In this chapter we will use the density of quantum states in the conduction band and the density of quantum
states in the valence band along with the Fermi-Dirac probability function to determine the concentration of
electrons and holes in the conduction and valence bands.
We will also apply the concept of the Fermi energy to the semiconductor material.
the semiconductor in equilibrium.
This chapter deals with
Equilibrium, or thermal equilibrium, implies that no external forces such as voltages, electric fields, magnetic
fields, or temperature gradients are acting on the semiconductor.
All properties of the semiconductor will be independent of time in this case. We will initially
consider the properties of an intrinsic semiconductor, that is, a pure crystal with no impurity atoms or defects.
We will see that the electrical properties of a semiconductor can be altered by adding controlled amounts of
specific impurity atoms, called dopant atoms to the crystal.
Depending upon the type of dopant atom added, the dominant charge carrier in the semiconductor will be
either electrons in the conduction band or holes in the valence band.
1
Charge Carriers in Semiconductors
Charge Carriers in Semiconductors
Current is the rate at which charge flows.
In a semiconductor, two types of charge carrier, the electron and the hole, can contribute to a current.
Since the current in a semiconductor is determined largely by the number of electrons in the conduction band
and the number of holes in the valence band, an important characteristic of the semiconductor is the density
of these charge carriers.
1.1
Equilibrium Distribution of Electrons and Holes
Equilibrium Distribution of Electrons and Holes
The distribution (with respect to energy) of electrons in the conduction band is given by the density of allowed
quantum states times the probability that a state is occupied by an electron.
This statement is written in equation form as
n(E) = gc (E)fF (E)
(1)
where gc (E) is the density of quantum states in the conduction band and fF (E) is the Fermi-Dirac probability
function. The total electron concentration per unit volume in the conduction band is found by integrating
Equation (1) over the entire conduction-band energy. The distribution (with respect to energy) of holes in the
valence band is the density of allowed quantum states in the valence band multiplied by the probability that a
state is not occupied by an electron:
p(E) = gv (E) [1 − fF (E)]
(2)
The total hole concentration per unit volume is found by integrating this function over the entire valence-band
energy. To find the thermal-equilibrium electron and hole concentrations, we need to determine the position
of the Fermi energy EF , with respect to the bottom of the conduction-band energy Ec and the top of the
valence-band energy Ev .
To address this question. we will initially consider an intrinsic semiconductor. An ideal intrinsic semiconductor
is a pure semiconductor with no impurity atoms and no lattice defects in the crystal (e.g., pure silicon).
For an intrinsic semiconductor at T = 0 K, all energy states in the valence band are filled with electrons and
all energy states in the conduction band are empty of electrons.
The Fermi energy must, therefore, be somewhere between Ec and Ev .
The Fermi energy does not need to correspond to an allowed energy.
above 0 K, the valence electrons will gain thermal energy.
As the temperature begins to increase
A few electrons in the valence band may gain sufficient energy to jump to the conduction band.
As an electron jumps from the valence band to the conduction band, an empty stale, or hole, is created in the
valence band.
1
In an intrinsic semiconductor, then ,electrons and holes are created in pairs by the thermal energy so that the
number of electrons in the conduction bond is equal to the number of boles in the valence band.
Figure 1 shows a plot of the density of states function in the conduction band gc (E), the density of states function
in the valence band gv (E), and the Fermi-Dirac probability function for T > 0 K when EF is approximately
halfway between Ec and Ev .
If we assume, for the moment, that the electron and hole effective masses are equal, then gc (E) and gv (E) are
symmetrical functions about the midgap energy. We noted previously that the function fF (E) for E > EF is
symmetrical to the function 1 − fF (E) for E < EF about the energy E = EF .
This also means that the function fF (E) for E = EF + dE is equal to the function 1 − fF (E) for E = EF − dE.
4 • 1 Charge Carriers in
105
I,
\
\
E,
\
E,
Area = nt) =
1!1cctron
\.
concentration
o
" , (F(E)
.'..
" \.
..
(b)
..... - -
g,(E)( 1 -
= p(I;)
\.
E,.
"
(1 - f,(£))
\
\
\,,
I,
Area · Po =
hole concentration
i,
h(E) = 1
(a)
(e)
Figure 4.11 (3) OCnS11y of !1;tate!; functions. Fermi-Dirac probability function. and area.... representing clectron and bole
concentrations
for the castfunctions,
when £,. is ncur
nlidgap
(b) expanded vicw
near the conduction
band energy;
Figure 1: (a) Density
of states
Fermi-Dirac
probability
function,
and areas
representing electron
and (c) expanded view near the valence band energy.
and hole concentrations for the case when EF is near the midgap energy energy. (b) Expanded view near the
conduction band energy, and (c) expanded view near the valence band energy.
lbat the number of electrons in the conduction bond is equal to the number of boles
in the valence band.
Figure 4.10
showsof
0 plot
the density
of states function
in the C(mduction
bandg (E) above the conduction band
Figure 1b is an expanded
view
theofplot
in Figure
1, showing
fF (E) an
c
8«£),lhe density of slates function in the valence band 8,.(E). and the fermi- Dirac
energy Ec .
probability function for T > 0 K when E r is approximalely halfway between E,. and
If we
assume,
For Ibe
that the eleclt'on
hole effective
masses
are conduction band given by EquaThe product of gcE,.
(E)
and
fF (E)
is moment,
the distribution
of and
electrons
n(E)
in the
equal, Ihen g,.(E) and 1f. (E) are symmetrical functions about the Illidgap energy (rhe
tion (1).
energy midway between E,. and E. ). We noted previously that the function fr(E)
for E > £,.
symmetrical
This product is plotted
inisFigure
1a.to the function 1 - fr(E) for E < £1' about the energy
E EF. This also means that the function j,-(£) for £ = £1' + dE is equal 10 the
Figure 1c is an expanded
plot
in Figure 1a showing [1 − fF (E)] and gv (E) below the valence band
function I - view
IAE) of
for the
E = £1'
- dE.
energy Ev . The product of gv (E) and [1 − [fF (E)] is the distribution of holes p(E) in the valence band given
by Equation (2).
This product is also plotted in Figure1a.
The areas under these curves are then the total density of electrons in the conduction band and the total density
of holes in the valence band.
From this we see that if gc (E) and gv (E) are symmetrical, the Fermi energy must be at the midgap energy in
order obtain equal electron and hole concentrations
If the effective masses of the electrons and hole are not exactly equal, then the effective density of states functions
gc (E) and gv (E) will not be exactly symmetrical about the midgap energy.
The Fermi energy for the intrinsic semiconductor will then shift slightly from the midgap energy in order to
obtain equal electron and hole concentrations.
2
1.2
The n0 and p0 Equations
The n0 and p0 Equations
We will assume initially that the Fermi level remains within the bandgap energy.
The equation for the thermal-equilibrium concentration of electrons may be found by integrating Equation 1
over the conduction band energy, or
Z
n0 = gc (E)fF (E)dE
(3)
The lower limit of integration is EC and the upper limit of integration should be the top of the allowed conduction
band energy.
However, since the Fermi probability function rapidly approaches zero with increasing energy as indicated in
Figure1a, we can take the upper limit of integration to be infinity.
We are assuming that the Fermi energy is within the forbidden-energy bandgap.
For electrons in the conduction band, we have E Ec .
If (Ec − EF ) kT then (E − EF ) kT , so that the Fermi probability function reduces to the Boltzmann
approximation, which is
1
−(E − EF )
fF (E) =
≈ exp
(4)
E − EF
kT
1 + exp
kT
Applying the Boltzmann approximation to Equation (3), the thermal-equilibrium density of electrons in the
conduction band is found from
Z ∞
−(E − EF )
4π(2m∗n )3/2 p
E − Ec exp
dE
(5)
n0 =
h3
kT
Ec
The integral of Equation (5) may be solved more easily by making a change of variable :
η=
E − Ec
kT
(6)
Then Equation (5) becomes
Z ∞
4π(2m∗n kT )3/2
−(Ec − EF )
n0 =
exp
η 1/2 exp(−η)dη
h3
kT
0
The integral is the gamma function, with a value of
Z ∞
1√
η
η 1/2 exp(−η)dη =
2
0
(7)
(8)
Then Equation (7) becomes
n0 = 2
2πm∗n kT
h2
3/2
exp
−(Ec − EF )
kT
(9)
We may define a parameter Nc as
Nc = 2
2πm∗n kT
h2
3/2
(10)
The value of n0
Thermal-equilibrium electron concentration in the conduction band
−(Ec − EF )
n0 = Nc exp
kT
(11)
The parameter Nc is called the effective density of states function in the conduction band.
