Carrier Action

Prof. Tae Hoon Lee

 Outline
Carrier Action – part 1

• Thermal motion

• Scattering

• Drift current – electric field induced motion

• Charge carrier mobility

• Summary

2
 Thermal motion

In thermal equilibrium, mobile electrons in the conduction

band (or holes in the valence band) will be in random
thermal motion, resulting in zero net current.

Kinetic energy thermal energy

k = Boltzmann’s constant = 8.617×10-5 eV/K = 1.38×10-23 J/K

mn = conductivity effective mass of electron
vth = average thermal velocity ~ 107 cm/s at 300 K
3
Scattering mechanisms for electrons

4
 Thermal equilibrium

Most processes such as transport from source to drain of a transistor and

recombination occur over much longer times or distances than this mean free time
or mean free distance.

This enables us to treat non-equilibrium situations as perturbations of thermal

equilibrium.

5
Net motion of electrons under E field

drift velocity

6
 Drift current

Drift current: flow rate of charge carriers in response to an electric field

∆𝑄 𝑞𝑝𝑙𝐴 𝑞𝑝𝑣𝑑 ∆𝑡𝐴

𝐼𝑝_𝑑𝑟𝑖𝑓𝑡 = = = = 𝑞𝑝𝑣𝑑 𝐴
∆𝑡 ∆𝑡 ∆𝑡
𝑱𝒑_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑝𝒗𝒅
∆𝑄 −𝑞𝑛𝑙𝐴 −𝑞𝑛(−𝑣𝑑 )∆𝑡𝐴
𝐼𝑛_𝑑𝑟𝑖𝑓𝑡 = = = = 𝑞𝑛𝑣𝑑 𝐴
∆𝑡 ∆𝑡 ∆𝑡
𝑱𝒏_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑛𝒗𝒅

𝑱𝒅𝒓𝒊𝒇𝒕 = 𝑞(𝑛 + 𝑝)𝒗𝒅

7
 Drift velocity vs E field

for low to moderate electric fields

for high electric fields

𝛽 = 1 for holes and 2 for electrons in Si

𝜇: constant of proportionality → mobility [cm2/Vs]

𝑱𝒑_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑝𝒗𝒅 = 𝑞𝑝𝜇𝑝 𝑬

for low to moderate electric fields
𝑱𝒏_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑛𝒗𝒅 = 𝑞𝑛𝜇𝑛 𝑬
8
Saturation in drift velocity

9
Drift velocity vs E field

10
Band bending: E-field and potential

11
 Scattering during drift

Motion of the electrons on the average is now in a direction opposite to the

electric field. This process is called DRIFT.

12
Charge carrier mobility

13
 Outline
Carrier Action – part 2

• Mobility limited by different scatterings

• Lattice scattering

• Ionized impurity scattering

• Mobility

• Resistivity

• Diffusion Current

• Einstein Relationship

• Summary

14
Mobility limited by different scatterings

15
 Lattice scattering (Phonon scattering)

Electron mobility vs temperature Hole mobility vs temperature

Temperature ↑ Lattice scattering↑ mobility↓

1 1
 phonon   phonon    T −3 / 2

phonon density  carrier thermal velocity T  T 1 / 2

16
 Ionized impurity scattering

Doping concentration↑ Ionized impurity scattering↑ mobility↓

17
Temperature dependence of impurity scattering

Boron Ion Electron

_
- -
Electron +
Arsenic
Ion

There is less change in the direction of travel if the electron zips by

the ion at a higher speed.
3 3/ 2
v T
impurity   th
Na + Nd Na + Nd

Modern Semiconductor Devices for Integrated Circuits (C. Hu) 18

Lattice scattering vs ionized impurity scattering

19
Total mobility

20
 Resistivity
Resistivity ρ [Ω · 𝑐𝑚]:
a material’s inherent resistance to current flow
a normalized resistance that does not depend on the physical dimensions

