Solid-State Electronics Textbook: “Semiconductor Physics and Devices” By Donald A. Neamen, 1997 Reference: “Advanced Semiconductor Fundamentals” By Robert F. Pierret 1987 “Fundamentals of Solid-State Electronics” By C.-T. Sah, World Scientific, 1994 Homework: 0% Midterm Exam: 60% Final Exam: 40% Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Contents

Chap. 1 Chap. 2 Chap. 3 Chap. 4 Chap. 5 Chap. 6 Chap. 7

Solid-State Electronics Chap. 1

Solid State Electronics: A General Introduction Introduction to Quantum Mechanics Quantum Theory of Solids Semiconductor at Equilibrium Carrier Motions: Nonequilibrium Excess Carriers in Semiconductors Junction Diodes

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Chap 1. Solid State Electronics: A General Introduction

Introduction Classification of materials Crystalline and impure semiconductors Crystal lattices and periodic structure Reciprocal lattice

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction

Solid-state electronic materials: –

Conductors, semiconductors, and insulators,

A solid contains electrons, ions, and atoms, ~1023/cm3. ⇒ too closely packed to be described by classical Newtonian mechanics.

Extensions of Newtonian mechanics: – –

Quantum mechanics to deal with the uncertainties from small distances; Statistical mechanics to deal with the large number of particles.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Classifications of Materials According to their viscosity, materials are classified into solids, liquid, and gas phases. Solid

Liquid

Gas

Low

Medium

High

Atomic density

High

Medium

Low

Hardness

High

Medium

Low

Diffusivity

Low diffusivity, High density, and High mechanical strength means that small channel openings and high interparticle force in solids.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Classification Schemes of Solids

Geometry (Crystallinity v.s. Imperfection) Purity (Pure v.s. Impure) Electrical Classification (Electrical Conductivity) Mechanical Classification (Binding Force)

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Geometry Crystallinity – Single crystalline, polycrystalline, and amorphous

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Geometry Imperfection – A solid is imperfect when it is not crystalline (e.g., impure) or its atom are displaced from the positions on a periodic array of points (e.g., physical defect). – Defect: (Vacancy or Interstitial)

– Impurity:

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Purity Pure v.s. Impure Impurity: – chemical impurities:a solid contains a variety of randomly located foreign atoms, e.g., P in n-Si. – an array of periodically located foreign atoms is known as an impure crystal with a superlattice, e.g., GaAs

Distinction between chemical impurities and physical defects.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Electrical Conductivity

Material type

Resistivity (Ω-cm)

Conduction Electron density (cm-3)

Examples

Superconductor

0 (low T) 0 (high T)

1023

Good Conductor

10-6 – 10-5

1022 – 1023

metals: K, Na, Cu, Au

Conductor

10-5 – 10-2

1017 – 1022

semi-metal: As, B, Graphite

Semiconductor

10-2 – 10-9

106 – 1017

Ge, Si, GaAs, InP

Semi-insulator

1010 – 1014

101 – 105

Amorphous Si

Insulator

1014 – 1022

1 – 10

Solid-State Electronics Chap. 1

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Sn, Pb Oxides

SiO2, Si3N4, Instructor: Pei-Wen Li Dept. of E. E. NCU

Mechanical Classification Based on the atomic forces (binding force) that bind the atom together, the crystals could be divided into: – Crystal of Inert Gases (Low-T solid): Van der Wall Force: dipole-dipole interaction – Ionic Crystals (8 ~ 10 eV bond energy): Electrostatic force: Coulomb force, NaCl, etc. – Metal Crystals Delocalized electrons of high concentration, (1 e/atom) – Hydrogen-bonded Crystals ( 0.1 eV bond energy) H2O, Protein molecules, DNA, etc.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Binding Force Bond energy is a useful parameter to provide a qualitative gauge on whether – The binding force of the atom is strong or weak; – The bond is easy or hard to be broken by energetic electrons, holes, ions, and ionizing radiation such as high-energy photons and x-ray.

In semiconductors, bonds are covalent or slightly ionic bonds. Each bond contains two electrons—electron-pair bond.A bond is broken when one of its electron is removed by impact collision (energetic particles) or x-ray radiation, —dangling bond.

Solid-State Electronics Chap. 1

12

Instructor: Pei-Wen Li Dept. of E. E. NCU

Semiconductors for Electronic Device Application

For electronic application, semiconductors must be crystalline and must contain a well-controlled concentration of specific impurities. Crystalline semiconductors are needed so the defect density is low. Since defects are electron and hole traps where e--h+ can recombine and disappear, short lifetime. The role of impurities in semiconductors: 1. To provide a wide range of conductivity (III- B or V-P in Si). 2. To provide two types of charge carriers (electrons and holes) to carry the electrical current , or to provide two conductivity types, n-type (by electrons) and p-type (by holes) Group III and V impurities in Si are dopant impurities to provide conductive electrons and holes. However, group I, II, and VI atoms in Si are known as recombination impurities (lifetime killers)when their concentration is low. Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Crystal Lattices A crystal is a material whose atoms are situated periodically on interpenetrating arrays of points known as crystal lattice or lattice points. The following terms are useful to describe the geometry of the periodicity of crystal atoms: – Unit cell; Primitive Unit Cell – Basis vectors a, b, c ; Primitive Basic vectors – Translation vector of the lattice; Rn = n1a +n2b +n3c – Miller Indices

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Basis Vectors The simplest means of representing an atomic array is by translation. Each lattice point can be translated by basis vectors, â, bˆ , ĉ.

