Carrier Transport Chapter 6

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Transcript Carrier Transport Chapter 6 

Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Chapter 6
Carrier Transport
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
DRIFT
Definition-Visualization
Drift is charge-particle motion in response to an applied electric field.
The relaxation time (πœπ‘š ) can be interpreted as mean free time (𝑑)
between collisions, if a particle reaches equilibrium by the collision once.
The probability that a particle experience collision during time dt: =
If there are n(t) particles,
Collision becomes less with time.
𝑑𝑛(𝑑) = βˆ’π‘›(𝑑)
Number of particles experience
a collision during dt.
Average time, 𝑑 =
∞
0 𝑑𝑛(𝑑)𝑑𝑑
∞
0 𝑛(𝑑)𝑑𝑑
Mean free time
𝑑𝑑
πœπ‘š
𝑑𝑑
πœπ‘š
= πœπ‘š
𝑑𝑛(𝑑)
𝑛(𝑑)
=βˆ’
𝑑𝑑
πœπ‘š
𝑙 = 𝑑 βˆ™ π‘£π‘‘β„Ž
Mean free path
𝑛(𝑑) = 𝑛 0 𝑒 βˆ’π‘‘/πœπ‘š
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
In steady-state, carriers drifts at a constant drift velocity by balancing between acceleration
by electric field and deceleration by collision.
𝑑𝑝π‘₯
𝑑𝑝π‘₯
|πœ€βˆ’π‘“π‘–π‘’π‘™π‘‘ +
|
=0
𝑑𝑑
𝑑𝑑 π‘π‘œπ‘™π‘™π‘–π‘ π‘–π‘œπ‘›
𝑑𝑝π‘₯
acceleration,
|
= βˆ’π‘›π‘žπœ€π‘₯
𝑑𝑑 πœ€βˆ’π‘“π‘–π‘’π‘™π‘‘
deceleration,
In 3-D,
< 𝑝π‘₯ >
π‘žπ‘‘
=
βˆ’
πœ€
π‘šπ‘›βˆ—
π‘šπ‘›βˆ— π‘₯
𝑣𝑑 = βˆ’
𝑣𝑑 =
βˆ—
π‘šπ‘›,𝑐
𝑑𝑝π‘₯ = βˆ’π‘π‘₯
π‘žπ‘‘
βˆ— πœ€ = πœ‡π‘› πœ€
π‘šπ‘›,𝑐
π‘žπ‘‘
βˆ— πœ€ = πœ‡π‘ πœ€
π‘šπ‘,𝑐
1
1
1
=3
+
+
π‘šπ‘™βˆ— π‘šπ‘‘βˆ— π‘šπ‘‘βˆ—
𝑑𝑑
𝑑
Momentum change due to
collision during dt
𝑑𝑝π‘₯
𝑝π‘₯
|π‘π‘œπ‘™π‘™π‘–π‘ π‘–π‘œπ‘› = βˆ’
𝑑𝑑
𝑑
Average momentum per electron,
< 𝑣π‘₯ >=
Total momentum at t
< 𝑝π‘₯ >=
𝑝π‘₯
𝑑𝑝π‘₯
=βˆ’
𝑑 = βˆ’π‘ž π‘‘πœ€π‘₯
𝑛
𝑛𝑑𝑑
1-D expression
for electron
βˆ—
βˆ—
, π‘šπ‘,𝑐
Where π‘šπ‘›,𝑐
are conductivity
effective masses.
for hole
βˆ’1
βˆ—
π‘šπ‘,𝑐
=
βˆ—
π‘šπ‘™β„Ž
3/2
βˆ—
+ π‘šβ„Žβ„Ž
3/2
βˆ— 1/2
βˆ— 1/2
π‘šπ‘™β„Ž
+ π‘šβ„Žβ„Ž
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Two effective masses of carrier
1) Density of state effective mass, π‘šπ‘‘βˆ— , in the density of state function
2) Conductivity (or mobility) effective mass, π‘šπ‘βˆ— , in the expression for mobility
The density of states effective mass for electrons and holes is given by,
βˆ—
π‘šπ‘›βˆ— = π‘šπ‘‘π‘’
= π‘šπ‘₯π‘₯ π‘šπ‘₯π‘₯ π‘šπ‘₯π‘₯
1/3
= π‘šπ‘₯π‘₯ for the 𝛀 -valley (= π‘šπ‘₯π‘₯ = π‘šπ‘¦π‘¦ = π‘šπ‘§π‘§ )
= π‘šπ‘™ π‘šπ‘‘2
π‘šπ‘βˆ—
=
βˆ— 3/2
π‘šβ„Žβ„Ž
+
βˆ— 3/2
π‘šπ‘™β„Ž
2/3
1/3
for the X or L -valley
βˆ—
π‘šπ‘‘β„Ž
3/2
βˆ—
= π‘šβ„Žβ„Ž
3/2
βˆ—
+ π‘šπ‘™β„Ž
3/2
The conductivity (or mobility) effective mass for electrons and holes is given by,
1
1 1
1
1
1
=
+
+
=
βˆ—
π‘šπ‘›,𝑐
3 π‘šπ‘₯π‘₯ π‘šπ‘¦π‘¦ π‘šπ‘§π‘§
π‘šπ‘₯π‘₯
=
1/2
1/2
βˆ—
βˆ—
1
π‘šπ‘™β„Ž
+ π‘šβ„Žβ„Ž
βˆ— =
βˆ— 3/2
βˆ— 3/2
π‘šπ‘,𝑐
π‘šπ‘™β„Ž
+ π‘šβ„Žβ„Ž
for the 𝛀 -valley (= π‘šπ‘₯π‘₯ = π‘šπ‘¦π‘¦ = π‘šπ‘§π‘§ )
1 1
2
+
3 π‘šπ‘™ π‘šπ‘‘
for the X or L -valley
βˆ—
𝐽𝑝 = π‘π‘žπœ‡π‘π‘œπ‘› πœ€ = π‘β„Žβ„Ž π‘žπœ‡β„Žβ„Ž + π‘π‘™β„Ž π‘žπœ‡π‘™β„Ž πœ€ ∝ π‘šβ„Žβ„Ž
βˆ—
= π‘šβ„Žβ„Ž
1/2
βˆ—
+ π‘šπ‘™β„Ž
1/2
βˆ—
= π‘šπ‘‘β„Ž
3/2
βˆ™
1
βˆ— =
π‘šπ‘,𝑐
3
2
βˆ™
1
βˆ—
βˆ— + π‘šπ‘™β„Ž
π‘šβ„Žβ„Ž
βˆ—
π‘šβ„Žβ„Ž
3/2
βˆ—
+ π‘šπ‘™β„Ž
1
βˆ—
π‘šπ‘™β„Ž
1
3/2
βˆ—
π‘šπ‘,𝑐
3/2
βˆ™
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Drift Current
The formal definition of current,
𝐼𝑃|π‘‘π‘Ÿπ‘–π‘“π‘‘ = π‘žπ‘π‘£π‘‘ 𝐴
In vector notation,
𝐽𝑃|π‘‘π‘Ÿπ‘–π‘“π‘‘ = π‘žπ‘π‘£π‘‘
Excluding situations involving large β„° fields,
𝑣𝑑 = πœ‡π‘ β„°
where πœ‡π‘ ,the hole mobility, is the constant of proportionality
constant
𝐽𝑃|π‘‘π‘Ÿπ‘–π‘“π‘‘ = π‘žπœ‡π‘ 𝑝ℰ
similarly,
𝐽𝑁|π‘‘π‘Ÿπ‘–π‘“π‘‘ = π‘žπœ‡π‘› 𝑛ℰ
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Mobility
The carrier mobility varies inversely with the amount of scattering taking
place within the semiconductor.