𝜌𝐿
𝑽 = 𝑅𝑰 ⇒ 𝑬𝐿 = ×𝐴×𝑱
𝐴
𝑬 = 𝜌𝑱
𝑱𝒑_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑝𝒗𝒅 = 𝑞𝑝𝜇𝑝 𝑬
𝑱𝒏_𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑛𝒗𝒅 = 𝑞𝑛𝜇𝑛 𝑬
𝑱𝒅𝒓𝒊𝒇𝒕 = 𝑞 𝜇𝑛 𝑛 + 𝜇𝑝 𝑝 𝑬
1
𝝆= 1
𝑞 𝜇𝑛 𝑛 + 𝜇𝑝 𝑝 conductivity 𝜎 = 𝜌 = 𝑞 𝜇𝑛 𝑛 + 𝜇𝑝 𝑝

1
𝝆= … 𝑛 − 𝑡𝑦𝑝𝑒 𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟
𝑞𝜇𝑛 𝑁𝐷
1
𝝆= … 𝑝 − 𝑡𝑦𝑝𝑒 𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟
𝑞𝜇𝑝 𝑁𝐴

measured ρ &
determine doping concentration
mobility-versus-doping data
21
 Resistivity vs doping concentration

1
𝝆= … 𝑛 − 𝑡𝑦𝑝𝑒 𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟
𝑞𝜇𝑛 𝑁𝐷
1
𝝆= … 𝑝 − 𝑡𝑦𝑝𝑒 𝑠𝑒𝑚𝑖𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟
𝑞𝜇𝑝 𝑁𝐴

doping concentration ↑ resistivity ↓

Impurity scattering ↑ mobility ↓ resistivity ↑ effect in region

22
Four point probes for resistivity measurement

𝑉
𝜌 = 2𝜋𝑆 Γ
𝐼
𝜋 𝑉
= ln2 𝐼 𝑡𝑜 if to << S and lateral dimension of samples >> S

S: probe to probe spacing

Γ: correction factor depends on the sample thickness(to) and on whether
the bottom of the semiconductor touches an insulator or a metal
23
 Diffusion(gradient driven motion)

The diffusing entity need not be charged and thermal motion, not inter-particle
repulsion, is the enabling action behind the diffusion process
24
Diffusion current

25
 Diffusion current
𝐹 = −𝐷𝛻𝜂 Fick’s first law
F : flux or particles/cm2s
𝜂 : the particle concentration
D : diffusion coefficient or diffusivity

𝐽 = ±𝑞𝐹

𝑑𝑝
𝐽𝑃|𝑑𝑖𝑓𝑓 = +𝑞(−𝐷𝑝 𝛻𝑝) = −𝑞𝐷𝑝 𝑖𝑛 1𝐷
𝑑𝑥
𝑑𝑛
𝐽𝑁|𝑑𝑖𝑓𝑓 = −𝑞(−𝐷𝑛 𝛻𝑛) = 𝑞𝐷𝑛 𝑑𝑥 𝑖𝑛 1𝐷

𝑱𝑷|𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑝𝒗𝒅 = 𝑞𝑝𝜇𝑝 𝑬

𝑱𝑵|𝒅𝒓𝒊𝒇𝒕 = 𝑞𝑛𝒗𝒅 = 𝑞𝑛𝜇𝑛 𝑬

𝑱𝑷 = 𝑱𝑷|𝒅𝒓𝒊𝒇𝒕 + 𝑱𝑷|𝒅𝒊𝒇𝒇 = 𝑞𝑝𝜇𝑝 𝑬 − 𝑞𝐷𝑝 𝛻𝑝

𝑱𝑵 = 𝑱𝑵|𝒅𝒓𝒊𝒇𝒕 + 𝑱𝑵|𝒅𝒊𝒇𝒇 = 𝑞𝑛𝜇𝑛 𝑬 + 𝑞𝐷𝑛 𝛻𝑛
𝑱 = 𝑱𝑵 + 𝑱𝑷

26
 Relating diffusion coefficients to mobility

Let’s derive the Einstein Relationship that

relates the diffusion coefficient to mobility
non-uniform doping
Assumptions:
• nondegenerate
𝐸𝑉 + 3𝑘𝑇 < 𝐸𝐹 < 𝐸𝐶 − 3𝑘𝑇
• non-uniformly doped
𝑛 = 𝑁𝐶 𝑒 (𝐸𝐹 −𝐸𝐶)/𝑘𝑇
• under equilibrium
𝑑𝐸𝐹 𝑑𝐸𝐹 𝑑𝐸𝐹
= = =0
𝑑𝑥 𝑑𝑦 𝑑𝑧
Fermi level inside a group of materials in intimate
contact is invariant as a function of position