Translation vectors: can be mathematically represented by the basis vectors. Rn = n1 â + n2 + n3 ĉ, where n1, n2, and n3 are integers. Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Unit Cell Unit cell: is a small volume of the crystal that can be used to represent the entire crystal. (not unique)

Primitive unit cell: the smallest unit cell that can be repeated to form the lattice. (not unique) Example: FCC lattice

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Miller Indices To denote the crystal directions and planes for the 3-d crystals. Plane (h k l) Equivalent planes {h k l}

Direction [h k l] Equivalent directions

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Miller Indices To describe the plane by Miller Indices – Find the intercepts of the plane with x, y, and z axes. – Take the reciprocals of the intercepts – Multiply the lowest common denominator = Mliller indices

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Example Use of Miller Indices Wafer Specification (Wafer Flats)

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

3-D Crystal Structures In 3-d solids, there are 7 crystal systems (1) triclinic, (2) monoclinic, (3) orthorhombic, (4) hexagonal, (5) rhombohedral, (6) tetragonal, and (7) cubic systems.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

3-D Crystal Structures In 3-d solids, there 14 Bravais or space lattices.

N-fold symmetry:

⇒6-fold symmetry

With 2π/n rotation, the crystal looks the same! Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Basic Cubic Lattice Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC)

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Surface Density

Consider a BCC structure and the (110) plane, the surface density is found by dividing the number of lattice atoms by the surface area; Surface density = 2 atoms (a1 )(a1 2 )

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Diamond Structure (Cubic System) Most semiconductors are not in the 7 crystal systems mentioned above. Elemental Semiconductos: (C, Si, Ge, Sn) lattice constant a v 1 1 1 v 1 1 1 a = (− , ,− ), b = ( , ,− ) 4 4 4v 4 4 4 v a •b 1 θ = cos −1 v v ,θ = cos −1( ) = 109.4o 3 a || b |

(0,0,0)

θ=109.4o

1 1 1 ( − , ,− ) 4 4 4

1 1 1 ( , ,− ) 4 4 4

The space lattice of diamond is fcc. It is composed of two fcc lattices displaced from each other by ¼ of a body diagonal, (¼, ¼, ¼ )a Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Diamond Structure Or the diamond could be visualized by a bcc with four of the corner atoms missing.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Zinc Blende Structure (Cubic system) Compound Semiconductors: (SiC, SiGe, GaAs, GaP, InP, InAs, InSb, etc) – Has the same geometry as the diamond structure except that zinc blende crystals are binary or contains two different kinds of host atoms.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Wurzite Structure (Hexagonal system) Compound Semiconductors (ZnO, GaN, ALN, ZnS, ZnTe) – The adjacent tetrahedrons in zinc blende structure are rotated 60o to give the wurzite structure. – The distortion changes the symmetry: cubic →hexagonal – Distortion also increase the energy gap, which offers the potential for optical device applications.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Reciprocal Lattice Every crystal structure has two lattices associated with it, the crystal lattice (real space) and the reciprocal lattice (momentum space). The relationship between the crystal lattice vector ( aˆ , bˆ, cˆ ) and reciprocal lattice vector ( Aˆ , Bˆ , Cˆ ) is ˆ ˆ ˆA = 2π bxcˆ ; Bˆ = 2π cˆxaˆ ; Cˆ = 2π aˆxb aˆ ⋅ bˆxcˆ aˆ ⋅ bˆxcˆ aˆ ⋅ bˆxcˆ

The crystal lattice vectors have the dimensions of [length] and the vectors in the reciprocal lattice have the dimensions of [1/length], which means in the momentum space. (k = 2π/λ) A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Example Consider a BCC lattice and its reciprocal lattice (FCC)

Similarly, the reciprocal lattice of an FCC is BCC lattice.

Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Chap 2. Introduction to Quantum Mechanics Principles of Quantum Mechanics Schrödinger’s Wave Equation Application of Schrödinger’s Wave Equation Homework

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction In solids, there are about 1023 electrons and ions packed in a volume of 1 cm3. The consequences of this highly packing density : – Interparticle distance is very small: ~2x10-8 cm. ⇒the instantaneous position and velocity of the particle are no longer deterministic. Thus, the electrons motion in solids must be analyzed by a probability theory. Quantum mechanics ⇔Newtonian mechanics Schrodinger’s equation: to describe the position probability of a particle.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction – The force acting on the j-th particle comes from all the other 1023-1 particles. – The rate of collision between particles is very high, 1013 collisions/sec ⇒average electron motion instead of the motion of each electron at a given instance of time are interested. (Statistical Mechanics) equilibrium statistical mechanics: Fermi-Dirac quantum-distribution ⇔Boltzmann classical distribution

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Principles of Quantum Mechanics Principle of energy quanta Wave-Particle duality principle Uncertainty principle

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Energy Quanta Consider a light incident on a surface of a material as shown below:

Classical theory: as long as the intensity of light is strong enough ⇒photoelectrons will be emitted from the material. Photoelectric Effect: experimental results shows “NOT”. Observation: – as the frequency of incident light ν < νo: no electron emitted. – as ν > νo:at const. frequency, intensity↑, emission rate↑, K.E. unchanged. at const. intensity, the max. K. E. ∝ the frequency of incident light. Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Quanta and Photon Planck postulated that thermal radiation is emitted from a heated surface in discrete energy called quanta. The energy of these quanta is given by E = hν, h = 6.625 x 10-34 J-sec (Planck’s constant) According to the photoelectric results, Einstein suggested that the energy in a light wave is also contained in discrete packets called photon whose energy is also given by E = hν. The maximum K.E. of the photoelectron is Tmax = ½mv2 = hν - hνo The momentum of a photon, p = h/λ

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Wave-Particle Duality de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. de Broglie suggested that the wavelength of a particle is expressed as λ = h /p, where p is the momentum of a particle Davisson-Germer experimentally proved de Broglie postulation of “Wave Nature of Electrons”.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Davisson-Germer Experiment Consider the experimental setup below:

Observation: – the existence of a peak in the density of scattered electrons can be explained as a constructive interference of waves scattered by the periodic atoms. – the angular distribution of the deflected electrons is very similar to an interference pattern produced by light diffracted from a grating. Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Conclusion In some cases, EM wave behaves like particles (photons) and sometimes particles behave as if they are waves. ⇒Wave-particle duality principle applies primarily to SMALL particles, e.g., electrons, protons, neutrons. For large particles, classical mechanics still apply.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Uncertainty Principle Heisenberg states that we cannot describe with absolute accuracy the behavior of the subatomic particles. 1. It is impossible to simultaneously describe with the absolute accuracy the position and momentum of a particle. ∆p ∆x ≥ ħ. (ħ = h/2π = 1.054x10-34 J-sec) 2. It is impossible to simultaneously describe with the absolute accuracy the energy of a particle and the instant of time the particle has this energy. ∆E ∆t ≥ ħ The uncertainty principle implies that these simultaneous measurements are in error to a certain extent. However, ħ is very small, the uncertainty principle is only significant for small particles.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Schrodinger’s Wave Equation Based on the principle of quanta and the wave-particle duality principle, Schrodinger’s equation describes the motion of electrons in a crystal. 1-D Schrodinger’s equation, − h 2 ∂ 2 Ψ ( x, t ) ∂Ψ ( x, t ) V x x t j ⋅ + ( ) Ψ ( , ) = h ∂x 2 ∂t 2m

Where Ψ(x,t) is the wave function, which is used to describe the behavior of the system, and mathematically can be a complex quantity. V(x) is the potential function. Assume the wave function Ψ(x,t) = ψ(x)φ(t), then the Schrodinger eq. Becomes − h2 ∂ 2ψ ( x) ∂φ (t ) 2m Solid-State Electronics Chap. 2

φ (t )

∂x 2

+ V ( x)ψ ( x)φ (t ) = jhψ ( x) 11

∂t

Instructor: Pei-Wen Li Dept. of E. E. NCU

Schrodinger’s Wave Equation − h 2 1 ∂ 2ψ ( x) 1 ∂φ (t ) + V ( x ) = j h =E 2 φ (t ) ∂t 2m ψ ( x) ∂x

where E is the total energy, and the solution of the eq. is and the time-indep. Schrodinger equation can be written as φ (t ) = e − j ( E / h )t ∂ 2ψ ( x) 2m + 2 ( E − V ( x))ψ ( x) = 0 h ∂x 2

The physical meaning of wave function: – Ψ(x,t) is a complex function, so it can not by itself represent a real physical quantity. – |Ψ2(x,t)| is the probability of finding the particle between x and x+dx at a given time, or is a probability density function. – |Ψ2(x,t)|= Ψ(x,t) Ψ*(x,t) =ψ(x)* ψ(x) = |ψ(x)|2 -- indep. of time Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Boundary Conditions

∫

∞

2

ψ ( x) dx = 1 since |ψ(x)|2 represents the probability density function, then for a single particle, the probability of finding the particle somewhere is certain. If the total energy E and the potential V(x) are finite everywhere, 2. ψ(x) must be finite, single-valued, and continuous. 3. ∂ψ(x)/∂x must be finite, single-valued, and continuous.

