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Prof. Ming-Jer ChenProf. Ming-Jer Chen

Department of Electronics EngineeringDepartment of Electronics Engineering

National Chiao-Tung UniversityNational Chiao-Tung University

09/24/201309/24/2013

DEE4521 Semiconductor Device PhysicsDEE4521 Semiconductor Device Physics

Lecture 2:Lecture 2:

Band StructureBand Structure

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According to Heisenberg’s Uncertainty Principle, According to Heisenberg’s Uncertainty Principle,

We have a We have a 6 dimensionality space at a time 6 dimensionality space at a time for a for a Semiconductor Semiconductor in a in a RealReal x-y-z Space; and x-y-z Space; and at at each point (x,y.z),each point (x,y.z),

ElectronsElectrons, , HolesHoles, , Phonons,Phonons, and and PhotonsPhotons

are all better dealt with in another space: are all better dealt with in another space:

kkxx-k-kyy-k-kzz Space Space

or Wavevector Spaceor Wavevector Space

or Momentum Space or Momentum Space

Electron Distribution Function f(x, y, z, kx, ky, kz, t)

by Analogy

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• De Broglie’s Wave and Particle Duality

• Degree of Freedom (DOF) – Kinetic Energy

• Potential Energy and its Reference

Effective Mass m*

Crystal momentum Ek = ħ2kx

2/2m*

Ball’s Mass m in x direction

Ball’s Momentum mvx

Ball’s Kinetic Energy mvx2/2

Electron Effective Mass mx* in xdirection

Crystal Momentum ħkx

(kx: wave vector in x direction)

Electron Momentum ħ(kx-kxo)

Electron Kinetic Energy Ek = ħ2(kx-kxo)2/2mx*

1. kxo: a point in k space around which electrons are likely found.

2. Crystal momentum (global) must be conserved in k space, not Electron Momentum (local).

A ball in the air Electrons in Solid

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+ ħ2kz2/2m*

+ ħ2kx2/2m*

Effective mass approximation: m* (to reflect electron confinement in solid)Ek = ħ2(ky – kcy)2/2m*

Si Conduction-Band Structure in wave vector k-space

(silicon)

Kcy 0.85 (2/a); Longitudinal Effective Mass m* (or ml*)= 0.92 mo

Transverse Effective Mass m* (or mt*)= 0.197 mo

a: Lattice Constant

6-fold valleys

Ellipsoidal energy surface

(Constant-Energy Surfaces in k-space)

E = Ek + Ec

total electron energyPotential energy

Kinetic energy

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(by Prof. Robert F. Pierret)

Effective Masses of Commonly Used Materials

Ge Si GaAsml*/mo 1.588 0.916mt*/mo 0.081 0.190me*/mo 0.067mhh*/mo 0.347 0.537 0.51mlh*/mo 0.0423 0.153 0.082mso*/mo 0.077 0.234 0.154

Electron and hole effective mass are anisotropic, depending on the orientation direction.

Electron (not hole) effective mass is isotropic, regardless of orientation.

Rest mass of electron mo = 0.9110-30 kg

(You may then find that these effective masses are far fromthe rest mass. This is just one of the quantum effects.)

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Electron Energy Electron Energy E-k Relation E-k Relation in a Crystal in a Crystal

m* 2 d2 E

dK2

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K 0

Diamonda = 5.43095 Å

Diamonda = 5.64613 Å

Zinc blendea = 5.6533 Å

( )2/a3 /2Quasi-Classical Approximation

Bottom of valley

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k-Space Definition

The zone center (Gamma at k = 0)The zone end along <100>

The zone end along <111>

<100> (in-plane)

<001>(out-of-plane)

<010>(in-plane)

(001)

Length = 2/a (Gamma to X)

Length =( )2/a (Gamma to L)

3-D View

(Principal-axis x, y, and z coordinate system usually aligned to match the k coordinate system)

On (001) Wafer

3 /2

a: Lattice Constant

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Electron E-k Diagram

Indirect gapDirect gap

EG: Energy Gap

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Conduction Band

8-fold valleys along <111>(half-ellipsoid in Brillouin)

6-fold valleys along <100> (ellipsoid)

one valley at the zone center(sphere)

(Constant-Energy Surface)

Comparisons between Different Materials

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Valence-Band Structure

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Conduction-Band Electrons andValence-Band Holes and Electrons

Hole: Vacancy of Valence-Band Electron

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No Electrons in Conduction Bands

All Valence Bands are filled up.

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(Electron Affinity) (= 4.05 eV for Si)Work Function

EcE

x