程序代写 Semiconductor Fundamentals – (III)

Semiconductor Fundamentals – (III)
2.5 Boltzmann approximation & E , n, p F
2.6 Carrier drift and diffusion
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Semiconductor Fundamentals – (III)
2.5 Boltzmann approximation & E , n, p F
2.6 Carrier drift and diffusion

Last lecture:
Negative charges:
Conduction electrons (density = n)
Ionized acceptor atoms (density = NA-) immobile
Positive charges:
Holes (density = p) mobile
Ionized donor atoms (density = ND+)
The net charge density (C/cm3) in a semiconductor is =q(p−n+N+ −N−)
DA Law of Mass Action: n×p = ni2
How to deduce the relationship between EF and n/p?

2.5 Boltzmann approximation & EF, n, p
➢ Fermi function and Fermi level ➢ Density of States
➢ Boltzmann Approximation
➢ Electron and Hole Concentrations

Thermal Equilibrium
No external forces are applied: electric field = 0, magnetic field = 0 mechanical stress = 0
Dynamic situation in which every process is balanced by its inverse process
Electron-hole pair (EHP) generation rate = EHP recombination rate
Thermal agitation→electrons and holes exchange
energy with the crystal lattice and each other
→ Every energy state in the conduction band and valence band has a certain probability of being occupied by an electron

Statistical Thermodynamics: Fermi energy
⚫ The Fermi energy, EF, is the energy associated with a particle, which is in thermal equilibrium with the system of interest. The energy is strictly associated with the particle and does not consist even in part of heat or work. This same quantity is called the electro- chemical potential, m, in most thermodynamics texts.
⚫ http://hyperphysics.phy- astr.gsu.edu/Hbase/solids/fermi.html#c2
⚫ http://hyperphysics.phy- astr.gsu.edu/Hbase/solids/fermi.html#c1

Probability that a state at energy level, E, is occupied
Fermi function and Fermi level
by one electron is, f(E)= 1
1+exp(E−EF ) kT
f(E): Fermi-Dirac function
An increase in E will reduce f(E)
EF — Fermi-level
When E = EF, f(E=EF) = 0.5.
textbook P.66
The first order approximation
f(E)=0 for E>EF f(E)=1 for E> 1
If EF −E3kT, f(E)1−e(E−EF )/kT
Probability that a state is empty:
1− f(E)e(E−EF )/kT =e−(EF −E)/kT
Probability that a state is occupied by a hole

2.5 Boltzmann approximation & EF, n, p
➢ Fermi function and Fermi level
➢ Density of States
➢ Boltzmann Approximation
➢ Electron and Hole Concentrations

Equilibrium Distribution of Electrons
⚫ Obtainn(E)bymultiplyinggc(E)andf(E)
Energy band diagram
Density of States
Probability of occupancy
Carrier distribution
gc(E) gv(E)

Equilibrium Electron Concentrations Integrate n(E) over all the energies in the
conduction band to obtain n:
n = topof conductionband gc(E)f(E)dE
By using the Boltzmann approximation, and extending the integration limit to , we obtain
−(E −E )/kT n=Nce c F

whereNc=2 n   h2 
2m*kT 3/2  

Equilibrium Distribution of Holes
⚫ Obtain p(E) by multiplying gv(E) and 1-f(E)
Energy band diagram
Density of States
Probability of occupancy
Carrier distribution
gc(E) gv(E)

Equilibrium Hole Concentrations Integrate p(E) over all the energies in the
valence band to obtain p:
p= Ev g(E)1−f(E)dE
bottomof valenceband
By using the Boltzmann approximation, and extending the integration limit to -, we obtain
−(E −E )/kT p=Nve F v

whereNv=2 h2  
 2m* kT   p 

Intrinsic Carrier Concentration
np=(Nce−(Ec−EF)/kT )(Nve−(EF−Ev)/kT ) =NcNve−(Ec−Ev)/kT =NcNve−Eg /kT
Law of Mass Action
n= NNe−Eg/2kT icv

Electron and hole concentrations
n=N exp−(EC −EF) C  kT 
p=N exp−(EF −EV) V  kT 
n p = n2 i
At RT p = NA
Ec At RT
p.83,ref1, E
n=N exp−(EC−Ei) i C  kT 
n=N exp−(Ei−EV)
iVkT E Ec
n=nexp(EF −Ei) i  kT 
p=nexp−(EF −Ei) i  kT 

HW3: Energy-band diagram Question: Where is EF for n = 1017 cm-3 ?
n=nexp(EF −Ei) i  kT 

Semiconductor Fundamentals – (III)
2.5 Boltzmann approximation & E , n, p F
2.6 Carrier drift and diffusion

2.6 Carrier drift and diffusion
➢ Carrier scattering
➢ Carrier drift:
▪ Carrier mobility
▪ Conductivity & Resistivity ▪ Energy band model
➢ Carrier diffusion Reading: Chapter 2.6

Thermal Motion
In thermal equilibrium, carriers are not sitting still:
undergo collisions with vibrating Si atoms (Brownian motion)
electrostatically interact with charged dopants and with each other
Characteristic time constant of thermal motion
mean free time between collisions: tc ≡ collision time [s]
In between collisions, carriers acquire high velocity: vth ≡ thermal velocity [cm/s]
…but get nowhere! (on average)
Characteristic length of thermal motion:
l ≡ mean free path [cm], l= vth tc

Carrier Scattering
electron 5
random motion 4
Mobile electrons and atoms in the Si lattice are always in
random thermal motion.
Average velocity of thermal motion for electrons in Si:
~107 cm/s @ 300K
Electrons make frequent “collisions” with the vibrating
“Lattice Scattering” or “Phonon Scattering”
Other scattering mechanisms:
deflection by ionized impurity atoms
deflection due to Coulombic force between carriers
The average current in any direction is zero, if no electric field is applied.

