3.15
Carrier Fundamentals
C.A. Ross, Department of Materials Science and Engineering
Reference: Pierret, chapters 1-2.
Electron and hole charge: e = 1.6 10-19 C
Effective mass: m*, rest mass mo
F = -eE = mo dv/dt
in vacuum
F = -eE = m* dv/dt
in solid
in Si, mn*/mo = 1.18, mh*/mo = 0.81 at 300K.
Intrinsic properties
in Si, n = p = 1010 cm-3 at 300K
N = 5 1022 atoms cm-3
Extrinsic properties
Donors – group V
Acceptors – group III
B
Al
Ga
In
Tl
C
Si
Ge
Sn
Pb
N
P
As
Sb
Bi
O
S
Se
Te
Po
Band diagrams: Ec = conduction band edge, Ev = valence band edge
band gaps: Si
1.12 eV
diamond
5.4 eV
silica
5 eV
energies of dopant levels, in meV, in silicon (kT = 26 meV @ 300K)
P
45
B
45
As 54
Al
67
Ga 72
Handout 1
Carrier distributions (intrinsic)
g(E) dE = density of electron states cm-3 in the interval (E, E+dE),
units #/eV.cm-3
gc (E) dE = mn*√(2mn*(E – Ec)) / (π2 h 3) dE
gv (E) dE = mp*√(2mp*(Ev – E)) / (π2 h 3) dE
In these states, the electrons distribute
! according to Fermi function
f(E) = 1/ {1 + exp (E – Ef)/kT }
!
Number of electrons in the interval (E, E+dE) is therefore f(E)g(E)dE.
In a doped semiconductor, the position of Ef with respect to the band gap determines whether there are more electrons or holes.
Total number of electrons: by integrating f(E)g(E)dE
n = ni exp (Ef - Ei)/kT
p = ni exp (Ei - Ef)/kT
where
ni = Nc exp (Ei - Ec)/kT
Nc = 2{2πmn*kT/h2}3/2 = ‘effective density of conduction band states’
Ei is the position of the Fermi level in the intrinsic case.
Similarly for Nv.
Hence
np = ni2 at equilibrium
ni2 = Nc Nv exp (Ev - Ec)/kT = Nc Nv exp (-Eg)/kT
Intrinsic case:
Ei = (Ev + Ec)/2 + 3/4 kT ln (mp*/ mn*)
In a doped material, where n ~ ND or p ~ NA
Ef - Ei = kT ln (n/ ni) = - kT ln (p/ ni)
~ kT ln (ND / ni) or - kT ln (NA / ni)
n-type
p-type
Handout 1
Properties
Si
SiO2
GaAs
Ge
Mayer and
Lau,
Electronic
Materials
Science
2.27a
amorphous
Atoms/cm3, molecules/cm3 x 1022
Structure
5.0
diamond
4.42
zincblende
Lattice constant (nm)
0.543
0.565
Density (g/cm3)
Relative dielectric constant, er
2.33
11.9
5.32
13.1
2.27a
3.9
Permittivity, e = ereo (farad/cm) x 10-12
1.05
1.16
0.34
Expansion coefficient (dL/LdT) x (10-6 K)
Specific Heat (joule/g K)
2.6
0.7
6.86
0.35
0.5
1.0
Thermal conductivity (watt/cm K)
1.48
0.46
0.014
Thermal diffusivity (cm2/sec)
0.9
0.44
0.006
Energy Gap (eV)
1.12
1.424
~9
0.67
Drift mobility (cm2/volt-sec)
Electrons
1500
8500
Holes
450
400
2.8
0.047
1.04
0.7
1.45 x 1010
1.79 x 106
Effective density of states
(cm-3) x 1019
Conduction band
Valence band
Intrinsic carrier concentration (cm-3)
Properties of Si, GaAs, SiO2, and Ge at 300 K
Figure by MIT OCW.
Energy Gap and Lattice Constants
2.6
0.517
GaP
2.2
0.563
AlAs
0.620
2.0
Energy gap (eV)
1.8
0.689
AlSb
1.6
0.775
GaAs (D)
1.4
1.2
0.885
InP(D)
1.033
Si
1.0
1.240
0.8
0.6
2.067
Indirect band gap
0.4
1.550
GaSb (D)
Ge
3.100
InAs (D)
Direct band gap (D)
0.2
0
0.54
Wavelength (microns)
2.4
0.477
AlP
6.200
InSb (D)
0.55
0.56
0.57
0.58
0.59
0.60
0.61
0.62
0.63
0.64
0.65
Lattice constant (nm)
Handout 1
Figure by MIT OCW.