To theoretically characterize mobility it is therefore necessary to consider the different
types of scattering events that can take place inside a semiconductor.
(i)
(ii)
(iii)
(iv)
(v)
Phonon (lattice) scattering
Ionized impurity scattering
Scattering by neutral impurity atoms and defects
Carrier-carrier scattering
Piezoelectric scattering
For the typically dominant phonon and ionized impurity scattering, single-component
theories yield, respectively, to first order
πœ‡πΏ ∝ 𝑇 βˆ’3/2
πœ‡πΌ ∝ 𝑇 βˆ’3/2 /𝑁𝐼
where 𝑁𝐼 = 𝑁𝐷+ + π‘π΄βˆ’
Matthiessen’s Rule
Noting that each scattering mechanism gives rise to a β€œresistance-to-motion” which is
inversely proportional to the component mobility, and taking the β€œresistance” to be simply
additive (analogous to a series combination of resistors in an electrical circuit), one obtains
1
1
1
=
+
+β‹―
πœ‡π‘› πœ‡πΏπ‘› πœ‡πΌπ‘›
1
1
1
=
+
+β‹―
πœ‡π‘ πœ‡πΏπ‘ πœ‡πΌπ‘
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Doping/Temperature Dependence
The Si carrier mobility versus doping and temperature plots presented respectively in Figs
6.5 and 6.6 were constructed employing the empirical-fit relationship
πœ‡ = πœ‡π‘šπ‘–π‘› +
πœ‡0
1 + (𝑁/π‘π‘Ÿπ‘’π‘“ )𝛼
where πœ‡: carrier mobility
N: doping density(either NA or ND)
All other quantities are fit parameters that exhibit a temperature
dependence of the form
𝑇 Ξ·
𝛼 = 𝐴0 (
)
300
where 𝐴0 : temperature-independent constant
T: temperature in Kelvin
η ∢ temperature exponent for the given fit parameter
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
πœ‡πΏ ∝ 𝑇 βˆ’3/2 from first order theory
Experimental values for lightly doped Si,
πœ‡ β‰ˆ πœ‡πΏ ∝ 𝑇 βˆ’2.3±0.1
∝ 𝑇 βˆ’2.2±0.1
for electron
for hole
Advanced Semiconductor Fundamentals
For GaAs,
Chapter 6. Carrier Transport
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
High-Field/Narrow-Dimensional Effects
Under low electric field
i)
ii)
Carrier gains energy from the electric field and loses the energy through collisions
with low energy acoustic phonons or impurities.
3
The averages energy of the electrons β‰ˆ π‘˜π‘‡at thermal equilibrium (𝑻𝒆 β‰ˆ π‘»π’π’‚π’•π’•π’Šπ’„π’† )
2
iii) Drift velocity 𝑣𝑑 ∝ πœ€ and current density 𝐽 ∝ πœ€.
Velocity Saturation under high electric field
i)
ii)
Electrons gain energy from the field faster than they can lose it to the lattice.
The electron distribution can be characterized by effective temperature, 𝑇𝑒 .
(𝑻𝒆 > π‘»π’π’‚π’•π’•π’Šπ’„π’† : hot electron effect)
iii) Drift velocity 𝑣𝑑 and current density 𝐽 are no longer linear with πœ€. (nonohmic)
iv) Electrons can transfer energy to the lattice by the generation of high energy optical
phonons. This causes saturated drift velocity (𝒗𝒅𝒔𝒂𝒕 ).
In Si at 300 K, π‘£π‘‘π‘ π‘Žπ‘‘ β‰ˆ 107 cm/sec for both electrons
and holes at πœ€ β‰ˆ 107 V/cm.
Temperature dependence of π‘£π‘‘π‘ π‘Žπ‘‘ for electrons in Si can
be modeled by the empirical-fit expression.
0
π‘£π‘‘π‘ π‘Žπ‘‘
π‘£π‘‘π‘ π‘Žπ‘‘ =
1 + 𝐴𝑒 𝑇/𝑇𝑑
0
π‘£π‘‘π‘ π‘Žπ‘‘
= 2.4 × 107 π‘π‘š/𝑠𝑒𝑐
A = 0.8
Td = 600 K
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Intervalley Carrier Transfer
For GaAs
ellipsoidal constant
energy surface
βˆ†Ξ“πΏ = 0.29
spherical constant
energy surface
eV
𝑣𝑑
βˆ—
Electrons in Ξ“; π‘šπ‘›Ξ“,𝑐
= π‘šπ‘›βˆ— = 0.063 π‘š0
Electrons in L; π‘šπ‘™βˆ— = 1.9 π‘š0 , π‘šπ‘‘βˆ— = 0.075 π‘š0
βˆ—
π‘šπ‘›πΏ,𝑐
= 0.55 π‘š0
πœ€
3
πœ€π‘ = 3.3 × 10 𝑉/π‘π‘š
βˆ—
βˆ—
π‘šπ‘›πΏ,𝑐
β‰ˆ 10π‘šπ‘›Ξ“,𝑐
Under normal circumstances, the Ξ“ βˆ’valley is the only one occupied, but for an applied field of ~ 3.5 KV
electrons begin to be transferred to the L-valley. The resulting negative differential conductance occurs
when the carriers are transferred from low mass, high velocity states to high mass, low velocity states is
referred to as the β€œGunn Effect”.