27
Derivation of the Einstein relationship

𝑛 = 𝑛𝑖 𝑒 (𝐸𝐹 −𝐸𝑖)/𝑘𝑇
1 𝑑𝐸𝑖 𝑑𝐸𝐹
𝐸=𝑞 , = 0 𝑢𝑛𝑑𝑒𝑟 𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚
𝑑𝑥 𝑑𝑥
𝑑𝑛 𝑛 𝑑𝐸𝑖 𝑞
= − 𝑘𝑇𝑖 𝑒 (𝐸𝐹 −𝐸𝑖)/𝑘𝑇 = − 𝑘𝑇 𝑛𝐸
𝑑𝑥 𝑑𝑥
non-uniform doping

𝑑𝑛
𝐽𝑁 = 𝐽𝑁|𝑑𝑟𝑖𝑓𝑡 + 𝐽𝑁|𝑑𝑖𝑓𝑓 = 𝑞𝑛𝜇𝑛 𝐸 + 𝑞𝐷𝑛
𝑑𝑥
𝑞 𝐽𝑁|𝑑𝑖𝑓𝑓
𝑞𝑛𝐸 𝜇𝑛 − 𝑞𝑛𝐸 𝐷 =0
𝑘𝑇 𝑁 𝐽𝑁|𝑑𝑟𝑖𝑓𝑡

𝐷𝑁 𝑘𝑇
=
𝜇𝑛 𝑞 turns out to be valid even under
non-equilibrium conditions
𝐷𝑃 𝑘𝑇
=
𝜇𝑝 𝑞

28
 Outline
Carrier Action – part 3

• Recombination and generation

• E-k diagram for momentum consideration

• Photon- and phonon-assisted transitions

• Photogeneration

• Indirect thermal recombination-generation

• Minority carrier lifetime

29
 Recombination and generation
After a semiconductor is perturbed from the equilibrium state, the carrier
excess or deficit inside the semiconductor is stabilized (if the perturbation is
maintained) or eliminated (if the perturbation is removed) by recombination and
generation (R-G) process.

Recombination : a process whereby electrons and holes (carriers) are

annihilated or destroyed
Generation : a process whereby electrons and holes are created

R-G event occurs at localized positions in the crystal.

Therefore, the R-G action does not lead to charge transport directly.
Rather, the R-G acts to change the carrier concentrations, indirectly affecting
the current flow.

So, we are interested in the time rate of change in the carrier concentrations.

𝜕𝑛 𝜕𝑝
,
𝜕𝑡 𝜕𝑡
30
 Recombination

• R-G centers near the midgap

• Normally low concentration of R-G centers
31
 Generation

• photogeneration
• direct thermal generation

• inverse of Auger recombination

• high E-field regions

32
 Dominant R-G processes
All of the various R-G processes occur at all times in all semiconductors – even
under equilibrium conditions.

In low E- field regions of a non-degenerately doped semiconductor at room

temperature, thermal band-to-band R-G and R-G via R-G centers dominate.

The rates at which the various processes are occurring is important. 33

 E-k diagram for momentum consideration
The momentum of an electron in an energy band can assume only certain
quantized values like the electron energy.
The crystal momentum like energy must be conserved in any R-G process.
The momentum conservation plays an important role in setting the process rate.

E-k plots are efficient to discuss momentum aspects of R-G processes.

k is a wave vector, a parameter proportional to the electron crystal momentum.

Minimum Ec and maximum Ev at the same k Minimum Ec and maximum Ev at the different k
Direct semiconductors Indirect semiconductors
34
E-k diagrams for semiconductors

35
 Photon- and phonon-assisted transitions
Photons carry very little momentum and a photon assisted transition is
essentially vertical on the E-K plot.
2𝜋 2𝜋
𝒌𝒑𝒉𝒐𝒕𝒐𝒏 = ≪
𝜆 𝑎

The thermal energy associated with lattice vibrations (phonons) is very small
(10~50 meV range), whereas the phonon momentum is comparatively large.
A phonon-assisted transition is essentially horizontal.

