Chapter 3: Carrier Action
This is the band diagram of solar cell
having p-type and n-type material
fused together to make a device. We
will see how charge carriers are
generated and how they are
transported within the semiconductor
in this chapter
Haris Mehmood, Hisham Nasser, Engin Ozkol, et al. “Simulation of an efficient silicon
heterostructure solar cell concept featuring molybdenum oxide carrier‐selective contact” Int J
Energy Res., Volume 42, pp.1563–1579, 2018. (https://doi.org/10.1002/er.3947)
Types of carrier action
Drift
Diffusion
Recombination-generation
3.1.1. Drift
• Movement of charged particle in response to an applied
electric field
• Collision with lattice atoms and interruption in carrier
acceleration give rise to scattering of charge carriers
3.1.2. Drift Current
3.1.2. Drift Current in p-type Semiconductor
3.1.2. Hole Drift (Current) & (Current Density)
hole drift current density
3.1.2. Drift Velocity in Pure Silicon
As the applied electric field increases, the carrier velocity no longer
increases because the carriers lose energy through increased levels of
interaction with the lattice, by emitting phonons or photons
3.1.2. Drift Current Densities
hole drift current density
electron drift current density
𝝁𝒏 = electron mobility
𝝁𝒑 = hole mobility
3.1.3. Mobility
Mobility is the measure of ease of carrier movement in crystal
Scattering of carriers in lattice has inverse relationship with mobility
𝜇 = 𝑞 < 𝜏 >/𝑚∗
where 𝜏 is the mean free time between collisions and m* is the effective
mass
𝒄𝒎𝟐 /𝑽𝒔 = standard unit of mobility
3.1.3. Doping Dependence Si @ T=300 K
𝑵𝑨 𝒐𝒓 𝑵𝑫 = dopant concentration
Lattice scattering dominant for < 1015 cm-3 (scattering of carriers by interaction with atoms in a lattice)
Impurity scattering dominant for > 1015 cm-3 (scattering of charge carriers by dopant atoms in lattice
3.1.3. Doping Dependence GaAs@T=300 K
𝑵𝑨 𝒐𝒓 𝑵𝑫 = dopant concentration
Why mobility µ of GaAs is larger than Si? Because effective mass of carriers in GaAs is lighter
than in Si
3.1.3. Temperature Dependence ( 𝜇𝑛 𝑣𝑠. 𝑇)
For electrons
Lattice scattering is a dominant
scattering mechanism in low
doping range
Decreasing the system
temperature causes an ever-
decreasing thermal agitation
of semiconductor lattice
atoms
So mobility increases
3.1.3. Temperature Dependence ( 𝜇𝑝 𝑣𝑠. 𝑇)
For holes
3.1.4. Resistivity (𝜌)
Resistivity (𝜌): material’s inherent property to resist flow of current
Where µn is an electron mobility and ND is donor concentration (or amount of n-type dopant atoms introduced in the material)
Where µp is the hole mobility and NA is an acceptor concentration (or amount of p-type dopant atoms introduced in the material)
3.1.4. Resistivity vs. 𝑁𝐴 𝑜𝑟 𝑁𝐷
• Considering the mobilities of
electrons and holes for Si at 300 K
temperature (see Slide 10)
• The resistivity decreased with an
increasing dopant concentration
because of the inverse relationship
between resistivity and dopant
concentration as shown in the
expression in the previous slide
• The decrease in resistivity means
increase in conductivity of the
semiconductor
3.1.4. How to measure resistivity?
Commonly used technique is 4-point probe method
V – voltage
I - current
s is the distance between any two
probes
Γ is the correction factor that depends
on semiconductor thickness and
whether the bottom of semiconductor is
touching metal or an insulator
3.1.4. Four-point probe system
This equipment tells us the resistivity reading
Semiconductor wafer placed on insulating medium
You can see the four probes here with uniform distance s
3.1.5. Band bending
K.E. = how farther an
electron is away from the
K.E. of electrons = E – EC conduction band edge and
vice versa for hole
K.E. of holes = EV – E P.E. = distance between
reference level and
conduction band
P.E. of electrons = EC – Eref Total E = K.E. + P.E.
P.E. of holes = Eref – EV
The potential graph is just
the invert of energy band
1
𝑉 = − (𝐸𝐶 − 𝐸𝑟𝑒𝑓 ) curve
𝑞
𝑑𝑉 1 𝑑𝐸𝑐 1 𝑑𝐸𝑉 1 𝑑𝐸𝑖
𝐸 = − = = =
𝑑𝑥 𝑞 𝑑𝑥 𝑞 𝑑𝑥 𝑞 𝑑𝑥
The electric field is the slope of the energy band curve
Example 3.2
Example 3.2
Solution Example 3.2 (a)
Solution Example 3.2 (b)
Solution Example 3.2 (c)
Solution Example 3.2 (d)
3.2. Diffusion
Spreading of particles due to either
1) Random thermal motion
2) Macroscopic movement from high concentration region to low
concentration region
3.2.1. Diffusion Example
With the passage of time,
number of particles in
different compartments
becomes almost same. An
example of diffusion where
particles try to distribute in
medium uniformly.
3.2.1. Diffusion in Semiconductor Devices
The hole diffusion current
density is in the same direction
as hole movement
Diffusion of holes and electrons in P-
type and N-type media respectively
from high concentration to low
concentration region
The electron diffusion
current density is in
opposite direction to that
of electron movement
3.2.2. Hot point probe measurement
A technique to determine type (P or N) of semiconductor material
It also provides information about diffusion process
3.2.3. Flux (flow) of Particles in Diffusion
ℱ = −𝐷 𝛻𝜼
ℱ = flux of particles (particles per cm2 second)
𝛻𝜼 = concentration gradient of particles
𝑫 = Diffusion coefficient
3.2.3. Flux (flow) of Particles in Diffusion
Fick’s Law describes diffusion as the flux, F, of particles in our case, is
proportional to the gradient in concentration.
ℱ = −𝐷 𝛻𝜼
ℱ = flux of particles (particles per cm2 second)
𝛻𝜼 = concentration gradient of particles
𝑫 = Diffusion coefficient
3.2.3. Diffusion Current Density
Diffusion current density due to holes
Diffusion current density due to electrons
𝛻𝒑 = concentration gradient of holes
𝛻𝒏 = concentration gradient of holes
Dp = Diffusion coefficients of holes
Dn = Diffusion coefficients of electrons
3.2.3. Total Current
3.2.4. Connection between ‘D’ and 𝜇𝑛
𝐷𝑁 = diffusion coefficient of electrons
𝑘 = Boltzman constant = 1.38 × 10−23 J. K −1
𝑇 = temperature K
𝜇𝑛 = electron mobility
𝑞 = charge on electron
3.2.4. Connection between ‘D’ and 𝜇𝑃
𝐷𝑁 = diffusion coefficient of holes
𝑘 = Boltzman constant = 1.38 × 10−23 J. K -1
𝑇 = temperature K
𝜇𝑛 = electron mobility
𝑞 = charge on electron
Exercise 3.4
Solution
