Solid-State Electronics

Solid-State Electronics Chap. 1

Instructor: Pei-Wen LiDept. of E. E. NCU

1

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%



2

Contents

Chap. 1 Solid State Electronics: A General IntroductionChap. 2 Introduction to Quantum Mechanics Chap. 3 Quantum Theory of SolidsChap. 4 Semiconductor at EquilibriumChap. 5 Carrier Motions: Chap. 6 Nonequilibrium Excess Carriers in SemiconductorsChap. 7 Junction Diodes　



3

Chap 1. Solid State Electronics: A General Introduction

IntroductionClassification of materialsCrystalline and impure semiconductorsCrystal lattices and periodic structureReciprocal lattice



4

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.



5

Classifications of Materials

According to their viscosity, materials are classified into solids, liquid, and gas phases.

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

Hardness

Atomic density

Diffusivity

High

High

Low

Solid

Medium

Medium

Medium

Liquid

Low

Low

High

Gas



6

Classification Schemes of Solids

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



7

Geometry

Crystallinity– Single crystalline, polycrystalline, and amorphous



8

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:



9

Purity

Pure v.s. ImpureImpurity:

– 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.



10

Electrical Conductivity

SiO2, Si3N4, 1 – 101014 – 1022Insulator

Amorphous Si101 – 1051010 – 1014Semi-insulator

Ge, Si, GaAs, InP106 – 101710-2 – 10-9Semiconductor

semi-metal: As, B, Graphite

1017 – 102210-5 – 10-2Conductor

metals: K, Na, Cu, Au1022 – 102310-6 – 10-5Good Conductor

Sn, PbOxides

10230 (low T)0 (high T)

Superconductor

ExamplesConduction Electron density (cm-3)

Resistivity (Ω-cm)

Material type



11

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 CrystalsDelocalized electrons of high concentration, (1 e/atom)

– Hydrogen-bonded Crystals ( 0.1 eV bond energy)H2O, Protein molecules, DNA, etc.



12

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.



13

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.



14

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



15

Basis Vectors

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

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

b



16

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



17

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 <h k l>



18

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



19

Example Use of Miller Indices

Wafer Specification (Wafer Flats)



20

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.



21

3-D Crystal Structures

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

⇒6-fold symmetryN-fold symmetry:

With 2π/n rotation, the crystal looks the same!



22

Basic Cubic Lattice

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



23

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 2

11 aa



24

Diamond Structure (Cubic System)

Most semiconductors are not in the 7 crystal systems mentioned above.Elemental Semiconductos: (C, Si, Ge, Sn)

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

lattice constant a

θ=109.4oobaba

b

4.109)31(cos,

|||cos

)41,

41,

41(),

41,

41,

41(

11 ==•

−=−−

−− θvv

vv

v

)0,0,0( )41,

41,

41( −−

)41,

41,

41( −

a

=

=

θ

v



25

Diamond Structure

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



26

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.



27

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.



28

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 ( ) and reciprocal lattice vector ( ) is

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.

cba ˆ ,ˆ ,ˆCBA ˆ ,ˆ ,ˆ

cbabaC

cbaacB

cbacbA

ˆxˆˆ

ˆxˆ2ˆ ;ˆxˆˆ

ˆxˆ2ˆ ;ˆxˆˆ

ˆxˆ2ˆ⋅

=⋅

=⋅

= πππ



29

Example

Consider a BCC lattice and its reciprocal lattice (FCC)

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



1

Chap 2. Introduction to Quantum Mechanics

Principles of Quantum MechanicsSchrödinger’s Wave EquationApplication of Schrödinger’s Wave EquationHomework



2

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 mechanicsSchrodinger’s equation: to describe the position probability of a particle.



3

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



4

Principles of Quantum Mechanics

Principle of energy quantaWave-Particle duality principleUncertainty principle



5

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.



6

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



7

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”.



8

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.



9

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.



10

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 ≥ ħThe 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.



11

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,

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

ttxjtxxV

xtx

m ∂Ψ∂

=Ψ+∂Ψ∂

⋅− ),(),()(),(2 2

22

hh

ttxjtxxV

xxt

m ∂∂

=+∂

∂− )()()()()()()(2 2

22 φψφψψφ hh



12

Schrodinger’s Wave Equation

where E is the total energy, and the solution of the eq. isand the time-indep. Schrodinger equation can be written as

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

Ett

tjxV

xx

xm=

∂∂

=+∂

∂− )()(

1)()()(

12 2

22 φφ

ψψ

hh

tEjet )/()( h−=φ

0)())((2)(22

2

=−+∂

∂ xxVEmx

x ψψh



13

Boundary Conditions

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)(2

=∫∞

∞−dxxψ



14

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

the solution is given by

0)(2)(22

2

=+∂

∂ xmEx

x ψψh

2212 where,sincos)(h

mEKKxAKxAx =+=ψ



15

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.

So the time-indep. Wave equation is given by

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,

1)(2

=∫∞

∞−dxxψ

aAKxdxA

a 21sin 22

0

22 =⇒=⇒ ∫

...3,2,1 where)sin(2)( == na

xna

x πψ

integer positive a isn where2 2

222

manEE nπh

==



16

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.



17

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

In region II, V = Vo, if we assume E < Vo, then

0)(2)(122

12

=+∂

∂ xmEx

x ψψh

211112 where)0( )( 11

h

mEKxeBeAx xjKxjK =≤+= −ψ

0)()(2)(222

22

=−−∂

∂ xEVm

x

xo ψψ

h



18

The Step Potential Function

The general solution is in the form

Boundary Conditions:– 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.,

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

22222)(2 where)0( )( 22

h

EVmKxeBeAx oxKxK −=≥+= +−ψ

)0( )( 222 ≥= − xeAx xKψ

0

21

0

1

== ∂∂

=∂∂

xx xxψψ

221111 AKBjKAjK −=−⇒



19

The Potential Barrier

Consider the potential barrier function as shown:Assume the total energy of an incident particleE < Vo, as before, we could solve the

Schrodinger’s equations in each region, and obtain

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.

xjKxjK eBeAx 11111 )( −+=ψ

2221)(2 and 2 where

hh

EVmKmEK o −== )( 22 222

xKxK eBeAx −+=ψxjKxjK eBeAx 11

333 )( −+=ψ



20

The Potential Barrier

The results implies that there is a finite probability that a particle will penetrate the barrier, that is so called “tunneling”.The transmission coefficient is defined byIf E<<Vo,

This phenomenon is called “tunneling” and it violates classical mechanics.

