Lecture 2 Fundamentals

8/11/2019 Lecture 2 Fundamentals

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KEEE 2224 Electronic Devices

Week 2:

Fundamentals of Electronic Devices

Lecturer : Dr. Sharifah Fatmadiana Wan Muhd Hatta

To arrange for appointments for small group discussions :

Email: [email protected]
mailto:[email protected]:[email protected]


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Week 2: learning outcomes

By the end of this week, student should be able

to :

1. Describe the mechanism of charge carrier

mechanism

2. Describe on energy band diagrams, density

states band diagram.

2KEEE 2224 Electronics Devices


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Basic Semiconductor Physics

3

Periodic Table and the Semiconductor Materials

Types of Solids

Atomic Bonding

Imperfections and Impurities

Doping

Electrical Conductions in Solids :

Electron and holes

Energy-band model

Density of States Function

Semiconductor in equilibrium

Charge carriers in Semiconductor

Position of Fermi Energy Level

Carrier Transport

Carrier drift, Carrier DiffusionConductivity, Resistivity

KEEE 2224 Electronics Devices


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Band Diagram: Potential vs Kinetic

Energy

KEEE 2224 Electronics Devices 4


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Electrostatic Potential



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Energy Bands


Basic convention:

EC

Ev

Eref

K.E.

P.E.

+E

+V

Kinetic energy:

Potential Energy:

CEEEK ..

refC

refC

EEq

V

EEqVEP

1

..

Electric field:dx

dE

qdx

dVV C

1D1inor,


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Electric Field



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Examples of energy band structures:


Si GaAs

Based on the energy band structure, semiconductors can beclassified into:

Indirect band-gap semiconductors (Si, Ge)

Direct band gap semiconductors (GaAs)


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Direct VS Indirect Bandgaps


Photogenerationband-diagramatic description:

Momentum and energy conservation:

E

k

Eg

Phonon emission

Phonon absorption

Indirect band-gap Semics.

Virtualstates

Ec

EV

E

kDirect band-gap Semics.

Eg


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Fermi-Dirac Distribution Function


The conduction band in a piece of semiconductor consists of many

available, allowed, empty energy levels. When calculating how many

electrons will fill these, we consider two factors:

How many energy levels are there within a given range of energy,

How likely is it that each level will be populated by an electron.

The likelihood in the second item is given by a probability function

called the Fermi-Dirac distribution function. f(E) is the probability

that a level with energy E will be filled by an electron, and the

expression is:


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Density of States


for electrons in the conduction band

for holes in the conduction band

The integration of the product of the density of states g(E) and the Fermi-

Dirac distribution function f(E) gives the carrier distribution over the

energy range above or below the bandgap (see next slide)

g(E)gives the

distribution ofenergy levels

(states) as a

function of energy.


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Density of States

12

Intrinsic semiconductor. (a) Schematic band diagram. (b) Density of

states. (c) Fermi distribution function. (d) Carrier concentration.


or g(E)


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The energy band diagram with the position of the

Fermi level EFas a function of the doping type


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Boltzmann Approximation


C

V


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Equilibrium Carrier Concentration


Density of states functionin the CB. If we assume

m*n=m0, hence at T=300K,

Nc=2.8x1019cm-3

C i d E ilib i C i


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Continued Equilibrium Carrier

Concentration


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Exercise: Fermi LevelThe probability that a state is filled at the conduction band edge (Ec) is precisely

equal to the probability that a state is empty at the valence band edge (Ev). Where

is the Fermi level located?

Answer: The Fermi function,f(E), specifies the probability of electrons occupying

states at a given energy E. The probability that a state is empty (not filled) at a

given energy E is equal to 1 f(E). Here we are told

f(Ec) = 1f(Ev)

Since

And,

Hence, we can conclude

or



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Quasi Fermi Level


Describes the population of electronsseparately in the conduction

bandand valence band, when their populationsare displaced

from equilibrium.

This displacement could be caused by the application of an external

voltage, or by exposure to light of energy, which alter the populations

of electrons in the conduction band and valence band.