Advanced Semiconductor Fundamentals
1 2π‘šΞ“βˆ—
𝑛Γ = 2
2πœ‹
ℏ2
1 2π‘šΞ“βˆ— π‘˜π‘‡π‘’
=
4 πœ‹β„ 2
3/2
1
𝐸 2 𝑒π‘₯𝑝
0
3/2
𝑒π‘₯𝑝
1 2π‘šπΏβˆ—
𝑛𝐿 = 4 βˆ™ 2
2πœ‹
ℏ2
n𝐿
π‘šπΏβˆ— 𝑇𝑒𝐿
=4
nΞ“
π‘šΞ“βˆ— 𝑇𝑒
∞
3/2
Chapter 6. Carrier Transport
βˆ’ 𝐸 βˆ’ 𝐸𝐹
π‘˜π‘‡π‘’
𝐸𝐹
π‘˜π‘‡π‘’
∞
βˆ†Ξ“πΏ
3/2
𝑒π‘₯𝑝 βˆ’
1
𝐸 2 𝑒π‘₯𝑝
𝑑𝐸
for nondegenerated semiconductor
where Te is an electron temperature
βˆ’ 𝐸 βˆ’ 𝐸𝐹
π‘˜π‘‡π‘’πΏ
𝑑𝐸
βˆ†Ξ“πΏ
𝐸𝐹 1
1
𝑒π‘₯𝑝
βˆ’
π‘˜π‘‡π‘’πΏ
π‘˜ 𝑇𝑒𝐿 𝑇𝑒
=
2π‘šΞ“βˆ— π‘˜π‘‡π‘’
πœ‹β„ 2
3/2
𝑒π‘₯𝑝 βˆ’
βˆ†Ξ“πΏ
𝐸𝐹
𝑒π‘₯𝑝
π‘˜π‘‡π‘’πΏ
π‘˜π‘‡π‘’πΏ
If TeL = Te is an electron temperature,
𝑛𝐿 β‰ˆ 𝑛Γ at Te = 950 K.
For temperature higher than this, the upper valley has a higher density of states occupied.
Thus when an electron initially in the Ξ“-valley at energy of E = βˆ†π›€πΏ is scattered, it is more
likely to undergo an intervalley scattering to L-valley.
The total conductivity for carriers in the two set of valleys,
𝜎 = nΞ“ π‘žπœ‡Ξ“ + n𝐿 π‘žπœ‡πΏ where n = nΞ“ +n𝐿
The change in the conductivity with electric field, assuming πœ‡ is only a very weak function.
π‘‘πœŽ
𝑑nΞ“
𝑑n𝐿
𝑑nΞ“
β‰ˆ π‘žπœ‡Ξ“
+ π‘žπœ‡πΏ
= πœ‡Ξ“ βˆ’ πœ‡πΏ
π‘‘πœ€
π‘‘πœ€
π‘‘πœ€
π‘‘πœ€
𝑑nΞ“
n𝐿
=βˆ’
π‘‘πœ€
π‘‘πœ€
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
From the current density equation, 𝐽 = πœŽπœ€
The differential conductivity,
𝑑𝐽
π‘‘πœ–
𝑑𝐽
π‘‘πœŽ
𝑑nΞ“
=𝜎+πœ€
= nΞ“ π‘žπœ‡Ξ“ + n𝐿 π‘žπœ‡πΏ + π‘žπœ€ πœ‡Ξ“ βˆ’ πœ‡πΏ
π‘‘πœ€
π‘‘πœ€
π‘‘πœ€
if
𝑑𝐽
π‘‘πœ–
< 0,
(-)function
nΞ“ π‘žπœ‡Ξ“ + n𝐿 π‘žπœ‡πΏ < βˆ’π‘žπœ€ πœ‡Ξ“ βˆ’ πœ‡πΏ
𝑑nΞ“
π‘‘πœ€
βˆ’
πœ‡Ξ“ βˆ’ πœ‡πΏ πœ€ 𝑑nΞ“
n𝐿 n π‘‘πœ€ > 1
πœ‡Ξ“ + πœ‡πΏ Ξ“
nΞ“
𝝁πšͺ > 𝝁𝑳 πœ‡ β‰ˆ 7000 π‘π‘š2 /𝑉 βˆ™ 𝑠𝑒𝑐
Ξ“
πœ‡L β‰ˆ 100 π‘π‘š2 /𝑉 βˆ™ 𝑠𝑒𝑐
also,
βˆ’
for GaAs.
𝑑nΞ“ nΞ“
>
π‘‘πœ€
πœ€
π‘šπΏβˆ—
where 𝑑nΞ“ = βˆ’ n𝐿 =
βˆ’4
π‘‘πœ€
π‘‘πœ€
π‘šΞ“βˆ—
3/2
𝑒π‘₯𝑝 βˆ’
βˆ†Ξ“πΏ 𝑑nΞ“
𝑑
βˆ†Ξ“πΏ
+ nΞ“
𝑒π‘₯𝑝 βˆ’
π‘˜π‘‡π‘’ π‘‘πœ€
π‘‘πœ€
π‘˜π‘‡π‘’
n𝐿
π‘šπΏβˆ— 𝑇𝑒𝐿
=4
nΞ“
π‘šΞ“βˆ— 𝑇𝑒
3/2
𝑒π‘₯𝑝 βˆ’
βˆ†Ξ“πΏ
𝐸𝐹 1
1
𝑒π‘₯𝑝
βˆ’
π‘˜π‘‡π‘’πΏ
π‘˜ 𝑇𝑒𝐿 𝑇𝑒
Advanced Semiconductor Fundamentals
𝑑nΞ“
π‘šπΏβˆ—
=βˆ’4
π‘‘πœ€
π‘šΞ“βˆ—
=βˆ’
3/2
𝑒π‘₯𝑝 βˆ’
n𝐿 𝑑nΞ“
βˆ†Ξ“πΏ 𝑑𝑇𝑒
+ nΞ“ 2
nΞ“ π‘‘πœ€
π‘˜π‘‡π‘’ π‘‘πœ€
Chapter 6. Carrier Transport
βˆ†Ξ“πΏ
π‘˜π‘‡π‘’
𝑑nΞ“
βˆ†Ξ“πΏ 𝑑𝑇𝑒
+ nΞ“ 2
π‘‘πœ€
π‘˜π‘‡π‘’ π‘‘πœ€
=βˆ’
n𝐿 𝑑nΞ“
βˆ†Ξ“πΏ 𝑑𝑇𝑒
βˆ’ n𝐿 2
nΞ“ π‘‘πœ€
π‘˜π‘‡π‘’ π‘‘πœ€
nΞ“ + n𝐿 𝑑nΞ“
βˆ†Ξ“πΏ 𝑑𝑇𝑒
= βˆ’n𝐿 2
nΞ“
π‘‘πœ€
π‘˜π‘‡π‘’ π‘‘πœ€
βˆ’
𝑑nΞ“
βˆ†Ξ“πΏ nΞ“ n𝐿 𝑑𝑇𝑒
nΞ“
=
>
π‘‘πœ€
π‘˜π‘‡π‘’ 𝑇𝑒 nΞ“ + n𝐿 π‘‘πœ€
πœ€
βˆ’
𝑑nΞ“ nΞ“
>
π‘‘πœ€
πœ€
n𝐿
βˆ†Ξ“πΏ πœ€ 𝑑𝑇𝑒
>1
nΞ“ + n𝐿 π‘˜π‘‡π‘’ 𝑇𝑒 π‘‘πœ€
Assuming that the electron temperature increases linearly with electric field,
πœ€ 𝑑𝑇𝑒
β‰ˆ1
𝑇𝑒 π‘‘πœ€
βˆ†Ξ“πΏ
nΞ“
1 π‘šπΏβˆ—
>1+
=1+
π‘˜π‘‡π‘’
𝑛𝐿
4 π‘šΞ“βˆ—
βˆ’3/2
βˆ†Ξ“πΏ
𝑒π‘₯𝑝
π‘˜π‘‡π‘’
simple transcendental equation
n𝐿
π‘šπΏβˆ—
=4
nΞ“
π‘šΞ“βˆ—
3/2
𝑒π‘₯𝑝 βˆ’
βˆ†Ξ“πΏ
π‘˜π‘‡π‘’
Advanced Semiconductor Fundamentals
βˆ†Ξ“πΏ
nΞ“
1 π‘šπΏβˆ—
>1+
=1+
π‘˜π‘‡π‘’
𝑛𝐿
4 π‘šΞ“βˆ—
There are two regions of
βˆ†Ξ“πΏ
π‘˜π‘‡π‘’
Chapter 6. Carrier Transport
βˆ’3/2
𝑒π‘₯𝑝
βˆ†Ξ“πΏ
π‘˜π‘‡π‘’
where this inequality is not satisfied:
i) At very high electron temperature( not of interest)
ii) At low electron temperature of interest
Upper limit of
βˆ†Ξ“πΏ
π‘˜π‘‡π‘’
β‰ˆ 5.8.