*11

*33

AAAAT⋅⋅

=

( )aKVE

VET

oo22exp116 −

−

≅



21

One-Electron Atom

Consider the one-electron atom potential function due to the coulomb attraction between the proton and electron:

Then we can generalize the Schrodinger’s eq. to 3-D in spherical coordinates:

Assume the solution to the equation can be written as

Then the solution Φ is of the form, Φ = ejmφ, where m is an integer.

rerV

oπε4)(

2−=

0))((2)(sinsin1

sin1)(1

2222

2

222

2 =−+∂∂⋅

∂∂

⋅+∂∂⋅+

∂∂

∂∂

⋅ ψθψθ

θθφψ

θψ rVEm

rrrr

rro

h

)()()(),,( φθφθψ Φ⋅Θ⋅= rRr



22

One-Electron Atom

Similarly, we can generate two additional constants n and l for the variables θ and r. n, l, and m are known as quantum numbers (integers)

, each set of quantum numbers corresponds to a quantum state which the electron may occupy.

The solution of the wave equation is designated by ψnlm. For the lowest energy state (n=1, l=0, m=0),

The electron energy E is quantized,

0,...,1, −=

0,...,3,2,1 −−−=

llmnnnl

,...3,2,1=n

angstrom 529.0 where11 /2/3

100 =

⋅= −

oar

o

aea

o

πψ

( ) 222

4

24 nemE

o

on

hπε−

=



23

One Electron Atom

The probability density function, or the probability of finding the electron at a particular distance form the nucleus, is proportional to ψ100ψ*100 and also to the differential volume of the shell around the nucleus.

The electron is not localized at a given radius.



24

Homework

2.12.152.23



1

Chap 3. Introduction to Quantum Theory of Solids

Allowed and Forbidden Energy Bandsk-space DiagramsElectrical Conduction in SolidsDensity of State FunctionsStatistical MechanicsHomework



2

Preview

Recall from the previous analysis that the energy of a bound electron is quantized. And for the one-electron atom, the probability of finding the electron at a particular distance from the nucleus is not localized at a given radius. Consider two atoms that are in close proximity to each other. The wave functions of the two atom electrons overlap, which means that the two electrons will interact. This interaction results in the discrete quantized energy level splitting into two discrete energy levels.



3

Formation of Energy Bands

Consider a regular periodic arrangement of atoms in which each atoms contains more than one electron. If the atoms are initially far apart, the electrons in adjacent atoms will not interact and will occupy the discrete energy levels. If the atoms are brought closer enough, the outmost electrons will interact and the energy levels will split into a band of allowed energies.



4

Formation of Energy Bands



5

Kronig-Penny Model

The concept of allowed and forbidden energy levels can be developed by considering Schrodinger’s equation.

Kronig-Penny Model



6

Kronig-Penny Model

The Kronig-Penny model is an idealized periodic potential representing a 1-D single crystal.We need to solve Schrodinger’s equation in each region. To obtain the solution to the Schrodinger’s equation, we make use of Bloch theorem. Bloch states that all one-electron wave functions, involving periodically varying potential energy functions, must be of the form, ψ(x) = u(x)ejkx, u(x) is a periodic function with period (a+b) and k is called a constant of the motion.The total wave function Ψ(x,t) may be written as Ψ(x,t) = u(x)ej(kx-(E/ħ)t).

In region I (0 < x < a), V(x) = 0, then Schrodinger’s equation becomes

2 , 0)()()(2)(2

21

22112

h

mExukdx

xdujkdx

xud≡=−−+ αα



7

Kronig-Penny Model

The solution in region I is of the form,

In region II (-b < x < 0), V(x) = Vo, and apply Schrodinger’s eq.

The solution for region II is of the form,

Boundary conditions:

axBeAexu xkjxkj <<+= +−− 0for )( )()(1

αα

2 , 0)()()(2)(2

222

22222

hooVmxuk

dxxdujk

dxxud

−≡=−−+ αββ

0for )( )()(2 <<+= +−− x-bDeCexu xkjxkj ββ

0)0()0( 21 =−−+⇒= DCBAuu

( ) ( ) ( ) ( ) 00

2

0

1 =++−−+−−⇒===

DkCkBkAkdxdu

dxdu

xx

ββαα

0)()( )()()()(21 =−−+⇒−= +−−+−− bkjbkjakjakj DeCeBeAebuau ββαα

( ) ( ) ( ) ( ) 0)()()()(21 =++−−+−−⇒= +−−+−−

−==

bkjbkjakjakj

bxax

DekCekBekAekdxdu

dxdu ββαα ββαα



8

Kronig-Penny Model

There is a nontrivial solution if, and only if, the determinant of the coefficients is zero. This result is

The above equation relates k to the total energy E (through α) and the potential function Vo (through β). The allowed values of E can be determined by graphical or numerical methods.

( )1)(cos))(cos(cos))(sin(sin

2)/(1

22≤+=+

+−=≡≤− bakbabaVEf o βαβα

αββαξ



9

Kronig-Penny Model

Recall -1≤cosk(a+b)≤1, so E-values which cause f(ξ) to lie in the range -1≤ f(ξ) ≤1 are the allowed system energies.—The ranges of allowed energies are called energy bands; the excluded energy ranges (|f(ξ)|≥1) are called the forbidden gaps or bandgaps .The energy bands in a crystal can be visualized by

Energy

123

4



10

E-k Diagram



11

k-space Diagram

Consider the special case for which Vo = 0, (free particle case)⇒ cosα(a+b) = cosk(a+b), i.e., α = k,

,where p is the particle momentum and k is referred as a wave number.

We can also relate the energy and momentum as E = k2ħ2/2m

kpmvmmE====⇒

)2

(22αhhh 22

21



12

E-k diagram

More interesting solution occur for E < Vo (β = jγ), which applies to the electron bound within the crystal. The result could be written as

Consider a special case, b→0, Vo →∞, but bVo is finite, the above eq. becomes

The solution of the above equation results in a band of allowed energies.

)(cos))(cosh(cos))(sinh(sin2

22bakbaba +=+

− γαγααγαγ

≡=+ 2' ,coscossin'

h

bamVPkaaa

aP oααα



13

E-k diagram

Consider the function of graphically, aaPaf ααα cossin')( +=aα



14

E-k diagram

E-k diagram could be generated from the above figure.

This shows the concept of the allowed energy bands for the particle propagating in the crystal.



15

Reduced k-space



16

Electrical Conduction in Solids

the Bond Model

Energy Band

E-K diagram of a semiconductor



17

Drift Current

If an external force is applied to the electrons in the conduction band and there are empty energy states into which the electrons can move, electrons can gain energy and a net momentum.

The drift current due to the motion of electrons is

where n is the number of electrons per volume and vi is the electron velocity in the crystal.

