Post on 06-Jul-2020
Modern Electronics: F1 semiconductors 1
Semiconductors – a brief introduction
• Band structure – from atom to crystal
• Fermi level – carrier concentration
• Doping
• Transport (drift-diffusion)
Hyperphysics (link on course homepage)basic introduction to semiconductors(almost no equations)
Reading: (Sedra/Smith 7th edition)
1.7-1.9
Modern Electronics: F1 semiconductors
Atomic energy levels
2
E
x
K
L
M
N
K
L
MN
0.1 nm
Quantum mechanics:
Wavefunction gives describes
probablility to find electron
0.1 nm
Modern Electronics: F1 semiconductors
2-atomic molecule – Pauli principle
3
E
x
E E
x x
0.1 nm 0.2 nm 0.1 nm
Overlapping
Valence electrons
Atoms share
valence electrons
0.2
nm
Modern Electronics: F1 semiconductors
16-atomic molecule
4
E
x
E
x
0.1 nm
E
0.4 nm
0.4
nm
x,y,z
0.4 nm
Modern Electronics: F1 semiconductors
1023-atomic molecule – energy bands
5
x
0.1 nm
x,y,z
1 cm
……
1 c
m
E
~ 1023
levels
Modern Electronics: F1 semiconductors
Valence and conduction bands
6
x,y,z
……
E
Conduction band
Valence band
Metal: The valence band is partially filled with electrons
Semiconductor / Insulator: The valence band is filled
Valence Band: The highest band that has electrons
Conduction Band: The next band with higher energy
Modern Electronics: F1 semiconductors
metals – semiconductors - insulators
7
Valence band
Conduction band
Energ
y (
eV
)
Valence band
Conduction band
Metal
Valence band
Conduction band
Valence band
Conduction band
semiconductors0 < Eg < 4 eV
Si: Eg=1.12 eVGe: Eg=0.67 eV
GaN: Eg=3.42 eVinsulatorsEg > 4eV
SiO2: Eg=9 eVDiamond (C): Eg=5.5 eV
x
EgEg
Modern Electronics: F1 semiconductors
What materials are semiconductors?
8
C
Si
Ge
N
P
As
B
Al
Ga
In SbSn
IV VIIITwo atom basis (4 neighbours)
4 own valence electrons
total 8 shared electrons in valence shell -> filled!
4+14=83+5=8
C GaP Si Ge InAs Sn
Eg 5.5 eV 2.24 eV 1.12 eV 0.67 eV 0.34 eV 0
Type insulator Semi cond uc tors metal
Modern Electronics: F1 semiconductors
Thermal excitation
9
Ener
gy (
eV)
Eg
• Each electrons gets (average) kinetic energy Ekin=3/2∙kT
• An electron can be excited to the conduction band
• Higher T or smaller Eg -> more electrons
• electron density in conduction band = n (cm-3)
• electron density in valence band = 1024 - n (cm-3)
• p (cm-3): hole density in valence band
• n=p without doping
Modern Electronics: F1 semiconductors
Thermal excitation – Fermi level
10
Energ
y (
eV
)
Eg
Fermi-Dirac distribution:The probability of an electron at an energy level E.
Higher T – higher probability that a level in the conduction band has an electron.
Symmetrical about EF (Fermi energy).
50% chance to have electron at EF
Excited electron leave a positive hole in valence band
EF
probability0 1
EC
EV
1)/)exp((
1)(
kTEEEf
F
Modern Electronics: F1 semiconductors
Holes vs electrons
11
electron (negative)hole (positive)
T > 0 K
Instead of describing all electrons remaining in valence band, positive holes (missing electrons) are introduced and treated as particles.
e
EC
EV
Modern Electronics: F1 semiconductors
Electron transport
12
T ≈ 0 K
- Apply voltage -> electric field moves charged electrons- Filled band: ½ of electrons move in one direction, ½ of electrons in the other. - Need to change velocity of some electrons to get current but all states are filled i.e. need
electrons in conduction band.