Lower limit of electron temperature, 𝑇𝑒 β‰ˆ 600 𝐾
𝑛𝐿
β‰ˆ 0.15
nΞ“
So that negative differential conductivity sets when as little as 15 % of the
electrons transferred to the upper valleys.
𝑣
increasing βˆ†Ξ“πΏ
πœ€
Read β€œBallistic transport/velocity overshoot”.
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Related Topics
Resistivity/Conductivity
𝜺 = ρ𝑱
or
1
𝜌
𝑱 = 𝜎𝜺 = 𝜺
In a homogeneous material,
𝑱 = π‘±π‘‘π‘Ÿπ‘–π‘“π‘‘ = 𝑱𝑁|π‘‘π‘Ÿπ‘–π‘“π‘‘ + 𝑱𝑃|π‘‘π‘Ÿπ‘–π‘“π‘‘ = q(πœ‡π‘› n +πœ‡π‘ p)𝜺
∴ resistivity , 𝜌 =
1
q(πœ‡π‘› n + πœ‡π‘ p)
[Ξ©βˆ™ π‘π‘š]
conductivity, 𝜎 = q(πœ‡π‘› n + πœ‡π‘ p)
1
qπœ‡π‘› 𝑁𝐷
1
𝜌=
qπœ‡π‘ 𝑁𝐴
𝜌=
for n-type semiconductor
for p-type semiconductor
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Sheet Resistance
𝑅𝑠 =
𝜌
𝑑
[Ξ©/βŽ•]
𝑅=𝜌
𝐿
𝐿
𝐿
=𝜌
= 𝑅𝑠
A
Wβˆ™t
W
Four-point probe technique
1) For thick sample (s << t)
At probe 1,
πœ€ π‘Ÿ = 𝜌 𝐽 π‘Ÿ = βˆ’π‘Ÿ
𝑉 π‘Ÿ =βˆ’
=
πœ•π‘‰(π‘Ÿ)
πœ•π‘Ÿ
1
D
𝐼
where 𝐽 π‘Ÿ = π‘Ÿ
2πœ‹π‘Ÿ 2
𝜌𝐼
π‘‘π‘Ÿ + 𝐢1
2πœ‹π‘Ÿ 2
𝐼𝜌
+ 𝐢1
2πœ‹π‘Ÿ
𝑉21
𝑉31 =
𝐼𝜌
+ 𝐢1
2πœ‹(2𝑠)
t
I
In spherical coordinate system
π‘Ÿ
𝐼𝜌
=
+ 𝐢1
2πœ‹π‘ 
2 3 4
π‘Ÿ
π‘Ÿ
Due to symmetry of current path,
V is not function of πœƒ and βˆ…
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
At probe 4,
I
𝜌𝐼
π‘‘π‘Ÿ + 𝐢2
2πœ‹π‘Ÿ 2
𝐼𝜌
=βˆ’
+ 𝐢2
2πœ‹π‘Ÿ
𝑉 π‘Ÿ =
𝑉24 = βˆ’
π‘Ÿ
π‘Ÿ
π‘Ÿ
𝐼𝜌
𝐼𝜌
+ 𝐢2 𝑉34 = βˆ’
+ 𝐢2
2πœ‹(2𝑠)
2πœ‹π‘ 
𝐼𝜌
𝐼𝜌
𝐼𝜌
+ 𝐢1 βˆ’
+ 𝐢2 =
+ 𝐢1 + 𝐢2
2πœ‹π‘ 
2πœ‹ 2𝑠
2πœ‹ 2𝑠
𝐼𝜌
𝐼𝜌
𝐼𝜌
=
+ 𝐢1 βˆ’
+ 𝐢2 = βˆ’
+ 𝐢1 + 𝐢2
2πœ‹ 2𝑠
2πœ‹π‘ 
2πœ‹ 2𝑠
𝑉2 = 𝑉21 + 𝑉24 =
𝑉3 = 𝑉31 + 𝑉34
𝑉 = 𝑉2 βˆ’ 𝑉3 =
𝐼𝜌
2πœ‹π‘ 
∴ 𝜌 = 2πœ‹π‘ 
If t >> s, 𝐹1 β†’ 1,
𝑑/𝑠
If t << s, 𝐹1 β†’
,
2𝑙𝑛2
𝑉
𝐹
𝐼 1
where 𝐹1 : thickness correction factor
𝑑/𝑠
𝐹1 =
𝑑
sinh
𝑉
𝑠
2ln
𝜌 = 2πœ‹π‘ 
𝑑
𝐼
sinh
2𝑠
πœ‹π‘‘ 𝑉
𝜌=
𝑙𝑛2 𝐼
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
2) For thin sample (s >> t)
At probe 1,
πœ€ π‘Ÿ = 𝜌𝐽 π‘Ÿ = βˆ’π‘Ÿ
𝑉 π‘Ÿ =βˆ’
𝑉21 = βˆ’
πœ•π‘‰(π‘Ÿ)
𝐼
where 𝐽 π‘Ÿ = π‘Ÿ
πœ•π‘Ÿ
2πœ‹π‘Ÿπ‘‘
𝜌𝐼
𝐼𝜌
π‘‘π‘Ÿ + 𝐢1 = βˆ’
π‘™π‘›π‘Ÿ + 𝐢1
2πœ‹π‘Ÿπ‘‘
2πœ‹π‘‘
𝐼𝜌
𝑙𝑛𝑠 + 𝐢1
2πœ‹π‘‘
𝑉31 = βˆ’
1
2 3 4
D
t
𝐼𝜌
𝑙𝑛2𝑠 + 𝐢1
2πœ‹π‘‘
I
π‘Ÿ
At probe 4,
𝑉 π‘Ÿ =
𝐼𝜌
π‘™π‘›π‘Ÿ + 𝐢2
2πœ‹π‘‘
In cylinderical coordinate system
𝐼𝜌
𝐼𝜌
𝑙𝑛2𝑠 + 𝐢2 𝑉34 =
𝑙𝑛𝑠 + 𝐢2
2πœ‹π‘‘
2πœ‹π‘‘
𝐼𝜌
𝐼𝜌
𝐼𝜌
𝑉2 = 𝑉21 + 𝑉24 = βˆ’
𝑙𝑛𝑠 + 𝐢1 +
𝑙𝑛2𝑠 + 𝐢2 =
𝑙𝑛2 + 𝐢1 + 𝐢2
2πœ‹π‘‘
2πœ‹π‘‘
2πœ‹π‘‘
𝐼𝜌
𝐼𝜌
𝐼𝜌
𝑉3 = 𝑉31 + 𝑉34 = βˆ’
𝑙𝑛2𝑠 + 𝐢1 +
𝑙𝑛𝑠 + 𝐢2 = βˆ’
𝑙𝑛2 + 𝐢1 + 𝐢2
2πœ‹π‘‘
2πœ‹π‘‘
2πœ‹π‘‘
𝐼𝜌
𝑉 = 𝑉2 βˆ’ 𝑉3 =
𝑙𝑛2
πœ‹π‘‘
𝑉24 =
Advanced Semiconductor Fundamentals
∴𝜌=
Chapter 6. Carrier Transport
