‐Penney Model The concept of allowed and forbidden energy

bands can be developed more rigorously byconsidering quantum mechanics andSchrodinger’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.


Kronig‐Penney Model For a single crystalline lattice, the Kronig Penney

model gives the relation between the wave number parameter k=2π/λ, total energy E (through the parameter α2=2mE/ħ2), and the potential barrier bV0.

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 Vo = amplitude of the potential barrier.



Electrical Condition in Solids Covalent bonding of Silicon determines howthe Silicon crystal is formed.

As the temperature increases some valenceelectrons of the Si atom can break thecovalent bond structure and jump into theconduction band.


Electrical Condition in Solids In terms of the k‐space diagram:


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 velocityVd(cm/s) Drift Current Density a collection of positively charged ions with a

volume density N (cm−3) and an average drift velocity vd (cm/s), then the drift current density would be:


Electron Effective Mass Movement of electrons in a lattice affects the mass

of electrons, which results in a different movementof 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:


Quantum Theory of Solids in 3D Particular characteristics of three dimensional

crystals in terms of E versus k plots, band gapenergy and effective mass are studied.

The distance between atoms varies as thedirection through the crystal changes, for e.g. in[100] planes and in [110] plane directions.

Different directions encounter different potentialpatterns and thus different k space boundaries.

For crystal lattices, the E versus k diagram isplotted such as [100] direction is along the +kaxis and [111] direction is along the –k axis.


Quantum Theory of Solids in 3D Direct Band Gap Semiconductors = semiconductor

lattice whose minimum conduction band energy andmaximum valence band energy occurs at the same k.Example is GaAs.

Transition between a valence band state andconduction band state occurs without a change inCrystal Momentum.

These materials are better suited for semiconductorlasers and optical devices.

Indirect Band Gap Semiconductors = semiconductorlattice whose minimum conduction band energy andmaximum valence band energy occurs at different k.Example are Si, Ge, GaP, AlAs.


Quantum Theory of Solids in 3D


Density of States Function Aim We want to find density of carriers in a semiconductor

1st find the number of available states at each energy level.

2nd find the number of electrons by multiplying number of states with the probability of occupancy.


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.


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 < aand V(x, y, z) = ∞ otherwise,(a free electron confined to three-dimensional infinite

potential well),Using wave number k = nπ/a, and therefore n = nx +

ny + nz,


Density of States Function Now, distance between two quantum states:

Volume Vk of a single quantum state:

Differential volume is (4πk2)dk because total volume = 4/3 πk3. 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 kx, ky, kz , 4πk2dk is the differential volume, and (π/a)3 = volume of one quantum state.


Density of States Function Substitute k2, k and dk/dE as

To find:

This gives the total number of Quantum States between E and dE. Then dividing by the volume a3 gives the density of quantum states as a function of energy.


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:‐


Statistical Mechanics There are three distribution laws determining

the distribution of particles among energy states:1. Maxwell‐Boltzmann Particles are considered to be distinguishable and

numbered.2. Bose‐Einstein Particles are indistinguishable with no limit to the number of

particles per energy state.3. Fermi‐Dirac Probability Function Particles in a crystalline lattice are indistinguishable

and also only one particle is allowed per each quantum state.

Electrons in a crystal obey the Fermi‐Dirac function.


Statistical Mechanics

is the Fermi‐Dirac distribution function where: N(E) = total # of electrons per unit volume g(E) = # of quantum states per unit volume EF = Fermi energy level. fF(E) = ratio of filled to total quantum states.


Statistical Mechanics

An approximation to Fermi‐Dirac function is Maxwell‐Boltzmann where:‐


Exercise1. Let T = 300K. Determine the probability of

finding an electron at an energy level of 3kT higher (above) than the Fermi energy EF of the electron.

2. Assume the Fermi energy level is 0.3 eVbelow the conduction band energy Ec. Assume T = 300K.

a) Determine the probability of a state being occupied by an electron at E = Ec + kT/4.

b) Find the probability of a state being occupied by an electron at E = Ec + kT.


Solution1. Using Fermi‐Dirac function at 3kT higher

energy level:


Solution2. Using Fermi‐Dirac function at E = Ec + kT/4

and at E = Ec + kT where EF = Ec – 0.3eV

Substituting, we find, a) 7.26*10‐6

and b) 3.43*10‐6.


Picture Credits Semiconductor Physics and Devices, Donald Neaman, 4th Edition, McGraw Hill Publications.

Spin up and spin down of Lithium atom in 1D array of tubes, Courtsey: Professor Randall G. Hulet, Rice University


Documents

Physical Electronics Slides of Chapter 3blogs.cae.tntech.edu/.../Physical-Electronics-Slides-of-Chapter-3.pdf · and Schrodinger’s wave equation to ... Tennessee Technological University