No available energy states to move to leads to zero current
e
metal semiconductor
T >> 0 K
EC
EV
semiconductor
e
Partially filled band
EC
EV
Modern Electronics: F1 semiconductors
Carrier concentration
13
Fermi-Dirac distribution x density of states = carrier concentration
probability of occupying a state
number of available states
population of conduction band
Ec
x =
energ
y
Modern Electronics: F1 semiconductors
Carrier concentration - simplified
14
Energ
y (
eV
)
Eg EF
kT
EENp
kT
EENn
Fvv
cFc
exp
exp
• Ignore real density of states, introduceNC, NV – effective density of states at band edges (describes how dense levels are)
(Si: Nc=3.2*1019 cm-3 / Nv=1.8*1019 cm-3)
EC
EV
Modern Electronics: F1 semiconductors
Intrinsic carrier concentration
15
kT
ENNpnn
g
vci2
exp
T=300K
SiEg=1.11eVni=11010 cm-3
GeEg=0.67 eVni= 21013 cm-3
Each electron excited from the valence band to the conduction band become a free carrier available for conduction
Intrinsic semiconductor:- n=p=ni (intrinsic carrier concentration)- EF is in the middle of the band gap
Modern Electronics: F1 semiconductors
Doping with donor atoms: n-type
16
x
Ec
Ev
E
+1
P – atom
5 valence electrons
C
Si
Ge
N
P
As
B
Al
Ga
In SbSn
IV VIII
Donates electron to the conduction band (mobile)Ionized atom – positive (immobile)
Modern Electronics: F1 semiconductors
Doping with acceptor atoms: p-type
17
x
Ec
Ev
E
-1
Al – atom
3 valence electrons
C
Si
Ge
N
P
As
B
Al
Ga
In SbSn
IV VIII
Captures electron -> extra hole to the valence band (mobile)Ionized atom – negative (immobile)
Modern Electronics: F1 semiconductors
Doping – extrinsic semiconductor
18
- ND: donor atom density (cm-3) / NA: acceptor atom density (cm-3)
- For ND >> ni (NA=0) the doping dominates majority carrier concentration
n=ND and p=ni2/ND
)exp(
)exp(
kT
EENp
kT
EENn
Fvv
cFc
2exp i
g
vc nkT
ENNnp
2
inpn Mass action law Independent of doping
Modern Electronics: F1 semiconductors 19
doping – carrier density
Fermi-Dirac Carrier conc.
Intrinsic
n-type
(donors)
p-type
(acceptors)
Ec
Ev
electrons
holes
Eg
0 1
n
p
i
AFiVFp
i
DFiVFn
n
NkTEEE
n
NkTEEE
ln
ln
FiE
Modern Electronics: F1 semiconductors 20
doping – carrier density
Fermi-Dirac Carrier conc.
Intrinsic
n-type
(donors)
p-type
(acceptors)
Ec
Ev
electrons
holes
Eg
0 1
n
p
i
AFiFp
i
DFiFn
n
NkTEE
n
NkTEE
ln
lnFnE
FiE
Modern Electronics: F1 semiconductors 21
doping – carrier density
Fermi-Dirac Carrier conc.
Intrinsic
n-type
(donors)
p-type
(acceptors)
Ec
Ev
electrons
holes
Eg
0 1
n
p
i
AFiFp
i
DFiFn
n
NkTEE
n
NkTEE
ln
lnFnE
FiE
Modern Electronics: F1 semiconductors
Transport - drift
22
pn
nn
pp
JJJ
εqnJ
εqpJ
Hole current density (A/cm2)
Electron current density (A/cm2)
e
+V
pnq
pnq
pn
pn
1
EC
EV
J
µn / µp– electron / hole mobility (cm2/Vs)
n / p – electron / hole concentration (cm-3)
=1/=conductivity (S/m)
= resistivity (Ωm)
vd
Modern Electronics: F1 semiconductors 23
Transport – effect of doping
endn qnµqnvJ
n: controlled by doping µn: determined by intrinsic semiconductor properties + scattering
Ionized impurity scattering
phonon scattering
p-typen-type
Modern Electronics: F1 semiconductors
Velocity saturation
24
- At high electric fields (εc ≈ 1.5x106 V/m) electrons can emit optical phonons (lattice vibrations) -> velocity saturates
satcndc
ndc
c
nd
nd
vµv
µvµv
qnµqnvJ
eee
eee
ee
e
ee
/1
)(
Modern Electronics: F1 semiconductors 25
Transport - diffusion
- Random thermal motion gives movement of particles from high to low concentration
- No external forces - No particle interaction- Rate depends on concentration gradient
Modern Electronics: F1 semiconductors 26
Transport - diffusion
dx
xdnµqV
dx
xdnµ
q
kTq
dx
xdnqDJ nTnnn
)()()(
x
n (
m-3
)
n(x)
n(0)=n0
n(L)=nL
Diffusion current given by gradient of electron (or hole) concentration
No addition/removal of carriers between 0 < x < L -> constant current -> linear concentration decrease
Jn
Modern Electronics: F1 semiconductors
2 min exercise – drift/diffusion
27
n
position
n
position
Later....