If m∗n = m0 then the value of the effective density of states function at T = 300 K is Nc = 2.5 × 1019 cm−3 ,
which is the order of magnitude of Nc for most semiconductors.
If the effective mass of the electron is larger or smaller than m0 then the value of the effective density of states
function changes accordingly, but is still of the same order of magnitude.
3
Example 1. Calculate the probability that a state in the conduction band is occupied by an electron and
calculate the thermal equilibrium electron concentration in Silicon at T = 300 K. Assume the Fermi energy is
0.25 eV below the conduction band. The value of NC for silicon at T = 300 K is Nc = 2.8 × 1019 cm−3 .
−(Ec − EF )
1
≈ exp
fF (Ec ) =
Ec − EF
kT
1 + exp
kT
or
fF (Ec ) = exp
−0.25
0.0259
= 6.43 × 10−5
Example 2. The electron concentration is given by
−0.25
−(Ec − EF )
= 2.8 × 1019 exp
n0 = Nc exp
kT
0.0259
or
n0 = 1.8 × 1015 cm−3
The probability of a state being occupied can be quite small, but the fact that there are a large number of
states means that the electron concentration is a reasonable value.
The thermal-equilibrium concentration of holes in the valence band is found by integrating Equation (2) over
the vaIence band energy, or
Z
p0 = gv (E) [1 − fF (E)] dE
(12)
We may note that
1 − fF (E) =
1
EF − E
1 + exp
kT
(13)
For energy states in the valence band E < Ev .
If (EF − Ev ) kT then we have slightly different form of the Boltzmann approximation :
1
−(EF − E)
≈ exp
1 − fF (E) =
EF − E
kT
1 + exp
kT
(14)
Applying the Boltzmann approximation of Equation (14) to Equation (12), we find the thermal-equilibrium
concentration of holes in the valence band is
Z Ev
4π(2m∗p )3/2 p
−(EF − E)
p0 =
E
−
E
exp
dE
(15)
v
h3
kT
−∞
where the lower limit of integration is taken as minus infinity instead of the bottom of the valence band.
The exponential term decays fast enough so that this approximation is valid.
more easily by again making a change of variable:
Equation (15) may be solved
Ev − E
kT
(16)
Z 0
−4π(2m∗p kT )3/2
−(EF − Ev )
(η 0 )1/2 exp(−η 0 )dη 0
exp
h3
kT
+∞
(17)
η0 =
the Equation (15) becomes
p0 =
where the negative sign comes from the differential dE = −kT dη 0 .
Note that the lower limit of η 0 becomes +∞ when E = −∞.
If we change the order of integration, we introduce another minus sign.
From Equation (8), Equation (17) becomes
p0 = 2
2πm∗p kT
h2
3/2
4
−(EF − Ev )
exp
kT
(18)
We may define a parameter Nv as
Nv = 2
2πm∗p kT
h2
3/2
(19)
The value of p0 is
Thermal-equilibrium holes concentration in the valence band
−(EF − Ev )
p0 = Nv exp
kT
(20)
The parameter Nv is called the effective density of states function in the valence band.
The magnitude of Nv is also on the order of 1019 cm−3 at T = 300 K for most semiconductors.
The effective density of states functions, Nc and Nv , are constant for a given semiconductor material at a fixed
temperature.
Table 1 gives the values of the density of states function and of the effective masses for silicon, gallium arseinide
and germanium.
Note that the value of Nc for gallium arsenide is smaller than the typical 1019 cm−3 value.
This difference is due to the small electron effective mass in gallium arsenide.
Nc (cm−3 )
Silicon
Gallium Arsenide
Germanium
Nv (cm−3 )
19
19
2.8 × 10
4.7 × 1017
1.04 × 1019
1.04 × 10
7.0 × 1018
6.0 × 1018
m∗n /m0
m∗p /m0
1.08
0.067
0.55
0.56
0.48
0.37
Table 1: Effective density of states function and effective mass values
1.3
The lntrinsic Carrier Concentration
The lntrinsic Carrier Concentration
For an intrinsic semiconductor, the concentration of electrons in the conduction band is equal to the concentration of holes in the valence band.
We may denote ni and pi as the electron and hole concentrations, respectively. in the intrinsic semiconductor.
These parameters are usually referred to as the intrinsic electron concentration and the intrinsic hole concentration.
However ni = pi so normally we use only the parameter ni which refer to either intrinsic electron or hole
concentration.
The Fermi energy level for the intrinsic semiconductor is called the intrinsic Fermi energy, or EF = EF i .
If we apply Equations (11) and (20) to the intrinsic semiconductor, then we can write
−(Ec − EF i )
n0 = ni = Nc exp
kT
and
−(EF i − Ev )
p0 = pi = ni = Nv exp
kT
If we take the product of Equations (21) and (22), we obtain
−(Ec − EF i )
−(EF i − Ev )
2
ni = Nc Nv exp
exp
kT
kT
or
n2i
(21)
−(Ec − Ev )
−Eg
= Nc Nv exp
= Nc Nv exp
kT
kT
(22)
(23)
where Eg is the bandgap energy.
5
(24)
For a given semiconductor material at a constant temperature, the value of ni is a constant, and independent
of the Fermi energy. The intrinsic carrier concentration for silicon at T = 300 K may be calculated by using
the effective density of states function values from Table 1.
The value of ni calculated from Equation (24) for Eg = 1.12 eV is ni = 6.95 × 1019 cm−3 .
The commonly accepted value of ni for silicon at T = 300 K is approximately 1.5 × 1010 cm−3 .
discrepancy may arise from several sources.
This
First, the values of the effective masses are determined at a low temperature where the cyclotron resonance
experiments are performed.
Since the effective mass is an experimentally determined parameter, and since the effective mass is a measure
of how well a particle moves in a crystal, this parameter may be a slight function of temperature.
Next,
the density of states function for a semiconductor was obtained by generalizing the model of an electron in a
three-dimensional infinite potential well.
This theoretical function may also not agree exactly with experiment.
However, the difference between the theoretical value and the experimental value of ni is approximately a factor
of 2, which, in many cases, is not significant. Table 2 lists the commonly accepted values of ni for silicon,
gallium arsenide, and germanium at T = 300 K.
The intrinsic carrier concentration is a very strong function of temperature.
1.5 × 1010 cm−3
1.8 × 106 cm−3
2.4 × 1013 cm−3
Silicon
Gallium Arsenide
Germanium
Table 2: Commonly accepted values of ni at T = 300 K.
Figure 2 is a plot of ni from Equation (24) for silicon, gallium arsenide, and germanium as a function of
temperature.
As seen in the figure, the value of ni for these semiconductors may easily vary over several orders of magnitude
... 1 Cl"'..arge Carriers in Semiconductors
113
as the temperature changes over a reasonable range.
•
10'
,.
1500
nO
e)
, l!(lOU 500
100 100
27 0-2o
•
10'
,
10'
•
,
10'
• C;.:
10'
,
•
10'
10'
.,
'E 10'
,
':;;;
v
u
'0
=
'0
oS
10'
$i
,
,
- \"
•
10•
10'
10'
)07
10"
0.5 1.0 1.5 2.0 25 3.0 35 4.0
IOOO/ T(K - ')
Figll.re 4.21 The imrill sic carrier
Figure 2: The intrinsic
carrier concentration
a function of temperature.
c()l1cenlration
of Ge. Si. <Ind GaAsas
as a
funcli on of tcmper-dlure.
{ Fmm S:t'/ IJI. }
6
TEST YOUR UNDERSTANDING
E4.J Find the intrinsic carrierconccntralion in silit:on at (c,) T = 200 K illld {b) T = 400 K.
I,_wo ,,01 x gel (q) ",_UI> ,.0 ) x 89'L ( O) 'su'o]
1.4
The Intrinsic Fermi-Level Position
The Intrinsic Fermi-Level Position
We have qualitatively argued that the Fermi energy level is located near the center of the forbidden bandgap
for the intrinsic semiconductor.