Direct semiconductors Indirect semiconductors

Band-to-band recombination is efficient Band-to-band recombination is very slow.
Photogeneration is possible for LED & lasers R-G center recombination is dominant
36
 Photogeneration
Assume the light to be monochromatic with wavelength 𝜆 and frequency 𝜈.
If the photon energy (ℎ𝜈) is greater than the band gap energy, then the light will
be absorbed and electron-hole pairs will be created as the light passes through
the semiconductor.

The light intensity 𝐼 is given by

𝑰 = 𝑰𝒐 𝒆−𝜶𝒙
𝐼𝑜 : light intensity at x=0+
𝛼: absorption coefficient, a function of material and 𝜆.
37
 Photogeneration
𝑰 = 𝑰𝒐 𝒆−𝜶𝒙
1/𝛼 [cm] average depth of penetration of the light into a material.

1
(i) ℎ𝜈 ≈ 𝐸𝐺 → 𝑠𝑚𝑎𝑙𝑙 𝛼 → 𝑙𝑎𝑟𝑔𝑒 ≈ 1 cm in Si → 𝑒 −𝛼𝑥 ≈ 1 → 𝐺𝐿 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 →
𝛼
𝑡ℎ𝑒 𝑙𝑖𝑔ℎ𝑡 𝑖𝑠 𝑢𝑛𝑖𝑓𝑜𝑟𝑚𝑙𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑
1
(ii) 𝜆 = 0.4 𝜇𝑚 → 𝑙𝑎𝑟𝑔𝑒 𝛼 𝑆𝑖 ≈ 105 𝑐𝑚−1 → 𝑠𝑚𝑎𝑙𝑙 ≈ 100 𝑛𝑚 →
𝛼
𝑡ℎ𝑒 𝑙𝑖𝑔ℎ𝑡 𝑖𝑠 𝑚𝑜𝑠𝑡𝑙𝑦 𝑎𝑏𝑠𝑜𝑟𝑏𝑒𝑑 𝑎𝑡 𝑡ℎ𝑒 𝑠𝑢𝑟𝑓𝑎𝑐𝑒

One photon creates a pair of one electron and one hole.

𝜕𝑛 𝜕𝑝
ቚ = ቚ = 𝐺𝐿 (𝑥, 𝜆)
𝜕𝑡 𝑙𝑖𝑔ℎ𝑡 𝜕𝑡 𝑙𝑖𝑔ℎ𝑡

𝐺𝐿 𝑥, 𝜆 = 𝐺𝐿0 𝑒 −𝛼𝑥 [ number/cm3s]

38
Indirect thermal recombination-generation
𝑛0 , 𝑝0 … carrier concentrations in the material under equilibrium conditions
𝑛, 𝑝 … carrier concentrations in the material under arbitrary conditions
Δ𝑛 ≡ 𝑛 − 𝑛0 …
Δ𝑝 ≡ 𝑝 − 𝑝0 … deviations in the carrier concentrations from their equilibrium values.
can be both positive and negative, where a positive deviation
corresponds to a carrier excess and a negative deviation
corresponds to a carrier deficit.
𝑁𝑇 … number of R-G centers/cm3

Low-level injection implies

Δ𝑝 ≪ 𝑛0 , 𝑛 ≅ 𝑛0 𝑖𝑛 𝑎𝑛 𝑛 − 𝑡𝑦𝑝𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙
Δ𝑛 ≪ 𝑝0 , 𝑝 ≅ 𝑝0 𝑖𝑛 𝑎 𝑝 − 𝑡𝑦𝑝𝑒 𝑚𝑎𝑡𝑒𝑟𝑖𝑎𝑙

ex) If 𝑁𝐷 = 1014 /𝑐𝑚3 and Δ𝑝 = Δ𝑛 = 109 /𝑐𝑚3 , low-level injection?