πœ‹π‘‘ 𝑉
𝐹
𝑙𝑛2 𝐼 2 where 𝐹2 : size correction factor
𝑙𝑛2
𝐷
( )3 +3
𝑠
𝑙𝑛2 + 𝑙𝑛 𝐷
( )3 βˆ’3
𝑠
πœ‹π‘‘ 𝑉
If D >> s, 𝐹2 β†’ 1, 𝜌 = 𝑙𝑛2 𝐼
𝐹2 =
: same as before
summary
i) If t >> s and D >> s,
𝑉
𝐼
ii) If the condition for t >> s and D >> s is not satisfied,
𝜌 = 2πœ‹π‘ 
𝜌 = 2πœ‹π‘ 
𝑉
𝐹𝐹
𝐼 1 2
iii) In most cases, t << s and D >> s,
𝜌=
πœ‹π‘‘ 𝑉
𝑙𝑛2 𝐼
𝑅𝑠 =
𝐹1 β†’
𝑑/𝑠
, 𝐹2 β†’ 1,
2𝑙𝑛2
πœ‹ 𝑉
𝑉
= 4.53
𝑙𝑛2 𝐼
𝐼
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Hall Effect
Lorenz force in the sample
assuming p-type,
𝐹 = π‘ž 𝑣𝑑 × π΅ + π‘ž πœ€
π‘₯𝐹π‘₯
𝑦𝐹𝑦
𝑧𝐹𝑧
π‘₯
= π‘ž 𝑣𝑑π‘₯
0
𝑦
𝑣𝑑𝑦
0
π‘₯πœ€π‘₯
𝑧
𝑣𝑑𝑧 + π‘ž π‘¦πœ€π‘¦
𝐡𝑧
π‘§πœ€π‘§
𝐹π‘₯ = π‘žπ‘£π‘‘π‘¦ 𝐡𝑧 + π‘žπœ€π‘₯
𝐹𝑦 = βˆ’π‘žπ‘£π‘‘π‘₯ 𝐡𝑧 + π‘žπœ€π‘¦
Lorenz force in y-direction must be balanced under steady state.
𝐹𝑦 = βˆ’π‘žπ‘£π‘‘π‘₯ 𝐡𝑧 + π‘žπœ€π‘¦ = 0
Moreover, 𝐽π‘₯ = π‘žπ‘π‘£π‘‘π‘₯ β†’ 𝑣𝑑π‘₯ =
Hall coefficient, 𝑅𝐻 =
𝐽π‘₯
π‘žπ‘
πœ€π‘¦
1
=
𝐽π‘₯ 𝐡𝑧 π‘žπ‘
𝑅𝐻 = βˆ’
1
π‘žπ‘›
βˆ΄βˆ’
for p-type
for n-type
𝐽π‘₯ 𝐡𝑧
+ πœ€π‘¦ = 0
π‘žπ‘
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
With measurable quantity,
πœ€π‘¦
1
𝑉𝐻 /𝑑
𝑉𝐻 𝑀 108 𝑉𝐻 𝑀
𝑅𝐻 =
=
=
=
=
𝐼
π‘žπ‘ 𝐽π‘₯ 𝐡𝑧
𝐡𝐼
𝐡𝐼
𝐡
𝑀𝑑
If VH is given in volts, w in cm, B in gauss,
I in amps, and RH in cm3/coul.
The resistance of the bar is just VA/I.
𝑅=
𝑉𝐴
𝑙
=𝜌
𝐼
𝑀𝑑
β†’
𝜌=
1
𝑉𝐴 𝑀𝑑
=
π‘žπ‘πœ‡π»
𝐼 𝑙
The Hall mobility,
πœ‡π» =
1 1
𝑅𝐻
βˆ™ =
π‘žπ‘ 𝜌
𝜌
More exacting analysis gives,
π‘Ÿπ»
π‘žπ‘
π‘Ÿπ»
𝑅𝐻 = βˆ’
π‘žπ‘›
𝑅𝐻 =
for p-type
Hall factor π‘Ÿπ» β‰ˆ 1
for n-type
The relationship between hall mobility and drift mobility
πœ‡π» = π‘Ÿπ» πœ‡π‘‘π‘Ÿπ‘–π‘“π‘‘
For GaAs, π‘Ÿπ» > 1 β†’ πœ‡π‘‘π‘Ÿπ‘–π‘“π‘‘ < πœ‡π»
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Anisotropic Conductivity
The equations, so far, for the mobility, conductivity, and Hall constant are applicable for
electrons in spherical band minima.
The situation is somewhat more complicated, when the carrier transport in an ellipsoidal
minima.
For nonspherical energy surface with one ellipsoidal conduction band minimum at 𝛀 (π‘˜ = 0)
π‘˜π‘¦
transverse
π‘˜π‘₯
π‘˜π‘¦2
ℏ2 π‘˜ 2 ℏ2 π‘˜π‘₯2
π‘˜π‘§2
𝐸(π‘˜) βˆ’ 𝐸𝐢 β‰…
=
+
+
2π‘šπ‘’βˆ—
2 π‘šπ‘₯π‘₯ π‘šπ‘¦π‘¦ π‘šπ‘§π‘§
π‘˜π‘§
longitudinal
π‘ž 2 𝑛𝑑
𝐽π‘₯ =
πœ€ = π‘žπ‘›πœ‡π‘₯ πœ€π‘₯
π‘šπ‘₯π‘₯ π‘₯
π‘ž 2 𝑛𝑑
𝐽𝑦 =
πœ€ = π‘žπ‘›πœ‡π‘¦ πœ€π‘¦
π‘šπ‘¦π‘¦ 𝑦
π‘ž 2 𝑛𝑑
𝐽𝑧 =
πœ€ = π‘žπ‘›πœ‡π‘§ πœ€π‘§
π‘šπ‘§π‘§ 𝑧
The total current density is therefore,
𝐽 = π‘ž 2 𝑛𝑑
1
βˆ™πœ€
π‘šβˆ—
where
1
π‘šβˆ—
is the effective mass tensor.