Sketch carrier distribution later with
1. No electric field
2. E-field in pos. direction (->)
Modern Electronics: F1 semiconductors
Summary
28
- Discrete atomic energy levels become bands when atoms are joined in a crystal.
- Semiconductors = filled valence band, no electrons in conduction band (at T=0). Band gap with no allowed states.
- Carrier concentration (n) = Fermi-dirac distribution * density of states
- Doping: • Donors (group V): adds mobile negative electrons and positive static ions.
Moves Fermi level towards conduction band.• Acceptors (group III): adds mobile positive holes and negative static ions.
Moves Fermi level towards valence band.
- Carrier transport:• Drift: electric field moves charged particles. Depends on field strength.• Diffusion: random motion moves particles from high to low concentration.
Depends on concentration gradient.
dx
dnqDqnJJJ nndiffndriftnn ε,,
2
inpn n=ND and p=ni2/ND for ND>>ni
)exp(kT
EENn cFc
Modern Electronics: F1 semiconductors
pn- junctions
29
Charge distribution -> electric field -> potential
Current transport
Breakdown mechanisms
Small-signal model
Depletion / diffusion capacitances
Reading: (Sedra/Smith 7th edition)
1.10-1.12, 3.1 - 3.3
Modern Electronics: F1 semiconductors
Why pn-junctions?
30
diode
LED
Solar cell
N-type P-typeN-type P N-type
Base
CollectorEmitter
P-typeN-type N-type
Gate
Source Drain
Substrate
BJT
MOSFET
Modern Electronics: F1 semiconductors
N - type P - type
31
Ec
Ev
electrons
holes
E
Eg
ND – donor concentration
nn0 – electron (majority) concentration
pn0 – hole (minority) concentration
Electrons: mobile, negative
Ionized donors: not mobile, positive
Ec
Ev
electrons
holes
E
Eg
NA – acceptor concentration
pp0 – hole (majority) concentration
np0 – electron (minority) concentration
holes: mobile, positive
Ionized acceptors: not mobile,
negative
Modern Electronics: F1 semiconductors
Recombination
32
EC
Ev
N
2
00 inpn
If there is an excess number free carrriers np
> ni2 electrons can recombine with holes to
reach equilibrium
p > p0
Three electrons recombine
leaving three positive ionized
donor atoms
nn0=ND+
nn0 <
ND+
In equilibrium Important for base current in BJTs
Modern Electronics: F1 semiconductors
PN-junction – band structure
33
E
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
+ Positive donor
- Negative Acceptor
Free electrons
Free holes
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
N-type P-type dx
dnJn
Large conc. difference -> large
diffusion current
No e-field – no drift current
dx
xdnVnqAI Tnn
)(ε
Modern Electronics: F1 semiconductors
PN-junction – band structure
34
E
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
+ Positive donor
- Negative Acceptor
Free electrons
Free holes
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
e
N-type P-type
dx
xdnVnqAI Tnn
)(ε
Modern Electronics: F1 semiconductors
PN-junction – band structure
35
E
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
+ Positive donor
- Negative Acceptor
Free electrons
Free holes
Ec
Ev
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
e
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
e
W
N-type P-type
No free carriers in depletion region,
charge neutral outside
Modern Electronics: F1 semiconductors
charge density -> electric field -> potential
36
x=0
x=-W1
X=W2
𝜌 [C/m-3]
V0+ + + +
+ + + +
- - - -
- - - -
Poisson equation
Depletion width
𝜌(𝑥)
𝜖𝑠=𝑑𝜀(𝑥)
𝑑𝑥= −
𝑑2𝑉(𝑥)
𝑑𝑥2 - 𝜌 charge density [C/m-3]
- 𝜖𝑠 permitivity [F/m]
- e electric field [V/m]
- V potential [V]
- V0 built-in potential [V]
x=0
V(x) [V]
x=W2
x=-W1
− න𝑑𝑥
e [V/m]
න𝑑𝑥
W=W1+W2
𝑊 =2𝜖𝑠𝑞
1
𝑁𝐴+
1
𝑁𝐷𝑉0
Q+=AqNDW1
Modern Electronics: F1 semiconductors
Built-in potential
37
E
Ec
Ev
qV0
EFn
EFp
20 lnlnln
in
NNkT
n
N
n
NkT AD
i
A
i
D
Ec
Ev
W=W1+W2
depletion region
(built-in) Potential barrier for electrons and holes!