39
Indirect thermal recombination-generation

recombination rate is proportional to electron-filled R-G centers and holes

𝜕𝑝
→ 𝜕𝑡 ቚ = −𝑐𝑝 𝑁𝑇 𝑝
𝑅

generation rate is proportional to empty R-G centers

𝜕𝑝 𝜕𝑝 𝜕𝑝
→ 𝜕𝑡 ቚ = 𝜕𝑡 ቚ = 𝜕𝑡 ቚ = 𝑐𝑝 𝑁𝑇 𝑝𝑜
𝐺 𝐺−𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚 𝑅−𝑒𝑞𝑢𝑖𝑙𝑖𝑏𝑟𝑖𝑢𝑚

40
 Indirect thermal recombination-generation
𝜕𝑝 𝜕𝑝 𝜕𝑝
ቚ = ቚ + ቚ = −𝑐𝑝 𝑁𝑇 𝑝 − 𝑝𝑜 = −𝑐𝑝 𝑁𝑇 ∆𝑝 For holes in n-type material
𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑅 𝜕𝑡 𝐺
𝑅−𝐺
𝜕𝑛 𝜕𝑛 𝜕𝑛
ቚ = ቚ + ቚ = −𝑐𝑛 𝑁𝑇 𝑛 − 𝑛𝑜 = −𝑐𝑛 𝑁𝑇 ∆𝑛 For electrons in p-type material
𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑅 𝜕𝑡 𝐺
𝑅−𝐺
𝑐𝑛 , 𝑐𝑝 : 𝑐𝑎𝑝𝑡𝑢𝑟𝑒 𝑐𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 1 1
𝜏𝑝 = 𝑐 𝜏𝑛 = 𝑐
𝑝 𝑁𝑇 𝑛 𝑁𝑇

𝜕𝑝 ∆𝑝
ቤ =− for minority carrier holes in n-type material under low level injection
𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜏𝑝
𝑅−𝐺

𝜕𝑛 ∆𝑛
ቤ =− for minority carrier electrons in p-type material under low level injection
𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜏𝑛
𝑅−𝐺

𝜕𝑝 𝜕𝑛 𝑛𝑖2 − 𝑛𝑝 for both carrier types and

ቤ = ቤ = arbitrary injection levels
𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑖−𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜏𝑝 𝑛 + 𝑛1 + 𝜏𝑛 (𝑝 + 𝑝1 )
𝑅−𝐺 𝑅−𝐺 in a nondegenerate semiconductor
where 𝑛1 ≡ 𝑛𝑖 𝑒 (𝐸𝑇 −𝐸𝑖 )/𝑘𝑇 , 𝑝1 ≡ 𝑛𝑖 𝑒 (𝐸𝑖 −𝐸𝑇 )/𝑘𝑇

ex) Does the general R-G equation reduce to the equation for the case of
minority carriers & low-level injection under the assumed special-case conditions?
41
 Minority carrier lifetime 𝜏𝑛 , 𝜏𝑝
Minority carrier lifetime:
Average time an excess minority carrier will live in a sea of majority carriers.
< 𝒕 > = 𝝉𝒏 (𝒐𝒓 𝝉𝒑 )
Depends on the often poorly controlled R-G center concentration (𝑁𝑇 ).
𝑁𝑇 changes even within a given sample during fabrication.
Gettering reduces the R-G center concentration, leading to 𝜏𝑛 (𝑜𝑟 𝜏𝑝 ) of 1 msec in Si.
Gold introduction to Si increases R-G centers, leading to 𝜏𝑛 (𝑜𝑟 𝜏𝑝 ) of ~1 nsec in Si.

Minority carrier lifetime measurement:

42
 Outline
Carrier Action – part 4

• Continuity equations

• Minority carrier diffusion equations

• Common diffusion equation simplifications

• Sample problems

• Minority carrier diffusion lengths

• Quasi-Fermi levels

43
 Continuity equations
𝜕𝑛 𝜕𝑛 𝜕𝑛 𝜕𝑛 𝜕𝑛
= ቤ + ቤ + ቤ + ቤ
𝜕𝑡 𝜕𝑡 𝑑𝑟𝑖𝑓𝑡 𝜕𝑡 𝑑𝑖𝑓𝑓 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
𝜕𝑝 𝜕𝑝 𝜕𝑝 𝜕𝑝 𝜕𝑝
= ቤ + ቤ + ቤ + ቤ
𝜕𝑡 𝜕𝑡 𝑑𝑟𝑖𝑓𝑡 𝜕𝑡 𝑑𝑖𝑓𝑓 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)