∑=

−=n

iiveJ

1



18

Electron Effective Mass

The movement of an electron in a lattice will be different than that of an electron in free space. There are internal forces in the crystal due to the positively charged ions or protons and electrons, which willinfluence the motion of electrons in the crystal. We can write

Since it is difficult to take into account of all of the internal forces, we can write

m* is called the effective mass which takes into account the particle mass and the effect of the internal forces.

maFFF exttotal =+= int

amFext*=



19

Effective mass, E-k diagram

Recall for a free electron, the energy and momentum are related by

– So the first derivative of E w.r.t. k is related to the velocity of the particle.In addition,

– So the second derivative of E w.r.t. k is inversely proportional to the massof the particle.

In general, the effective mass could be related to

vmp

dkdE

mp

mk

dkdE

mk

mpE ==⇒==⇒==

h

hhh 122

2222

mdkEd

mdkEd 11

2

2

2

2

2

2

=⇒=h

h

2

2

2*

11dk

Edm h

=



20

Effective mass, E-k diagram

m* >0 near the bottoms of all band; m* <0 near the tops of all bandsm* <0 means that, in response to an applied force, the electron will accelerate in a direction opposite to that expected from purely classical consideration.In general, carriers are populated near the top or bottom band edge in a semiconductor—the E-k relationship is typically parabolic and, therefore,

thus carriers with energies near the top or bottom of an energy band typically exhibit a CONSTANT effective mass

edge2

2

near E ...Econstant =dk

Ed



21

Concept of Hole



22

Extrapolation of Concepts to 3-D

Brilliouin Zones: is defined as a Wigner-Seitz cell in the reciprocal lattice.

Γ point: Zone center (k = 0) ⇒ (0 0 0 )X point: Zone-boundary along a <1 0 0 >direction ⇒ 6 symmetric points(1 0 0) (-1 0 0) (0 1 0) (0 -1 0) (0 0 1) (0 0 -1)L point: Zone-boundary along a <1 1 1>direction ⇒ 8 symmetric points

Γ, X, and L points are highly symmetric ⇒ energy stable states ⇒ carriers accumulate near these points in the k-space.

)21,

21,

21(2

aπ

)0,0,1(2aπ



23

E-k diagram of Si, Ge, GaAs



24

Energy Band

Valence Band: – In all cases the valence-band maximum occurs at the zone center, at k = 0– is actually composed of three subbands. Two are degenerate at k = 0,

while the third band maximizes at a slightly reduced energy. The k = 0 degenerate band with the smaller curvature about k = 0 is called “heavy-hole” band, and the k = 0 degenerate band with the larger curvature is called “light-hole” band. The subband maximizing at a slightly reduced energy is the “split-off” band.

– Near k = 0 the shape and the curvature of the subbands is essentially orientation independent.



25

Energy Band

Conduction band:– is composed of a number of subbands. The various subbands exhibit

localized and abssolute minima at the zone center or along one of the high-symmetry diirections.

– In Ge the conduction-band minimum occurs right at the zone boundary along <111> direction. ( there are 8 equivalent conduction-band minima.)

– The Si conduction-band minimum occurs at k~0.9(2π/a) from the zone center along <100> direction. (6 equivalent conduction-band minima)

– GaAs has the conduction-band minimum at the zone center directly over the valence-band maximum. Morever, the L-valley at the zone boundary <111> direction lies only 0.29 eV above the conduction-band minimum. Even under equilibrium, the L-valley contains a non-negligible electron population at elevated temp. The intervalley transition should be taken into account.



26

Metal, Semiconductor, and Insulator

Insulator Semiconductor Metal



27

The k-space of Si and GaAs

Direct bandgap: the valence band maximum and the conduction band minimum both occur at k = 0. Therefore, the transition between the two allowed bands can take place without change in crystal momentum.



28

Constant-Energy Surfaces

A 3-D k-space plot of all the allowed k-values associated with a given energy E. The geometrical shapes, being associated with a given energy, are called constant-energy surfaces (CES).Consider the CES’s characterizing the conduction-band structures near Ec in Ge, Si, and GaAs.

(a) Constant-energy surfaces (b) Ge surface at the Brillouin-zone boundaries.



29

Constant-Energy Surfaces of Ec

For Ge, Ec occurs along each of the 8 equivalent <111> directions; a Si conduction band minimum, along each of 6 equivalent <100> directions. For GaAs, Ec is positioned at the zone center, giving rise to a single constant-energy surface.For energy slightly removed from Ec:

E-Ec ≈ Ak12+Bk2

2+Ck32,

where k1, k2, k3 are k-space coordinates measured from the center of a band minimum along principle axes.For example: Ge, the k1, k2, k3 coordinate system would be centered at the [111] L-point and one of the coordinate axes, say k1-axis, would be directed along the kx-ky-kz [111] direction.For GaAs, A = B = C, exhibits spherical CES; For Ge and Si, B=C, the CES’s are ellipsoids of revolution.



30

Effective Mass

In 3-D crystals the electron acceleration arising from an applied force is analogously by

where

For GaAs, , so mij = 0 if i≠j, and

therefore, we can define mii=me*, that is the the effective mass tensor reduces

to a scalar, giving rise to an orientation-indep. equation of motion like that of a classical particle.

Fmdt

dv⋅= *

1

=−−−

−−−

−−−

111

111

111

*1

zzzyzx

yzyyyx

xzxyxx

mmmmmmmmm

mzyxji

kkEm

jiij ,,,.. 1 2

21 =

∂∂∂

=−

h

)( 222zyxc kkkAEE ++=− 2

111 2h

Ammm zzyyxx === −−−

)(2

2222

2

zyxe

c kkkm

EE ++=−⇒h



31

Effective Mass

For Si and Ge: E-Ec = Ak12+B(k2

2+k32)

so mij = 0 if i≠j, and

Because m11 is associated with the k-space direction lying along the axis of revolution, it is called the longitudinal effective mass ml*.Similarly, m22 = m33, being associated with a direction perpendicular to the axis of revolution, is called the transverse effective mass mt*.

211 2

h

Bmm zzyy == −−,22

1

h

Amxx =−

)(22

23

222

2212

2

kkm

km

EEtl

c ++=−⇒hh



32

Effective Mass

The relative sizes of ml* and mt

* can be deduced by inspection of the Si and Ge constant-energy plots.

For both Ge and Si, ml* > mt

*. Further, ml*/mt

* of Ge > ml*/mt

* of Si.The valence-band structure of Si, Ge, and GaAs are approximately spherical and composed of three subbands. Thus, the holes in a given subband can be characterized by a single effective mass parameter, but three effective mass (mhh

*, mlh*, and mso

*) are required to characterize the entire hole population. The split-off band, being depressed in energy, is only sparsely populated and is often ignored.