Effective Mass
⚫ Under an externally applied force, Fext, the movement of electrons (or holes) is influenced by the positively charged protons and by negatively charged electron in the lattice. So, the movement in the crystal is different from that in vacuum.
⚫ The total force Ftotal
F =F +F =ma
total ext int
where a is the acceleration, Fint is the
internal force. We can write
where m* is called effective mass.
Notation: mn* for electrons, m*p
for holes,

Electrons as Moving Particles In vacuum In semiconductor
F = (-q)E = moa Fext = (-q)E = mn*a where mn* is the electron effective mass.
If tcn is electron mean free time between collisions, |a| = dv/dt ≈ ve/tcn  v = qtcnE, v = qtcpE
|a|=qE/m* e * h * n mn mp
average drift velocity

2.6 Carrier drift and diffusion
➢ Carrier scattering
➢ Carrier drift:
▪ Carrier mobility
▪ Conductivity & Resistivity ▪ Energy band model
➢ Carrier diffusion

Carrier Drift
When an electric field (e.g., due to an externally applied voltage) is applied to a semiconductor, mobile charge- carriers will be accelerated by the electrostatic force. This force superimposes on the random motion of electrons:
Electrons drift in the direction opposite to the E-field →Current flows
electron 5
❖ Because of scattering, electrons in a semiconductor do not achieve constant acceleration. However, they can be viewed as classical particles moving at a constant average drift velocity.

Carrier Drift
The process in which charged particles move because of an electric field is called drift.
Charged particles within a semiconductor move with an average velocity proportional to the electric field.
The proportionality constant is the carrier mobility. +-
→→ Holevelocity vh =p E
→→ Electron velocity ve = −n E

Carrier Drift
ve =qtcnE, vh =qtcpE  m* m*
n =qtcn , p =qtcp
→→ Holevelocity vh =p E
→→ Electron velocity ve = −n E
p  hole mobility (cm2/V·s)
n  electron mobility (cm2/V·s)

1/ =1/L +1/I
I T+3/2 /NI

Drift Velocity and Carrier Mobility
Mobile charge-carrier drift velocity is proportional to applied E-field:
 is the mobility
(Units: cm2/V•s)
Note: Carrier mobility depends on total dopant concentration (ND +NA)!

Drift Current
⚫ Drift current is proportional to the carrier velocity and carrier concentration:
1) p—hole density
2) q =1.610-19 C
— One electron charge
3) Charges passing through ‘A’ per second
— The definition of current.
P-type sem.
vh t A = volume from which all holes cross plane in time t
p vh t A = # of holes crossing plane in time t
q p vh t A = charge crossing plane in time t
q p vh A = charge crossing plane per unit time = hole current
➔ Hole current per unit area (i.e. current density) Jp,drift = q p vh

Electrical Conductivity s
When an electric field is applied, current flows due to
drift of mobile electrons and holes:
electroncurrentdensity: Jn =(−q)nve =qnnE
Negatively charged electron
Direction of electron drift
holecurrentdensity:
Jp =(+q)pvh =qppE E
J=Jn +Jp =(qnn +qpp)E
Units: (W•cm)-1 35
total current density:
conductivity
s qnn +qpp

Electrical Resistivity 
s qnn +qpp
  1 qp p
for n-type material
for p-type material (Units: ohm•cm)

⚫ Estimate the resistivity of a Si sample doped with phosphorus to a
concentration of 1015 cm-3 and boron to a
concentration of 1017 cm-3.
⚫ The electron mobility and hole mobility are 700 cm2/Vs and 350 cm2/Vs, respectively. (Why??)

Consider a Si sample doped with 1016/cm3 Boron.
What is its resistivity? Answer:
NA = 1016/cm3 , ND = 0 (NA >> ND →p-type) → p  1016/cm3 and n  104/cm3
=11 qnn +qpp qpp
= (1.6 10
−1 )(10 )(450)
−19 16 Fromvs.(NA +ND )plot
= 1.4 W  cm

Consider a Si sample doped with 1016/cm3 Boron.
What is its resistivity?
NA = 1016/cm3 , ND = 0 (NA >> ND →p-type)
→ p  1016/cm3 and n  104/cm3
=11 qnn +qpp qpp
= (1.6 10
−1 )(10 )(450)
−19 16 Fromvs.(NA +ND )plot
= 1.4 W  cm