𝜕𝑛 𝜕𝑛 1
ቤ + ቤ = 𝛻 ∙ 𝐽𝑁
𝜕𝑡 𝑑𝑟𝑖𝑓𝑡 𝜕𝑡 𝑑𝑖𝑓𝑓 𝑞 change in carrier concentrations if an
imbalance exists between the total carrier
𝜕𝑝 𝜕𝑝 1 currents into and out of the region.
ቤ + ቤ = − 𝛻 ∙ 𝐽𝑃
𝜕𝑡 𝑑𝑟𝑖𝑓𝑡 𝜕𝑡 𝑑𝑖𝑓𝑓 𝑞

𝜕𝑛 1 𝜕𝑛 𝜕𝑛
= 𝛻 ∙ 𝐽𝑁 + ቤ + ቤ
𝜕𝑡 𝑞 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
𝜕𝑝 1 𝜕𝑝 𝜕𝑝
= − 𝛻 ∙ 𝐽𝑃 + ቤ + ቤ
𝜕𝑡 𝑞 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
44
 Continuity equations

𝜕𝑛 1 𝜕𝑛 𝜕𝑛
= 𝛻 ∙ 𝐽𝑁 + ቤ + ቤ
𝜕𝑡 𝑞 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
𝜕𝑝 1 𝜕𝑝 𝜕𝑝
= − 𝛻 ∙ 𝐽𝑃 + ቤ + ቤ
𝜕𝑡 𝑞 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
45
 Minority carrier diffusion equations
Simplifying assumptions:
① 1D, ② Minority carriers, ③ ℰ ≈ 0, ④ 𝑛0 ≠ 𝑛0 𝑥 , 𝑝0 ≠ 𝑝0 𝑥 ,
⑤ Low-level injection, ⑥ Indirect thermal recombination-generation is dominant,
⑦ No “other processes” except possibly photogeneration

𝜕𝑛 1 𝜕𝑛 𝜕𝑛
= 𝛻 ∙ 𝐽𝑁 + ቤ + ቤ
𝜕𝑡 𝑞 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
1 1 𝜕𝐽𝑁
𝛻 ∙ 𝐽𝑁 → by ①
𝑞 𝑞 𝜕𝑥
𝜕𝑛
𝐽𝑁 = 𝑞𝜇𝑛 𝑛ℰ + 𝑞𝐷𝑁 𝜕𝑥 ( 𝑛ℰ → 0 due to small n by ② and small ℰ by ③)
𝜕𝑛 𝜕𝑛 𝜕𝑛0 𝜕∆𝑛 𝜕∆𝑛
≅ 𝑞𝐷𝑁 𝜕𝑥 ( 𝜕𝑥 = + = by ④)
𝜕𝑥 𝜕𝑥 𝜕𝑥
𝜕∆𝑛
= 𝑞𝐷𝑁 𝜕𝑥
1 𝜕2 ∆𝑛
𝛻 ∙ 𝐽𝑁 → 𝐷𝑁 2
𝑞 𝜕𝑥

𝜕𝑛 𝜕 2 ∆𝑛 𝜕𝑛 𝜕𝑛
= 𝐷𝑁 + ቤ + ቤ
𝜕𝑡 𝜕𝑥 2 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
46
 Minority carrier diffusion equations
Simplifying assumptions:
① 1D, ② Minority carriers, ③ ℰ ≈ 0, ④ 𝑛0 ≠ 𝑛0 𝑥 , 𝑝0 ≠ 𝑝0 𝑥 ,
⑤ Low-level injection, ⑥ Indirect thermal recombination-generation is dominant,
⑦ No “other processes” except possibly photogeneration

𝜕𝑛 𝜕 2 ∆𝑛 𝜕𝑛 𝜕𝑛
= 𝐷𝑁 + ቤ + ቤ
𝜕𝑡 𝜕𝑥 2 𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝜕𝑡 𝑜𝑡ℎ𝑒𝑟 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝑅−𝐺 (𝑙𝑖𝑔ℎ𝑡, 𝑒𝑡𝑐.)
𝜕𝑛
ቚ 𝑜𝑡ℎ𝑒𝑟 = 𝐺𝐿 by ⑦
𝜕𝑡
𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠
𝜕𝑛 ∆𝑛
ቚ =−𝜏 by ②, ⑤, ⑥
𝜕𝑡 𝑡ℎ𝑒𝑟𝑚𝑎𝑙 𝑛
𝑅−𝐺
𝜕𝑛 𝜕𝑛0 𝜕∆𝑛 𝜕∆𝑛 𝜕𝑛0
= + = by equilibrium conditions where =0
𝜕𝑡 𝜕𝑡 𝜕𝑡 𝜕𝑡 𝜕𝑡