The displacement from equilibrium is such that the carrier populationscan no longer be described by a single Fermi level, however it is possible

to describe using separate quasi-Fermi levels for each band.
http://en.wikipedia.org/wiki/Electronhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Valence_bandhttp://en.wikipedia.org/wiki/Populationshttp://en.wikipedia.org/wiki/Thermodynamic_equilibriumhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Energyhttp://en.wikipedia.org/wiki/Thermodynamic_equilibriumhttp://en.wikipedia.org/wiki/Populationshttp://en.wikipedia.org/wiki/Valence_bandhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Conduction_bandhttp://en.wikipedia.org/wiki/Electron


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Example: Quasi Fermi Level



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d h


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Types of Solids

Atomic Bonding


Doping


Electron and holes

Energy-band model





Carrier Transport

Carrier drift, Carrier Diffusion

Conductivity, ResistivityKEEE 2224 Electronics Devices


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Intrinsic Carrier Concentration, ni



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Intrinsic Fermi Level, Ei



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Charge Carrier Concentration

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For Si, ni = 1e10 cm-3



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Thermal Equilibrium



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Majority and Minority Carriers

N-type material, the electron is called majority carrier and hole the minority

carrier

P-type material, the hole is called majority carrier and electron the minority

carrier.


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Example: Extrinsic Semic

Silicon at T=300K is doped with arsenic atoms

such that the concentration of the electrons is

n0=7 x 1015 cm-3.

(a) Find Ec-EF

(b) Finc EF-Ev

(c) po

(d) Which carrier is the minority carrier?

(e)Ev-EF



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Anwers


d) Holes


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Example A new semiconductor is to be designed. The

semiconductor is to be p-type and doped with 5 x 1015

cm-3 acceptor atoms. Assume complete ionization and

assume Nd=0. The effective density of states functions

are Nc=1.2 x 10

18

cm

-3

and Nv=1.8 x 10

19

cm

-3

at T=300Kand vary as T2. A particular semiconductor device

fabricated with this material requires the hole

concentration to be no greater than 5.08 x 1015cm-3at

T=350K. What is the minimum bandgap energyrequired in this new material?



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Answer


B i S i d t Ph i


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32


Types of Solids

Atomic Bonding


Doping


Electron and holes

Energy-band model





Carrier Transport




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Free Carriers in Semiconductors

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Three primary types of carrier action occur

inside a semiconductor:

drift

diffusionrecombination-generation



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Electron as Moving Particles

34

Where mn* is the electron effective mass



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Example: Kinetic Energy of ElectronThe initial velocity of an electron is 107 cm/s. If the kinetic

energy of the electron increases by E=10-12 eV, determine theincrease in velocity. Hint:



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Carrier Effective Mass

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In an electric field, , an electron or a hole accelerates:

Electron and hole conductivity effective masses:



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Carrier Scattering

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Mobile electrons and atoms in the Si lattice are

always in random thermal motion.

lattice scattering or phonon scattering

increases with increasing temperature

Average velocity of thermal motion forelectrons: ~107 cm/s @ 300K

Other scattering mechanisms:

deflection by ionized impurity atoms

deflection due to Coulombic force betweencarriers

only significant at high carrier concentrations

The net current in any direction is zero, if no electric

field is applied.KEEE 2224 Electronics Devices


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Example: Scattering Three scattering mechanisms are present in a particular

semiconductor material. If only the first scattering mechanismwere present, the mobility would be 1=2000 cm

2/V-s, is only

the second mechanism were present, the mobility would be

2=1500 cm2/V-s and if only the third mechanism were

present, the mobility would be 3=500 cm2

/V-s . What is thenet mobility?



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Carrier Drift

When an electric field (e.g. due to an externally applied voltage)

is applied to a semiconductor, mobile charge carriers will be

accelerated by the electrostatic force. This force superimposes

on the random motion of electrons:

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Electrons driftin the direction opposite to the electric field(current flows). Hence, because of scattering, electrons in a

semiconductor do not achieve constant acceleration.

However, they can be viewed as quasi-classical particles

moving at a constant average drift velocity Vd.KEEE 2224 Electronics Devices


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Small devices=> non-stationary transport

velocity overshoot=> faster devices (smaller transit time)

0

5x106

1x107

1.5x107

2x107

2.5x107

3x107

0 0.5 1 1.5 2 2.5 3

E=1kV/cm

E=4kV/cm

E=10kV/cmE=20kV/cm

E=40kV/cm

E=100kV/cm

Drif

tvelocity

[cm

/s]

time [ps]

T = 300 K

Silicon

Velocity overshoot effect


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Diffusion

Particles diffuse from regions of higher

concentration to regions of lower concentration

region, due to random thermal motion.



