PHYSICAL ELECTRONICS(ECE3540) CHAPTER 3 INTRODUCTION TO THE QUANTUM - - PowerPoint PPT Presentation

PHYSICAL ELECTRONICS(ECE3540)

Brook Abegaz, Tennessee Technological University, Fall 2013

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CHAPTER 3 – INTRODUCTION TO THE QUANTUM THEORY OF SOLIDS

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.

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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

• f an electron in the crystal to an applied

external force, such as an electric field.

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Allowed and Forbidden Energy Bands

• The

energy

• f

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.

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Allowed and Forbidden Energy Bands

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Allowed and Forbidden Energy Bands

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Formation of Energy Bands

• The wave functions of the two atom electrons
• verlap, 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.

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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.

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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.

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Friday, September 13, 2013 Tennessee Technological University

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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.

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Electrical Condition in Solids

• In terms of the k‐space diagram:

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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:

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Electron Effective Mass

• Movement of electrons in a lattice affects the mass
• f electrons, which results in a different movement
• f 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:

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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.

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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.

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Quantum Theory of Solids in 3D

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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.

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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
• f 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.

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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 = nπ/a, and therefore n = nx + ny + nz,

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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.

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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.

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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:‐

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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.

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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.

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Statistical Mechanics

• An approximation to Fermi‐Dirac function is

Maxwell‐Boltzmann where:‐

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Exercise

• 1. 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 eV

below the conduction band energy Ec. Assume T = 300K.

a)

Determine the probability of a state being

• ccupied by an electron at E = Ec + kT/4.

b)

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

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Solution

• 1. Using Fermi‐Dirac function at 3kT higher

energy level:

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Solution

• 2. 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.

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Picture Credits

• Semiconductor Physics and Devices, Donald

Neaman, 4th Edition, McGraw Hill Publications.

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

Rice University

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