Semiconductor Device Physics
1
Outline
1. Introduction
2. Energy level and energy band
3. Direct and indirect semiconductor
4. Intrinsic and extrinsic semiconductor
5. Carrier transmission
2
Introduction
Solid-state materials are divided into three types via resistivity
or electrical conductivity :
-Insulators => ρ > 108 Ω-cm
-Semiconductor => 108 Ω-cm > ρ > 10-4 Ω-cm
-Conductor => ρ < 10-4 Ω-cm
Where ρ(Ω-cm ; resistivity) ≡ σ -1 (σ (S/cm ; conductivity))
3
1
2
He
H
氫
氦
3 Li
4 Be
5
鋰
鈹
硼
碳
氮
氧
氟
氖
11 Na
12 Mg
13 Al
14 Si
15 P
16 S
17 Cl
18 Ar
鈉
鎂
鋁
矽
磷
硫
氯
氬
19 K
20 Ca
21 Sc
22 Ti
23 V
24 Cr
25 Mn
26 Fe
27 Co
28 Ni
29 Cu
30 Zn
31 Ga
32 Ge
33 As
34 Se
35 Br
36 Kr
鉀
鈣
鈧
鈦
釩
鉻
錳
鐵
鈷
鎳
銅
鋅
鎵
鍺
砷
硒
溴
氪
37 Rb
38 Sr
39 Y
40 Zr
41 Nb
42 Mo
43 Tc
44 Ru
45 Rh
46 Pd
47 Ag
48 Cd
49 In
50 Sn
51 Sb
52 Te
53 I
54 Xe
銣
鍶
釔
鋯
鈮
鉬
鎝
釕
銠
鈀
銀
鎘
銦
錫
銻
碲
碘
氙
55 Cs
56 Ba
57 La
72 Hf
73 Ta
74 W
75 Re
76 Os
77 Ir
78 Pt
79 Au
80 Hg
81 Tl
82 Pb
83 Bi
84 Po
86 Rn
銫
鋇
鑭
鉿
鉭
鎢
錸
鋨
銥
鉑
金
汞
鉈
鉛
鉍
釙
氡
88 Ra
89 Ac
105Db
106Sg
107Bh
108Hs
109Mt
110
111
112
鐳
錒
Uun
Uuu
Uub
65
Tb
66
Dy
67
Ho
68 Er
69 Tm
71 Lu
鋱
鏑
鈥
鉺
銩
70
Y
b
62
Sm
鑭系
元素
58 Ce
59 Pr
60 Nd
鈰
鐠
釹
錒系
元素
90 Th
91 Pa
92 U
93 Np
釷
鏷
鈾
錼
B
6
C
7
N
8
O
63 Eu
64 Gd
銪
釓
94 Pu
95 Am
96 Cm
98 Cf
99 Es
100Fm
101Md
103Lr
鈽
鋂
鋦
鉲
鑀
鐨
鍆
鐒
釤
4
鎰
鎦
9
F
10 Ne
Semiconductor Materials
Semiconductor materials are divided into two types via the
construction of element :
-Element semiconductor => Si ,Ge…. composed by column IV
in periodic table.
*Si is the well-behaved element semiconductor for its stability
at room temperature and prone to oxidize
-Compound semiconductor (二元,三元,四元..)=> GaN, GaAs,
ZnO, ZnSe….compose by Column III-V or II-VI in periodic
table
*GaAs is widely employed as high speed and microwave
devices
*GaN related materials is widely employed as blue LED to
achieve white LED.
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6
7
8
9
Energy level and energy band
Energy Band
Energy level for an independent atom
•As atoms separated by long distance => each energy level obeyed
“Quantum Mechanics” => ground state, first excited state…..
ex. : for the “hydrogen atomic model”
EH = -13.6 / n2 eV ; n is principal quantum number
•As atoms had getting approaching => the identical energy level of
each atom would interact and spilt
into “N level”.
•As N is increased, the “energy level”
would degrade to the so called “energy
band”.
•Therefore, the discrete energy levels
in atoms degenerated into continuous
energy bands in solid state.
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The “energy level” and “energy band” for silicon
•The electronic configuration of 14Si : 1s22s22p63s23p2 => 4 valence
electrons
Forming the covalent bonds
Conduction band
Bandgap
Degenerate energy
Valence band
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12
Direct and indirect semiconductor
K-space diagram (能量-動量圖)
•The energy and momentum
for a free carrier :
v
r
r
r
r
1
P
∵
and
E = m0 V
P = m0V ⇒ V =
2
m0
∴
E
=
v
P
2
2m0
where, m0 is the carrier mass
P vs. E is a function of
“Parabolic” curve
13
2
v
 P
1
⇒ E = m0 
m
2
 0




2
K-space diagram (能量-動量圖)
•The energy and momentum for the solid state materials :
Since the atoms interacted, the free carrier mass replaced by effective
v 2
mass
P
E =
∴
Differential
∴
mn
2 mn
d  dE 
d 2E
1
=
=