Consider the same Si sample, doped additionally
with 1017/cm3 Arsenic. What is its resistivity? Answer:
NA = 1016/cm3, ND = 1017/cm3 (ND>>NA→n-type) → n  9×1016/cm3 and p  1.1×103/cm3
=11 qnn +qpp qnn
)(9 10 )(700) = 0.10 W  cm
= (1.6 10
The sample is converted to n-type material by adding more donors than
acceptors, and is said to be “compensated”. Fromvs.(NA +ND )plot

Consider the same Si sample, doped additionally
with 1017/cm3 Arsenic. What is its resistivity?
NA = 1016/cm3, ND = 1017/cm3 (ND>>NA→n-type)
→ n  9×1016/cm3 and p  1.1×103/cm3 p
=11 qnn +qpp qnn
= (1.6 10
)(9 10 )(700) = 0.10 W  cm
The sample is converted to n-type material by adding more donors than acceptors, and is said to be “compensated”.

Electrons and Holes (Band Model)
Electrons and holes tend to seek lowest- energy positions
Electrons tend to fall
Holes tend to float up (like bubbles in water)
electron kinetic energy

hole kinetic energy
Increasing hole energy
Increasing electron energy

Potential vs. Kinetic Energy electron kinetic energy

Ec represents the electron potential energy: P.E.=Ec −Ereference
hole kinetic energy
increasing electron energy
increasing hole energy

Electrostatic Potential, V 0.7V
The potential energy of a particle with charge -q is related
to the electrostatic potential V(x): V(x)
– Ec(x) Ef(x)
P.E.=−qV V=1(E x −E)
Ereference is 0

Electric Field, e 0.7V
Variation of E Ec
V(x) q reference
– Ec(x) Ef(x)
e = − dV = 1 dEc
e = − dV dx
with position is called “band bending.” x

HW 5: Carrier Drift (Band Diagram Visualization)
e = − dV = 1 dEc dx qdx
Q1: what is the direction of electric field?
Q2: what is the direction of carriers’ drift?

2.6 Carrier drift and diffusion
➢ Carrier scattering
➢ Carrier drift:
▪ Carrier mobility
▪ Conductivity & Resistivity ▪ Energy band model
➢ Carrier diffusion

⚫ Diffusion occurs when there exists a concentration
⚫ In the figure below, imagine that we fill the left
chamber with a gas at temperate T
⚫ If we suddenly remove the divider, what happens?
⚫ The gas will fill the entire volume of the new chamber.
⚫ How does this occur?
•••••• •• • • •
••••• ••••••

⚫ Particles diffuse from higher concentration to lower concentration locations.

Carrier Diffusion
Due to thermally induced random motion, mobile particles tend to move from a region of high concentration to a region of low concentration.
Analogy: ink droplet in water
Current flow due to mobile charge diffusion is
proportional to the carrier
concentration gradient
The proportionality constant is the diffusion constant.
J =−qD dp p pdx
Dp  hole diffusion constant (cm2/s)
Dn  electron diffusion constant (cm2/s)

Carrier Diffusion
Current flow due to mobile charge diffusion is proportional to the carrier concentration gradient.
The proportionality constant is the diffusion constant.

Diffusion Examples
Linear concentration profile Non-linear concentration profile →constant diffusion current →varying diffusion current
p = N1− x   L 
− x p = N e x p L d
=−qD dp J p dx
=−qD dp p dx
=qDpNexp−x pL LdLd

Total Diffusion Current
⚫ Due to the non-uniform distribution of carriers
⚫ Dn — electron diffusion constant
⚫ Driving force: thermal energy, not electric field
⚫ dn/dx—densitygradient
⚫ Total diffusion current ➢ J = Jn + Jp

Total Diffusion Current
Diffusion current within a semiconductor consists of hole and electron components:
p,diff tot,diff
=−qD dp J =qD dn p dx n,diff n dx
=q(D dn−D dp) n dx p dx

Total current
The total current flowing in a semiconductor is the sum of drift current and diffusion current:
Jtot =Jp,drift +Jn,drift +Jp,diff +Jn,diff
Jp,drift =qppE,
Jn,drift =qnnE
=−qD dp, p dx
J =qD n,diff n

The characteristic constants for drift and diffusion are related:
Note that kT  26mV at room temperature (300K) q
This is often referred to as the “thermal voltage”.

Important Constants
⚫ Electronic charge, q = 1.610-19 C
⚫ Permittivity of free space, eo = 8.85410-
⚫ Boltzmann constant, k = 8.6210-5 eV/K
⚫ Planck constant, h = 4.1410-15 eV•s
⚫ Free electron mass, mo = 9.110-31 kg
⚫ Thermal voltage kT/q = 26 mV, at T=300K

HW3: Energy-band diagram Question: Where is EF for n = 1017 cm-3 ?
n=nexp(EF −Ei) i  kT 

⚫ The electron mobility and hole mobility are 700 cm2/Vs and 350 cm2/Vs, respectively.

HW5: Carrier Drift (Band Diagram Visualization)
Q1: what is the direction of electric field? Q2: what is the direction of carriers’ drift?

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