=

revolution of axis thelar toperpendicu ellipsoid theof widthmax.

revolution of axis thealongelliosoid theof length

*

*

t

l

mm



33

Effective Mass measurement

The near-extrema point band structure, multiplicity and orientation of band minima, etc. were all originally confirmed by cyclotron resonance measurement.Resonance experiment is performed in a microwave resonancecavity at temperature 4K. A static B field and an rf E-field oriented normal to B are applied across the sample. The carriers in the sample will move in an orbit-like path about the direction of B and the cyclotron frequency ωc = qB/mc. When the B-field strength is adjusted such that ωc = the ω of the rf E-field, the carriers absorb energy from the E-field (in resonance). ⇒m= qB/ωc

Repeating the different B-field orientations allows one to separate out the effective mass factors (ml* and mt*)



34

Effective Mass of Si, Ge, and GaAs



35

Density of State Function

To calculate the electron and hole concentrations in a material, we must determine the density of these allowed energy states as a function of energy.Electrons are allowed to move relatively freely in the conduction band of a semiconductor but are confined to the crystal.To simulate the density of allowed states, consider an appropriate

model: A free electron confined to a 3-D infinite potential well, where the potential well represents the crystal. The potential of the well is defined as V(x,y,z) = 0 for 0<x<a, 0<y<a, 0<z<a, and V(x,y,z) = ∞ elsewhereSolving the Schrodinger’s equation, we can obtain

++=++==⇒ 2

22222222

2 )(2a

nnnkkkkmEzyxzyx

πh



36


The volume of a single quantum state is Vk =(π/a)3, and the differential volume in k-space is 4πk2dkTherefore, we can determine the density of quantum states in k-space as

– The factor, 2, takes into account the two spin states allowed for each quantum state; the next factor, 1/8, takes into account that we are considering only the quantum states for positive values of kx, ky, and kz.

32

2

3

24812)( adkk

a

dkkdkkgT ⋅=

=

πππ



37


Recall that

We can determine the density of states as a function of energy E by

Therefore, the density of states per unit volume is given by

Extension to semiconductors, the density of states in conduction band is modified as

and the density of states in valence band is modified as

323

3 )2(4)( adEEmh

dEEgT ⋅⋅⋅⋅=π

22 2

h

mEk =

dEEmh

dEEgT ⋅⋅⋅= 23

3 )2(4)( π

cnc EEmh

Eg −⋅⋅= 23*

3 )2(4)( π

vpv EEmh

Eg −⋅⋅= 23*

3 )2(4)( π



38


mn* and mp

* are the electron and hole density of states effective masses. In general, the effective mass used in the density of states expression must be an average of the band-structure effective masses.



39

Density of States Effective Mass

Conduction Band--GaAs: the GaAs conduction band structure is approximately spherical and the electronss within the band are characterized by a single isotropic effective mass, me

*, ⇒Conduction Band--Si, Ge: the conduction band structure in Si and Ge is characterized by ellipsoidal energy surfaces centered, respectively, at points along the <100> and <111> directions in k-space.

Valence Band--Si, Ge, GaAs: the valence band structures are al characterized by approximately spherical constant-energy surfaces (degenerate).

( )( ) ...Ge 4

...Si 63

13

2

31

32

2***

2***

tln

tln

mmm

mmm

=

=

...GaAs**en mm =

( ) ( )[ ] 32

23

23 ***

lhhhp mmm +=



40

Density of States Effective Mass



41

Statistics Mechanics

In dealing with large numbers of particles, we are interested only in the statistical behavior of the whole group rather than in the behavior of each individual particle.There are three distribution laws determining the distribution of particles among available energy states.Maxwell-Boltzmann probability function:

– Particles are considered to be distinguishable by being numbered for 1 to N with no limit to the number of particles allowed in each energy state.

Bose-Einstein probability function:– Particles are considered to be indistinguishable and there is no limit to the

number of particles permitted in each quantum state. (e.g., photons)Fermi-Dirac probability function:

– Particles are indistinguishable but only one particle is permitted in each quantum state. (e.g., electrons in a crystal)



42

Fermi-Dirac Distribution

Fermi-Dirac distribution function gives the probability that a quantum state at the energy E will be occupied by an electron.

the Fermi energy (EF) determine the statistical distribution of electrons and does not have to correspond to an allowed energy level.At T = 0K, f(E < EF) = 1 and f(E >EF ) = 0, electrons are in the lowest possible energy states so that all states below EF are filled and all states above EF are empty.

)exp(1

1)(

kTEEEf

F−+

=



43

Fermi-Dirac Distribution, at T=0K



44

Fermi-Dirac Distribution

For T > 0K, electrons gain a certain amount of thermal energy so that some electrons can jump to higher energy levels, which means that the distribution of electrons among the available energy states will change.

For T > 0K, f(E = EF) = ½



45

Boltamann Approximation

Consider T >> 0K, the Fermi-Dirac function could be approximated by

which is known as the Maxwell-Boltzmann approximation.

−−

≈−

+=

kTEE

kTEEEf F

F

)(exp)exp(1

1)(



46

Homework

3.53.83.16



1

Chap 4. Semiconductor in Equilibrium

Carriers in Semiconductors Dopant Atoms and Energy LevelsExtrinsic SemiconductorStatistics of Donors and AcceptorsCharge NeutralityPosition of Fermi Energy



2

Equilibrium Distribution of Electrons and Holes

The distribution of electrons in the conduction band is given by the density of allowed quantum states times the probability that a state will be occupied.

The thermal equilibrium conc. of electrons no is given by

Similarly, the distribution of holes in the valence band is given by the density of allowed quantum states times the probability that a state will not be occupied by an electron.

And the thermal equilibrium conc. Of holes po is given by

)()()( EfEgEn c=

)](1)[()( EfEgEp v −=

)()(∫∞

=cE co EfEgn

∫ ∞−−= vE

vo EfEgp )](1)[(



3

Equilibrium Distribution of Electrons and Holes



4

The no and po eqs.

Recall the thermal equilibrium conc. of electrons

Assume that the Fermi energy is within the bandgap. For electrons in the conduction band, if Ec-EF >>kT, then E-EF>>kT, so the Fermi probability function reduces to the Boltzmann approximation,

Then

We may define , (at T =300K, Nc ~1019 cm-3), which

is called the effective density of states function in the conduction band

)()(∫∞

=cE co EfEgn

kTEEEf F )]([exp)( −−

≅

( ) −−

=

−−

−= ∫∞

kTEE

hkTmdE

kTEEEE

hmn FcnF

cEn

oc

(exp22)(exp2423

2

*

3

23* ππ

23

2

*22

=

hkTmN n

cπ

)



5

The no and po eqs.