We can specifically calculate the intrinsic Fermi-Ievel position.
Since the electron and hole concentrations m equal, setting Equations (21) and (22) equal to each other, we
have
−(EF i − Ev )
−(Ec − EF i )
= Nv exp
(25)
Nc exp
kT
kT
If we take the natural log of both sides of this equation and solve for EF i we obtain
1
1
Nv
(Ec + Ev ) + kT ln
2
2
Nc
EF i =
(26)
From the definitions for Nc and Nv given by Equations (10) and (19) Equation (26) may be written as
EF i =
m∗p
1
3
(Ec + Ev ) + kT ln ∗
2
4
mn
(27)
so that
Intrinsic Fermi level
m∗p
3
EF i = Emidgap + kT ln ∗
4
mn
(28)
If the electron and hole effective masses are equal so that m∗n = m∗p then the intrinsic Fermi level is exactly in
the center of the bandgap.
If m∗p > m∗n the intrinsic Fermi level is slightly above the center, and if m∗p < m∗n , it is slightly below the center
of the bandgap. The density of states function is directly related to the carrier effective mass; thus a larger
effective mass means a larger density of stales function.
The intrinsic Fermi level must shift away from the band with the larger density of states in order to maintain
equal numbers of electrons and holes.
2
Dopant Atoms and Energy Levels
Dopant Atoms and Energy Levels
The intrinsic semiconductor may be an interesting material, but the real power or semiconductors is realized
by adding small, controlled amounts of specific dopant, or impurity, atoms.
The doped semiconductor, called an extrinsic material, is the primary reason we can fabricate the various
semiconductor devices.
2.1
Qualitative Description
Qualitative Description
In the previous chapter we discussed the covalent bonding of silicon and considered the simple two-dimensional
representation of the single-crystal silicon lattice as shown in Figure 3.
Now consider adding a group V element, such as phosphorus, as a substitutional impurity.
The group V element has five valence electrons.
Four of these will contribute to the covalent bonding with the silicon atoms, leaving the fifth more loosely bound
to the phosphorus atom.
This effect is schematically shown in Figure 4.
We refer to the fifth valence electron as a donor electron.
The phosphorus atom without the donor electron is positively charged.
7
, conrrolledFigure
amounts
4.4. of
Wespecific
refer todopant,
the fifthorvalence electron as a donor electro
ribed hriefly in Chapter \. can greally aller
."
nductor. The doped semiconductor, L::llled
.
we can fabricate the various
..
""
" semiconduc"
"
"
hapters.
::: Si = Si = Si = Si = Si
II
II
II
=Si =Si =
=
lJ
II
S1 =
Si =
lJ
Si ...
representati on of the intrinsic silicon lattice.
"
Si ...
lJ
Si _..
II
Si =
II
II
:: : Si
Si ...
/I
Si =
=
II
'"
Si
"
II
Si
Si = 5i
lJ
=
Ii.-
=
Si
=
=
II
..
..
.
Si =
Si =
Si
1\
II
II
Si
=
V
P
II
=
.--
= Si -- II
=
...
Si = Si = Si = Si . _-
i ---
"
figutc 4.4 (Two-dimen!'ional
representation of the silicol1 1anice doped
thephosphorus
donor electron is bound
to the phosphorus atom.
with a
atom.
attice. At very low temperatures,
Figure 4: Two-dimensional representation of the silicon lattice doped with a phosphorus atom.
However, by intuition, it should seem clear that the energy required to elevate the donor electron into the
conduction band is considerably less than that for the electrons involved in the covalent bonding.
Figure 5 shows the energy-band diagram that we would expect.
The energy level, Ed , is the energy state of the donor electron.
If a small amount of energy, such as thermal energy, is added to the donor electron, it can be elevated into the
conduction band, leaving behind a positively charged phosphorus ion.
The electron in the conduction band can now move through the crystal generating a current, while the positively
charged ion is fixed in the crystal. This type of impurity atom donates an electron to the conduction band
and so is called a donor impurity atom.
8
II
:: : Si
onding of silicon and IIconsidered
II
IIIhe sim1/
II
/I _..
silicoll lanice
shown
= Si as
=Si
=Siill = Si = Si
V e.lemellt. such ,'s phosphorus.
II
II asII a subII
II
\I
has five valence electrons.
of=Si
these =S; -Si --:: : Si = SiFour
= S1
h the silicon atoms, leaving"the fifth mOre
" shown ill
"
"
This efleci is schematically
Figure 3: Two-dimensional
representation of the intrinsic silicon lattice.
ectron as a donor electron
. TW'o-dimensional
Figure
4.31
." .."
Si
"
Si ...
II
'"
Si
figutc
represe
with a
puritypect.
[0 silicon.
Thelevel
group
three
vadonor
lence electron.
electrons. which are "II
The energy
, Ed,IIIis clement
!he energyhas
statc
of !he
taken up in
covalent
bonding.
Assuch
shown
in rm
Figure
4.6a, isone
covalent
If the
a small
amount
of energy.
as the
al energy,
added
to the bonding
donor eler·po·
it cantobebe
eJevated
theelectron
conduction
band,
a positively
chargl'dilS
sitiontron.
appears
empty.into
(I' an
were
[ 0 leaving
occupybehind
[his "empty"
p<lsition.
phosphorus ion. The ",cctron in Ihe conduction band can now move through therTj"
tal generating a current. while the poSilively charged ion is fixed in the crys tal. Thi,
type of impurit
y atomband
donates an elec tron (0 the conduction band and so is called a
COnduction
- dOllor
--E,
,
dOl1or impurity a/()II1. The
impurit)'
<ldd elc(.'uoos 10 the conduclionEb2m1
-to Ed
- E"band. The resulting
without creating holes in the valence
as an
... material is...referred
!!n-f)'pe
'"
e!'
semicond uctor (II for the negatively charged electron).
•c
• Now consider adding a group III element. such as boron. as a substitutional '"'purity [0 silicon. The group III clement has three va lence electrons. which are "II
E.
o
...
E"
t
)
t-}:)...
g
'.
-----------
taken up in the covalent bonding. As shownain Figure 4.6a, one covalent bonding po·
sition appears to be empty. (I' an electron were
[0
occupy [his "empty" p<lsition. ilS
(.)
t
(b)
JThe COnduction
energy-band
showing
Ih eenergy
di screle
energy
sHiftof a donor
FigureFigure
5: The 4.5
energy-band
diagram showing
(a) the
discrete (,l)
donor
statedonor
and (b)
the effect
banddiagram
E
state being ionized.
- - - - - - - - - - - E,
,
)
E"ionized.t-}:)
..... ... ..
..
..... - :: - Ed ::
!!•
'"
e!'
c
The donor •
impurity atom add electrons to the conduction band without creating holes in the
g
valence band.
'.
II
: :: Si =
Si= Sj =
Si = Si =
Si :::
II
II
5 i = Si = Si = 5i =
-----------
::: Si =
Si :::
E.
... (n for the negatively charged electron).
E"
The resultingomaterial is referred to as an n-type semiconductor
II
II
II
a II
II
II
II
II
II
Now consider adding a group III element, such as boron, as a substitutional impurity to to silicon.
=
=
::: group
Si = III5ielementSihas
= three
Si,:-:-:,
B electrons,
Si :::which are all taken up in the(b)
The
valence
covalent bonding.
(.)
II
II
11----11
II
\I
'"
rr=
= "' = "i:>1: = "'
As shown in Figure 6a, one covalent bonding position appears to be empty.
Figure
4.5
JThe energy-band
diagram
showing (,l) Ih e di screle donor energy sHift
:::an $;=
=to
Si=
Si =
Si :::
If
electronSi
were
occupySi=
this empty
position
its energy would have to be greater than that of the valence
II
II
II
5i =
Si
II
II
'" fI; =
II
occupying
= "'..However.. the electron
::
::
electrons, since the net charge state of the boron atom would now be negative.
: : : Si =
: :: Si =
II
::: Si =
=
5i
=
Si
=
Si :::
Si= Sj =
Si = Si =
Si :::
II
II
II
II
Si =
Si,:-:-:, B
5i
=
II
=
ionized. ..
::: Si =
::: Si =
II
Si :::of a
..
Si =
Si ': a -!Si =
..