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Diffusion Current

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D is the diffusion constant, or diffusivity.KEEE 2224 Electronics Devices


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Total Current = Drift + Diffusion



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Einstein Relationship between D and


Under equilibrium conditions,JN= 0 andJP= 0


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Example : Diffusion Constant

What is the hole diffusion constant in a sample

of silicon with p= 410 cm2 / V s ?

Answer:


WherebykT/q = 26 mV at room temperature with

boltzman constant , k = 1.3806488 10-23m2kg s-2K-1 ,

T=300K, and q= 1.6e19 C.



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46


Types of Solids

Atomic Bonding


Doping


Electron and holes

Energy-band model





Carrier Transport




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Carrier Mobility (Electron Momentum)

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With every collision, the electron loses momentum

Between collisions, the electron gains momentum

where mn = average time between scattering

events



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Carrier Mobility


if


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Drift Current

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vd t A = volume from which all holes cross plane in time t

p vd

t A = # of holes crossing plane in time t

q p vdt A = charge crossing plane in time t

q p vdA = charge crossing plane per unit time = hole current

Hole current per unit areaJ = q p vdKEEE 2224 Electronics Devices

C d i i


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Conductivity

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The conductivity of

a semiconductor isdependent on

the carrier

concentrations

and mobilities


El i l R i


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Electrical Resistance



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EXAMPLE : Resistivity

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Consider a Si sample doped with 1016/cm3 Boron. What is itsresistivity?

Answer:

And for Si, ni = 1e10 cm-3



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Example: Dopant Compensation

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Consider the same Si sample, doped additionally with1017/cm3Arsenic. What is its resistivity?

Answer:


R i i f E ti i F d t l


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Revision of Equation in Fundamental

Electronic Physics


Basic equations for Semic. device operation:

Maxwells equations Minority-carrier equations

Continuity equations

Poissons equation


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Relaxation to Equilibrium State

Consider a semiconductor with no current flow

in which thermal equilibrium is disturbed by the

sudden creation of excess holes and electrons.

The system will relax back to the equilibriumstate via R-G mechanism:


Mi it C i E ti


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Minority Carrier Equation


Consider minority hole carrier injection at x=0

C ti it E ti


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Continuity Equations


The continuity equation describes a basic concept, namely that a change in carrier density

over time is due to the difference between the incoming and outgoing flux of carriers plus

the generation and minus the recombination.

Poisson Equation


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Poisson Equation



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Important Constants


Important formulae that govern the conductivity of a


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p f g y f

semiconductor :


The intrinsic carrier concentration: ii pn

(same density of free electrons and holes in an intrinsic semiconductor).

With n, prespectively the electron and hole density ; ND,Arespectively the donor

and acceptor doping density (concentration).

Withn, pthe mobility of electrons, resp. holes.

m

q

With qthe charge, tthe average time between collisions (scattering), mthe mass.



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p f g y f

semiconductor :

Ev

vthe drift velocity and Ethe electric field. Electrons and holes have

opposite velocities.

In a homogeneously doped semiconductor or a semiconductor with a constant

carrier density, applying an electric field will cause drift currents to flow.

WithAthe cross sectional area perpendicular to the current flow, Eis the

applied electric field

When carrier gradients exist in a semiconductor, diffusion currents will occur.

With Dn,pthe diffusion constant of electrons respectively holes,xis the

direction of carrier propagation.



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The Einstein equation gives the relationship between the diffusion constant and

the mobility of the carrier:

p f g y f

semiconductor :

e

kTD

with, kthe Boltzman constant, Tthe temperature in Kelvin.

In the general case where both concentration gradients and electric fields arepresent the total current is the sum of both drift and diffusion currents:



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Another basic equation in semiconductor devices is the Poisson equation

p f g y f

semiconductor :

With Vthe electrostatic potential, the charge density as a function ofx;p& n

the free hole, resp. electron density which are both a function ofxas well as of Vand N-A and N

+Dthe concentration of ionised doping atoms which are a function

ofx.

Lecture 2 Fundamentals

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