MSE-630 Week 2

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MSE
-
630 Week 2

Conductivity, Energy Bands and
Charge Carriers in
Semiconductors

Objectives:


To understand conduction, valence energy
bands and how bandgaps are formed


To understand the effects of doping in
semiconductors


To use Fermi
-
Dirac statistics to calculate
conductivity and carrier concentrations


To understand carrier mobility and how it is
influenced by scattering


To introduce the idea of “effective mass”


To see how we can use Hall effect to determine
carrier concentration and mobility



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3


Ohm's

Law:

D
V = I R

voltage drop (volts)

resistance (Ohms)

current (amps)


Resistivity,
r

and
Conductivity,
s
:


--
geometry
-
independent forms of Ohm's Law

E: electric

field

intensity

resistivity

(Ohm
-
m)

J: current density

conductivity

• Resistance:

ELECTRICAL CONDUCTION

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Resistivity and Conductivity as
charged
particles

mobility
,
m

=

Where

is the average
velocity

is the average distance between
collisions,

divided by the average time between
collisions,

t

d

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• Electrical Conductivity given by:

11

# electrons/m
3

electron mobility

# holes/m
3

hole mobility

• Concept of electrons and holes:

CONDUCTION IN TERMS OF ELECTRON AND
HOLE MIGRATION

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4

• Room T values (Ohm
-
m)

-
1

CONDUCTIVITY
:
COMPARISON

As the distance between
atoms decreases, the
energy of each orbital
must split, since
according to Quantum
Mechanics we cannot
have two orbitals with
the same energy.


The splitting results in “bands” of
electrons. The energy difference
between the conduction and valence
bands is the “gap energy” We must
supply this much energy to elevate an
electron from the valence band to the
conduction band. If Eg is < 2eV, the
material is a semiconductor.

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• Metals:

--

Thermal energy puts


many electrons into


a higher energy state.

• Energy States:

--

the cases below


for metals show


that nearby


energy states


are accessible


by thermal


fluctuations.

CONDUCTION & ELECTRON TRANSPORT

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• Insulators:


--
Higher energy states not


accessible due to gap.

• Semiconductors:


--
Higher energy states


separated by a smaller gap.

ENERGY STATES: INSULATORS AND
SEMICONDUCTORS

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10

• Data for
Pure Silicon
:


--
s

increases with T


--
opposite to metals

electrons

can cross

gap at

higher T

material

Si

Ge

GaP

CdS

band gap (eV)

1.11

0.67

2.25

2.40

PURE SEMICONDUCTORS: CONDUCTIVITY VS T

Simple representation of silicon atoms bonded in a crystal.
The dotted areas are covalent or shared electron bonds.
The electronic structure of a single Si atom is shown
conceptually on the right. The four outermost electrons are
the valence electrons that participate in covalent bonds.


Portion of the periodic table relevant
to semiconductor materials and
doping. Elemental semiconductors
are in column IV. Compound
semiconductors are combinations of
elements from columns III and V, or
II and VI.

Electron (
-
) and hold (+) pair
generation represented b a broken
bond in the crystal. Both carriers are
mobile and can carry current.

Intrinsic carrier concentration vs.
temperature.

Doping of group IV semiconductors
using elements from arsenic (As, V)
or boron (B, III)

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Intrinsic
:


# electrons = # holes (n = p)


--
case for pure Si


Extrinsic
:


--
n ≠ p


--
occurs when impurities are added with a different


# valence electrons than the host (e.g., Si atoms)


N
-
type

Extrinsic: (n >> p)


P
-
type

Extrinsic: (p >> n)

INTRINSIC VS EXTRINSIC CONDUCTION

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Equations describing
Intrinsic
and
Extrinsic

conduction

Using the Fermi
-
Dirac equation, we can find the number of charge carrier per
unit volume as:



N
e

= N
o
exp
(
-
Eg/2kT)


N
o

is a preexponential function,

E
g

is the band
-
gap energy and

k

is Boltzman’s constant (8.62 x 10
-
5 eV/K
)


If


Eg

> ~2.5 eV



the material is an insulator

If

0 <
Eg

< ~2.5 eV



the material is a semi
-
conductor


Semi
-
conductor conductivity can be expressed by:


s
(T)

=
s
o
exp(
-
E*/nkT)



E* is the relevant gap energy (Eg, Ec
-
Ed or Ea)


n is 2 for intrinsic semi
-
conductivity and 1 for extrinsic semi
-
conductivity

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• Data for Doped Silicon:


--
s

increases doping


--
reason:
imperfection sites


lower the activation energy to


produce mobile electrons.

• Comparison:

intrinsic

vs


extrinsic conduction...


--
extrinsic doping level:


10
21
/m
3

of a n
-
type donor


impurity (such as P).


--
for T < 100K: "freeze
-
out"


thermal energy insufficient to


excite electrons.


--
for 150K < T < 450K: "extrinsic"


--
for T >> 450K: "intrinsic"

DOPED SEMICON: CONDUCTIVITY VS T

Dopant designations and
concentrations

Resistivity as a function of
charge mobility and number

When we add carriers by doping, the number of additional carrers, Nd, far
exceeds those in an intrinsic semiconductor, and we can treat conductivity as

s

= 1/
r

= q
m
d
N
d

Simple band and bond representations of pure
silicon. Bonded electrons lie at energy levels
below Ev; free electrons are above Ec. The
process of intrinsic carrier generation is
illustrated in each model.