The thermal equilibrium conc. of holes in the valence band is given by

For energy states in the valence band, E<Ev. If (EF-Ev)>>kT,

Then,

We may define , (at T =300K, Nv ~1019 cm-3), which

is called the effective density of states function in the valence band

∫ ∞−−= vE

vo EfEgp )](1)[(

kTEEEf F )]([exp)(1 −−

≅−

( )

−−

=

−−

−= ∫ ∞− kTEE

hkTm

dEkT

EEEEhm

p vFpFv

E po

v )(exp2

2)(exp24

23

2

*

3

23* ππ

23

2

*22

=

hkTm

N pv

π



6

nopo product

The product of the general expressions for no and po are given by

⇒ for a semiconductor in thermal equilibrium, the product of no and po is always a constant for a given material and at a given temp.

Effective Density of States Function

−=

−−

⋅

−−

=kTE

NNkT

EENkT

EENpn gvc

vFv

Fccoo exp)(exp )(exp



7

Intrinsic Carrier Concentration

For an intrinsic semiconductor, the conc. of electrons in the conduction band, ni, is equal to the conc. of holes in the valence band, pi. The Fermi energy level for the intrinsic semiconductor is called the intrinsic Fermi energy, EFi.For an intrinsic semiconductor,

For an given semiconductor at a constant temperature, the value of ni is constant, and independent of the Fermi energy.

−−

=

−−

==

kTEEN

kTEE

hkTmnn Fic

cFicn

io)(exp)(exp22

23

2

*π

−−

=

−−

==

kTEEN

kTEE

hkTm

pp vFiv

vFipio

)(exp)(exp2

223

2

*π

energy bandgap theis where,exp2g

gvci E

kTE

NNn

−=⇒



8

Intrinsic Carrier Conc.

Commonly accepted values of ni at T = 300 KSilicon ni = 1.5x1010 cm-3

GaAs ni = 1.8x106 cm-3

Germanium ni = 1.4x1013 cm-3



9

Intrinsic Fermi-Level Position

For an intrinsic semiconductor, ni = pi,

Emidgap =(Ec+Ev)/2: is called the midgap energy. If mp

* = mn*, then EFi = Emidgap (exactly in the center of the bandgap)

If mp* > mn

*, then EFi > Emidgap (above the center of the bandgap)If mp

* < mn*, then EFi < Emidgap (below the center of the bandgap)

)ln(43)(

21)ln(

43)(

21

])(exp[])(exp[

*

*

n

pvc

c

vvcFi

vFiv

Ficc

mm

kTEENNkTEEE

kTEEN

kTEEN

++=++=⇒

−−=

−−⇒



10

Dopant and Energy Levels



11

Acceptors and Energy Levels



12

Ionization Energy

Ionization energy is the energy required to elevate the donor electron into the conduction band.



13

Extrinsic Semiconductor

Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons and holes in the material, and therefore, the Fermi energy position will change correspondingly.Recall

−−

=

−−

=kT

EENkT

EENn vFiv

Ficci

)(exp)(exp

−+−−

=

−−

=kT

EEEENkT

EENn FiFFicc

Fcco

)()(exp)(exp

−+−−

=

−−

=kT

EEEENkT

EENp FivFiFv

vFvo

)()(exp)(exp

−−

=

−

=⇒kT

EEnpkT

EEnn FiFio

FiFio

)(exp and exp



14

Extrinsic Semiconductor

When the donor impurity atoms are added, the density of electrons is greater than the density of holes, (no > po) ⇒ n-type; EF > EFiWhen the acceptor impurity atoms are added, the density of electrons is less than the density of holes, (no < po) ⇒ p-type; EF < EFi



15

Degenerate and Nondegenerate

If the conc. of dopant atoms added is small compared to the density of the host atoms, then the impurity are far apart so that there is no interaction between donor electrons, for example, in an n-material.⇒nondegenerate semiconductor If the conc. of dopant atoms added increases such that the distance between the impurity atoms decreases and the donor electrons begin to interact with each other, then the single discrete donor energy will split into a band of energies. ⇒EF move toward Ec

The widen of the band of donor states may overlap the bottom of the conduction band. This occurs when the donor conc. becomes comparable with the effective density of states, EF ≥ Ec

⇒degenerate semiconductor



16

Degenerate and Nondegenerate



17

Statistics of Donors and Acceptors

The probability of electrons occupying the donor energy state was given by

where Nd is the conc. of donor atoms, nd is the density of electrons occupying the donor level and Ed is the energy of the donor level. g =2 since each donor level has two spin orientation, thus each donor level has two quantum states.Therefore the conc. of ionized donors Nd

+ = Nd –nd

Similarly, the conc. of ionized acceptors Na- = Na –pa, where

factor degeneracy: ,)exp(11

g

kTEE

g

NnFd

dd −

+=

GaAs and Siin levelacceptor for the 4 ,)exp(11

=−

+= g

kTEE

g

NpaF

aa



18

Complete Ionization

If we assume Ed-EF>> kT or EF-Ea >> kT (e.g. T= 300 K), then

that is, the donor/acceptor states are almost completely ionized and all the donor/acceptorimpurity atoms have donated an electron/holeto the conduction/valence band.

ddddFd

dd NnNNkT

EENn ≅−=⇒

−−

≈ +)(exp2

aaaaaF

aa NpNNkT

EENp ≅−=⇒

−−

≈ −)(exp4



19

Freeze-out

At T = 0K, no electrons from the donor state are thermally elevated into the conduction band; this effect is called freeze-out.At T = 0K, all electrons are in their lowest possible energy state; that is for an n-type semiconductor, each donor state must contain an electron, therefore, nd = Nd or Nd

+ = 0, which means that the Fermi level must be above the donor level.



20

Charge Neutrality

In thermal equilibrium, the semiconductor is electrically neutral. The electrons distributing among the various energy states creating negative and positive charges, but the net charge density is zero.Compensated Semiconductors: is one that contains both donor and acceptor impurity atoms in the same region. A n-type compensated semiconductor occurs when Nd > Na and a p-type semiconductor occurs when Na > Nd. The charge neutrality condition is expressed by

where no and po are the thermal equilibrium conc. of e- and h+ in the conduction band and valence band, respectively. Nd

+ is the conc. Of positively charged donor states and Na

- is the conc. of negatively charged acceptor states.