𝜕∆𝑛𝑝 𝜕 2 ∆𝑛𝑝 ∆𝑛𝑝

= 𝐷𝑁 − + 𝐺𝐿 Minority carrier diffusion equations are valid
𝜕𝑡 𝜕𝑥 2 𝜏𝑛
𝜕∆𝑝𝑛 𝜕 2 ∆𝑝𝑛 ∆𝑝𝑛 only for minority carriers, electrons in p-type
= 𝐷𝑃 − + 𝐺𝐿 and holes in n-type.
𝜕𝑡 𝜕𝑥 2 𝜏𝑝
47
Common diffusion equation simplifications
𝜕∆𝑛𝑝 𝜕 2 ∆𝑛𝑝 ∆𝑛𝑝
= 𝐷𝑁 − + 𝐺𝐿
𝜕𝑡 𝜕𝑥 2 𝜏𝑛
𝜕∆𝑝𝑛 𝜕 2 ∆𝑝𝑛 ∆𝑝𝑛
= 𝐷𝑃 − + 𝐺𝐿
𝜕𝑡 𝜕𝑥 2 𝜏𝑝

48
Common special-case diffusion equation solutions

49
 Sample problem No. 1
Problem: A uniformly donor-doped silicon wafer maintained at room
temperature is suddenly illuminated with light at time t=0. Assuming ND=1015
/cm3, 𝜏𝑝 = 10−6 sec, and a light-induced creation of 1017 electrons and holes
per cm3-sec throughout the semiconductor, determine ∆pn(t) for t > 0.
Solution: Assumptions:
𝜕∆𝑝𝑛 𝜕 2 ∆𝑝𝑛 ∆𝑝𝑛 ✓ ① 1D,
= 𝐷𝑃 − + 𝐺𝐿 ✓ ② Minority carriers,
𝜕𝑡 𝜕𝑥 2 𝜏𝑝
✓ ③ ℰ ≈ 0,
𝜕∆𝑝𝑛 ∆𝑝𝑛
=− + 𝐺𝐿 ✓ ④ 𝑛0 ≠ 𝑛0 𝑥 , 𝑝0 ≠ 𝑝0 𝑥 ,
𝜕𝑡 𝜏𝑝
✓ ⑤ Low-level injection,
b.c.(boundary condition): ∆𝑝𝑛 (𝑡)|𝑡=0 = 0
✓ ⑥ Indirect thermal recombination-
∆𝑝𝑛 𝑡 = 𝐺𝐿 𝜏𝑝 + 𝐴𝑒 −𝑡/𝜏𝑝 generation is dominant,
By applying b.c., 𝐴 = −𝐺𝐿 𝜏𝑝 ✓ ⑦ No “other processes” except
∆𝑝𝑛 𝑡 = 𝐺𝐿 𝜏𝑝 (1 − 𝑒 −𝑡/𝜏𝑝 ) possibly photogeneration

𝑛0 = 𝑁𝐷 = 1015 /𝑐𝑚3
𝑛𝑖2
𝑝0 = = 105 /𝑐𝑚3
𝑁𝐷
∆𝑝𝑛 ∞ = 𝐺𝐿 𝜏𝑝 < 𝑛0

50
 Light-on/light-off measurements

When the light is turned off,

𝜕∆𝑛𝑝 𝜕 2 ∆𝑛𝑝 ∆𝑛𝑝
= 𝐷𝑁 − + 𝐺𝐿
𝜕𝑡 𝜕𝑥 2 𝜏𝑛

51
 Sample problem No. 2
Problem: The x=0 end of a uniformly doped semi-infinite bar of silicon with
ND=1015 /cm3 is illuminated so as to create ∆𝑝𝑛0 = 1010 /𝑐𝑚3 excess holes at
x=0. The wavelength of the illumination is such that no light penetrates into the
interior (x>0) of the bar. Determine ∆𝑝𝑛 𝑥 .

