Semiconductor Basic and Electronic Concepts

Lecture-1

Outline

Basic Concept Energy band gaps, Concept of Hole, Free Carriers Effective Mass, Density of States (DOS) Fermi-Dirac Function Carrier Concentration, Charge Neutrality

Electronic Concept Drift

Drude model, Scattering Conductivity and Resistivity

Diffusion Drift & Diffusion Current Continuity Equation

2

Energy Band-gap

Eg = Ec - Ev

Ec -- the lowest possible conduction band energy Ev -- the highest possible valence band energy Eg (band gap energy ): the energy takes to break a bond in the spatial view of the crystal

Eg = 1.42 eV (GaAs) =1.12 eV (Si) @T= 300K

3

Concept of Hole

A hole is defined as an empty state in the valence band

A pure semiconductor (ideally, T ----- 0K ) contains a completely filled (with electrons) valence band and a completely empty conduction band. Thus no current can flow:

no electrons at all in the conduction band no empty states (i.e. states containing no

electrons) in the valence band to which electrons inside this band can move.

4

Free Carriers

Current can flow (charge transport ) though existing free electrons or holes, so-called charge carriers (free carriers) Free carriers generated by excitation or doping

1- Excitation

If semiconductor is excited by energy (light, temperature or electric fields), the electrons in valence band can jump to the conduction band and take part in a current flow. Also Known as electron-hole pair (intrinsic) generation.

5

Free Carriers 2- Doping

If Si is doped with Group V atoms, i.e. they have one more valence electron, Nd impurity atoms are called donors.

Similarly, Na impurity atoms are called acceptors (Group-III atoms) 6

Effective Mass Second law of Newton for crystals that are large compared to atomic dimensions

-- effective mass of an electron, and -- group velocity of the wave packet that describes the electron motion.

Similarly, the empty states in the valence band, have an effective mass of

7

e.g. conductivity effective mass and density of states effective mass - in Si

= 0.26 m0 and 1.18 m0

Density of States (DOS)

DOS function, N(E), describes the distribution of energy states, i.e. number of states per unit energy and unit volume

8

Fermi-Dirac Function

Fermi-Dirac function f(E) specifies how many existing states at energy (E) will be filled with an electron. formally, f(E) specifies, under equilibrium conditions, probability that an available state at an energy E will be occupied by an electron.

9

Carrier Concentration

The carrier concentration between the energy levels E1 and E2, n, can be expressed as the integral

electron concentration in the conduction band at equilibrium (n0)

where ET is the upper band edge of the conduction band

10

Carrier Concentration

Similarly

11

Charge Neutrality A material is doped with Nd donors and Na acceptors, at RT all impurities are ionized, so-called complete ionization, so that there exist Nd positive ions and Na negative ions. Thus, for charge neutrality can be written as:

NOTE: This is based on assumption that all impurities are ionized

In most practical cases the net dopant concentration >> intrinsic concentration

12

Carrier Action

There are three kinds of carrier action; Drift, Diffusion and Recombination and Generation.

1- Drift is the mechanism in operation when an external electric field is applied to the semiconductor; charged particles respond to the electric field by moving, depending on the charge, along the field or opposite to it.

13

Carrier Action

2-Diffusion is the process whereby particles tend to spread out or redistribute as a result of their random thermal motion, migrating on a macroscopic scale from regions of high particle concentration into regions of low particle concentration.

Mathematically the diffusion of electrons in a non-uniformly doped semiconductor can be written as

14

Carrier Action

3-Recombination-Generation(R-G) is not manifested through carrier transport, but rather affects current densities by changing the carrier concentration. Unlike drift and diffusion the terms recombination and generation do not refer to a single process: there are several processes based on R-G.

Either the R-G process goes directly from band to band or it passes through some localized allowed energy state, an R-G center, sometimes referred to as a defect state or trap. 15

Carrier Action Recombination-Generation(R-G)

16

Energy Bandgap In the context of R-G, it is possible to notice the differentiation between direct and indirect band-gap energies of semiconductors.