+− +=+ doao NpNn



21

Compensated Semiconductor



22


If we assume complete ionization, Nd+ = Nd and Na

- = Na, then

If Na = Nd = 0, (for the intrinsic case), ⇒no = po

If Nd >> Na, ⇒no = Nd

If Na > Nd, is used to

calculate the conc. of holes in valence band

o

iodoao n

npNpNn2

recall , =+=+

( ) 22

222

22

0)(

iadad

o

ioadodn

iao

nNNNNn

nnNNnNnnNn

+

−

+−

=⇒

=−−−⇒+=+

( ) 22

22 idada

o nNNNNp +

−

+−

=⇒



23




24

Position of Fermi Level

The position of Fermi level is a function of the doping concentration and a function of temperature, EF(n, p, T). Assume Boltzmann approximation is valid, we have

( ) ( )

ln and lnor

ln and ln

exp and exp

=−

=−

=−

=−⇒

−−=

−−=

i

ovFi

i

oFiF

o

vvF

o

cFc

vFvo

Fcco

npkTEE

nnkTEE

pNkTEE

nNkTEE

kTEENpkT

EENn



25

EF(n, p, T)



26

EF(n, p, T)



27

Homework

4.184.204.24



1

Chap 5. Carrier Motion

Carrier DriftCarrier DiffusionGraded Impurity DistributionHall EffectHomework



2

Carrier Drift

When an E-field (force) applied to a semiconductor, electrons and holes will experience a net acceleration and net movement, if there are available energy states in the conduction band and valence band. The net movement of charge due to an electric field (force) is called “drift”.Mobility: the acceleration of a hole due to an E-field is related by

If we assume the effective mass and E-field are constants, the we can obtain the drift velocity of the hole by

where vi is the initial velocity (e.g. thermal velocity) of the hole and t is the acceleration time.

qEdtdvmF p == *

EtvmeEtv i

pd ,* ∝+=



3

Mobility

E = 0

In semiconductors, holes/electrons are involved in collisions with ionized impurity atoms and with thermally vibration lattice atoms. As the hole accelerates in a crystal due to the E-field, the velocity/kinetic energy increases. When it collides with an atom in the crystal, it lose s most of its energy. The hole will again accelerate/gain energy until is again involved in a scattering process.



4

Mobility

If the mean time between collisions is denoted by τcp, then the average drift velocity between collisions is

where µp (cm2/V-sec) is called the hole mobility which is an important parameter of the semiconductor since it describes how well a particle will move due to an E-field. Two collision mechanisms dominate in a semiconductor:

– Phonon or lattice scattering: related to the thermal motion of atoms; µL ∝T-3/2

– Ionized impurity scattering: coulomb interaction between the electron/hole and the ionized impurities; µI ∝T3/2/NI., : total ionized impurity conc. ↑, µI ↓If T↑, the thermal velocity of hole/electron ↑⇒carrier spends less time in the

vicinity of the impurity. ⇒ less scattering effect ⇒ µI ↑

EEme

v pp

cpd µ

τ≡

= * *

p

cpdpp m

eE

v τµ ==

−+ += adI NNN



5

Mobility

Electron mobility Hole mobility



6

Drift Current Density

If the volume charge density of holes, qp, moves at an average drift velocity vdp, the drift current density is given by

Jdrfp = (ep) vdp = eµppE.Similarly, the drift current density due to electrons is given by

Jdrfn = (-en) vdp = (-en)(-µnE)=eµnnEThe total drift current density is given by Jdrf = e(µnn+µpp) E



7

Conductivity

The conductivity σ of a semiconductor material is defined by Jdrf ≡ σ E, so σ= e(µnn+µpp) in units of (ohm-cm)-1

The resistivity ρ of a semiconductor is defined by ρ ≡ 1/ σ



8

Resistivity Measurement

Four-point probe measurement

factor correction: ;2 cc FFIVsπρ =



9

Velocity Saturation

So far we assumed that mobility is indep. of E-field, that is the drift velocity is in proportion with the E-field. This holds for low E-filed. In reality, the drift velocity saturates at ~107 cm/sec at an E-field ~30 kV/cm. So the drift current density will also saturate and becomes indep. of the applied E-field.



10

Velocity Saturation of GaAs

For GaAs, the electron drift velocity reaches a peak and then decreases as the E-field increases. ⇒negative differential mobility/resistivity, which could be used in the design of oscillators.This could be understood by considering the E-k diagram of GaAs.



11

Velocity Saturation of GaAs

In the lower valley, the density of state effective mass of the electron mn

* = 0.067mo. The small effective mass leads to a large mobility. As the E-field increases, the energy of the electron increases and can bescattered into the upper valley, where the density of states effective mass is 0.55mo. The large effective mass yields a smaller mobility. The intervalley transfer mechanism results in a decreasing average drift velocity of electrons with E-field, or the negative differential mobility characteristic.



12

Carrier Diffusion

Diffusion is the process whereby particles flow from a region of high concentration toward a region of low concentration. The net flow of charge would result in a diffusion current.



13

Diffusion Current Density

The electron diffusion current density is given by Jndif = eDndn/dx,where Dn is called the electron diffusion coefficient, has units of cm2/s.The hole diffusion current density is given by Jpdif = -eDpdp/dx,where Dp is called the hole diffusion coefficient, has units of cm2/s.The total current density composed of the drift and the diffusion current density.

1-D

or 3-D

dxdpeD

dxdneDEepEenJ pnxpxn −++= µµ

peDneDEepEenJ pnxpxn ∇−∇++= µµ



14

Graded Impurity Distribution

In some cases, a semiconductors is not doped uniformly. If the semiconductor reaches thermal equilibrium, the Fermi level is constant through the crystal so the energy-band diagram may qualitatively look like:

Since the doping concentration decreases as x increases, there will be a diffusion of majority carrier electrons in the +x direction. The flow of electrons leave behind positive donor ions. The separation of positive ions and negative electrons induces an E-field in +x direction to oppose the diffusion process.



15

Induced E-Field

The induced E-field is defined asthat is, if the intrinsic Fermi level changes as a function of distance through a semiconductor in thermal equilibrium, an E-field exists.If we assume a quasi-neutrality condition in which the electron concentration is almost equal to the donor impurity concentration, then

So an E-field is induced due to the nonuniform doping.

dxdE

edxeEd

dxdE FiFi

x1))/((

=−

−=−=φ

dxxdN

xNekTE

dxxdN

xNkT

dxEd

dxEEd

nxNkTEExN

kTEEnn

d

dx

d

d

iiF

i

diFd

iFio

)()(

1

)()(

)()(

)(ln)(exp

−=⇒

=−

=−

⇒

=−⇒≈

−

≈



16

Einstein Relation

Assuming there are no electrical connections between the nonuniformly doped semiconducotr, so that the semiconductor is in thermal equilibrium, then the individual electron and hole currents must be zero.

Assuming quasi-neutrality so that n ≈ Nd(x) and

Similarly, the hole current Jp = 0

dxdneDEenJ nxnn +==⇒ µ0

relation-Einstein ---

)()()(

1)(0

)()(0

ekTD

dxxdNeD

dxxdN

xNekTxNe

dxxdNeDExeNJ

n

n

dn

d

ddnn

dnxndn

=⇒

+

−=⇒

+==

µ

µ

µ

ekTD

p

p =⇒µ



17

Einstein Relation

Einstein relation says that the diffusion coefficient and mobility are not independent parameters.

