Question of Interest
• How many electrons are there in the conduction band? • How many holes are there in the valence band?
• It depends on:
- Temperature of the crystal - Size of the bandgap
- Number of available states in the conduction band and valence band
Finding Number of Electrons
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer EngineeringNumber of states in an energy band
• A state is a place to hold oneelectron and it becomes a hole when there is no electron there
Density of States
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer EngineeringEC
EV Density of states
D(E)
• The number of states in a band is a function of energy given by the function D(E)
EC Energy at the bottom of
the conduction band
Energy at the top of the valence band
EV
• The total number available states in a band is obtained by
integration of the entire band
! " =
8% 2 ℎ( )*
∗(,
" − ". /01 " ≥ ". 8% 2
ℎ( )*∗(, "3 − " /01 " ≤ "3
No need to know
Fermi-Dirac Statistics [1]
Carrier distribution
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
• the carrier distribution follows a Fermi-Dirac function fe(E) that define the probably of a state being filled by a electron and
Without supply of energy
With surrounding energy source (e.g. room temperature)
kT E E
e F
e E
f
-+ =
1
1 )
(
fe(E)
E EF
1 0 fe(E)
E
1 0
EF
• EF is defined as the Fermi energy level where the probability to find a electron in a state is ½
Fermi-Dirac Statistics [2]
For material with a bandgap
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
EC
EV
e
-0 1
EF states
states
fe(E) • Due to symmetry, EF is located
somewhere in the middle of the bandgap
• at 0K, all electrons stay below EF (or EV) • above 0K, electrons have finite
probability to stay at the conduction band
• The probably to find a “hole” in a state is given by fh(E) where
fh(E) =1− fe(E) = 1
1+e
Ef−E kT
EC
EV
0 1
EF states
states
Fermi-Dirac Statistics [3]
Number of Electrons/Holes
• the total number of electron is given by the number of available states multiplied by the probability that a state is filled
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
EC
EV
EC
EV e
-0
1 h+
EC
EV Density of states Fermi-Dirac # of e- and h+
D(E)
fe(E)
fe(E)
D(E)
ne
( )
E = D E( )
× fe( )
E#e− = ni = D E
( )
11+e
E−EF kT EConduction
∫
dE #h+ = #e− = ni = D E
( )
11+e EF−E
kT EValence
∫
dE
Equivalent density of states
Equivalent Density of States [1]
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
• We only care about the total density of states, but do not care where they are located
• To simplify the discussion, we assume all states are located at the edge of the band, EC and EV
EC EV
EC EV
NC
NV
•The equivalent density of states at the conduction band and valence band are denoted by NC and NV respectively
• the “equivalent” density of states at the conduction and valence band edges are
Equivalent Density of States [2]
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering 2 / 3 2 2 2 ÷ ø ö ç è æ = h kT m N e C
p 3/2
2 2 2 ÷ ø ö ç è æ = h kT m N h V p ; -me: effective mass of e- (1.02x10-30kg)
-mh: effective mass of h+ (5.25x10-31kg)
• once the temperature is know, the number of states at the edge of the conduction and valence band become more or less constant
• For silicon at room temperature, they are
3 19cm 10
8 .
2 ´
-=
C
N NV =1.09×1019cm−3
-h: Planck constant (6.63x10-34Js)
-k: Boltzmann constant (8.62x10-5eV/K)
-T: absolute temperature
Simplified Carrier Statistics [1]
Simplification with the equivalent density of state
• the total number of electrons is given by the number ofavailable states multiplied by the probability that a state is filled
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
EC
EV
EC
EV
0
1 h+
EC
EV
Fermi-Dirac # of e- and h+
NC
fe(E)
fe(EC) NC
#e− = ni = NC fe
( )
EC = NC 1+eEC−EF kT
#h+ = ni = NV fh
( )
EV = NV1+e
EF−EV kT
EC EV
Density of states
De(E)
NV
Simplified Carrier Statistics [2]
Boltzmann Approximation
• the Fermi-Dirac function is not easy to use and the Boltzmann approximation is often applied
fe(E)= 1 1+e
E−EF kT
≈ e− E−EF
kT
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering E −EF >> kT
for
kT ≈ 0.025eV at room temperature
and
EC
EV
e
-0 1
EF
€
fe
( )
E ≈ e−E−EF kT
fe(E) = 1
1+ e E−EF
kT
fh
( )
E ≈ e−EF−E kT
similarly
note: fh(E)=1-fe(E)
Simplified Carrier Statistics [3]
Number of Electrons and Holes
• After Boltzmann’s approximation, the number of elections and holes in intrinsic silicon can be found by
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
ni ≈ NC exp− EC − Ei kT ⎛
⎝
⎜ ⎞ ⎠
⎟ pi = ni ≈ NV exp− Ei −EV
kT
⎛ ⎝
⎜ ⎞ ⎠ ⎟
;
np = ni2 ≈ NCNV exp− EG
kT
# $
% & ' (
• At room temperature, ni=1.45x1010cm-3 (or simply 1010cm-3)
-n: number of carriers in the conduction band
-p: number of carriers in the valence band
-ni/pi: intrinsic number of carriers in conduction/valence band
-EF: Fermi-level of a material
-Ei: Fermi-level of intrinsic silicon
Meaning of Fermi-Dirac Statistics [1]
Visualizing the Fermi-Dirac Statistics
• To understand the concept of Fermi-level (EF), we can borrow the water analogy again
• At thermal equilibrium (no battery voltage), the Fermi-level outside the semiconductor is the same
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering
conduction band
electron as water vapor
valence band holes as bubbles
EC
EV
EF EG=1.1eV
water
water table
• Take the forbidden gap as a soft impermeable material • Take the Fermi energy level (EF), as the water table that
represent the water pressure Level depends
on battery
semiconductor
external reference (battery)
external reference (battery)
Meaning of Fermi-Dirac Statistics [2]
Under external voltage
ansun
M
ansunM
Hong Kong University of Science & Technology, Department of Electronic & Computer Engineering• Along the channel, the “water level” gradually become lower causing the “soft material” floating on it to slope downward • The carriers will move from one end to the other causing
conduction
water
semiconductor
V
-+
• You may consider a battery define the Fermi-level at the two sides of the
semiconductor
• In static system, you may imagine the system is
further divided into many
small sections to be lowered by different amount