1.

−∞

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Applications of Schrodinger’s Eq. The infinite Potential Well

In region I, III, ψ(x) = 0, since E is finite and a particle cannot penetrate the infinite potential barriers. In region II, the particle is contained within a finite region of space and V = 0. 1-D time-indep. Schrodinger’s eq. becomes ∂ 2ψ ( x) 2mE + 2 ψ ( x) = 0 h ∂x 2

the solution is given by

ψ ( x) = A1 cos Kx + A2 sin Kx, where K = Solid-State Electronics Chap. 2

14

2mE h2

Instructor: Pei-Wen Li Dept. of E. E. NCU

Infinite Potential Well Boundary conditions: 1. ψ(x) must be continuous, so that ψ(x = 0) = ψ(x = a) = 0 ⇒A1 = A2sinKa ≡ 0 ⇒ K = nπ/a, where n is a positive integer. 2.

∞

2

a 2 2 2 ψ ( x ) dx = 1 sin 1 ⇒ = ⇒ = A Kxdx A 2 2 ∫−∞ ∫

a

0

So the time-indep. Wave equation is given by ψ ( x) =

2 nπx sin( ) where n = 1,2,3... a a

The solution represents the electron in the infinite potential well is in a standing waveform. The parameter K is related to the total energy E, therefore, h 2 n 2π 2 E = En =

Solid-State Electronics Chap. 2

2ma

2

where n is a positive integer

15

Instructor: Pei-Wen Li Dept. of E. E. NCU

Infinite Potential Well That means that the energy of the particle in the infinite potential well is “quantized”. That is, the energy of the particle can only have particular discrete values.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Step Potential Function Consider a particle being incident on a step potential barrier:

In region I, V = 0, And the general solution of this equation is

∂ 2ψ 1 ( x) 2mE + 2 ψ 1 ( x) = 0 2 ∂x h

ψ 1 ( x) = A1e jK x + B1e − jK x ( x ≤ 0) where K1 = 1

1

2mE h2

In region II, V = Vo, if we assume E < Vo, then ∂ 2ψ 2 ( x) ∂x Solid-State Electronics Chap. 2

2

−

2m h

2

(Vo − E )ψ 2 ( x) = 0 17

Instructor: Pei-Wen Li Dept. of E. E. NCU

The Step Potential Function The general solution is in the form ψ 2 ( x) = A2e − K 2 x + B2e + K 2 x ( x ≥ 0) where K 2 =

Boundary Conditions:

2m(Vo − E ) h2

ψ 2 ( x) = A2e − K 2 x ( x ≥ 0)

– ψ2(x) must remain finite, ⇒B2 ≡ 0 ⇒ – ψ(x) must be continuous, i.e., ψ1(x = 0) = ψ2(x = 0) ⇒A1+B1 = A2 – ∂ψ(x)/ ∂x must be continuous, i.e., ∂ψ 1 ∂x

= x =0

∂ψ 21 ∂x

⇒ jK1 A1 − jK1 B1 = − K 2 A2 x =0

A1, B1, and A2 could be solved from the above equations.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Potential Barrier Consider the potential barrier function as shown: Assume the total energy of an incident particle E < Vo, as before, we could solve the Schrodinger’s equations in each region, and obtain ψ 1 ( x) = A1e jK x + B1e − jK x 1

1

ψ 2 ( x) = A2e K 2 x + B2e − K 2 x

where K1 =

ψ 3 ( x) = A3e jK x + B3e − jK x 1

1

2m(Vo − E ) 2mE and K = 2 h2 h2

We can solve B1, A2, B2, and A3 in terms of A1 from boundary conditions: – B3 = 0 , once a particle enters in region III, there is no potential changes to cause a reflection, therefore, B3 must be zero. – At x = 0 and x = a, the corresponding wave function and its first derivative must be continuous. Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Potential Barrier

The results implies that there is a finite probability that a particle will penetrate the barrier, that is so called “tunneling”. * A ⋅ A 3 3 The transmission coefficient is defined by T = A1 ⋅ A1* If E

1

Instructor: Pei-Wen Li Dept. of E. E. NCU

Contents

Chap. 1 Chap. 2 Chap. 3 Chap. 4 Chap. 5 Chap. 6 Chap. 7

Solid-State Electronics Chap. 1

Solid State Electronics: A General Introduction Introduction to Quantum Mechanics Quantum Theory of Solids Semiconductor at Equilibrium Carrier Motions: Nonequilibrium Excess Carriers in Semiconductors Junction Diodes

2

Instructor: Pei-Wen Li Dept. of E. E. NCU

Chap 1. Solid State Electronics: A General Introduction

Introduction Classification of materials Crystalline and impure semiconductors Crystal lattices and periodic structure Reciprocal lattice

Solid-State Electronics Chap. 1

3

Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction

Solid-state electronic materials: –

Conductors, semiconductors, and insulators,

A solid contains electrons, ions, and atoms, ~1023/cm3. ⇒ too closely packed to be described by classical Newtonian mechanics.

