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P.Ravindran, PHY02E – Semiconductor Physics, 17 January 2014 : Carriers Concentration in Semiconductors - IV

http://folk.uio.no/ravi/semi2013

Prof.P. Ravindran, Department of Physics, Central University of Tamil

Nadu, India

Carriers Concentration in Semiconductors - IV

1


Bandgap Energy: Energy required to remove a valence electron and allow it to freely conduct.

Intrinsic Semiconductor: A “native semiconductor” with no dopants. Electrons in the

conduction band equal holes in the valence band. The concentration of

electrons (=holes) is the intrinsic concentration, ni.

Extrinsic Semiconductor: A doped semiconductor. Many electrical properties controlled by the

dopants, not the intrinsic semiconductor.

Donor: An impurity added to a semiconductor that adds an additional electron not found in

the native semiconductor.

Acceptor: An impurity added to a semiconductor that adds an additional hole not found in the

native semiconductor.

Dopant: Either an acceptor or donor.

N-type material: When electron concentrations (n=number of electrons/cm3) exceed the hole

concentration (normally through doping with donors).

P-type material: When hole concentrations (p=number of holes/cm3) exceed the electron

concentration (normally through doping with acceptors).

Majority carrier: The carrier that exists in higher population (i.e. n if n>p, p if p>n)

Minority carrier: The carrier that exists in lower population (i.e. n if n<p, p if p<n)

Summary of Important terms and symbols


Increasing conductivity by temperature

150 200 250 300 350 400 450 500100

1 103

1 104

1 105

1 106

1 107

1 108

1 109

1 1010

1 1011

1 1012

1 1013

1 1014

1 1015

1 1016

1 1017

Carrier Concentration vs Temp (in Si)

Temperature (K)

Intr

insi

c C

once

ntra

tion

(cm

^-3

)

niT

T

Therefore the conductivity of a semiconductor is influenced by temperature

As temperature increases, the number of free electrons and holes created increases exponentially.


The conductivity of the semiconductor material increases when the temperature increases.

This is because the application of heat makes it possible for some electrons in the valence band to move to the conduction band.

Obviously the more heat applied the higher the number of electrons that can gain the required energy to make the conduction band transition and become available as charge carriers.

This is how temperature affects the carrier concentration.

Another way to increase the number of charge carriers is to add them in from an external source.

Doping or implant is the term given to a process whereby one element is injected with atoms of another element in order to change its properties.

Semiconductors (Si or Ge) are typically doped with elements such as Boron, Arsenic and Phosphorous to change and enhance their electrical properties.

Increasing conductivity


Extrinsic MaterialBy doping, a crystal can be altered so that it has a predominance of either

electrons or holes. Thus there are two types of doped semiconductors, n-type (mostly electrons) and p-type (mostly holes). When a crystal is doped such that the equilibrium carrier concentrations n0 and po are different from the intrinsic carrier concentration ni, the material is said to be

extrinsic.

Donor impurities (elements of

group V): P, Sb, As

Acceptor elements (group III):

B, Al, Ga, In

The valence and conduction bands of silicon

with additional impurity energy levels within

the energy gap.

When impurities or lattice defects are introduced, additional levels are created in the energy bands structure, usually within the band gap.

Total number of electrons

III – Al – 13

IV – Si – 14

V - P - 15


Extrinsic Material – donation of electrons

An impurity from column V introduces an energy level very near the conduction band in Ge or Si. This level is filled with electrons at 0 K, and very little thermal energy is required to excite these electrons to the conduction band. Thus, at about 50-100 K nearly all of the electrons in the impurity level are "donated" to the conduction band. Such an impurity level is called a donor level, and the column V impurities in Ge or Si are called donor impurities. From figure we note that the material doped with donor impurities can have a considerable concentration of electrons in the conduction band, even when the temperature is too low for the intrinsic EHP concentration to be appreciable. Thus semiconductors doped with a significant number of donor atoms will have n0>>(ni,p0) at room temperature. This is n-type material.

Donation of electrons from a donor

level to the conduction band

n-type material


Extrinsic Material – acceptance of electrons

Acceptance of valence band electrons by

an acceptor level, and the resulting

creation of holes.

Atoms from column III (B, Al, Ga, and In) introduce impurity levels in Ge or Si near the valence band. These levels are empty of electrons at 0 K. At low temperatures, enough thermal energy is available to excite electrons from the valence band into the impurity level, leaving behind holes in the valence band. Since this type of impurity level "accepts" electrons from the valence band, it is called an acceptor level, and the column III impurities are acceptor impurities in Ge and Si. As figure indicates, doping with acceptor impurities can create a semiconductor with a hole concentration p0 much greater than the conduction band electron concentration ni (this is p-type material).

P-type material


The motion of electrons in a crystal can be visualized and described in a

quasi-classical manner.

In most instances

The electron can be thought of as a particle.

The electronic motion can be modeled using Newtonian mechanics.

The effect of crystalline forces and quantum mechanical properties are

incorporated into the effective mass factor.

m* > 0 : near the bottoms of all bands

m* < 0 : near the tops of all bands

Carriers in a crystal with energies near the top or bottom of an energy

band typically exhibit a constant (energy-independent) effective mass.

` : near band edge

Effective Mass Approximation

constant2

2

dk

Ed


Effective Mass (I)

An electron moving in respond to an applied electric field.

E

E

within a Vacuum within a semiconductor crystal

dt

dvmEqF 0

dt

dvmEqF n

It allow us to conceive of electrons and holes as quasi-classical particles and to employ classical particle

relationships in semiconductor crystals or in most device analysis.


Carrier Movement Within the Crystal

Density of States Effective Masses at 300 K

Ge and GaAs have “lighter electrons” than Si which results in faster devices


Effective Mass (II)

Electrons are not free but interact with periodic potential of the lattice.

Wave-particle motion is not as same as in free space.

Curvature of the band determine m*.

m* is positive in CB min., negative in VB max.


Effective Masses

Curvature of the band determines the effective mass of the

carriers in a crystal, which is different from the free electron

mass.

Smaller curvature heavier mass

Larger curvature lighter mass

• For parabolic bands, the components of the effective mass

tensor are calculated according to:

jiijkk

E

m

2

2*

11

Si

*

*

*

*

100

01

0

001

1

zz

yy

xx

m

m

m

m


• From the knowledge of the energy band structure, one can construct the plot for the

allowed k-values associated with a given energy => constant energy surfaces

*

2

*

2

*

22

2ziyixi m

k

m

k

m

kEE zyx

iC

Si Ge

Note: The electron effective mass in GaAs is isotropic, which leads to spherically

symmetric constant energy surfaces.


Due to the p-like symmetry and mixing of the V.B. states, the constant energy surfaces

are warped spheres:

The hh-band is most warped

The lh- and so-band are more spherical

212222222422

2

2)( xzzyyx kkkkkkCkBAk

mkE

A

mm

CBA

mm

CBA

mm o

soo

lho

hh

*

22

*

22

*,

6/,

6/

Valence

bands

Constant energy

surfaces

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