Si =
5i :::
5i =
Si =
Si =
5i =
Si :::
II
II
II
II
II
(a)
with a boron at..
rr=
II
II
II
11----11 \I
'"
and (b) l\howing the ioniZalion of the boron atom resulting
in a =
h()le."' = "i:>1: = "'
::: $;= Si = Si= Si= Si = Si :::
II
: : : Si =
II
II
5i =
Si
=
II
5i
=
II
Si
=
'" fI; =
II
::: Si =
Si :::
= "'
Si =
Si ': a -!Si =
..
Si =
5i :::
of a (a) doped with (a)
withand
a boron
at.. the
Figure 6: Two dimensional representation of a silicon lattice
a boron atom
(b) showing
andof(b)
the ioniZalion
the boron atom resulting in a h()le.
ionization
thel\howing
boron atom
resulting inof
a hole.
this empty position does not have sufficient energy to be in the conduction band, so its energy is far smaller
than the conduction-band energy.
Figure 6b shows how valence electrons may gain a small amount of thermal energy and move about in the
crystal.
The ”empty” position associated with the boron atom becomes occupied, and other valence electron positions
become vacated.
These other vacated electron positions can be thought of as holes in the semiconductor material.
Figure 7 shows the expected energy state of the ”empty” position and also the formation of a hole in the valence
band. The hole can move through the crystal generating a current, while the negatively charged boron atom
is fixed in the crystal.
9
• •2
Dopant Atoms and Energy Levels
t -,.- -----..-.-
t
band
'
..
(a)
E,
--+)-)+----,-1)--+ +
+
(b)
Figure Figure
7: The 4.7
energy-band
diagram showing
(a) showing
the discrete
energy.cceplor
states and
(b) theSlOIC
effect or an
I The enctgy.band
diagram
(0)acceptor
the discrele
cnctg)'
acceptor
state
being
ionized.
and (h) th.e effect or an <lccepto( state being ioni zed.
Theenergy
group would
III atomhave
accepts
an greater
electron from
and soelectrons,
is referred to
as anthe
acceptor
impurity
to be
thallthe
thatvalence
of theband
valence
since
net charge
atom.
srale of the boron atom would now be negative. Howevcr. the electron occupying tbi.'
The acceptor atom can generate holes in the valence band without generating electrons in the
"empty" position docs
nO{ have sufficient energy to be in the conduction band, so its
conduction band.
This type of semiconductor material is referred to as a p-type material (p for the
energycharged
is far smaller
tlt"Jl the conduclion·balld eJlergy. Figure 4.6b shows how va·
positively
hole).
Thelence
pure electrons
single-crystal
semiconductor
material
is calledof
anthermal
intrinsic energy
material.
may
gain a small
amount
and
move about in the
Adding
controlled
amounts of
dopant atoms,
either donors
or acceptors,
createsbecomes
a material occupied.
called an extrinsic
cry lal.
The "empty"
position
associated
with Ihe
boron alom
and
semicondctor.
other valence electron positions become vacated. These other vacated electron posiAn extrinsic semiconductor will have either a preponderance of electrons (n type) or a preponderance of holes
tions can be thought of as holes in the semiconductor material.
(p type).
Figure 4.7 shows the expected energy state of the "empty" position and also the
fonnation of a hole in the valence band. The hole can move through the crystal gen3 The Extrinsic Semiconductor
erating a current. while the negatively charged boron atom is fixed in the crystal. The
group
III alom
accepts an electron from the valence band and so is refcrred to as an
The
Extrinsic
Semiconductor
mom. The acceptor
atOm
generate
holes
ininthe
Weacceptor
defined animpurity
intrinsic semiconductor
as a material
with can
no impurity
atoms
present
the valence
crystal. band
electrons
conduction inband.
type amounts
of semiconductor
ma- or
An without
extrinsicgenerating
semiconductor
is definedinasthe
a semiconductor
whichThis
controlled
of specific dopant
impurity
atoms
have
been
added
so
that
thermal
equilibrium
electron
and
hole
concentrations
are
different
terial is referred to as a ,,·type materi,,1 (p for Ihe
chnrgcd hole).
from the intrinsic carrier concentration.
The pure singlc·crYSlal semiconductor material is called an intrinsic materia1.
One type of carrier will predominate in an extrinsic semiconductor.
Adding controlled amounls of dopant atoms, either donors or acceptors. creates a
called an Distribution
extrinsic semicondcH,'IQr.
Anand
eXlrinsic
3.1material
Equilibrium
of Electrons
Holes semiconductor will have either a preponderance of electrons (n type) or a preponderance of holes (I' eype).
Equilibrium Distribution of Electrons and Holes
Adding donor or acceptor impurity atoms to a semiconductor will change the distribution of electrons and holes
4.2.2
Ionization Energy
in the
material.
Since the Fermi energy is related to the distribution function, the Fermi energy will change as dopant atoms
We can calculate Ihe approxirnilte distance of the donor electron from the donor imare added.
purity ion, and al 0 the approXimate energ), required to elevate the donor electron
If the Fermi energy changes from near the midgap value, the density of electrons in the conduction band and
the ofconduction
is referred to as the ionization energy. We will
theinto
density
holes in the band.
valenceThis
band energy
will change.
useeffects
the Bohr
modelinof
the atO
m 9.
for
These
are shown
Figures
8 and
these calculations. The justification for using this
modei
is that
of antheelectron
from
Figure
8 shows
thethe
casemost
for EFprObable
> EF i anddistance
Figure 9 shows
case for E
F < Ethe
F i . nucieus in a hydro·
When
concentration
larger than theishole
whenradius.
EF < EThe
genEatom,
from
quantumis mechanics,
theconcentration,
same as theand
Bohr
F > Edetermined
F i , the electron
F i , the
hole concentration is larger than electron concentration.
energy levels in the hydrogen atom determined from quantum mechanics are ,,1so the
When the density of electrons is greater than the density of holes, the semiconductor is n type; donor impurity
same as obtained from the Bohr theory.
atoms have been added.
In the case of the donor impurity atom, we may visualize the donor electron orbiting Ihe donor ion. which is embedded in the semiconduclor maleri,,1. We will neeel
10 material in the calculations rather than
to use the pcnniHivity of the semiconductor
Fermi energy changes from near lhe midgap value, the density of electrons in lhe con·
duction band anel Ihe density of holes in the valence band will change. These effects
are shown in Figures 4.8 and 4.9. Figure 4.8 shows the case foJ' E" > En and
Figure 4.9 shows the caSe for EF < E Pi, When E F > EF; , the electron coneentra·
lion is Jarger rhan Ihe hole concentration, and when EF < EF;, the hole concentration
Area = ,,,,
=::
elec{101'I
concentration
j,.{E}
EF j
Area = Po.
!l ole concentration
{,(E) =O
/,{e)· I
Figure 4.81 Density of states functions. Fenni-Dirac
function.probability
and areas re))rc5cming
clectron and areas representing electron and hole
Figure 8: Density of states functions, probability
Fermi-Dirac
function,
and hole concchtf<ltions for the case when £ F above
concentrations for the case when EF ilheisiJ1[(jnsit:
abovef:urn;
theenergy_
intrinsic Fermi energy.
122
CHAPTER 4
The Semioonductor in Ec,u .Ib
,
m
.
\
Area =
eF;
=
declron
\ fr(£)
concentration
---\---------------
""
F.••
- - - -'..-.......- - - --
",
Area
= 1'0=
hole COIICenlrdt.io o
Me} - I
("I.e) - 0
Figure 4.9\ Density of states functions. f'emli- Dirac
probability function.probability
and areas f('.presenting
electron and
Figure 9: Density of states functions, Fermi-Dirac
function,
and areas representing electron and hole
hole concentrations for the case when £,.. is below tbe
concentrations for the case when EF i intrinsic
is below
the
intrinsic
Fermi
energy.