Extensions of Newtonian mechanics: – –

Quantum mechanics to deal with the uncertainties from small distances; Statistical mechanics to deal with the large number of particles.

Solid-State Electronics Chap. 1

4

Instructor: Pei-Wen Li Dept. of E. E. NCU

Classifications of Materials According to their viscosity, materials are classified into solids, liquid, and gas phases. Solid

Liquid

Gas

Low

Medium

High

Atomic density

High

Medium

Low

Hardness

High

Medium

Low

Diffusivity

Low diffusivity, High density, and High mechanical strength means that small channel openings and high interparticle force in solids.

Solid-State Electronics Chap. 1

5

Instructor: Pei-Wen Li Dept. of E. E. NCU

Classification Schemes of Solids

Geometry (Crystallinity v.s. Imperfection) Purity (Pure v.s. Impure) Electrical Classification (Electrical Conductivity) Mechanical Classification (Binding Force)

Solid-State Electronics Chap. 1

6

Instructor: Pei-Wen Li Dept. of E. E. NCU

Geometry Crystallinity – Single crystalline, polycrystalline, and amorphous

Solid-State Electronics Chap. 1

7

Instructor: Pei-Wen Li Dept. of E. E. NCU

Geometry Imperfection – A solid is imperfect when it is not crystalline (e.g., impure) or its atom are displaced from the positions on a periodic array of points (e.g., physical defect). – Defect: (Vacancy or Interstitial)

– Impurity:

Solid-State Electronics Chap. 1

8

Instructor: Pei-Wen Li Dept. of E. E. NCU

Purity Pure v.s. Impure Impurity: – chemical impurities:a solid contains a variety of randomly located foreign atoms, e.g., P in n-Si. – an array of periodically located foreign atoms is known as an impure crystal with a superlattice, e.g., GaAs

Distinction between chemical impurities and physical defects.

Solid-State Electronics Chap. 1

9

Instructor: Pei-Wen Li Dept. of E. E. NCU

Electrical Conductivity

Material type

Resistivity (Ω-cm)

Conduction Electron density (cm-3)

Examples

Superconductor

0 (low T) 0 (high T)

1023

Good Conductor

10-6 – 10-5

1022 – 1023

metals: K, Na, Cu, Au

Conductor

10-5 – 10-2

1017 – 1022

semi-metal: As, B, Graphite

Semiconductor

10-2 – 10-9

106 – 1017

Ge, Si, GaAs, InP

Semi-insulator

1010 – 1014

101 – 105

Amorphous Si

Insulator

1014 – 1022

1 – 10

Solid-State Electronics Chap. 1

10

Sn, Pb Oxides

SiO2, Si3N4, Instructor: Pei-Wen Li Dept. of E. E. NCU

Mechanical Classification Based on the atomic forces (binding force) that bind the atom together, the crystals could be divided into: – Crystal of Inert Gases (Low-T solid): Van der Wall Force: dipole-dipole interaction – Ionic Crystals (8 ~ 10 eV bond energy): Electrostatic force: Coulomb force, NaCl, etc. – Metal Crystals Delocalized electrons of high concentration, (1 e/atom) – Hydrogen-bonded Crystals ( 0.1 eV bond energy) H2O, Protein molecules, DNA, etc.

Solid-State Electronics Chap. 1

11

Instructor: Pei-Wen Li Dept. of E. E. NCU

Binding Force Bond energy is a useful parameter to provide a qualitative gauge on whether – The binding force of the atom is strong or weak; – The bond is easy or hard to be broken by energetic electrons, holes, ions, and ionizing radiation such as high-energy photons and x-ray.

In semiconductors, bonds are covalent or slightly ionic bonds. Each bond contains two electrons—electron-pair bond.A bond is broken when one of its electron is removed by impact collision (energetic particles) or x-ray radiation, —dangling bond.

Solid-State Electronics Chap. 1

12

Instructor: Pei-Wen Li Dept. of E. E. NCU

Semiconductors for Electronic Device Application

For electronic application, semiconductors must be crystalline and must contain a well-controlled concentration of specific impurities. Crystalline semiconductors are needed so the defect density is low. Since defects are electron and hole traps where e--h+ can recombine and disappear, short lifetime. The role of impurities in semiconductors: 1. To provide a wide range of conductivity (III- B or V-P in Si). 2. To provide two types of charge carriers (electrons and holes) to carry the electrical current , or to provide two conductivity types, n-type (by electrons) and p-type (by holes) Group III and V impurities in Si are dopant impurities to provide conductive electrons and holes. However, group I, II, and VI atoms in Si are known as recombination impurities (lifetime killers)when their concentration is low. Solid-State Electronics Chap. 1