Typical mobility and diffusion coefficient values at T=300K(µ = cm2/V-sec and D = cm2/sec)

µn Dn µp Dp

Silicon 1350 35 480 12.4GaAs 8500 220 400 10.4Germaium 3900 101 1900 49.2



18

Hall Effect

The hall effect is a consequence of the forces that are exerted on moving charges by electric and magnetic fields. We can use Hall measurement to

– Distinguish whether a semiconductor is n or p type– To measure the majority carrier concentration– To measure the majority carrier mobility



19

Hall Effect

A semiconductor is electrically connected to Vx and in turn a current Ixflows through. If a magnetic field Bz is applied, the electrons/holes flowing in the semiconductor will experience a force F = q vx x Bz in the (-y) direction. If this semiconductor is p-type/n-type, there will be a buildup of positive/negative charge on the y = 0 surface. The net charge will induce an E-field EH in the +y-direction for p-type and -y-direction for n-type. EH is called the Hall field.In steady state, the magnetic force will be exactly balanced by the induced E-field force. F = q[E + v x B] = 0 ⇒ EH = vx Bz and the Hall voltage across the semiconductor is VH = EHWVH >0 ⇒ p-type, VH < 0 ⇒ n-type



20

Hall Effect

VH = vx W Bz, for a p-type semiconductor, the drift velocity of hole is

for a n-type,

Once the majority carrier concentration has been determined, we can calculate the low-field majority carrier mobility.

For a p-semiconductor, Jx = epµpEx.

For a n-semiconductor,

( )( ) H

zxzxH

xxdx edV

BIpepd

BIVWdep

IepJv =⇒=⇒==

H

zx

edVBIn −=

WdepVLI

x

xp =⇒ µ

WdenVLI

x

xn =⇒ µ



21

Hall Effect

Hall-bar with “ear” van deer Parw configuration



22

Homework

5.145.20



23



1

Chap 6. Nonequilibrium Excess Carriers in Semiconductor

Carrier Generation and RecombinationContinuity EquationAmbipolar TransportQuasi-Fermi Energy LevelsExcess-Carrier LifertimeSurface Effects



2

Nonequilibrium

When a voltage is applied or a current exists in a semiconductor device, the semiconductor is operating under nonequilibrium conditions.Excess electrons/holes in the conduction/valence bands may be generated and recombined in addition to the thermal equilibrium concentrations if an external excitation is applied to the semiconductor.Examples: 1. A sudden increase in temperature will increase the thermal generation rate of electrons and holes so that their concentration will change with time until new equilibrium reaches.2. A light illumination on the semiconductor (a flux of photons) can also generate electron-hole pairs, creating a nonequilibrium condition.



3

Generation and Recombination

In thermal equilibrium, the electrons are continually being thermal generated from the valence band (hereby holes are generated) to conduction band by the random thermal process. At the same time, electrons moving randomly through the crystal may come in close proximity to holes and recombine. The rate of generation and recombination of electrons/holes are equal so the net electron and hole concentrations are constant (independent of time).



4

Excess Carrier Generation and Recombination

When high-energy photons are incident on a semiconductor, electron-hole pairs are generated (excess electrons/holes) ⇒ the concentration of electrons in the conduction band and of holes in the valence band increase above their thermal-equilibrium value. n = no +δn, p = po+ δp where no/po are thermal–equilibrium concentrations, and δn/δp are the excess electron/hole concentrations. np ≠ nopo = ni

2 ( nonequilibrium)

For the direct band-to-band generation, the generation rates (in the unit of #/cm3-sec) of electrons and holes are equal; gn’ = gp’ (may be functions of the space coordinates and time)



5


An electron in conduction band may “fall down” into the valence band and leads to the excess electron-hole recombination process.

Since the excess electrons and holes recombine in pairs so the recombination rates for excess electrons and holes are equal, Rn’ = Rp’. (in the unit of #/cm3-sec). ⇒ δn(t) = δp(t)The direct band-to-band recombination is spontaneous, thus the probability of an electron and hole recombination is constant with time. Rn’ = Rp’ ∝ the electron and hole concentration.



6

Recombination Process

Band-to-Band: direct thermal recombination.This process is typically radiative, with the excess energy releasedduring the process going into the production of a photon (light)

R-G Center: Induced by certain impurity atoms or crystal defects.Electron and hole are attracted to theR-G center and lead to the annihilationof the electron-hole pair.

Or a carrier is first captured at the R-Gsite and then makes an annihilatingtransition to the opposite carrier band.This process is indirect thermal recombination (nonradiative). Thermal energy (heat) is released during the process (lattice vibrations, phonons are produced)



7


Recombination via Shallow Levels:—induced by donor or acceptor sites. At RT, if an electron is captured at a donor site,however, it has a high probability of being re-emitted into the conduction band before completing the recombination process. Therefore, the probability of recombination via shallow levels is quite low at RT. It should be noted that the probability of observing shallow-level processes increases with decreasing system temperature.

Recombination involving Excitons:It is possible for an electron and a hole to become boundtogether into a hydrogen-atom-like arrangement which moves as a unit in response to applied forces. This coupled e-h pair is called an “exciton”. The formation of an exciton can be viewed as introducing a temporary level into the bandgap slightly above or below the band edge.



8


Recombination involving Excitons: Recombination involving excitons is a very important mechanism at low temperatures and is the major light-producing mechanism in LED’s. Auger Recombinations:In a Auger process, band-to-band recombinationat a bandgap center occurs simultaneously with the collision between two like carriers. The energy released by the recombination or trappingsubprocess is transferred during the collision to the surviving carrier. Subsequently, this high energetic carrier “thermalizes”-loses energy through collisions with the semiconductor lattice.Auger recombination increases with carrier concentration, becoming very important at high carrier concentration. Therefore, Auger recombination mmust be considered in treating degenerately doped regions (like solar cell, junction lasers, and LED’s)



9

Generation Process

Band-to-Band generation:

R-G center generation:

Photoemission from band gap centers:



10

Generation Process

Impact-Ionization:An e-h pair is produced as a result of the energy released when a highly energetic carrier collides with the crystal lattice. The generation of carriers through impact ionizationroutinely occurs in the high e-filed regions of devices and is responsible for the avalanche breakdown in pn junctions.



11

Momentum Consideration

In a direct semiconductor where the k-values of electrons and holes are all bunched near k = 0, little change is required for the recombination process to proceed. The conservation of both energy and crystal momentum is readilymet by the emission of a photon.

In a indirect semiconductor, there isa large change in crystal momentumassociated with the recombination process. The emission of a photon will conserve energy but cannot simultaneously conserve momentum. Thus for band-to-band recombination to proceed in an indirect semiconductor a phonon must be emitted coincident with the emission of a photon.