Fermi energy.
is larger than the electron concentration. When the density of electrons i grearer lhan
(he density of holes. the semiconductor is n type; donor impurity atoms have been
added. When th e density of holes is greater than the density of electrons. the semiWhen the density of holesconductor
is greater
than the density of electrons, the semi-conductor is p type; acceptor
is p type; acceptor impurity atoms have been added. The Fermi energy :
impurity atoms have been le\'el
added.
in a semiconductor changes the electron and hole conce.ntralions change and,
again. the Fermi energy changes as donor Or acceptor impurities are added. The
The Fermi energy level in achange
semiconductor
changes
as of
donor
and
acceptorwill
impurity
are added. The expressions
in Ihe Fermi level
as a function
impurilY
concentrations
be
previously derived for the thermal-equilibrium
concentration electrons and holes, given by Equations (11) and
in Section 4.6.
previously
the thermal-equilibrium
concentration
(20) are general equations for TI,e
n0 expre
and sions
p0 in
termsderived
of theforFermi
energy.
electrons and holes. given by F<luati ons (4.11) and (4. 19) are general equations fi
These equation are again Iro and Po in terms of the Fermi energy. These equation ' arc again given as
−(E. c-−
EF )
[-(E
E
f')]
n0 =noN=cN
exp
.. exp
kT
kT
11
(29)
and
p0 = Nv exp
−(EF − Ev )
kT
(30)
In an n-type semiconductor, electrons are referred to as the majority carrier and holes as the minority carrier.
In a p-type semiconductor it will be the inverse.
We may derive another form of the equations for the thermal-equilibrium concentrations of electrons and holes.
If we add and subtract an intrinsic Fermi energy in the exponent of Equation (11), we can write
−(Ec − EF i ) + (EF − EF i )
n0 = Nc exp
kT
or
−(Ec − EF i )
(EF − EF i )
n0 = Nc exp
exp
kT
kT
(31)
(32)
The intrinsic carrier concentration is given by
ni = Nc exp
−(Ec − EF i )
kT
(33)
so that the thermal-equilibrium electron concentration can be written as
Thermal-equilibrium electron concentration (expressed in terms of intrinsic concentration)
EF − EF i
n0 = ni exp
(34)
kT
Similarly, if we add and subtract an intrinsic Fermi energy in the exponent of Equation (20) we obtain
Thermal-equilibrium hole concentration (expressed in terms of intrinsic concentration)
−(EF − EF i )
p0 = ni exp
kT
(35)
As we will see, the Fermi level changes when donors and acceptors are added, but Equations (34) and (35) show
that, as the Fermi level changes from the intrinsic Fermi level, n0 and p0 change from the ni value.
If EF > EF i then we will have n0 > ni and p0 < ni , thus n0 > p0 .
Similarly, in a p-type semiconductor, EF < EF i so that p0 > ni and n0 < ni , thus p0 > n0
3.2
The n0 p0 product
The n0 p0 product
We may take the product of the general expressions for n0 and p0 as given in Equations (11) and (20), respectively:
−(Ec − EF )
−(EF − Ev )
n0 p0 = Nc Nv exp
exp
(36)
kT
kT
which may be written as
−Eg
ni = Nc Nv exp
kT
(37)
As Equation (37) was derived for a general value of Fermi energy, the values of n0 and p0 are not necessarily
equal.
However, Equation (37) is exactly the same as Equation (24), which we derived for the case of an intrinsic
semiconductor.
We then have that, for the semiconductor in thermal equilibrium,
n0 p0 Equation
n0 p0 = n2i
12
(38)
Equation (38) states that the product of n0 and p0 is always a constant for a given semiconductor material at
a given temperature.
Although this equation seems very simple, it is one of the fundamental principles of semiconductors in thermal
equilibrium.
It is important to keep in mind that Equation (38) was derived using the Boltzmann approximation.
If the Boltzmann approximation is not valid, then Equation (38) is not valid. An extrinsic semiconductor
in thermal equilibrium does not, strictly speaking, contain an intrinsic carrier concentration, although some
thermally generated carriers are present.
The intrinsic electron and hole carrier concentrations are modified by the donor or acceptor impurities.
3.3
Degenerate and Nondegenerate Semiconductors
Degenerate and Nondegenerate Semiconductors
In our discussion of adding dopant atoms to a semiconductor, we have implicitly assumed that the concentration
of dopant atoms added is small when compared to the density of host or semiconductor atoms.
The small number of impurity atoms are spread far enough apart so that there is no interaction between donor
electrons for example in an n-type material.
We have assumed that the impurities introduce discrete noninteracting donor energy states in the n-type
semiconductor and discrete, noninteracting acceptor states in the p-type semiconductor.
These types of semiconductors are referred to as nondegenerate semiconductors. If the impurity concentration
increases, the distance between the impurity atoms decreases and a point will be reached when donor electrons,
for example, will begin to interact with each other.
When this occurs, the single discrete donor energy will split into a band of energies. As the donor concentration
further increases, the band of donor states widens and may overlap the bottom of the conduction band.
This overlap occurs when the donor concentration becomes comparable with the effective density of states.
When the concentration of electrons in the conduction band exceeds the density of states Nc , the Fermi energy
lies within the conduction band.
This type of semiconductor is called a degenerate n-type semiconductor. In a similar way, as the acceptor
doping concentration increases in a p-type semiconductor, the discrete acceptor encegy states will split into a
band of energies and may overlap the top of the valence band.
The Fermi energy will lie in the valence band when the concentration of holes exceeds the density of states Nv .
This
type semiconductor is called a degenerate p-type semiconductor. Schematic models of the energyC HAP T E R 4 l ite SemiccnduCtor in Equilibrium
band diagrams for a degenerate n-type and degenerate p-type semiconductors are shown in Figure 10.
t --,-
- - .,.. E .
f
E,
'
Filled
stales
{c1cclro ns)
- - - - ' - - - - ' - - - - - f.',
V..a}cnccband
<al
t
!il
Conduction band
!
Empty ".tes
L (holes)
g --+,---------------
];; - -.-..,.,
...,.....,_.- -.-
£f
V OlICI\cc! b.md
(b)
Figure 4.11 I Simplified energy,band diagrams for degenerately doped (a) ".Iype and
Figure 10: Simplified energy-band diagrams for degenerately doped (a) n-type and (b) p-type semiconductors.
(b)
se miconduclors.
The energy states below EF are mostly filled with electrons and the energy states above EF are mostly empty.
y ofn-type
statessemiconductor,
Nt" . the Fermi
energy
lies within
the
conduction
This ty thus,
Inthe
the densit
degenerate
the states
between
EF and E
mostly filled band.
with electrons,
c are
the
electron
concentration
in
the
conduction
band
is
very
large.
of semiconductor is called a degenerate n-type semiconductor.
Similarly,
degenerate
semiconductor,
energyconcentration
states between Evincreases
and Ef areinmostly
empty:
fn ina the
similar
way,p-type
as the
acceptOr the
doping
a p_ly
thus, the hole concentration in the valence band is very large.
5emiconductor. the discrete acceptor encrgy states will split into a band of energi
and may overl ap the top of the valence band,
13 The Fenni energy will lie in the vale
band when the concentration of holes exceeds the de nsity of states N, .. This type
4
Statistics of Donors and Acceptors
Statistics of Donors and Acceptors
We discussed the Fermi-Dirac distribution function, which give the probability thaIta particular energy state
will be occupied by an electron.
We need to reconsider this function and apply the probability statistics to the donor and acceptor energy states.
4.1
Probability Function
Probability Function
One postulate used in the derivation of the Fermi-Dirac probability function was the Pauli exclusion principle,
which states that only one particle is permitted in quantum state.
The Pauli exclusion principle also applies to the donor and acceptor states. Suppose we have Ni electrons and
gi quantum states, where the subscript i indicates the i-th energy level.
There are gi ways of choosing where to put the first particle.
Each donor level has two possible spin orientations for the donor electron: thus each donor level has two quantum
states.
The insertion of an electron into one quantum state, however, precludes putting an electron into the second
quantum state. By adding one electron, the vacancy requirement of the atom is satisfied, and the addition of
a second electron in the donor level is not possible.
The distribution function of donor electrons in the donor energy states is then slightly different than the FermiDirac function. The probability function of electrons occupying the donor state is
nd =
1
1 + exp
2
Nd
Ed − EF
kT
(39)
where nd is the density of electrons occupying the donor level and Ed is the energy of the donor level.
The factor 1/2 is a direct result of the spin factor just mentioned.
The 1/2 factor is sometimes written as 1/g where g is called a degeneracy factor.
written in the form
nd = Nd − Nd+
where
Nd+
Equation (39) can also be
(40)
is the concentration of ionized donors.