=> dP  dP  dP 2 mn
 d 2E
=
 dP 2





−1
斜率變化的倒數有效質量
斜率變化曲率
曲率↑ R↓ mn↓
14
m0 ~
9.1E-31kg
Si
GaAs
mn/m0
0.26
0.063
mp/m0
0.69
0.57
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Direct and indirect semiconductor
•Direct semiconductor and indirect semiconductor definition :
Direct => the lowest of conduction band and highest valence
band was aligned in the K-space
Indirect => the lowest of conduction band and highest valence
band was not aligned in the K-space
•The influence on optical properties :
Direct => almost radiative recombination
Indirect => almost nonradiative recombination (need some
recombination centers to transfer the momentum
difference)
16
Direct and indirect semiconductor
 (1) effective mass => high speed devices
(2) direct and indirect => optical properties
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Intrinsic and doped semiconductor
•Definitions : these carriers (electron or hole) in semiconductor that
excited mostly by thermal treatment => the so called “intrinsic
semiconductor”.
•The number of occupied conduction band levels by electron (electron
concentration) in unit volume is :
n=
∫
E top
Ec
n( E )dE =
∫
E top
Ec
不同能量所允許的空位(N(E))
N ( E ) F ( E )dE
× 佔據機率(F(E))
where, n is in unit of cm-3;
N(E) is in unit of (cm3-eV)-1;
Ec and Etop is the min. and max. energy of the filled band level
(electron)
N(E) is the density of state function for unit volume;
F(E) is the occupation probability accorded to Fermi-Dirac
distribution function (also called Fermi distribution function)
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•Fermi distribution function is presented as :
F (E) =
1
1 + e ( E − E F ) / kT
where, k is Boltzmann constant;
T is Kelvin temperature;
EF is Fermi level (Fermi energy)
•Fermi level is defined as the
energy of the “electron”
occupation probability equal
to 1/2 (F(E) = 1/2).
•If E-EF > 3kT, the Fermi distribution function is simplified as :
F ( E ) ≈ e − ( E − E F ) / kT(for electron) and F ( E ) ≈ 1 − e − ( E − E F ) / kT (for hole)
T↑=> F(E)↑
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•For the density of state N(E) => N(E) ∞E1/2=>Parabolic function
Therefore, the electron concentration (above conduction band
and Ec ≡E0) is described as :
E top
− ( E − E F ) / kT
and
F
E
e
(
)
≈
n=
N ( E ) F ( E )dE
E0
•So, the electron carrier concentration and hole concentration can
be shown as :
導電帶電子濃度 : n = N c exp[− ( Ec − E F ) / kT ]
∫
價電帶電洞濃度 :
p = N v exp[( Ev − EF ) / kT ]
Nc and Nv are effective density of state for electron and hole, respectively.
T↑, n↑; and EF↑, n↑, for n type
20
Basic concept for intrinsic semiconductor
(a) At thermal equilibrium, electron concentration (n) in
conduction band is equal to hole (p) in valence band, therefore :
n = p = ni , ni is intrinsic carrier concentration
∴ n × p = ni2 = NcNvexp(-Eg/kT)
(1) this is so called “mass-action law
(質量作用定律)”
(2) for T↑ or Eg↓ => n & p↑
(3) ∵Eg(Si)<Eg(GaAs)<Eg(GaN)
∴ni(Si)>ni(GaAs)>ni(GaN)
(b) The Fermi level for intrinsic
semiconductor EF ≈ Eg/2 (at 0oK)  EF ≡ Ei = (Ec+Ev)/2 +
(kT/2)ln(Nv/Nc) ≈ Eg/2 (at 0oK)
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22
Intrinsic and extrinsic semiconductor
•Extrinsic semiconductor : when impurities are doped in semiconductor,
this semiconductor has become extrinsic semiconductor (外質半導體)
•Donor : impurities that donate electrons per atom => the
semiconductor presents n-type semiconductor
Available donor impurities for Si => the element of column V or VI (ie.
N, P , As, O, S….)
•Acceptor : impurities that accept electrons and generate holes per atom
=> the semiconductor presents p-type semiconductor
Available acceptor impurities for Si => the element of column III or
II (ie. B, Al, Ga, In, Mg….)
•Charge state of impurities :
Donors : Neutral or Positive (while activation)
Acceptor : Neutral or Negative (while activation)
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25
•The carrier concentration of extrinsic semiconductor:
-assume complete ionization (ie. electron = donor; hole = acceptor)
∴ n = Ncexp[-(Ec-EF)/kT] = ND => Ec-EF = kTln(Nc/ND) => ND↑, EF ↑
p = Nvexp[(Ev-EF)/kT] = NA => EF-Ev = kTln(Nv/NA) => NA↑, EF ↓
•At thermal equilibrium, the mass-action law still exists :
∵ n = Ncexp[-(Ec-EF)/kT] = Ncexp[-(Ec-Ei)/kT]exp[-(Ei-EF)/kT]
= niexp[-(Ei-EF)/kT)]
p = Nvexp[(Ev-EF)/kT] = Nvexp[(Ev-Ei)/kT]exp[(Ei-EF)/kT]
= niexp[(Ei-EF)/kT)]
∴n × p = ni2
 mass-action law is always valid at thermal equilibrium
26
Carrier transmission
•The overall carrier transmissions include : drift (漂移),