Advanced Semiconductor Fundamentals
1
=
π‘šβˆ—
1
π‘šπ‘₯π‘₯
0
0
0
1
π‘šπ‘¦π‘¦
0
0
1
π‘šπ‘§π‘§
0
𝐽π‘₯
𝐽𝑦
𝐽𝑧
π‘›π‘žπœ‡π‘₯
0
=
0
0
π‘›π‘žπœ‡π‘¦
0
Chapter 6. Carrier Transport
This can also be put in the form,
𝐽 = 𝜎 βˆ™πœ€
where 𝜎 is the conductivity tensor.
π‘›π‘žπœ‡π‘₯
0
𝜎 =
0
0
πœ€π‘₯
0 βˆ™ πœ€π‘¦
πœ€π‘§
π‘›π‘žπœ‡π‘§
0
π‘›π‘žπœ‡π‘¦
0
0
0
π‘›π‘žπœ‡π‘§
If πœ€ is off axis and three diagonal terms are not equal, the current is not in
same direction as πœ€.
β€œanisotropic conductivity”
y
πœ€ is off axis with ellipsoidal conduction band minima.
(π‘šπ‘₯π‘₯ β‰  π‘šπ‘§π‘§ β‰  π‘šπ‘¦π‘¦ , π‘žπ‘›πœ‡π‘₯ β‰  π‘žπ‘›πœ‡π‘¦ β‰  π‘žπ‘›πœ‡π‘§ )
πœ€
πœ€ is on axis.
𝐽
πœ€ is off axis but spherical conduction band minima.
(π‘šπ‘₯π‘₯ = π‘šπ‘¦π‘¦ = π‘šπ‘§π‘§ , π‘žπ‘›πœ‡π‘₯ = π‘žπ‘›πœ‡π‘¦ = π‘žπ‘›πœ‡π‘§ )
𝐽
πœ€
πœ€ is on axis.
x
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
For multiple equivalent ellipsoidal conduction band minimum at X or L (π‘˜ β‰  0)
ky
2
2 2
2
2
2
𝐸(π‘˜) =
kx
For example, Si:
6 equivalent conduction band minima in the directions
of π‘˜
kz
π‘˜π‘¦
ℏ π‘˜
ℏ
π‘˜π‘₯
π‘˜π‘§
=
+
+
2π‘šπ‘’βˆ—
2 π‘šπ‘₯π‘₯ π‘šπ‘¦π‘¦ π‘šπ‘§π‘§ (π‘š β‰  π‘š β‰  π‘š )
π‘₯π‘₯
𝑦𝑦
𝑦𝑦
πœ‹
= (1, 0, 0)
π‘Ž
πœ€π‘₯
Concentration of electron in each minima is n/6.
When the electric field in x-direction, the total current in the x-direction
π‘ž 2 𝑛𝑑 2
2
2
π‘›π‘žπœ€π‘₯
𝐽π‘₯ =
+
+
πœ€π‘₯ =
πœ‡π‘₯ + πœ‡π‘¦ + πœ‡π‘§ = πœŽπœ€π‘₯
6 π‘šπ‘₯π‘₯ π‘šπ‘¦π‘¦ π‘šπ‘§π‘§
3
Similar expressions can be obtained for y- and z-directions and for any electric field.
π‘ž 2 𝑛𝑑
Compare with the expression for the conductivity, 𝐽 = βˆ— πœ€ = π‘žπ‘›πœ‡π‘π‘œπ‘› πœ€ = πœŽπœ€
π‘šπ‘π‘œπ‘›
πœ‡π‘₯ + πœ‡π‘¦ + πœ‡π‘§
: conductivity mobility
πœ‡π‘π‘œπ‘› =
3
1
1 1
1
1
Isotropic conductivity
=
+
+
: conductivity
βˆ—
π‘šπ‘π‘œπ‘› 3 π‘šπ‘₯π‘₯ π‘šπ‘¦π‘¦ π‘šπ‘§π‘§
The current and the electric field
effective mass
are always in the same direction.
scalar quantity
Advanced Semiconductor Fundamentals
DIFFUSION
Definition-Visualization
Diffusion Current
SIMPLIFYING ASSUMPTIONS:
1) Carrier motion and concentration gradients are restricted to1-D.
2) All carriers move with the same velocity, 𝑣 = π‘£π‘‘β„Ž.
3) The distance traveled by carriers between collisions is a fixed
length, 𝑙 which corresponds to the mean distance traveled by
carriers between scattering events
Chapter 6. Carrier Transport
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Consider the p-type semiconductor bar of cross-sectional area A and the steady-state hole
concentration gradient shown in the figure.
If t is arbitrarily set equal to zero at the instant all of the carriers scatter, if follows that half of
the holes in a volume 𝑙𝐴 on either side of x = 0 will be moving in the proper direction so as to
cross the x = 0 plane prior to the next scattering event at 𝑙/𝑣.
𝑝 =
𝑝=
π»π‘œπ‘™π‘’π‘  π‘šπ‘œπ‘£π‘–π‘›π‘” 𝑖𝑛 π‘‘β„Žπ‘’ + π‘₯
𝐴
π‘‘π‘–π‘Ÿπ‘’π‘π‘‘π‘–π‘œπ‘› π‘€β„Žπ‘–π‘β„Ž π‘π‘Ÿπ‘œπ‘ π‘  π‘‘β„Žπ‘’ =
2
π‘₯ = 0 π‘π‘™π‘Žπ‘›π‘’ 𝑖𝑛 π‘Ž π‘‘π‘–π‘šπ‘’ 𝑙/𝑣
π»π‘œπ‘™π‘’π‘  π‘šπ‘œπ‘£π‘–π‘›π‘” 𝑖𝑛 π‘‘β„Žπ‘’ βˆ’ π‘₯
𝐴
π‘‘π‘–π‘Ÿπ‘’π‘π‘‘π‘–π‘œπ‘› π‘€β„Žπ‘–π‘β„Ž π‘π‘Ÿπ‘œπ‘ π‘  π‘‘β„Žπ‘’ =
2
π‘₯ = 0 π‘π‘™π‘Žπ‘›π‘’ 𝑖𝑛 π‘Ž π‘‘π‘–π‘šπ‘’ 𝑙/𝑣
0
𝑝 π‘₯ 𝑑π‘₯
βˆ’π‘™
𝑙
𝑝 π‘₯ 𝑑π‘₯
0
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Since 𝑙 is typically quite small, the first two terms in a Taylor series expansion of p(x) about x
= 0 will closely approximate p(x) for x values between βˆ’ 𝑙 and + 𝑙 .