13

Instructor: Pei-Wen Li Dept. of E. E. NCU

Crystal Lattices A crystal is a material whose atoms are situated periodically on interpenetrating arrays of points known as crystal lattice or lattice points. The following terms are useful to describe the geometry of the periodicity of crystal atoms: – Unit cell; Primitive Unit Cell – Basis vectors a, b, c ; Primitive Basic vectors – Translation vector of the lattice; Rn = n1a +n2b +n3c – Miller Indices

Solid-State Electronics Chap. 1

14

Instructor: Pei-Wen Li Dept. of E. E. NCU

Basis Vectors The simplest means of representing an atomic array is by translation. Each lattice point can be translated by basis vectors, â, bˆ , ĉ.

Translation vectors: can be mathematically represented by the basis vectors. Rn = n1 â + n2 + n3 ĉ, where n1, n2, and n3 are integers. Solid-State Electronics Chap. 1

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Unit Cell Unit cell: is a small volume of the crystal that can be used to represent the entire crystal. (not unique)

Primitive unit cell: the smallest unit cell that can be repeated to form the lattice. (not unique) Example: FCC lattice

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Miller Indices To denote the crystal directions and planes for the 3-d crystals. Plane (h k l) Equivalent planes {h k l}

Direction [h k l] Equivalent directions

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Miller Indices To describe the plane by Miller Indices – Find the intercepts of the plane with x, y, and z axes. – Take the reciprocals of the intercepts – Multiply the lowest common denominator = Mliller indices

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Example Use of Miller Indices Wafer Specification (Wafer Flats)

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3-D Crystal Structures In 3-d solids, there are 7 crystal systems (1) triclinic, (2) monoclinic, (3) orthorhombic, (4) hexagonal, (5) rhombohedral, (6) tetragonal, and (7) cubic systems.

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3-D Crystal Structures In 3-d solids, there 14 Bravais or space lattices.

N-fold symmetry:

⇒6-fold symmetry

With 2π/n rotation, the crystal looks the same! Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Basic Cubic Lattice Simple Cubic (SC), Body-Centered Cubic (BCC), and Face-Centered Cubic (FCC)

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Surface Density

Consider a BCC structure and the (110) plane, the surface density is found by dividing the number of lattice atoms by the surface area; Surface density = 2 atoms (a1 )(a1 2 )

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Diamond Structure (Cubic System) Most semiconductors are not in the 7 crystal systems mentioned above. Elemental Semiconductos: (C, Si, Ge, Sn) lattice constant a v 1 1 1 v 1 1 1 a = (− , ,− ), b = ( , ,− ) 4 4 4v 4 4 4 v a •b 1 θ = cos −1 v v ,θ = cos −1( ) = 109.4o 3 a || b |

(0,0,0)

θ=109.4o

1 1 1 ( − , ,− ) 4 4 4

1 1 1 ( , ,− ) 4 4 4

The space lattice of diamond is fcc. It is composed of two fcc lattices displaced from each other by ¼ of a body diagonal, (¼, ¼, ¼ )a Solid-State Electronics Chap. 1

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Diamond Structure Or the diamond could be visualized by a bcc with four of the corner atoms missing.

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Zinc Blende Structure (Cubic system) Compound Semiconductors: (SiC, SiGe, GaAs, GaP, InP, InAs, InSb, etc) – Has the same geometry as the diamond structure except that zinc blende crystals are binary or contains two different kinds of host atoms.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Wurzite Structure (Hexagonal system) Compound Semiconductors (ZnO, GaN, ALN, ZnS, ZnTe) – The adjacent tetrahedrons in zinc blende structure are rotated 60o to give the wurzite structure. – The distortion changes the symmetry: cubic →hexagonal – Distortion also increase the energy gap, which offers the potential for optical device applications.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Reciprocal Lattice Every crystal structure has two lattices associated with it, the crystal lattice (real space) and the reciprocal lattice (momentum space). The relationship between the crystal lattice vector ( aˆ , bˆ, cˆ ) and reciprocal lattice vector ( Aˆ , Bˆ , Cˆ ) is ˆ ˆ ˆA = 2π bxcˆ ; Bˆ = 2π cˆxaˆ ; Cˆ = 2π aˆxb aˆ ⋅ bˆxcˆ aˆ ⋅ bˆxcˆ aˆ ⋅ bˆxcˆ

The crystal lattice vectors have the dimensions of [length] and the vectors in the reciprocal lattice have the dimensions of [1/length], which means in the momentum space. (k = 2π/λ) A diffraction pattern of a crystal is a map of the reciprocal lattice of the crystal.

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Example Consider a BCC lattice and its reciprocal lattice (FCC)

Similarly, the reciprocal lattice of an FCC is BCC lattice.