Simple band and bond representations of doped
silicon. E
A

and E
D

represent acceptor and donor
energy levels, respectively. P
-

and N
-
type
doping are illustrated in each model, using As as
the donor and B as the acceptor

Behavior of free carrier concentration
versus temperature. Arsenic in silicon is
qualitatively illustrated as a specific
example (N
D

= 10
15

cm
-
3
). Note that at high
temperatures ni becomes larger than 10
15

doping and
n≈n
i
. Devices are normally
operated where
n

= N
D
+
.

Fabrication occurs
as temperatures where
n≈n
i

Fermi level position in an undoped (left),
N
-
type (center) and P
-
type (right)
semiconductor. The dots represent free
electrons, the open circles represent
mobile holes.


Probability of an electron occupying
a state. Fermi energy represents the
energy at which the probability of
occupancy is exactly ½.

The density of allowed states at an
energy
E
.

Integrating the product of the probability of occupancy with the density of
allowed states gives the electron and hole populations in a
semiconductor crystal.

Effective Mass

In general, the curve of Energy vs. k is non
-
linear, with E increasing as k increases.


E = ½ mv
2

= ½ p
2
/m = h
2
/4
p
m
k
2

We can see that energy varies inversely with
mass. Differentiating E wrt k twice, and
solving for mass gives:


Effective mass is significant because it
affects charge carrier mobility, and
must be considered when calculating
carrier concentrations or momentum

Effective mass and other semiconductor properties may be found in
Appendix A
-
4

Substituting the results from the previous slide into the expression for the
product of the number of holes and electrons gives us the equation above.
Writing NC and NV as a function of ni and substituting gives the equation
below for the number of holes and electrons:

In general, the number of electron
donors plus holes must equal the
number of electron acceptors plus
electrons

Fermi level position in the forbidden band for a
given doping level as a function of temperature.

The energy band gap gets smaller with
increasing temperature.

In reality, band structures are highly
dependent upon crystal orientation. This
image shows us that the lowest band gap
in Si occurs along the [100] directions,
while for GaAs, it occurs in the [111]. This
is why crystals are grown with specific
orientations.

The diagram showing the
constant energy surface
(3.10 (b)), shows us that
the effective mass varies
with direction. We can
calculate average effective
mass from:

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Allows flow of electrons in one direction only

(e.g., useful


to convert alternating current to direct current.

• Processing: diffuse P into one side of a B
-
doped crystal.

• Results:

--
No applied potential:


no net current flow.

--
Forward bias: carrier


flow through p
-
type and


n
-
type regions; holes and


electrons recombine at


p
-
n junction; current flows.

--
Reverse bias: carrier


flow away from p
-
n junction;


carrier conc. greatly reduced


at junction; little current flow.

P
-
N RECTIFYING JUNCTION

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512

Piezoelectrics

Field
produced
by stress:

Strain
produced
by field:

Elastic
modulus:



= electric field

s

= applied
stress

E
=Elastic
modulus

d

= piezoelectric
constant

g

= constant

2

• Created by current through a coil:

• Relation for the applied magnetic field, H:

applied magnetic field

units = (ampere
-
turns/m)

current

APPLIED MAGNETIC FIELD

In
Air
:

With
Magntic
cor:

m
o
1.257  10

6

Wb/(A

m)

4

• Measures the response of electrons to a magnetic


field.

• Electrons produce magnetic moments:

• Net magnetic moment:


--
sum of moments from all electrons.

• Three types of response...

Adapted from Fig.
20.4,
Callister 6e
.

MAGNETIC SUSCEPTIBILITY

Magntic domains align in
prsnc of magntic fild, H

Hysteresis Loop

Soft and Hard Magnetic Materials

Typical proprtis
of soft and hard
magntic matrials

9

• Information is stored by magnetizing material.

recording head

recording medium

• Head can...


--
apply magnetic field H &


align domains (i.e.,


magnetize the medium).


--
detect a change in the


magnetization of the


medium.

• Two media types:

--
Particulate: needle
-
shaped


g
-
Fe
2
O
3
. +/
-

mag. moment


along axis. (tape, floppy)

--
Thin film: CoPtCr or CoCrTa


alloy. Domains are ~ 10
-
30nm!


(hard drive)

MAGNETIC STORAGE

Magntic Forcs

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Sheet Resistivity

R =


=

=

=

=

r
s

is the
sheet resistivity

Sheet resistivity is the
resistivity divided by
the thickness of the
doped region, and is
denoted
W
/


L

w

If we know the area per square, the
resistance is

Conductivity

Charge carriers follow a
random path unless an
external field is applied.
Then, they acquire a drift
velocity that is dependent
upon their
mobility,

m
n

and the
strength of the field,




V
d

=
-
m
n



The average drift vel ocity,
v
av

is dependent

Upon the mean time between collisions,
2
t

Charge Flow and Current Density

Current density,
J
, is the rate at
which charges, cross any plane
perpendicular to the flow direction.

J =
-
n
q
v
d

= n
q
m
n
  s

n
is the number of charges, and


q
is the charge (1.6 x 10
-
19

C)

OHM’s Law:

V = IR



Resistance,
R(
W
F

is an
extrinsic

quantity. Resistivity,

r
(
W

m⤬


is the
corresponding
intrinsic

property.

r
= R*A/l

Conductivity,
s
Ii猠the牥捩p牯捡lof牥獩獴iit示
s
(
W


-
1

= 1/
r


The total current density depends upon the total charge
carriers, which can be ions, electrons, or holes


J = q(n
m
n

+ p
m
p
)