diffusion (擴散), recombination (復合), generation (產生),
thermionic emission (熱游子散射), tunneling (穿隧), and
impact ionization (衝擊離子化).
•The dominate carrier transmissions are “drift” and
“diffusion”.
•The basic equations that dominate the associated
semiconductor operation are “current density equation”
and “continuity equation”.
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Carrier drift
-As an external electric field apply => electron carrier is accelerated
with an addition velocity => the so called “drift velocity”
for F = ma ⇒ −qε = m a = m dvn
n n
n
dt
integral each side => − qετ c = mn vn
=> vn = −(
qτ c
)ε = − µ nε
mn
qτ c
µ
≡
where, n m is called electron carrier mobility
n
similarity, µ ≡ qτ c is hole carrier mobility
p
mp
μn ∞ τc and 1/mn
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Characteristics for carrier mobility
(A) Mobility (μ) is proportional to the mean free time (τc) (載子活期)
τc is influenced by each collision appears in semiconductor that
mainly results from the following scattering mechanisms :
(a) Lattice scattering τc, L(晶格散射) : original from the lattice
vibration induces from heat treatment => T↑, then τc, L↓ .
(b) Impurity scattering τc, I(雜質散射) : original from Coulomb
force induces from doping concentration => NT↑, then τc, I↓
(c) Since the influence of impurity is neglect-able at high
temperatures, the Lattice scattering is dominate at elevate
temperatures.
(d) μn(τc, L)∞T-3/2 and μn(τc, I)∞T3/2/NT
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Characteristics for carrier mobility
(e) The overall carrier mean free time τc :
1
1
1
1
1
1
=
+
=
+
τ c τ c , L τ c , I or µ µ, L µ I
(B) Mobility (μ) is inverse proportional to the effective mass (m)
Typically, electron mobility (μn) is large than hole mobility (μp)
resulted from the difference of effective mass (mp > mn)
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Drift current
• electric current density:
Jn
I
dQ / dt
=
=
=
A
A
n
∑ (−qv ) = −qnv
i
i =1
n
= qnµ n ε
similarity, J p = qpµ p ε
therefore, J = J p + J n = ( pµ p + nµ n )qε ≡ σε ≡ ε
ρ
where, Conductivity (傳導係數)σ ≡ (pμp + nμn) q
Resistivity (電阻係數) ρ ≡ 1/σ = 1 / (pμp + nμn) q
for an n-type => ρ ≈ 1 / nμnq
ρ ≈ 1 / pμpq
∴“σ” is proportional to
“mobility” and “carrier
concentration (more dominate)”
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Carrier diffusion
•Definition : electron carriers at high concentration region tend to
migrate to low concentration region => diffusion current (擴散電流)
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Carrier diffusion
•Derivation for electron diffusion current
-F1: mean velocity of electron in unit area electron flow from left to x=0
F2: same as F1 except right to x=0
=>
1
[ n(−l )] • l
1
F1 = [ n(−l )] • vth = 2
2
τ
1 c
[ n(l )] • l
1
F2 = [ n(l )] • vth = 2
τc
2
The probability of
electron move
where, l is mean free path
τc is mean free time
n(l) is carrier concentration at l
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-net electron current velocity in unit area at x=0
1
F = F1 − F2 = [ n(−l ) − n(l )] ⋅ vth , expand by “Taylor expanding” at x=0
2
dn  
1 
dn  
1 
dn 
dn
vth n(0) − l  − n(0) + l   + sec . odrer ≈ vth  − 2l
 = −vth .l
dx  
2 
dx  
2 
dx 
dx
=>
F=
=>
F ≡ − Dn
=> J n
dn
dx
, where Dn (diffusion coefficient, 擴散係數) ≡ vth × l
Why positive => electron current direction
dn
dn
= − qF = − q (− Dn
) = qDn
dx
dx
=> Dn ∞ vth and l
Concentration gradient
(濃度梯度)
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Einstein Relation
•Purpose : to describe the relation between “mobility (original from
“electric field”)” and “diffusion coefficient (original from
“concentration gradient”)”
qτ c
l
Dn = vth l
and µ n =
where, vth =
τc
mn
for
⇒ Dn = vth ⋅ vth ⋅ τ c =
vth2
⋅
µ n mn
The diffusion and drift is now limited in 1-D
q
1
1
2
mn vth = kT ⇒ mn vth2 = kT
for 1-dimension electron carrier =>
2
2
 kT 
=
µ n
 q 


∴ Dn
=> the so called “Einstein Relation”
• The Einstein Relation is depend on “temperature”
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Current Density Equation
•Purpose : to describe the current density that both the “electric field ”
and “concentration gradient” exit (ie “Drift current density” +
“diffusion current density”:
Why?
For electron carrier density :
For hole carrier density :
dn 

J n = q µ n nε + Dn

dx 

dp 

J p = q µ p pε + (−) D p

dx 

=> total current density : J = Jn + Jp (called “current density equation”)
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