In many applications we will be interested more in the concentration of ionized donors than in the concentration
of electrons remaining in the donor state.
If we do the same type of analysis for acceptor atoms. we obtain the expression
pa =
Na
= Na − Na−
1
EF − Ea
1 + exp
g
kT
(41)
where Na is the concentration of acceptor atoms, Ea is the acceptor energy level, pa is the concentration of
holes in the acceptor states and Na− is the concentration of ionized acceptors. A hole in an acceptor state
corresponds to an acceptor atom that is neutrally charged and still has an ”empty” bonding position.
The parameter g is, again a degeneracy factor.
The ground state degeneracy factor g is normally taken as four for the acceptor level in silicon and gallium
arsenide because of the detailed band structure.
4.2
Complete Ionization and Freeze-Out
Complete Ionization and Freeze-Out
The probability function for electrons in the donor energy state was just given by Equation (39).
14
If we assume that Ed − EF kT then
nd ≈
Nd
−(Ed − EF )
= 2Nd exp
Ed − EF
1
kT
exp
2
kT
(42)
If (Ed − EF ) kT , then the Boltzmann approximation is also valid for the electrons in the conduction band
so from Equation (11):
−(Ec − EF )
n0 = Nc exp
(43)
kT
We can determine the relative number of electrons in the donor state compared with the total number of
electrons, therefore we can consider the ratio of electrons in the donor state to the total number of electrons in
the conduction band plus donor state.
Using the expressions of Equations (43) and (11), we write
−(Ed − EF )
2Nd exp
nd
kT
=
−(Ed − EF )
−(Ec − EF )
nd + n0
2Nd exp
+ Nc exp
kT
kT
(44)
Dividing by the numerator term we obtain:
nd
=
nd + n0
1
Nc
−(Ec − Ed )
1+
exp
2Nd
kT
(45)
The factor Ec − Ed is just the ionization energy of the donor electrons. At room temperature, then, the donor
states are essentially completely ionized.
At room temperature, there is also also essentially complete ionization of the acceptor atoms.
This ionization effect and the creation of electrons and holes in4.
the
and valence
band, respec4 conduction band
c1 Donors
aoo Acceptors
tively, are shown in Figure 11.
The opposite of complete ionization occurs at T = 0 K: all electrons are
t) ) )
Conduction band
)
t
,
) ) ")
>., ++ T T T T T t l.'
>.
c
- - - - - - - - - - - - - - - - - - - - EFi
Conductiun band
--..::.:.....::.:.....::.:.....::.:...-- £..
-- - ... --------- - -----. E,..;
u
-=- -=- -=- -=- -=- -=- £
g -=) ) ) ) ) ) L;"
!
" ----,,-,---..,-,---C•.
Valence band .
II
iii
.... .,.-
+++++++
ValeDce band
(a )
'
..
..-...
(b)
4.12 I En('rgy·bnnd
diagrams
showi ng
complete
ioniZ:llion
of and
(a) donor
Figure 11: Energy-band
diagrams showing
complete
ionization
of (a)
donor states
(b) acceptor states.
and (b) acceptor slates.
in their lowest possible energy state; that is, for an n-type semiconductor, each donor state must contain an
electron. Therefore nd = Nd or Nd+ = 0.
t __
We must have, then, from Equation (39) that exp(Ed − EF )/kT = 0.
------------- E,
Conduction
b.1nd energy level must be
Since T = 0 K, this co_
will
exp(−∞)
= 0, which means that EF >
Ed . The Fermi
n_
doccur
""_lio_nfor
b_'n
_d
above the donor energy level at absolute zero.
t
In the case of a p-type semiconductor at absolute zero temperature. the impurity atom will not contain any
electrons, so that the Fermi energy level must be below the;",\acceptor energy state.
- - - - - - - - - - - - - - - - - - - . r.F;
u
2e ________
ValeD,,-e band
u
15
--- - - --------------- Ef ,
2E ::-.:=.:=.:=...:::....= if;.
iii _ - - - - - - - - - £'
:..:.:..!. ..
\'
and (b) acceptor slates.
t __
t
co_n_
d ""_lio_n b_'n_d
------------- E,
Conduction b.1nd
;",\
- - - - - - - - - - - - - - - - - - - . r.F;
u
u
2e ________
--- - - --------------- Ef ,
2E ::-.:=.:=.:=...:::....= if;.
ValeD,,-e band
iii _ - - - - - - - - - £'
<a)
(b)
:..:.:..!. ..
\'
Figure
Energy-band
diagram
0 (a)
K l'or
(a) and
n-t)'pC
(b)semiconductors.
V-tyJl<!
Figure4.131
12: Energy-band
diagrams
at T at
= 0T K=for
n-type
(b) and
p-type
semiconductors.
The distribution of electrons among the various energy states. and hence the Fermi energy, is a function of
temperature. A detailed analysis shows that at T = 0 K the Fermi energy is halfway between Ec and Ed for
The
upposite
of complete
occurs at l' = 0 K. At absolute lero dethe n-type
material
and halfway
between Eionization
a and Ev for the p-type material.
grees.
all electrons
are
Figure
12 shows
these effects.
in their lowest possible energy state; that is, for an n-type
each states
donor
must
contain
anconduction
electron.
therefore
= Nfreezed or
No semiconductor.
electrons from the donor
are Slate
thermally
elevated
into the
band;
this effect 11,1
is called
out.Nt = O. We must have. then. from Equation (4.50) that exp HEd - E,-) /kTI = O.
Similarly, when no electrons from the valance band are elevated into the acceptor states, the effecltis also called
Since T = 0 K. this will occur lor cxp (-00) = O. which means lhat E f" > Ed. The
freeze-out.
Fermi energy level muSt be above the donor energy level at absolute zero. In the case
Between T = 0 K, freeze-out, and T = 300 K, complete ionization, we have partial ionization of donor or
of a p-type
acceptor
atoms. semiconductor at absolu te zero temperature. the impurity at01m will not
contain any electrons. so that lhe Fermi energy level must be below the acceptor enstate. The
distribution of electrons among the various energy Slates. amI hence
5 ergy
Charge
Neutrality
the Fenni enc.rgy. is a function of temperature.
Charge A
Neutrality
detailed analysis. not given in this text. shows that at T = 0 K. the Fenni enIn thermal
the semiconductor
is electrically
neutral.
ergy is equilibrium
halfway between
E,. andcrystal
Ed fm
the n-type
material and halfway between E.
The eletrons are distributed among the various energy stats, creating negative and positive charges, but the net
and E. for the p-typc material. Figure 4.13 shows these effects . No electrons from
charge density is zero.
the donor State arc thennally elevated into the conduction band; this effect is called
This charge-neutrality condition is used determine the thermal-equilibrium electron and hole concentrations as
/reeze-olli.
when
nOelectrons from the valance band are elevated into the
a function
of theSimilarly.
impurity doping
concentration.
acceptor statcs. the effecl is also called freeze-out.
5.1
Compensated Semiconductor
Compensated Semiconductor
A compensated semiconductor is one that cotrains both donor and acceptor impurity atoms in the same region.
A
compensated semiconductor can be formed. for example. by diffusing acceptor impurities into an n-type
material or viceversa.
An n−type compensated semiconductor occurs when Nd > Na and a p-type compensated semiconductor occurs
when Na > Nd .
If Nd = Na we have a completely compensated semiconductor that has, as we will show, the characteristics of
an intrinsic material.
5.2
Equilibrium Electron and Hole Concentrations
Equilibrium Electron and Hole Concentrations
Figure 13 shows the energy-band diagram of a semiconductor when both donor and acceptor impurity atoms
are added to the same region to form a compensated semiconductor. The figure shows how the electrons and
16
4.5.2 Equilibrium Electron and Hole Concentrations
Figure 4.14 shows the energy-band diagram of a semiconductor when both donor
and acceptor impurity atoms are added to the same region [0 form a compensated
Tmal electron
concentration
I
DonOT
electrons
""
---+
E,
./
Ed
Un-iollized
Nt · (Nd
donors
-
tid)
loo;lcd donors
---------------En
=
Un·joni7.ctI
NI,)-
(Na - Pel)
Ionized acceptors
....
':' +++
.;
holes
+
...
Po
Thermal
-
=+
+
E.