N P
qV0
homework: calculate depletion width for Si pn-junction with ND=NA=1017 cm-3
at bias voltages V=0 V and V= -1 V.
i
AFiFp
i
DFiFn
n
NkTEE
n
NkTEE
ln
ln
𝑞𝑉0 = 𝐸𝐹𝑛 − 𝐸𝐹𝑝
𝑞𝑉0
Modern Electronics: F1 semiconductors
2 min exercise – assymetric pn-junction
• Consider a pn-junction with ND =5*NA (donors > acceptors)
• Sketch the
1) charge distribution
2) electric field
3) potential
38
N-type P-type
ND NA
ερ V
Modern Electronics: F1 semiconductors
Diode – forward bias
39
N Pe
- VD +DEpot=-qVD
eVD
N Pe
- VD +
eVD
- Only top of the electron distribution can pass over the barrier
- Increase bias -> exponentially increasing amount of electrons can pass
- Reduce electric field but counteracting built-in potential
- Depletion width is reduced
Modern Electronics: F1 semiconductors
Minority carrier density at depletion edges at forward bias
40
electron concentration
in p-side
hole concentration
in n-side
pn(x)
np(x)
forward bias
At equilibrium:
np0=ni2/NA
pn0=ni2/ND
-W1 W2
np(x)
0
deple
tion
regio
n
pn(x)
np / pn
np0pn0
conta
ct
conta
ct
kT
qVpWp Dnn exp)( 01
kT
qVnWn Dpp exp)( 02
Modern Electronics: F1 semiconductors
Diode – forward bias
41
N P
e
- VD +
DEpot=-eVD
eVD VD
I
J = JS(exp(V/VT)-1)
VD
dx
dnVnqJ Tnn ε
𝐽𝑠 = 𝑞𝐷𝑝𝐿𝑝
𝑝𝑛0 +𝐷𝑛𝐿𝑛
𝑛𝑝0
= 𝑞1
𝐿𝑝
𝑘𝑇
𝑞𝜇𝑝
𝑛𝑖2
𝑁𝐷+
1
𝐿𝑛
𝑘𝑇
𝑞𝜇𝑛
𝑛𝑖2
𝑁𝐴
Modern Electronics: F1 semiconductors
Diode – reverse bias
42
N Pe
VD
I
+ VD -
eVD
J ≈ 0
-VD
DEpot=-eVD
dx
dnVnqJ Tnn ε
Modern Electronics: F1 semiconductors
Diode – total current
𝐽 = 𝐽𝑠 𝑒 Τ𝑞𝑉 𝑛𝑘𝑇 − 1
Js = saturation current density
n = ideality factor (1-2)
diffusion current
drift current
diffusion current diffusion current
drift current drift current
N Pe
zero bias forward biasreverse bias
43
𝐽𝑠 = 𝑞𝐷𝑝𝐿𝑝
𝑝𝑛0 +𝐷𝑛𝐿𝑛
𝑛𝑝0
= 𝑞1
𝐿𝑝
𝑘𝑇
𝑞𝜇𝑝
𝑛𝑖2
𝑁𝐷+
1
𝐿𝑛
𝑘𝑇
𝑞𝜇𝑛
𝑛𝑖2
𝑁𝐴
Modern Electronics: F1 semiconductors
1 min excercise – Si vs Ge diode
44
I
VD
Si (Eg=1.11 eV) pn-junction
I
VD
Ge (Eg=0.67 eV) pn-junction
???