Solution:
𝜕∆𝑝𝑛 𝜕 2 ∆𝑝𝑛 ∆𝑝𝑛
= 𝐷𝑃 − + 𝐺𝐿
𝜕𝑡 𝜕𝑥 2 𝜏𝑝
𝜕 2 ∆𝑝𝑛 ∆𝑝𝑛
0 = 𝐷𝑃 − 𝑓𝑜𝑟 𝑥 > 0
𝜕𝑥 2 𝜏𝑝 Assumptions:
b.c.: ∆𝑝𝑛 |𝑥=0+ = ∆𝑝𝑛 |𝑥=0 = ∆𝑝𝑛0 ✓ ① 1D,
✓ ② Minority carriers,
∆𝑝𝑛 ቚ =0
𝑥→∞ ✓ ③ ℰ ≈ 0 ← small ∆𝑝𝑛 |𝑚𝑎𝑥 and
∆𝑝𝑛 𝑥 = 𝐴𝑒 −𝑥/𝐿𝑝 + 𝐵𝑒 𝑥/𝐿𝑝 𝑤ℎ𝑒𝑟𝑒 𝐿𝑝 ≡ 𝐷𝑝 𝜏𝑝 electron redistribution
✓ ④ 𝑛0 ≠ 𝑛0 𝑥 , 𝑝0 ≠ 𝑝0 𝑥 ,
By applying b.c., 𝐵 = 0 𝑎𝑛𝑑 𝐴 = ∆𝑝𝑛0
✓ ⑤ Low-level injection,
∆𝑝𝑛 𝑥 = ∆𝑝𝑛0 𝑒 −𝑥/𝐿𝑝
✓ ⑥ Indirect thermal recombination-
generation is dominant,
✓ ⑦ No “other processes” except
possibly photogeneration

52
 Minority carrier diffusion lengths
𝐿𝑝 ≡ 𝐷𝑝 𝜏𝑝 for minority carrier holes in an n-type material
𝐿𝑛 ≡ 𝐷𝑛 𝜏𝑛 for minority carrier electrons in a p-type material

Represent the average distance minority carriers can diffuse into a sea of
majority carriers before being annihilated.

ex) T=300K, ND=1015 /cm3 doped Si, 𝜏𝑝 = 10−6 sec

𝑘𝑇
𝐿𝑝 ≡ 𝐷𝑝 𝜏𝑝 = 𝜇𝑝 𝜏𝑝 = 0.0259 458 10−6 1/2
𝑞
= 3.44 × 10−3 𝑐𝑚

53
 Quasi-Fermi levels
Quasi-Fermi levels are energy levels used to specify the carrier concentrations
under non-equilibrium conditions.

equilibrium non-equilibrium

𝑛
𝑛0 = 𝑛𝑖 𝑒 (𝐸𝐹 −𝐸𝑖)/𝑘𝑇 𝑛 = 𝑛𝑖 𝑒 (𝐹𝑁 −𝐸𝑖)/𝑘𝑇 or 𝐹𝑁 = 𝐸𝑖 + 𝑘𝑇𝑙𝑛(𝑛 )
𝑖
𝑝
𝑝0 = 𝑛𝑖 𝑒 (𝐸𝑖−𝐸𝐹 )/𝑘𝑇 𝑝 = 𝑛𝑖 𝑒 (𝐸𝑖−𝐹𝑃 )/𝑘𝑇 or 𝐹𝑃 = 𝐸𝑖 − 𝑘𝑇𝑙𝑛(𝑛 )
𝑖

𝑛0 = 𝑁𝐷 = 1015 /𝑐𝑚3 𝑛 ≅ 𝑛0 = 𝑁𝐷 = 1015 /𝑐𝑚3

𝑛𝑖2 ∆𝑝𝑛 𝑡 = 𝐺𝐿 𝜏𝑝 = 1011 /𝑐𝑚3
𝑝0 = = 105 /𝑐𝑚3
𝑁𝐷 𝑝 = 𝑝0 + ∆𝑝 ≅ 1011 /𝑐𝑚3

54
Currents derived from quasi-Fermi levels

55