𝑝 π‘₯ β‰ˆπ‘ 0 βˆ’
𝑑𝑝
| π‘₯
𝑑π‘₯ 0
1
1 𝑑𝑝 𝑙2
𝑝 = 𝐴𝑙𝑝 0 βˆ’ 𝐴 |0 ,
2
2 𝑑π‘₯ 2
. . . . . βˆ’π‘™ ≀ π‘₯ ≀ 𝑙
1
1 𝑑𝑝 𝑙2
𝑝 = 𝐴𝑙𝑝 0 + 𝐴 |0
2
2 𝑑π‘₯ 2
The net number of + x directed holes that cross the x = 0 plane in a time 𝑙/𝑣.
𝑑𝑝 𝑙2
𝑝 βˆ’ 𝑝 = βˆ’π΄ |0
𝑑π‘₯ 2
The net number crossing the x = 0 plane per unit time due to diffusion, dropping the β€œ|0 ”.
𝐼𝑃|𝑑𝑖𝑓𝑓 =
π‘ž(𝑝 βˆ’ 𝑝)
1
𝑑𝑝
= βˆ’ π‘žπ΄π‘£π‘™
𝑙/𝑣
2
𝑑π‘₯
𝑣𝑙 𝑑𝑝
𝐽𝑃|𝑑𝑖𝑓𝑓 = βˆ’π‘ž( )
2 𝑑π‘₯
𝑣𝑙
𝑣𝑙
(exact three dimensional analysis leads to 𝐷𝑃 ≑
)
2
3
𝑑𝑝
= βˆ’π‘žπ·π‘
𝑑π‘₯
Finally, introducing 𝐷𝑃 ≑
𝐽𝑃|𝑑𝑖𝑓𝑓
In three dimension,
𝑱𝑃|𝑑𝑖𝑓𝑓 = βˆ’π‘žπ·π‘ƒ 𝛻𝑝
𝑱𝑁|𝑑𝑖𝑓𝑓 = βˆ’π‘žπ·π‘ 𝛻𝑛
𝐷𝑝 , 𝐷𝑛 : The hole and electron diffusion
coefficient with standard unit of
cm2/sec.
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Einstein Relationship
Consider a nununiformly doped semiconductor under equilibrium conditions, as shown below.
𝑑𝐸𝐹
=0
𝑑π‘₯
𝛻𝐸𝐹 = 0
under equilibrium condition
Nonzero electric field is established inside nonuniformly doped semiconductors under
1
1
1
equilibrium conditions.
πœ€ = 𝛻𝐸 = 𝛻𝐸 (π‘₯) = 𝛻𝐸 (π‘₯)
π‘ž
π‘ž
𝐢
π‘ž
𝑉
𝑱𝑁|π‘‘π‘Ÿπ‘–π‘“π‘‘ + 𝑱𝑁|𝑑𝑖𝑓𝑓 = π‘žπœ‡π‘› π‘›πœ€ + π‘žπ·π‘ 𝛻𝑛 = 0
∴ π‘žπœ‡π‘› π‘›πœ€ + π‘žπ·π‘ 𝛻𝑛 = π‘ž πœ€ πœ‡π‘› 𝑛 βˆ’
𝑛 = 𝑁𝐢 β„±1/2 η𝐢
𝛻𝑛 =
where η𝐢 = (𝐸𝐹 βˆ’ 𝐸𝑐 )/π‘˜π‘‡
π‘ž πœ•π‘›
𝐷
π‘˜π‘‡ πœ•Ξ·πΆ 𝑁
πœ•π‘›
1
πœ•π‘›
π‘ž πœ•π‘›
𝛻η𝐢 = βˆ’
𝛻𝐸𝐢 (π‘₯)
= βˆ’
πœ€
πœ•Ξ·πΆ
π‘˜π‘‡
πœ•Ξ·πΆ
π‘˜π‘‡ πœ•Ξ·πΆ
∴ 0 = π‘žπœ‡π‘› π‘›πœ€ + π‘žπ·π‘ 𝛻𝑛 = π‘ž πœ€ πœ‡π‘› 𝑛 βˆ’
π‘ž πœ•π‘›
𝐷
π‘˜π‘‡ πœ•Ξ·πΆ 𝑁
𝐷𝑁 π‘˜π‘‡ 𝑛
=
πœ‡π‘›
π‘ž πœ•π‘›
πœ•Ξ·πΆ
generalized form
of Einstein
relationship
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
In the nondegenerate semiconductor,
𝑛
𝑛 β†’ 𝑁𝐢 𝑒π‘₯𝑝 η𝐢 ,
β†’1
πœ•π‘›
πœ•Ξ·πΆ
𝐷𝑁 π‘˜π‘‡
=
πœ‡π‘›
π‘ž
Einstein relationship for electrons
𝐷𝑁 π‘˜π‘‡
=
πœ‡π‘›
π‘ž
Einstein relationship for holes
Similar argument for holes,
(Home work)
Prove that Einstein relationship is also valid even under nonequilibrium conditions (𝛻𝐸𝐹 β‰  0).
Use the relation for ℱ𝑗 η𝐢 ,
πœ•β„±π‘— η𝐢
= β„±π‘—βˆ’1 η𝐢
πœ•Ξ·πΆ
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
EQUATIONA OF STATE
Current Equations
Carrier Currents
𝐽𝑃 = π‘žπœ‡π‘ 𝑝ℰ βˆ’ π‘žπ·π‘ƒ 𝛻𝑝
𝐽 = 𝐽𝑃 + 𝐽𝑁
𝐽𝑁 = π‘žπœ‡π‘› 𝑛ℰ + π‘žπ·π‘ 𝛻𝑛
: total particle current at steady state
Dielectric Displacement Currents
The change in polarization may be viewed as given rise to a nonparticle
current, the dielectric displacement current.
Ex) Current flow through a capacitor under a.c. and transient conditions
𝐽𝐷 =
πœ•π·
πœ•π‘‘
For a linear dielectric,
𝐷 = 𝐾𝑠 πœ–0 β„°
βˆ΄π‘— = 𝐽𝑃 + 𝐽𝑁 +
πœ•π·
πœ•π‘‘
: total current under a.c. and transient conditions
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Quasi-Equilibrium and Quasi-Femi Energies
For nondegenerate semiconductors,
𝑛 = 𝑛𝑖 𝑒 (𝐸𝐹 βˆ’πΈπ‘– )
𝑛 = 𝑛𝑖 𝑒 (𝐹𝑁 βˆ’πΈπ‘– )
due to perturbation
𝑝 = 𝑛𝑖 𝑒 (𝐸𝑖 βˆ’πΈπΉ )
equilibrium
𝑛 = 𝑛𝑖 𝑒 (𝐸𝐹 βˆ’πΈπ‘– )
𝑝 = 𝑛 𝑒 (𝐸𝑖 βˆ’πΈπΉ )
𝑖
𝑝 = 𝑛𝑖 𝑒 (𝐸𝑖 βˆ’πΉπ‘ƒ )
nonequilibrium with small perturbation
Carrier distributions remain almost unchanged from
the equilibrium distributions even when small
perturbation applied to the semiconductor.