Physical Constants, Conversions,
and Useful Combinations
PHYSICAL CONSTANTS, CONVERSIONS, AND USEFUL COMBINATIONS
Physical Constants
Physical
Constants
Avogadro
constant
Avogadro constant
Boltzmann
constant
Boltzmann constant
Elementary
charge
Elementary
charge
constant
PlanckPlanck
constant
Speed of light
SpeedPermittivity
of light (free space)
Electron mass
Permittivity
(free
space)
Coulomb
constant
Atomic
mass unit
Electron
mass
Coulomb constant
Useful Combinations
Atomic
mass energy
unit (300 K)
Thermal
Photon energy
Useful Combinations
Coulomb constant
N,
NA = 6.022 x 1023 particles/mole
= 6.022 x
particles/mole
k =8.617
8.617
10-5 eV/K= =1.38
1.38x x 10-23J/K
J/K
k =
x x10-'eV/K
-19
e =1.602
1.602
10 coulomb
coulomb
e =
X x
10-l9
h = 4.136 x 10-IS-15
eV . s .
h =6.626
4.136
10 joule
eV ss
=
x xlo-"
c =
2.998 xx 10
10"-34cm/s
joule .s
= 6.626
co = 8.85 X 10-l410farad/cm
c = 2.998 x 10 cm/s
rn = 9.1095 x lo-" kg
8.85xx lo9
10-14
farad/cm
kc ε=0 =
8.988
ne~ton-m~/(coulomb)~
u =
x x 10-31
kg kg
m =1.6606
9.1095
.
kc = 8.988 x 109 newton-m2/(coulomb)2
1.6606eVx =
101-27
kg
kTu== 0.0258
eV/40
E = 1.24eVatX = 1 pm
kce2 = 1.44 eV . nm~
_
kTe,e,
= 0.0258
1 eV/40
E =
= 1.05eV
x 10-l2
faradlcm
q, E==55.3e/V
1.24 eV. pm
at λ = µm
Thermal
energy (300
Permittivity
(Si) K)
PhotonPermittivity
energy (free space)
Coulomb constant
kce2 1.44 eV . nm
Prefixes
-12
Permittivity
(Si)
1.05=x10'
10
k = kilo
= I@; M = mega = εI =
d ;εG
; T =farad/cm
tera = 1012
rε0==giga
.
n = nano
Permittivity
(free
µm= IW9;
m = milli
= space)p = micm ε=
0 = 55.3e/V
p
=
pica
=
lo-''
Prefixes
k = kilo = 103; M = mega = 106; G = giga = 109; T = tera = 1012
Symbols-3for Units
m = milli
= 10 ; µ = micro = 10-6; n = nano = 10-9; p = pica = 10-12
Ampere (A), Coulomb (C), Farad (F)
Gram
(g), Joule (J), Kelvin (K)
Symbols for
Units
Meter (m), Newton (N), Ohm (a)
Ampere
(A), (s),
Coulomb
Farad
Second
Siemen (C),
(S), Tesla
(T)(F), Gram (g), Joule (J), Kelvin (K)
Watt (W),
Meter Volt
(m),(V),
Newton
(N), Weber
Ohm (Wb)
(Ω), Second (s), Siemen (S), Tesla (T)
Volt (V), Watt (W), Weber (Wb)
Conversions
Conversions
1 nm =
m = 10 A =
cm
1.602
X 10-l2
erg-9 Joule = 1.602 x 10-12 erg;
eV-9=m1.602
1 nm =110
= 10 xA =10-l9
10-7joule
cm; =1 eV
= 1.602
x 10
1 eV/particle
= 23.06 kcal/mol
1 eV/particle
= 23.06
kcal/mol; 1 newton = 0.102 kgforce;
1 newton2 = 0.102 kg,,
6
7 dyn/cm2 ; 1 µm = 10-4 cm 0.001 inch = 1 mil = 25.4 µm;
10 newton/m
= 146= psi
10=
146=psi
lo7 dyn/crn2
lo6 newton/m2
2cm
1 bar =110
= 105 N/m2; 1 weber/m2 = 104 gauss = 1 tesla;
Fm6 dyn/cm
=
-3 torr;
0.001
mil =
25.4
pm 1 erg = 10-7 joule = 1 dyn-cm
1 pascal
= 1 inch
N/m=2 =1 7.5
x 10
1 bar = lo6 dyn/cm2 = Id ~ / m ~
1 weber/m2 = lo4 gauss = 1 tesla
torr
1 pascal = 1 N/m2 = 7.5 X
Figure by MIT OCW.
1 erg = 10-'joule = 1 dyn-cm