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Chap 2. Introduction to Quantum Mechanics Principles of Quantum Mechanics Schrödinger’s Wave Equation Application of Schrödinger’s Wave Equation Homework

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction In solids, there are about 1023 electrons and ions packed in a volume of 1 cm3. The consequences of this highly packing density : – Interparticle distance is very small: ~2x10-8 cm. ⇒the instantaneous position and velocity of the particle are no longer deterministic. Thus, the electrons motion in solids must be analyzed by a probability theory. Quantum mechanics ⇔Newtonian mechanics Schrodinger’s equation: to describe the position probability of a particle.

Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Introduction – The force acting on the j-th particle comes from all the other 1023-1 particles. – The rate of collision between particles is very high, 1013 collisions/sec ⇒average electron motion instead of the motion of each electron at a given instance of time are interested. (Statistical Mechanics) equilibrium statistical mechanics: Fermi-Dirac quantum-distribution ⇔Boltzmann classical distribution

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Principles of Quantum Mechanics Principle of energy quanta Wave-Particle duality principle Uncertainty principle

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Energy Quanta Consider a light incident on a surface of a material as shown below:

Classical theory: as long as the intensity of light is strong enough ⇒photoelectrons will be emitted from the material. Photoelectric Effect: experimental results shows “NOT”. Observation: – as the frequency of incident light ν < νo: no electron emitted. – as ν > νo:at const. frequency, intensity↑, emission rate↑, K.E. unchanged. at const. intensity, the max. K. E. ∝ the frequency of incident light. Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Quanta and Photon Planck postulated that thermal radiation is emitted from a heated surface in discrete energy called quanta. The energy of these quanta is given by E = hν, h = 6.625 x 10-34 J-sec (Planck’s constant) According to the photoelectric results, Einstein suggested that the energy in a light wave is also contained in discrete packets called photon whose energy is also given by E = hν. The maximum K.E. of the photoelectron is Tmax = ½mv2 = hν - hνo The momentum of a photon, p = h/λ

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Wave-Particle Duality de Broglie postulated the existence of matter waves. He suggested that since waves exhibit particle-like behavior, then particles should be expected to show wave-like properties. de Broglie suggested that the wavelength of a particle is expressed as λ = h /p, where p is the momentum of a particle Davisson-Germer experimentally proved de Broglie postulation of “Wave Nature of Electrons”.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Davisson-Germer Experiment Consider the experimental setup below:

Observation: – the existence of a peak in the density of scattered electrons can be explained as a constructive interference of waves scattered by the periodic atoms. – the angular distribution of the deflected electrons is very similar to an interference pattern produced by light diffracted from a grating. Solid-State Electronics Chap. 2

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Conclusion In some cases, EM wave behaves like particles (photons) and sometimes particles behave as if they are waves. ⇒Wave-particle duality principle applies primarily to SMALL particles, e.g., electrons, protons, neutrons. For large particles, classical mechanics still apply.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Uncertainty Principle Heisenberg states that we cannot describe with absolute accuracy the behavior of the subatomic particles. 1. It is impossible to simultaneously describe with the absolute accuracy the position and momentum of a particle. ∆p ∆x ≥ ħ. (ħ = h/2π = 1.054x10-34 J-sec) 2. It is impossible to simultaneously describe with the absolute accuracy the energy of a particle and the instant of time the particle has this energy. ∆E ∆t ≥ ħ The uncertainty principle implies that these simultaneous measurements are in error to a certain extent. However, ħ is very small, the uncertainty principle is only significant for small particles.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Schrodinger’s Wave Equation Based on the principle of quanta and the wave-particle duality principle, Schrodinger’s equation describes the motion of electrons in a crystal. 1-D Schrodinger’s equation, − h 2 ∂ 2 Ψ ( x, t ) ∂Ψ ( x, t ) V x x t j ⋅ + ( ) Ψ ( , ) = h ∂x 2 ∂t 2m

Where Ψ(x,t) is the wave function, which is used to describe the behavior of the system, and mathematically can be a complex quantity. V(x) is the potential function. Assume the wave function Ψ(x,t) = ψ(x)φ(t), then the Schrodinger eq. Becomes − h2 ∂ 2ψ ( x) ∂φ (t ) 2m Solid-State Electronics Chap. 2

φ (t )

∂x 2

+ V ( x)ψ ( x)φ (t ) = jhψ ( x) 11

∂t

Instructor: Pei-Wen Li Dept. of E. E. NCU

Schrodinger’s Wave Equation − h 2 1 ∂ 2ψ ( x) 1 ∂φ (t ) + V ( x ) = j h =E 2 φ (t ) ∂t 2m ψ ( x) ∂x

where E is the total energy, and the solution of the eq. is and the time-indep. Schrodinger equation can be written as φ (t ) = e − j ( E / h )t ∂ 2ψ ( x) 2m + 2 ( E − V ( x))ψ ( x) = 0 h ∂x 2

