ch1 1 carrier_modeling
Post on 02-Jun-2018
217 Views
Preview:
TRANSCRIPT
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 1/25
Basic Theory of Semiconductors
and Junctions
References:(1) RF Pierret, “Semiconductor Device Fundamentals”, Addison Wesley, 1996.(2) RS Muller & TI Kamins, “Device Electronics for Integrated Circuits,” John Wiley &
Sons, 1986, 2nd Edition
Carrier Modeling
Carrier Action
PN Junction
MS Junction
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 2/25
Carrier Modeling
Quantization Concept
Hydrogen atom model, isolated Si atom
Semiconductor Models
Bonding model, energy band model, carriers
Distribution of CarriersCarrier numbers in intrinsic semiconductor, doping,carrier distribution
Carrier Concentration
Carrier concentration calculation, Fermi levelcalculation
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 3/25
ps
+14
The quantization concept
Isolated Si atom
Hydrogen atom model
- Niels Bohr, 1903
- Explained why the emitted light was observed at onlycertain discrete wavelength
- Assume orbiting electron could take only certain valuesof angular momentum
- Electron limited to certain energies inside H atom
n=1
2 Electrons
n=2
8 Electrons
n=3
Two allowed levelsat same energy
Six allowed levelsat same energy
- 10 electrons occupy very deep energy levels,
and these 10 electrons are treated as “core” ofatom along with nucleus due to tight bound
- Remaining 4 electrons, weakly bond,collectively called “valence electrons”
...3,2,1,6.13
)4(2 22
0
4
0 neV
nn
qm E H
2/h
+
n=2
-3.4eV
n=3
EH=-1.51eV
n=1
-13.6eV
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 4/25
Semiconductor Models: Bonding model
Line: a sharedvalence electron
Circle: the coreof a Si atom
Bonding model: Si atoms share one of valenceelectrons with the four nearest neighbors
Line refers to a shared valence electron
Circle refers to the core of a Si atom
Applications: can explain
A missing of atom
Breaking of atom-to-atom bond and freeing ofan electron
Drawback: cannot give more information, such asenergy-related aspects of events
Physical picture of a missingatom using bonding model
Physical picture of breaking of atom-to-atom
bond and freeing of an electron usingbonding model
-
Bonding model
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 5/25
Mostly filled
Mostly empty
Semiconductor Models: Energy band model
N IsolatedSi atom
ps n=3
4N electrons total8N electron states total
(6N p-states total)
(2N s-states total)
E
Crystal Si with N atoms
4N allowed states(conduction band)
4N allowed states(valence band)
No statesE
E (electron energy)
ETOP
EBOTTOM
EC
EVEg
Decreasing atom spacingIsolated Si atom Si lattice spacing
EC
EV
4N emptystates
2N + 2Nfilled states
Silicon semiconductor
E (electron energy)
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 6/25
Semiconductor Models: Applications
Electron and hole carrier
EC
EV
Empty
Completelyfilled
EC
EV
EC
EV
No carriersOne electron carrier One hole carrier
-
Bonding model
Energy band model
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 7/25
Semiconductor Models: Applications
Band gap and material classification
Insulator:
wide band-gap, difficult to excite at room temperature, poorconductor
Metal:
either very small or no band gap exists due to an overlap of thevalence and conduction bands
Semiconductor:
between insulator and metal
EC
EV
Eg
Insulator
EC
EV
Eg
Semiconductor
EC
EC
EV
EV
Metal
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 8/25
-e-
Carrier Distributions: Carrier properties
Carrier Properties
Charge:
One electron carries charge of “–q”; one hole carriescharge of “+q”
Effective mass: Physical meaning
Have multiple components due to the variety ofcarrier acceleration with different direction of travel
in a crystal Some examples of effective mass: density of
states effective mass, conductivity effectivemass, …
Material mn*/m0 Mp*/m0
Si 1.18 0.81
Ge 0.55 0.36
GaAs 0.066 0.52
Density of states effective masses at 300K
- a =Fext
mn
Fext
Ex
x
Vacuum
a =Fext
mn*
Fint
Ex
x
Crystal
++
+ +
+
+
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 9/25
Carrier Distribution: Intrinsic semiconductor
Carrier numbers in intrinsic semiconductor
Intrinsic:
Pure material, no impurity atom
n=p=ni (why n=p?)