This is referred to the quasi-equilibrium and the Fermistatistics developed at equilibrium still can be used with
quasi-Fermi energies, FN and FP, at quasi-equilibrium.
𝑛 = 𝑛𝑖 𝑒 (𝐹𝑁 βˆ’πΈπ‘– )
𝑝 = 𝑛 𝑒 (𝐸𝑖 βˆ’πΉπ‘ƒ )
𝑖
𝐹𝑁 = 𝐸𝑖 + π‘˜π‘‡ ln(𝑛/𝑛𝑖 )
𝐹𝑃 = 𝐸𝑖 βˆ’ π‘˜π‘‡ ln(𝑝/𝑛𝑖 )
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Differentiating the nonequilibrium carrier concentration,
𝛻𝑝 = 𝑛𝑖 /π‘˜π‘‡ βˆ™ 𝑒
𝐸𝑖
βˆ’πΉπ‘ƒ (𝛻𝐸 βˆ’ 𝛻𝐹 )
𝑖
𝑃
where β„° =
1
𝛻𝐸
π‘ž 𝑖
𝐽𝑃 = π‘ž(πœ‡π‘ βˆ’π‘žπ·π‘ƒ /π‘˜π‘‡)𝑝 β„° + (π‘žπ·π‘ƒ /π‘˜π‘‡)𝑝𝛻𝐹𝑃 where π‘žπ·π‘ƒ /π‘˜π‘‡ = πœ‡ 𝑝
𝐽𝑃 = πœ‡π‘ƒ 𝑝𝛻𝐹𝑃 = π‘žπœ‡π‘ƒ 𝑝
𝐽𝑁 = πœ‡π‘› 𝑛𝛻𝐹𝑁 = π‘žπœ‡π‘› 𝑛
𝛻𝐹𝑃
= πœŽπ‘ πœ€π‘’π‘“π‘“
π‘ž
𝛻𝐹𝑁
= πœŽπ‘› πœ€π‘’π‘“π‘“
π‘ž
Ohm’s law
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Continuity Equations
πœ•π‘› πœ•π‘›
πœ•π‘›
πœ•π‘›
πœ•π‘›
=
|π‘‘π‘Ÿπ‘–π‘“π‘‘ +
|𝑑𝑖𝑓𝑓 +
|π‘…βˆ’πΊ +
| π‘œπ‘‘β„Žπ‘’π‘Ÿ
πœ•π‘‘ πœ•π‘‘
πœ•π‘‘
πœ•π‘‘
πœ•π‘‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘ π‘’π‘ 
πœ•π‘ πœ•π‘
πœ•π‘
πœ•π‘
πœ•π‘
=
|π‘‘π‘Ÿπ‘–π‘“π‘‘ +
|𝑑𝑖𝑓𝑓 +
|π‘…βˆ’πΊ +
| π‘œπ‘‘β„Žπ‘’π‘Ÿ
πœ•π‘‘ πœ•π‘‘
πœ•π‘‘
πœ•π‘‘
πœ•π‘‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘ π‘’π‘ 
By introducing,
πœ•π‘›
𝑔𝑁 ≑
| π‘œπ‘‘β„Žπ‘’π‘Ÿ
πœ•π‘‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘ π‘’π‘ 
𝑔𝑃 ≑
πœ•π‘
| π‘œπ‘‘β„Žπ‘’π‘Ÿ
πœ•π‘‘ π‘π‘Ÿπ‘œπ‘π‘’π‘ π‘ π‘’π‘ 
and noting the particle divergence,
πœ•π‘›
πœ•π‘›
1
|π‘‘π‘Ÿπ‘–π‘“π‘‘ +
|𝑑𝑖𝑓𝑓 = 𝛻 βˆ™ 𝐽𝑁
πœ•π‘‘
πœ•π‘‘
π‘ž
πœ•π‘
πœ•π‘
1
|π‘‘π‘Ÿπ‘–π‘“π‘‘ +
|𝑑𝑖𝑓𝑓 = βˆ’ 𝛻 βˆ™ 𝐽𝑃
πœ•π‘‘
πœ•π‘‘
π‘ž
πœ•π‘› 1
= 𝛻 βˆ™ 𝐽𝑁 βˆ’ π‘Ÿπ‘ + 𝑔𝑁
πœ•π‘‘ π‘ž
πœ•π‘
1
= βˆ’ 𝛻 βˆ™ 𝐽𝑃 βˆ’ π‘Ÿπ‘ƒ + 𝑔𝑃
πœ•π‘‘
π‘ž
π‘Ÿπ‘ ≑ βˆ’
πœ•π‘›
|
πœ•π‘‘ π‘…βˆ’πΊ
π‘Ÿπ‘ƒ ≑ βˆ’
πœ•π‘
|
πœ•π‘‘ π‘…βˆ’πΊ
Advanced Semiconductor Fundamentals
Chapter 6. Carrier Transport
Minority Carrier Diffusion Equations
Assumptions: 1) 1-D
2) minority carrier only
3) πœ€ β‰ˆ 0
4) uniformly doped
5) low-level injection
6) no other processes except possibly photo generation
1
1 πœ•π½π‘
𝛻 βˆ™ 𝐽𝑁 β†’
π‘ž
π‘ž πœ•π‘₯
𝐽𝑁 = π‘žπœ‡π‘› π‘›πœ€ + π‘žπ·π‘
πœ•π‘›
πœ•π‘›
β‰ˆ π‘žπ·π‘
πœ•π‘₯
πœ•π‘₯
πœ•π‘› πœ•π‘›0 πœ•βˆ†π‘› πœ•βˆ†π‘›
=
+
=
πœ•π‘₯
πœ•π‘₯
πœ•π‘₯
πœ•π‘₯
π‘Ÿπ‘ =
βˆ†π‘›
πœπ‘›
𝑔𝑁 = 𝐺𝐿
πœ•π‘› πœ•π‘›0 πœ•βˆ†π‘› πœ•βˆ†π‘›
=
+
=
πœ•π‘‘
πœ•π‘‘
πœ•π‘‘
πœ•π‘‘
1
πœ• 2 βˆ†π‘›
𝛻 βˆ™ 𝐽𝑁 β†’ 𝐷𝑁
π‘ž
πœ•π‘₯ 2
πœ•βˆ†π‘›π‘
πœ• 2 βˆ†π‘› βˆ†π‘›π‘
= 𝐷𝑁
βˆ’
+ 𝐺𝐿
πœ•π‘‘
πœ•π‘₯ 2
πœπ‘›
πœ•βˆ†π‘π‘›
πœ• 2 βˆ†π‘ βˆ†π‘π‘›
= 𝐷𝑃
βˆ’
+ 𝐺𝐿
πœ•π‘‘
πœ•π‘₯ 2
πœπ‘›
Minority carrier diffusion equations