12


Low-level injection: the excess carrier concentration is much less than the thermal equilibrium majority carrier concentration, e.g., for a n-type semiconductor, δn = δp << no.High-level injection: δn ≈ no or δn >> no

For a p-type material (po >> no) under low-level injection, the excess carrier will decay from the initial excess concentration with time;

where τn0 is referred to as the excess minority carrier lifetime (τn0 ∝1/p0)

and the recombination rate of excess carriers Rn’ = Rp’=For a n-type material (no >> po) under low-level injection,

Rn’ = Rp’=

0

)(

n

tnτδ

0/)0()( ntetntn τδδ −==

0/)0()( pntetptp τδδ −==0

)(

p

tpτδ



13

Continuity Equations

Consider a differential volume element in which a 1-D hole flux, Fp+ (#

of holes/cm2-sec), is entering this element at x and is leaving at x+dx.

So the net change in hole concentration per unit time is

----continuity equation for holes

Similarly, the continuity equation for electron flux ispt

pp pgx

Ftp

τ−+

∂

∂−=

∂∂ +

ntn

n ngx

Ftn

τ−+

∂∂

−=∂∂ −



14

Ambipolar Transport

If a pulse of excess electrons and holes are created at a particular point due to an applied E-field, the excess e-s and h+s will tend to drift in opposite directions. However, any separation of e-s and h+s will induce an internal E-field and create a force attracting the e-s and h+s back.The internal E-field will hold the pulses of excess e -s and h+s together, then the electrons and holes will drift or diffuse together with a single effective mobility or diffusion coefficient. This is so called “ambipolar diffusion” or “ambipolar transport”.Fig. Show the above situation



15

Ambipolar Transport



16

Ambipolar Transport



17

Quasi-Fermi Levels

At thermal-equilibrium, the electron and hole concentrations are functions of the Fermi level by

Under nonequilibrium conditions, excess carriers are created in a semiconductor, the Fermi energy is strictly no longer defined. We may define a quasi-Fermi level, EFn, for electrons and a quasi-Fermi level, EFp, for holes that apply for nonequilibrium. So that the total electron and hole concentrations are functions of the quasi-Fermi levels.

exp and exp

−

=

−

=kT

EEnpkT

EEnn FFiio

FiFio

exp and exp

−

=+

−

=+kT

EEnppkT

EEnnn FFiio

FiFnio δ



18

Quasi-Fermi Levels

For a n-type semiconductor under thermal equilibrium, the band diagram is

Under low-level injection, excess carriers are created and the quasi-Fermi level for holes (minority), EFp, is significantly different from EF.



19

Excess-Carrier Lifetime

An allowed energy state, also called a trap, within the forbidden bandgap may act as a recombination center, capturing both electrons and holes with almost equal probability. (it means that the capture cross sections for electrons and holes are approximately equal)Acceptor-type trap:

– it is negatively charged when it contains an electron and it is neutrall when it does not contain an electron.

Donor-type trap:– it is positively charged when empty and neutral when filled with an electron



20

Shockley-Read-Hall Theory of Recombination

Assume that a single recombination center exists at an energy Et within the bandgap. And there are four basic processes that may occur at this single trap.Process 1: electron from the conduction band captured by an initially neutral empty trap.Process 2: electron emission from atrap into the conduction band.Process 3: capture of a hole from the valence band by a trap containing an electron.Process 4: emission of a hole from aneutral trap into the valence band.



21


In Process 1: the electron capture rate (#/cm3-sec): Rcn = CnNt(1-fF(Et))n

Cn=constant proportional to electron-capture cross sectionNt = total concentration in the conduction bandn = electron concentration in the conduction band

fF(Et)= Fermi function at the trap energyFor Process 2: the electron emission rate (#/cm3-sec):

Ren = EnNtfF(Et)En=constant proportional to electron-capture cross section Cn

In thermal equilibrium, Rcn = Ren, using the Boltzmann approximation for the Fermi function,

In nonequilibrium, excess electrons exist, ( )

ntc

cnn CkT

EENCnE

−−

== exp'

( )[ ])()(1 'tFtFtnencnn EfnEfnNCRRR −−=−=



22


In Process 3 and 4, the net rate at which holes are captured from the valence band is given by

In semiconductor, if the trap density is not too large, the excess electron and hole concentrations are equal and the recombination rates of electrons and holes are equal.

In thermal equilibrium, np = ni2 ⇒ Rn = Rp = 0

[ ]))(1()( 'tFtFtpp EfpEpfNCR −−=

( )

−−

=kT

EENp vtv exp'

RppCnnC

nnpNCCRR

ppCnnCpCnC

Ef

pn

itpnpn

pn

pntF

≡+++

−==

++++

=⇒

)'()'()(

and

)'()'('

)(

2



23

Surface Effects

Surface states are functionally equivalent to R-G centers localized at the surface of a material. However, the surface states (or interfacial traps) are typically found to be continuously distributed in energy throughout the semiconductor bandgap.



24

Surface Recombination Velocity

As the excess concentration at the surface becomes smaller than that in the bulk, excess carriers from the bulk region diffuse toward the surface where they recombine, and the surface recombination velocity increases. An infinite surface recombination velocity implies that the excess minority carrier concentration and lifetime are zero.



25

Homework

6.146.176.19



1

Chap 7. P-N junction

P-N junction FormationFermi Level AlignmentBuilt-in E-field (cut-in voltage)Homework



2

P-N junction

EC

EV

EF

EFi

P-SemiconductorEC

EV

EFEFi

N-Semiconductor

EC

EV

EF

EFi

P-Semiconductor

EFi

N-SemiconductorEC

EV

EF



3

P-N Junction

EC

EV

EF

EFi

P-Semiconductor

EFi

N-SemiconductorEC

EV

EF

Depletion region



4

Hetero-Junction

EC

EV

EF

EFi

Semiconductor AEC

EV

EFEFi

Semiconductor B

EV

EC∆EC < 20 meV

bulk Si

Eg = 1.17 eV

Strained Si0.8Ge0.2

Eg ~ 1.0 eV

∆EV ~ 0.15 eV

∆EC ~0.15 eV

∆EV ~ 0.05 eV

Relaxed Si0.7Ge0.3

Eg ~ 1.08 eVStrained Si

Eg ~ 0.88 eV

Type I Alignment Type II Alignment



5

Quantum Well

Hole Confinement

∆EC ~ 0.02 eV

relaxed Si0.7Ge0.3

Eg = 1.08 eV

Strained Si0.3Ge0.7

Eg ~ 0.72 eV

∆EV ~ 0.48 eV

Strained Si

Eg = 0.88 eV

∆EC ~0.18 eV

∆EV ~0.34 eV

Electron Confinement



6

. . . . . . . .. . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . .. . . . . . . .

bulk Si bulk Si

Relaxed Si1-xGex Relaxed Si1-xGex

Strained Si1-xGex Strained Si

misfit dislocation misfit dislocation

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