~1E10 cm-3 in Si at room temperature
Carrier numbers is very low, poor conductor
for example,
Si atom ~ 5E22 cm-3
Total bonds (or total valence electrons) ~ 2E23 cm-3
Ratio of breaking bonds / total ~ 1E10/2E23
~ 1E-13 very small value !!!
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 10/25
Carrier Distribution: Doping
Doping: manipulation of carrier numbers n- and p-type:
Physical picture (bonding model)
N-type: P+ atom replace Si atom the 5th
electron weakly bonded easily to be free P-type: B+ atom replace Si atom
Energy band model (to answer the question: how
easy to make electron free) Binding energy
Can be modeled using pseudo-hydrogenatom model
-
P+
Ec
Ev
ED0.045eV(P) (Si) ~ 1.12eV
Donors Bindingenergy
Acceptor Bindingenergy
Sb 0.039 eV B 0.045 eV
P 0.045 eV Al 0.067 eV
As 0.054 eV Ga 0.072 eV
Dopant-site binding energy
+
B-
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 11/25
Carrier Distribution: DOS
Density of States (DOS)
Carrier Distribution
How the carriers are distributed with energy levelsWhat we should know:
How many states are available at a given energy E Probability that an available state at energy E will be occupied by an electron
Analogy: The density of states can be likened to the description of the seating in a
football stadium, Number of seats in the stadium a given distance from the playing field Probability of seats to be occupied
gC(E)dE means no. of conduction band states in dE
gV(E)dE means no. of valence band states in dE
Calculated with quantum physics
Only valid for energies not too far from the band energies
E
Ev
Ec gC(E)
gV(E)
32
** )(2)(
C nn
c
E E mm E g
for E EC
32
** )(2)(
E E mm E g
V p p
v
for E EV
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 12/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Carrier Distribution: Fermi-Dirac Function
Fermi-Dirac Function
T=0 K
1/2
1
EF
E
T>0 K
E
1
EF
Some discussions
T=0K
T>0K
if E=EF, F(EF)=1/2
if EEF+3kT, f(E)exp[-(E-EF)/kT]
if EEF-3kT, f(E)1-exp[(E-EF)/kT]
A note on the meaning of “Fermi level”
For energy level below EF, it will beoccupied by electron
In thermal equilibrium, EF constantthrough the system
kT E E F e E f
/)(1
1)(
Where, EF is Fermi energy, k is Boltzmanconstant, T is Temperature in Kelvin (K)
f(E) represents probability of one state atenergy E to be occupied by electron
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 13/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Carrier Distribution: How carrier concentration looks like
Carrier Distribution
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 14/25
Carrier Distribution
Carrier Distribution: Dopant Effect
gc(E)
gv(E)
Ec
Ev
EF
1-f(E)
f(E)
gc(E)
gv(E)
Ec
Ev
EF
gc(E)f(E)
gv(E)[1-f(E)]
1-f(E)
f(E)
gc(E)
gv(E)
Ec
EvEF
1-f(E)
f(E)
Energy band Density of states Occupancy factors Carrier distributions
N-type
Intrinsic
P-type
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 15/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Since gc(E)dE represents the number of
conduction band states/cm3 lying in the E to E+dEenergy range
f(E) specifies the probability that an availablestate at an every E will be occupied by an
electron then, gc(E)f(E)dE gives the number of conduction
band electrons/cm3 lying in the E to E+dE energyrange
So, integration of gc(E)f(E)dE over all conduction
band energies must yield the total number ofelectrons in the conduction band.
