013
Chp2. Carrier Modeling
014
2. Carrier Modeling
carriers : electric charge transporters
electrons and holes
2.1 Quantization Concept
“ Particle – in – a – box ” Model (electron)
The energy of the electron is not
continuous but quantized.
015
. .
.
. .
+
proton
n=1
n=2
EN = - 13.6/n2 eV
n=
n = 1, 2, 3, …
eV = 1.6 * 10-19 J
Si은 14개의 전자를 갖고 있으므로 좀 더 복잡하다.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
016
2.2 Semiconductor Models
Bonding Model
/ Band Model
2.2.1 Bonding Model
4 nearest neighbors
Line represents a shared valence electron
Circle represents the core of a semiconductor (e.g. Si) atom
Figure 2.3 The bonding model
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
017
2.2.2 Energy Band Model
Energy – related aspects of events
( Bonding Model은 실제 공갂에서 일어나는 일을 예상)
Figure 2.5 Conceptual development of the energy band model starting with N isolated Si atoms on the top left
and concluding with a “dressed-up” version of the energy band model on the top right.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
018
2.2.3 Carriers (운반자)
(a) No Carriers
전류의 흐름을 맡는 졲재
(b) The electron
(c) The hole
Figure 2.7 Visualization of carriers using the bonding model (left) and the energy band model (right).
(a) No-carrier situation; (b) visualization of an electron; (c) visualization of a hole.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
019
2.2.4 Band Gap and Material Classification
Eg의 크기에 의해서 순수한 상태의 전기적
특성이 결정된다.
부도체
SiO2
Diamond
금 속
very small band gap or no band gap
반도체
Eg = 1.12 eV (Si)
0.67 eV (Ge)
1.42 eV (GaAs)
Eg
Eg
8 eV
5 eV
020
Figure 2.8 Explanation of the distinction between (a) insulators,
(b) semiconductors, and (c) metals using the energy band model
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
021
2.3 Carrier Properties
2.3.1 Charge
electrons and holes have same magnitude of
q = 1.6 * 10-19 Coul.
the carrier charge
2.3.2 Effective Mass
m*
different for electrons and holes
different in different materials
022
Figure 2.9 An electron moving in response to an applied electric field
(a) within a vacuum, and (b) within a semiconductor crystal.
within a vacuum



dv
F  q   m0
dt
within a semiconductor



* dv
F  q  me
dt
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
023
2.3.3 Carrier Numbers in Intrinsic Materials
“ intrinsic semiconductor ” = pure semiconductor
물질 고유의 성질 (ex. 전기전도도)을 보유하고 있다.
Let’s define.
n = number of electrons, cm-3
p = number of holes, cm-3
For intrinsic semiconductors at equilibrium
n = p = ni
intrinsic semiconductor의 정의
024
For Si
ni
1010cm-3, For Ge ni
1013cm-3 (암기할 것)
2.3.4 Manipulation of Carrier Numbers – Doping
Figure 2.10 Visualization of (a) donor and (b) acceptor action using the bonding
model. In (a) the Column Ⅴ element P is substituted for a Si atom; in (b) the
Column Ⅲ element B is substituted for a Si atom.
Doping : 특정한 불순물을 첨가하여 반도체의 전기전도도
를 제어하는 조작. 이 때 사용되는 불순물을 dopant
(impurity)라고 부른다.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
025
Figure 2.11 Pseudo-hydrogen atom model for the donor-site bond
Dopant (impurity)는 일반적으로 matrix atom (e.g. Si)
보다 가전자 수가 하나 많거나 (e.g. Ⅴ족 원소) 하나 작은
원소 (e.g. Ⅲ족 원소)를 선택한다.
Si내에 Ⅴ족 원소를 첨가하였을 때의 그림이 Fig2.11에
나타나있다. 이는 수소원자와 비슷한 형상이므로, 최외각
전자가 취할 수 있는 에너지도 수소원자 모델로부터 구할
수 있다.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
026
me* 1
EB 
13.6eV  0.1eV
2
m0 K S
(보통 반도체의 ε r 는 거의 10)
EB의 의미?
Donor and Acceptor ?
대체로 donor level은 CB 가까이
acceptor level은 VB 가까이 위치한다.
정의 : negatively charged when ionized - acceptor
positively charged when ionized - donor
027
실제 energy level은 어떻게 측정하는가 ? 온도를 올리면서 conductivity 측정
Figure 2.13 Visualization of (a) donor and (b) acceptor action using the energy band model.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
028
2.3.5 Carrier Related Terminology
•
•
•
•
•
•
•
•
•
Dopants ( Dopant impurities )
Intrinsic semiconductor
Extrinsic semiconductor
Donor
Acceptor
n-type materials
p-type materials
Majority carriers
Minority carriers
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2.4 States and Carrier Distributions
2.4.1 Density of States
distribution of energy states