.
. '- .
A('(ep{or
(
E,
+
•
holes.
Total hole
conCenlt31ion
Figure 4.141 Energy-band diagJ"'.lnl of a compensated
Figure 13: Energy-band diagram
of a compensated semiconductor showing ionized and un-ionized donors and
semiconductor showing ionized and un-ionized donors
acceptors.
and acceptors.
holes can be distribute among the various states.
The charge neutrality condition is expressed by equating the density of negative charges to the density of positive
charges:
n0 + Na− = p0 + Nd+
(46)
or
n0 + (Na − pa ) = p0 + (Nd − nd )
(47)
where n0 and p0 are the thermal-equilibrium concentrations of electrons and holes in the conduction band and
valence band.
The parameter nd is the concentration of electrons in the donor energy states, so Nd+ = Nd − nd is the
concentration of positively charged donor states.
Similar]y, pa is the concentration of holes in the acceptor states, so Na− = Na − pa is the concentration of
negatively charged acceptor states.
We have expressions for n0 , p0 , nd and pa in terms of the Fermi energy and temperature. If we assume complete
ionization, nd and pa are both zero, and Equation (47) becomes
n0 + Na = p0 + Nd
(48)
If we express p0 as n2i /n0 then Equation (48) can be written as
n 0 + Na =
n2i
+ Nd
n0
(49)
which in turn can be written as
n20 − (Nd − Na )n0 − n2i = 0
(50)
The electron concentration n0 can be determined using the quadratic formula
Equilibrium electron concentration in compensated semiconductor
s
2
(Nd − Na )
Nd − Na
n0 =
+
+ n2i
2
2
17
(51)
n'
-.1. =
Po
ho
(1.5 x lO in)'
2.25 x LO'
I X 10 16
CIll- l
• Comment
In this ell.an1ple. NJ » fI ; . sO that rhe lhcnnal-equilibriulO majority carrier electrUIl concenis essentiall
y equal
to themust
donorbe
impurity
maThe positive signtratiQn
in the
quadratic
formula
used, concentration.
since, jn the Thc
limitthcrmru-equilibriulIl
of an intrinsic semiconductor
when
jorityelectron
and minority
carrier concentrations
differ by quantity
(llaoy orders
Na = Nd = 0, the
concentration
must be can
a positive
or of
n0magoitude.
= ni . Equation (51) is used to
calculate the electron concentration in an n-type semiconductor, or when Nd > Na .
Although Equation We
(51)have
wasargued
derived
compensated
thethe
equation
valid for Na = 0.
in for
OUfadiscussion
and semiconductor,
we may note ffOOl
results isofalso
ExamThe concentration
of
electrons
in
the
conduction
band
increases
above
the
intrinsic
carrier
concentration
as we
ple 4.9Ih.t the concentnltion of electrons in the conduction band increases abO\'e the
add donor impurity
atoms.
intrinsic carricrconcenrration as We add donor impuri[y a(oms. At the same time, the
At the same time,
the minority
carrier
hole concentration
the intrinsic
carrier concentration as
minority
carrier hole
concentration
decreasesdecreases
below thebelow
intrinsic
carrier concentrawe add donor atoms.
tion as we add donor atoms. We must keep in mind Ihat we add donor impurity
theascorresponding
electrons.
is acorresponding
redistribution of
electrons
We must keep inatOmS
mindand
that
we add donordonor
impurity
atomsthere
and the
donor
electrons, there is a
.vailable
energy
Figurestales.
4.15 shows a schematic of this physical redisredistribution of among
electrons
among
available energy
A fewofofthis
the physical
donor electrons
will fall into Ihe empty states in (he valence
Figure ?? shows tribution.
a schematic
redistribution.
fn(rinsic
etcelJons
------
- - - - Cd
donuT';(,
Un-jonilL'd d(1I10fS
-- ---- -- - -- -- - -- --- ---- - .... _-- £J'j
>A few donor c\cctrons
annihilate some
hvlc..
,
"'r;"\+""'r;"\+""'r;"\+,-----:>------"!J '\!:..J V
+
.Intrinsic
. . -.
'
...
E,.
'
Figure 4.15 I Energy-band diagram showing the
Figure 14: Energy-band diagram showing the redistribution of electrons when donors are added.
redistrcbution of electrons when donors are ,lidded.
A few of the donor electrons will fall into the empty states in the valence band and, in doing so, will annihilate
some of the intrinsic holes.
The minority carrier hole concentration will therefore decrease.
At the same time, because of this redistribution, the net electron concentration in the conduction band is not
simply equal to the donor concentration plus the intrinsic electron concentration.
We have seen that the
intrinsic carrier concentration ni is a very strong function of temperature.
As the temperature increases, additional electron-hole pairs are thermally generated so that the n2i term in
Equation (51) may begin to dominate.
The semiconductor will eventually lose its extrinsic characteristics.
Figure 15 shows the electron concentration versus temperature in silicon doped with 5 × 1014 donors per cm3 .
As the temperature increases, we can see where the intrinsic concentration begins to dominate.
Also shown is the partial ionization, or the onset of freeze-out, at the low temperature. For the case of holes
we obtain in an analogous way
Equilibrium hole concentration in compensated semiconductor
s
2
(Na − Nd )
Na − Nd
p0 =
+
+ n2i
2
2
(52)
where the positive sign must be used.
Equation (52) is used to calculate the thermal equilibrium majority carrier hole concentration in a p-type
semiconductor, or when Na > Nd .
18
4.5
)Olh
Intrinsic
,
lOI S
-.
101.1
E
3-
•
•
E>:trins'ic
/
/
Partial
ionil.ation
10 1•1
I
I
lOll
T
Charge
I
I
I
I
/
/
/
,
/
/
" n,'
I
I
I
I
I
I
I
100
lOO
300
400
500
600
700
T (K)
Figure 4.16versus
I EleClrOIl
conceiliration
lelllperaturc
Figure 15: Electron concentration
temperature
showingversus
the three
regions: partial ionization, extrinsic
and intrinsic.
showing the three regions: punial ;onczalion. exrrinsic. and
intrinsic.
i
This equation also applies for Nd = 0. Equations (51) and (52) are used to calculate the majority carrier electron
concentration in? an n-type semiconductor and majority carrier hole concentration in a p-type semiconductor.
Using the
quadratic
fonnula, thejn hole
concentration
is given
The minority
carrier
hole concentratjon
an n-type
semiconductor
could by
theoretically be calculated from
Equation (52). However we would be substracting two numbers two big numbers which from a practical point
of view is not possible.
The minority carrier concentrations are calculated from n0 p0 = n2i once the majority carrier concetration has
(4.62)
been determined.
6
Position
of the
Fermi
Level
where the positive
sign.
again,Energy
must be used.
EquatioJl (4.62) is used to calculate the
thennal.equilibrium majority carrier hole concentration in a p·type semiconductor.
Position of the Fermi Energy Level
or when Na > N". This equation also applies for IV<I = O.
We discussed how the electron and hole concentrations change as the Fermi energy level moves through the
bandgap energy.
We calculated the electron and hole concentrations as a function of donor and acceptor impurity concentrations.
Objectiveand as
We can now determine the position of the Fermi energy level as a function of the doping concentrations
a function of temperature.
Tocalculale the 11lennal-equilibriunl electron and hole concentrations in a
6.1
lcmiconductor.
Mathematical Derivation
Consider a silicon
3. Assume Il i
Mathematical
Derivation
3 X lOIS<:111-
p-lype
al T = 300 K in which Nu = 10 1(0 cm- ·1 and Nd =
= 1.5 X 10 10 cm- ) .
If we assume the Boltzmann approximation to be valid, then from Equation (11) we have n0 = Nc exp[−(Ec −
EF )/kT• ].Solution
compensated
semiconductor
We canSince
solve N(;
for >EcNd.
− E(he
this equation
and obtain is p-Iype and the Iherrnal-equilibrium enaF from
as
jorit)' carner hole concentration is given by Equation N
(4.62)
c
Ec − EF = kT ln
(53)
n0
tOl t. - 3 X lOtS
101ft - 3 x
where n0 is given byp,,Equation
(51).