Similarly, the total number of holes in the valenceband can be calculated as
dE E f E gnTOP
C
E
E
c )()(
dE E f E g p
V
BOTTOM
E
E
v )](1[)(
Formulas for n and p
Carrier Concentration: Calculation
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 16/25
where
NC and NV are the effective density of conduction and valence band states
Some discussions
Detailed calculation is very complex, simplified calculation equations areoften used
Only valid for non-degenerate semiconductor (Ev+3kT<EF<Ec-3kT)
Carrier Concentration: Simplified Calculation
kT E E
V
kT E E
C
F V
C F
e N p
e N n/)(
/)(
Simplified calculation for n and p
2/3
2
*
]2
[2
kT m N n
C 2/3
2
*
]2
[2
kT m N
p
V
Degeneratesemiconductor
Degeneratesemiconductor
3kT
3kT
EC
EV
Non-degeneratesemiconductor
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 17/25
Alternative expression for n and p
Considering intrinsic semiconductor, where n=p=ni and EF=Ei. we have
Then, Nc and NV can be calculated as
Then, n and p can be solved and expressed as a function of ni and Ei, so
Carrier Concentration: Alternative expression for n and p
kT E E C i
C ie N n /)( kT E E
V iiV e N n /)(
kT E E iC
iC en N /)( kT E E
iV V ien N /)(
kT E E i
iF enn /)(
kT E E
iF ien p
/)(
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 18/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Carrier Concentration: Mass action law
kT E
V C
kT E E
V C iGV C e N N e N N n
//)(2
kT E
V C iGe N N n
2/
ni
2
innp
np product (mass action law)
kT E E
C iC ie N n
/)(
kT E E
V iiV e N n
/)(
kT E E
iiF enn
/)(
kT E E
iF ien p
/)(
Some discussions
Very important equations
Equations for n (p) and np product are valid fornon-degenerate semiconductor in equilibrium
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 19/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Charge neutrality relationship
For an uniformly doped semiconductor to be everywhere charge-neutral clearlyrequires
where
At room, it is very reasonable to assume all dopants ionized, so we have
Finally, one then obtains
03
A D qN qN qnqpcm
charge
0
A D N N n p
D D N N A A N N
0 A D
N N n p
D N = number of ionized (positively charged) donors/cm3
A N =number of ionized (negatively charged) acceptors/cm3
Carrier Concentration: Charge Neutrality
OR
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 20/25
Carrier Concentration: Two general equations
Carrier concentration calculation
2
innp 0 A D N N n p
2/122 ])2
[(2
i A D A D n N N N N n
2/1222
])2
[(2 i
D A D Ai n N N N N
n
n p
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 21/25
Carrier Concentration: Some special cases
Some special cases
Intrinsic semiconductor (N A=0, ND=0)
n = p = ni
Doped semiconductor where either ND - N A ~ ND>> ni or N A - ND ~N A >> ni
n=ND, p=ni2/ ND
p=N A, n=ni
2
/ N A
Doped semiconductor where ni >> |ND - N A|
n = p =ni
Compensated semiconductor (ND-N
A= 0)
like intrinsic material, n = p = ni
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 22/25
EE5502/Chunxiang Zhu/NUS/Sem I, 2010/11
Case I: Intrinsic semiconductor
In an intrinsic material
Carrier Concentration: Determination of Fermi level
pn kT Ei E V kT E EiC
V C e N e N /)(/)(
)ln(22 C
V V C i N
N kT E E E
2/3*
*
)(n
p
C
V
m
m
N
N
)ln(4
3
2 *
*
n
pV C i
m
mkT
E E E
EF=Ei in an intrinsic material
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 23/25
Doped semiconductors (non-degenerate, dopants totally ionized) N-type semiconductor
P-type semiconductor
Carrier Concentration: Determination of Fermi level
)/ln( iiF nnkT E E
)/ln( i DiF n N kT E E
)/ln( i AF i n N kT E E
n=N DkT E E
iiF enn
/)(
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 24/25
Carrier Concentration: Temperature Dependence
n=0 n=ND+ n=ND n=ni
0 K Low T Moderate T High T
8/11/2019 Ch1 1 Carrier_modeling
http://slidepdf.com/reader/full/ch1-1-carriermodeling 25/25
Summary of “ Carrier Modeling”
Semiconductor Models Bonding Model
Energy Band Model
Carrier Distribution Density of States
Fermi-Dirac Functions
Carrier Concentration Some important equations Mass action law
top related