Semiconductor Devices and Models I

woundcallousSemiconductor

Nov 1, 2013 (3 years and 9 months ago)

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November 1, 2013

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1

ECSE
-
6230

Semiconductor Devices and Models I

Lecture 4

Prof. Shayla Sawyer

Bldg. CII, Rooms 8225

Rensselaer Polytechnic Institute

Troy, NY 12180
-
3590

Tel. (518)276
-
2164

Fax. (518)276
-
2990

e
-
mail: sawyes@rpi.edu


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Outline



Carrier Concentration at Thermal Equilibrium


Introduction


Fermi Dirac Statistics


Donors and Acceptors


Determination of Fermi Level


Dopant Compensation


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Carrier Concentration Introduction


One of most important properties of a
semiconductor is that it can be doped with
different types and concentrations of
impurities


Intrinsic material
-
no impurities or lattice
defects


Extrinsic
-
doping, purposely adding impurities


N
-
type mostly electrons


P
-
type mostly holes


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Carrier Concentration Introduction


To calculate semiconductor electrical
properties, you must know the number of
charge carriers per cm
3

of the material


Must investigate distribution of carriers over
the available energy states


Statistics are needed to do so

Fermi
-
Dirac statistics


Distribution of electrons over a range of
allowed energy levels at thermal equilibrium

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Fermi
-
Dirac Distribution

Probability that an available energy

state at E will be occupied by an

electron at absolute temperature T



Mathematically,

E
F

(Fermi Energy) is the energy at
which f(E) = 1/2

The transition region in (E
-

E
F
)

from f(E) =1 to f(E) = 0 is within

3 k T.


When T


0,

E is discontinous at E = E
F
.

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Fermi
-
Dirac Distribution


To apply the Fermi
-
Dirac distribution, we must recall
that f(E) is the probability of occupancy of an
available
state

at E.


Where can we find available states?

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Carrier Concentration

At Thermal Equilbrium


Number of electrons (occupied conduction band
levels) given by:





Density of states g(E) can be approximated by
the
density near the bottom of the conduction band


where

M
C
is the number of equiv. minima

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Carrier Concentration

At Thermal Equlibrium


The integral can be evaluated as

Where N
C

is the effective
density of states in the
conduction band given by:

For the valence band, consider light and
heavy holes for the density of states
effective mass for holes (m
dh
)

and use similar equation

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Carrier Concentration

At Thermal Equlibrium: Intrinsic


For intrinsic material lies at some intrinsic level E
i
near the middle of the band gap, electron and hole
concentrations are





Law of mass action: product of maj. and min.
carriers is fixed

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Donors and Acceptors


Doping by substituting Si atoms with Column III or V of
the Periodic Table.


Very dilute doping level, typical 10
14

to 10
18

cm
-
3
, results
in discrete energy levels.


Donor level is neutral if filled with e
-
,

positively charged if empty.

e.g., P, As, and Sb in Si.


Acceptor level is neutral if empty,

negatively charged if filled with e
-
.

e.g., B and Al in Si.


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Donors and Acceptors

“Hydrogen
-
like” Model to describe
dopant atom ionization.

Hydrogen Atom





Ground state (n=1) ionization energy
of hydrogen is 13.6 eV.


To
estimate

ionization energy of
donors, replace m
0

with m* and



0

and

S

(e.g., 11.7

0

for Si).


E
D

= (

0

/

S

)
2

( m*/ m
0

) E
H


~
0.006 eV for Ge,


0.025 eV for Si,


0.007 eV for GaAs

E
A


~ 0.015 eV for Ge,



0.05 eV for Si,


0.05 eV for GaAs


http://gemologyproject.com/wiki/index.php?title=
The_Chemistry_of_Gemstones


kT~0.026eV

Comparable to thermal energies so ionization

is complete at room temperature

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Donors and Acceptor Levels



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Determination of Fermi Level

Intrinsic

Semiconductor

-

E
F

~ E
g

/ 2

Extrinsic

Semiconductor
-

E
F

adjusted to preserve





space charge neutrality

Space Charge Neutrality





n
0

+ N
A
-

= N
D
+

+ p
0







Total Neg. Charges = Total Positive Charges



electrons and ionized acceptors=holes and ionized donors






100% ionization assumed.

Ionized Concentration of Donors

When impurities are introduced:





where

g
D

is the ground state degeneracy of donor impurity


g
D

= 2 (i) electrons with either spin




(ii) no electrons at all

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Determination of Fermi Level

Ionized Acceptors



where

g
A

is the ground state degeneracy of acceptor impurity


g
A

= 4 for Ge, Si, and GaAs because


(i) Acceptor levels can receive electrons with either spin and


(ii) Valence band double degeneracy.


Space Charge Neutrality

N
-
type Semiconductor is assumed.

n=N
D
+
+p ~ N
D
+
therefore

Charge Neutrality


Since the material must balance
electrostatically, the Fermi level must adjust
such that charge neutrality remains.


The Fermi level therefore can be calculated for
a set given N
D
, E
D
, N
C
, and T

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Dopant Compensation

When both n
-

and p
-
type (donor and acceptor) impurities are present,

the space charge neutrality condition

n
0

+ N
A
-

= N
D
+

+ p
0

holds, even when the impurities are deep levels.

In an n
-
type semiconductor where N
D
>>>N
A








Fermi level can be obtained from







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Dopant Compensation

In an p
-
type semiconductor where N
A
>>>N
D






Fermi level can be obtained from







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Example Problem

A hypothetical semiconductor has an intrinsic carrier
concentration of 1.0 x 10
10

cm
-
3

at 300 K, it has a
conduction and valence band effective density of
states N
C

and N
V

both equal to 10
19

cm
-
3
.

a)
What is the band gap Eg?

b)
If the semiconductor is doped with N
d

= 1x10
16

donors/cm
3
, what are the equilibrium electron and
hole concentrations at 300K?

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Example Problem

A hypothetical semiconductor has an intrinsic carrier
concentration of 1.0 x 10
10

cm
-
3

at 300 K, it has a
conduction and valence band effective density of
states N
C

and N
V

both equal to 10
19

cm
-
3
.

c) If the same piece of semiconductor, already having N
d
= 1x10
16

donors/cm
3,
is also doped with N
a
= 2x10
16

acceptors/cm
3

, what are the new equiliblrium
electron and hole concentrations at 300 K?

a)
Consistent with your answer to part (c), what is the
Fermi level position with respect to the intrinsic
Fermi level, E
F



E
i
?