=
+
2
( 1.5 x to"')'
(
2
If we consider an n-type semiconductor in which Nd ni then n0 ≈ Nd so that
Nc
Ec − EF = kT ln
(54)
Nd
IOl.S)' +
We may note that if we have a compensated semiconductor, then the Nd term in Equation (54) is simply
replaced by Nd − Na or the net effective donor concentration.
19
EXA M
We may develop a slightly different expression for the position of the Fermi level.
We had from Equation (34) that n0 = ni exp[(EF − EF i )/kT ].
We can solve for EF − EF i as
EF − EF i = kT ln
n0
ni
(55)
Equation (55) can be used specifically for an n-type semiconductor, where n0 is given by Equation (51) to find
the difference between the Fermi level and the intrinsic Fermi level as a function of the donor concentration.
We may note that, if the net effective donor concentration is zero, that is, Nd − Na = 0 then n0 = ni and
EF = EF i .
A completely compensated semiconductor has the characteristics of an intrinsic material in terms of carrier
concentration and Fermi level position. For the case of a p-type semiconductor we obtain in a similar way the
formula
Nv
(56)
EF − Ev = kT ln
p0
which, in the case Na ni becomes
EF − Ev = kT ln
We have also
Nv
Na
EF i − EF = kT ln
CHAPTER 4
tEl SemiconductC'. in Equilibrium
These results are schematically shown in Figure 16
t ...
Ii,.
t
p0
ni
(57)
(58)
....... <'I'
- - - - - - - - - - - li,
g -------------------- Ef(
" ------------- £1'
E
E
£,.
iii
(a)
( b)
li1gure
4.171 of
Position
of Fermi
for an
n-Iypc: (Nd > N,,) and (I)) p-lype
Figure
16: Position
Fermi level
for an level
(a) n-type
(N(3)
d > Na ) and (b) p-type (Na > Nd ) semiconductor.
(N,; > NJ !iemiCOllducLOr.
6.2
Variation of EF with Doping Concentration and Temperature
EF; > EF . The Fenni level for a p·lyp. semicondo ctor is below £,.;. These resuln
Variation of EF with Doping Concentration and Temperature
are shown in Figure 4.17.
We may plot the position of the Fermi energy level as a function of the doping concentration.
Figure 17 shows the Fermi energy level as a function of donor concentration (n-type) and as a function of
acceptor concentration (p-type) for silicon at T = 300 K.
4.6.2 Variation of E f · with Doping Concentration
As the doping level increases the Fermi energy level moves closer to conduction band for the n-type material
and Temperature
and closer to the valence band for the p-type material. The intrinsic carrier concentration is a strong function
of temperature, so that EF is a function of temperature aIso.
We may plot th e position of th e Fermi energy level a$ a function of [he doping C<ln·
Figure 18 shows the variation of the Fermi energy level in silicon with temperature for several donor and acceptor
centration. Fig ure 4.18 hows the Fermi energy level as <I function o f donor ("'oncen,
concentrations.
(n type)increases,
and as nai increases
functionand
of Eacceptor
concentrali on (p type) for
As[ration
the temperature
At high
F moves closer to the intrinsic Fermi level.
temperature
material
begins
to lose its
beginscloser
to behave
T = 300the
K.semiconductor
As the doping
levcl$
incrca$e.
lheextrinsic
fermi characteristics
energy leveland
moves
to more
like an intrinsic semiconductor.
conduction band for the n·type material and closer to the valence band for the p·ty
material . Keep in mind that lhe cqumions for the Fenni energy level th at we have
rived assum e [hat the Boltzmann approximation
is va lid.
20
rived assum e [hat the Boltzmann approximation is va lid.
Nd(:m-")
10 12.
10 13
10 16
1015
JO I4
10 17
HI' t!
IO n
10"
I
ntype
P Iype
£
10"
10"
.f\'d
(em - 3)
Figure 17: Position
of Fermi
as a function
of donor
concentration
and acceptor concentration
Figure
4.18level
I Posilion
of Fermi
level
as a func (n-type)
tion of donor
(p-type).
concentration (n type) and acceplor concentration (p type).
1.0
0.8
Cunduclion l><Uld
0.6
0.4
;; 0.2
'"
'"
0
I
-0.2
-0.4
- 0.6
- 0.8
- 1.0
0
100
JOO
200
400
500
600
T(K)
r'1gureof4.191
r:crmiofletemperature
vel as a function
Of doping concentrations.
Figure 18: Position
Fermi Position
level as a of
function
for various
tcmpc:r.ltuT¢ for v(lriOllS doping conccntmlions.
rFmmS't)! {J3 /.)
At the very low temperature freeze-out occurs: the Boltzmann approximation is no longer valid and the equations
we derived for the Fermi-level position no longer apply.
AtFermi-level
the low temperature
whereno
freeze-out
the At
Fermi
goestempenlture
above Ed for the
n-typefreeze-out
material andcx,..
below
position
longeroccurs
apply.
thelevel
low
where
Ea for the p-type material.
curs. the Fenni leve l goes above Ed for the n-type material and below Eo for the
p-type material. At absolute zero degrees, all energy states below EF are full and all
6.3 Relevance of the Fermi Energy
energy states ahove Ef ' arc empty.
Relevance of the Fermi Energy
4_6_3
Relevauce ofthe Fermi Euergy
21
We have been calculating the position of the Fermi energy level as a function of dop-
An important point is that in thermal equilibrium, the Fermi energy level is a constant throughout a system.
We will not prove this statement. but we can intuitively see its validity by considering the following example.
,
Suppose we have a particular material, A, whose electrons are distributed in the energy states of an allowed
band as shown in Figure 19a.
Most of the energy states below EF A contain electrons and most of the energy states
EF A are empty of
4. 7 above
Summary
electrons.
£
==1
A
. II.wed
=
,--------1
..f
.
,:
(;rL_-",--_.J---:
,
2
Allowed
\!nergy
!>I:Ues
Ir
(b)
___
-----
-=--
--
-
t
-
,
=
(ej
-
F
----.
(d)
F'fgure 4.20 I The t--°enni energy of (a) materia( A ill thetinaC equifjbrium. (0) maleriar B
Figure 19: The Fermi
energy
of (a) material
A in
thermal
equilibrium,
(b)phlC'ed
material
B in ,thermal equilibrium,
in Ihem131
equilibrium,
(c) materials
A and
B at the
they ,ll'C
in conwcl
(c) materials A and
the instant
are placed
in contact
and (d) materials A and B in contact at thermal
andB(d)atma(tri:.ds
A andthay
B in cOntact
u(-therlllal
equilibrium.
equilibrium.
of B,
energy.
the same are
in the
two materials.
equilibrium
occurs band as shown
Consider anothera function
material,
whoseis electrons
distributed
in theThis
energy
states of<tate
an allowed
when
the
Fermi
energy
is
the
same
in
the
twO
materials
as
shown
in
Figure
4.20d.
in Figure 19b.
Thebelow
FermiEenergy.
in and
th e physics
of the
semiconductor,
alsomostly
provides
a
The energy states
are important
mostly full
the energy
states
above E
are
empty.
FB
FB
good pictorial represcmation of the characteristics of the semiconductor materials
If these two materials
are brought into contact, the electrons in the entire system will tend to seek the lowest
and devices.
possible energy. Electrons from material A will flow into the lower energy states of material B, as indicated in
Figure 19c, until thermal equilibrium is reached.
4.71 SUMMARY
Thermal equilibrium
occurs when the distribution of electrons a function of energy is the same in the two
materials.
• The concelllratl".lH of declrQlI s in tht cOlld llctk Hl band is the illtegral over Ihe conducti on
This equilibrium state
the Fermi
same
in the
twoconduc
materials
as shown
bandoccurs
energy when
of the product
of theenergy
densityisofthe
states
function
in the
tion band
and in Figure19d.
the Fermi- Dirac 11fobabiJil'j function,
• ThecQncc::ntration ofhvLes in the v:llencc hand i:. thc integral uver the valence bUIld
energy of the product oflhe dl!n si(y ofst;}(cs fun<.:tion in Ihe valence band and the
prob3bility of a state being empt y, which is II - JF (£) \'
• Using (he Ma.xwC:H-Bolumann approxi nllttion. Ihe fhermal equilibrium cOll(.'enlr.:ltion
of electrons in (he conduction band is given by
where N" is the effecti ve density of MalC!'. ill the conducti on bal\d,
22
1411