density of states
QM calculations
g c E  
g c E  
me* 2me* E  EC 
 
2 3
m*p 2m*p EV  E 
 
2 3
E  EC
E  EV
gc(E)dE : number of conduction band states/cm3 between E and E+dE (E ≧ Ec)
gv(E)dE : number of valence band states/cm3 between E and E+dE (E ≦ Ev)
030
Figure 2.14 General energy dependence of gc(E) and gv(E) near the band edges.
gc(E) and gv(E) are the density of states in the conduction and valence bands, respectively.
2.4.2 Fermi Function ( Fermi-Dirac Distribution Function)
전자의 state가 전자로 채워져 있을 확률 ( at thermal equilibrium )
EF : Fermi level
f E  
1
1  e  E  EF / kT
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
정의 : ① f(EF) = 0.5 ② Electrochemical potential energy of electrons
031
Figure 6.1 (a) The Fermi-Dirac distribution function for a Fermi energy of 2.5 eV and for temperatures of
0K, 600K, and 6000K. (b) The classical Maxwell-Boltzmann distribution function of energies for the same temperatures.
When T > 0K
⑴ E = EF f(EF) = 1/2
⑵E
(EF + 3kT)
states mostly empty
Boltzmann Distribution
 E  EF / kT 
f E   e
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
032
⑶E
(EF – 3kT)
states mostly filled
exp [(E- EF)/kT] << 1
f(E)
1 – f(E)
1 – exp [(E- EF)/kT]
exp [(E- EF)/kT]
exp [-(EF-E)/kT]
⑷ At R.T.
T=300ok
kT = 0.026 eV
3kT = 0.078 eV
Fermi level

<<
Eg
equilibrium에서 적용
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2.4.3 Equilibrium Distribution of Carriers
g(E)*f(E)
Available states * Occupation Probability
f(E)*gC(E):electrons in CB, [1-f(E)]*gV(E):holes in VB
Figure 7.17 (a) The two highest bands at T=0K. (b) The two highest bands at T>>0K. There are electrons
at the bottom of the conduction band, and holes at the top of the valence band.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
034
교과서 p47
Fig. 2.16 참조
① electron과 hole이 band edge에 가까이 분포되어 있다.
② EF의 위치에 따라 CB내의 전자 분포와 VB내의 hole
분포가 결정된다. majority carriers, minority carriers
③Thermal equil. 상태에서는 n이 증가하면 p는 감소한다.
Figure 2.18 “At a glance” representation of intrinsic (left), n-type (middle), and p-type (right)
Semiconductor materials using energy band diagram.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company
035
Figure 4.8 Density of states function, Fermi-Dirac
Probability function, and areas representing electron
and hole concentrations for the case when EF is
above the midgap energy.
Figure 4.9 Density of states function, Fermi-Dirac
Probability function, and areas representing electron
and hole concentrations for the case when EF is
below the midgap energy.
Figure reference: “Semiconductor Device Fundamentals”
Robert F. Pierret, Addison-Wesley Publiching Company