NP
N P
Modern Electronics: F1 semiconductors
Zener tunneling / Avalanche Breakdown
45
e
- High reverse bias gives enough e-field to enable tunneling
- High energy of electron/hole can be lost by creating new e-h pairs
through impact ionization
tunneling
e
avalanche multiplication
V
IBV
Modern Electronics: F1 semiconductors
small-signal model
46
- Want to replace non-linear components with linear ones to simplify circuit
calculations
- Constant VD + small varying vd(t) = total voltage vD (t).
t
= +
vD (t) VD
vd (t)
Linearize by
Taylor expansion
Modern Electronics: F1 semiconductors
Small-signal model of diode (3.3.7)
47
- Diode IV is non-linear so difficult to do
calculations
- Apply constant VD and small varying vd(t)
- Diode IV almost linear in a small region
𝐽 = 𝐽𝑠 𝑒 Τ𝑞𝑉𝐷𝑘𝑇 − 1 ≈ 𝐽𝑠𝑒
Τ𝑞𝑉𝐷𝑘𝑇
Small signal resistance
rd=VT/ID (VT=kT/q)
Modern Electronics: F1 semiconductors
depletion-region / junction capacitance
48
V
QC Definition: A
WWC r
21
0
eeer
W1+W2
+Q-Q
Example: paralell plate capacitor
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
W1 W2
+ + + + + + + + +
+ + + + + + + + +
+ + + + + + + + +
- - - - - - - - - - - - -
- - - - - - - - - - - - -
- - - - - - - - - - - - -
W1 W2
Non-linear relationship between
VR and Q -> C(VR)𝑊 ∝ 𝑉0 + 𝑉𝑅QJ = 𝐴 ∙ 𝑞 ∙ 𝑁𝐷 (𝑊1 +𝑊2)
Modern Electronics: F1 semiconductors
depletion region / junction capacitance
49
Definition Applied bias (VR) changes depletion width
Depletion width
Charge on
either side
Cj
V
Increasing VR
𝑊 =2𝜖𝑠𝑞
1
𝑁𝐴+
1
𝑁𝐷(𝑉0+ 𝑉𝑅)
𝐶𝑗 =𝑑𝑄𝐽𝑑𝑉𝑅
𝑄𝐽 = 𝐴𝑞𝑁𝐷𝑊1 = 𝐴𝑞𝑁𝐷𝑁𝐴
𝑁𝐴+𝑁𝐷W
𝐶𝐽 = 𝐴 2𝜖𝑠𝑞𝑁𝐷𝑁𝐴𝑁𝐴 +𝑁𝐷
1
𝑉0 +𝑉𝑅
Modern Electronics: F1 semiconductors
1 min exercise – pn-junction with forward bias
50
V
C
V
C
V
C
A B C
In forward bias there is a diffusion current flowing through the junction. How
does the CV curve behave?
Only Cj
No change in
capacitance
Larger capacitance
for forward bias
Lower capacitance
for forward bias
dV
dQC
Modern Electronics: F1 semiconductors
diffusion capacitance
51
W2-W1
Forward bias -> inject minority carriers in
neutral regions (electrons in p-region and
holes in n-region)-> extra charge dQ.
x
n
Wp
(length of p-region)
W2 (=0)
V
DQ
V+Dv
np0
dV
dQC
0
02
)(
exp)(
ppp
Dpp
nWn
kT
qVnWn
Modern Electronics: F1 semiconductors
Total capacitance
52
-5 -4 -3 -2 -1 0 10
10
20
30
40
50
60
70
80
Voltage (V)
Capacitance (
pF
)
Ctot
Cj
Cd
rd (VD)Cj (V) Cd (V)
Add capacitances in parallel
Cj: dominates for reverse bias
Cdiff: dominates for forward bias.
Cdiff ≈ 0 for reverse bias.
Ctot=Cj+Cd
Modern Electronics: F1 semiconductors
Summary – pn-junctions
• pn-junctions used in LEDs, solar cells, BJT, MOSFETs
• Poissons equations: charge distribution -> electric field -> potential
• Drift is balanced by diffusion in unbiased pn-junction
• Current given by ideal diode equation:
– Forward bias: current (diffusion) increases exponentially
– Reverse bias: current saturates
• Capacitances:
– Junction capacitance due to change in depletion width (dominates reverse bias)
– Diffusion capacitance due to change in charge in p/n region (dominates forward bias)
• Small-signal model (ok for V << VT): replace diode with resistor + capacitances
53
𝐽 = 𝐽𝑠 𝑒 Τ𝑞𝑉 𝑛𝑘𝑇 − 1