The physical meaning of wave function: – Ψ(x,t) is a complex function, so it can not by itself represent a real physical quantity. – |Ψ2(x,t)| is the probability of finding the particle between x and x+dx at a given time, or is a probability density function. – |Ψ2(x,t)|= Ψ(x,t) Ψ*(x,t) =ψ(x)* ψ(x) = |ψ(x)|2 -- indep. of time Solid-State Electronics Chap. 2

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Boundary Conditions

∫

∞

2

ψ ( x) dx = 1 since |ψ(x)|2 represents the probability density function, then for a single particle, the probability of finding the particle somewhere is certain. If the total energy E and the potential V(x) are finite everywhere, 2. ψ(x) must be finite, single-valued, and continuous. 3. ∂ψ(x)/∂x must be finite, single-valued, and continuous.

1.

−∞

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Applications of Schrodinger’s Eq. The infinite Potential Well

In region I, III, ψ(x) = 0, since E is finite and a particle cannot penetrate the infinite potential barriers. In region II, the particle is contained within a finite region of space and V = 0. 1-D time-indep. Schrodinger’s eq. becomes ∂ 2ψ ( x) 2mE + 2 ψ ( x) = 0 h ∂x 2

the solution is given by

ψ ( x) = A1 cos Kx + A2 sin Kx, where K = Solid-State Electronics Chap. 2

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2mE h2

Instructor: Pei-Wen Li Dept. of E. E. NCU

Infinite Potential Well Boundary conditions: 1. ψ(x) must be continuous, so that ψ(x = 0) = ψ(x = a) = 0 ⇒A1 = A2sinKa ≡ 0 ⇒ K = nπ/a, where n is a positive integer. 2.

∞

2

a 2 2 2 ψ ( x ) dx = 1 sin 1 ⇒ = ⇒ = A Kxdx A 2 2 ∫−∞ ∫

a

0

So the time-indep. Wave equation is given by ψ ( x) =

2 nπx sin( ) where n = 1,2,3... a a

The solution represents the electron in the infinite potential well is in a standing waveform. The parameter K is related to the total energy E, therefore, h 2 n 2π 2 E = En =

Solid-State Electronics Chap. 2

2ma

2

where n is a positive integer

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Instructor: Pei-Wen Li Dept. of E. E. NCU

Infinite Potential Well That means that the energy of the particle in the infinite potential well is “quantized”. That is, the energy of the particle can only have particular discrete values.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Step Potential Function Consider a particle being incident on a step potential barrier:

In region I, V = 0, And the general solution of this equation is

∂ 2ψ 1 ( x) 2mE + 2 ψ 1 ( x) = 0 2 ∂x h

ψ 1 ( x) = A1e jK x + B1e − jK x ( x ≤ 0) where K1 = 1

1

2mE h2

In region II, V = Vo, if we assume E < Vo, then ∂ 2ψ 2 ( x) ∂x Solid-State Electronics Chap. 2

2

−

2m h

2

(Vo − E )ψ 2 ( x) = 0 17

Instructor: Pei-Wen Li Dept. of E. E. NCU

The Step Potential Function The general solution is in the form ψ 2 ( x) = A2e − K 2 x + B2e + K 2 x ( x ≥ 0) where K 2 =

Boundary Conditions:

2m(Vo − E ) h2

ψ 2 ( x) = A2e − K 2 x ( x ≥ 0)

– ψ2(x) must remain finite, ⇒B2 ≡ 0 ⇒ – ψ(x) must be continuous, i.e., ψ1(x = 0) = ψ2(x = 0) ⇒A1+B1 = A2 – ∂ψ(x)/ ∂x must be continuous, i.e., ∂ψ 1 ∂x

= x =0

∂ψ 21 ∂x

⇒ jK1 A1 − jK1 B1 = − K 2 A2 x =0

A1, B1, and A2 could be solved from the above equations.

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Potential Barrier Consider the potential barrier function as shown: Assume the total energy of an incident particle E < Vo, as before, we could solve the Schrodinger’s equations in each region, and obtain ψ 1 ( x) = A1e jK x + B1e − jK x 1

1

ψ 2 ( x) = A2e K 2 x + B2e − K 2 x

where K1 =

ψ 3 ( x) = A3e jK x + B3e − jK x 1

1

2m(Vo − E ) 2mE and K = 2 h2 h2

We can solve B1, A2, B2, and A3 in terms of A1 from boundary conditions: – B3 = 0 , once a particle enters in region III, there is no potential changes to cause a reflection, therefore, B3 must be zero. – At x = 0 and x = a, the corresponding wave function and its first derivative must be continuous. Solid-State Electronics Chap. 2

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Instructor: Pei-Wen Li Dept. of E. E. NCU

The Potential Barrier

The results implies that there is a finite probability that a particle will penetrate the barrier, that is so called “tunneling”. * A ⋅ A 3 3 The transmission coefficient is defined by T = A1 ⋅ A1* If E