Direct Energy Bandgap

The energy minima of both the conduction and valence band occur at k = 0.

The transition occurs without any R-G center visited in between conduction and valence band, but not necessarily without a change in momentum.

17

Energy Bandgap

Indirect Energy Bandgap

The minimum of the conduction band is displaced to a non-zero momentum in the k- space.

In indirect transitions, they require an R-G center via which the R-G makes the transition. There is large probability of occurrence for two-particle interaction. e.g. a free carrier and a phonon that can take place if there are R-G centers into which electrons and holes can make transitions

18

Drift: Drude model

random scattering centers

e-

Electric field E ma F =

t

v m qE

=

E m

q v avg

t =

E m

nq nq vavg j

t 2 = =

{

m

{

s Caused by electric field

Electron density constant 19

Types of scattering

Electron-phonon Very temperature dependent Phonons are lattice vibrations At low temperatures, lattice is perfectly still

Impurity scattering Temperature independent Depends on impurity concentration

impurityphononelectrontotal ttt

111=

20

Drift Mechanism

The general description of an electron in an electric field now becomes

Where, I is mean free time , and vx is the electron velocity in x-direction, it is calculated by following relation

21

Drift Mechanism

The current density for n electrons with charge q as

Where, n is the mobility of the electron, and defined as

Similarly for hole, p is defined as

22

Conductivity and Resistivity

The definition of the conductivity of the semiconductor, , as

The current density can also be written as

Similarly, the resistivity is defined as;

An essential Ohms law can be written as in form of resistivity is as following

23

Example

Solution : Known parameters: q = 1.6x10-19 C A = 5 mm2

Ex =10 V/cm n= 8500 cm

2/Vs for GaAs p= 400 cm

2/Vs

24

Example -cont Sample-1

= 1.04x 10 7 cm 3

I = 7.1 nA

Sample-2

no = 3.8x105 cm 3 , and Po = 1.04x 10

7 cm 3

I = 0.59 nA.

These samples are not very good to conduct current due to the extremely light doping.

25

Diffusion current

dx

xdnqDJ

)(=

Now assume n depends on position:

nB

nq

TkD m=

dx

xdnJ

)(

Einstein relationship:

26

Diffusion Mechanism

Diffusion due to a doping gradient.

A sketch showing a situation where the semiconductor has a non-zero doping gradient and where has to exist diffusion of carriers that balances the difference in the concentration. Assuming electrons as carriers (means no electric field), the mean free path, lcn

lcn is the distance covered by an electron in the mean free time, cn

27

Diffusion Mechanism As each electron carries a charge , -q, the particle flow corresponds to a electron current density due to diffusion

------ Diffusion coefficient Where

Similarly for hole

28

Drift and Diffusion Current

dx

xdnDqEnqJ nnn

)(.. = m

E, n can depend on x!

dx

xdpqDEpqJ ppp

)(= m

Holes:

Electrons:

29

Continuity Equation

txAJq

)(1

)(xJ

)( xxJ

x

A

tdxxAJq

)(1

---- enter

---- Leave

How many electrons in blue box after a time t?

txAnn o

t---- recombine

txAGn ---- generated

txAGtxAnntdxxJxJAe

N no

=

t)()(

1

txA Divide by:

no Gnn

x

xJ

qt

n

=

t

)(130

nnn Gn

x

nE

x

nD

t

n

=

t

m

)(2

2

Neutrality Assumption

oo pppnnn =

For p-type materials:

ppp Gp

x

pE

x

pD

t

p

=

t

m

)(2

2

For n-type materials:

31

Steady state

t

n

x

nD =

2

2

==

nD

xxnxn

t exp)0()(

02

2

=

=

t

n

x

nD

t

nn

(Assume E=0)

length diffusionnDt

Typically about a micron.

32

Thank you!

33