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Semiconductor Physics
Part 1
Wei E.I. Sha (沙威)
College of Information Science & Electronic Engineering
Zhejiang University, Hangzhou 310027, P. R. China
Email: [email protected]
Website: http://www.isee.zju.edu.cn/weisha
Wei SHASlide 2/28
1. Energy Bandgap
2. Effective Mass
3. Doping
4. Mobility, Scattering, and Relaxation
5. Boltzmann Statistics
6. Recombination
7. Drift-Diffusion Equations
Course Overview
Ref:
Enke Liu, Semiconductor Physics, Chapter 1-5;
Shun Lien Chuang, Physics of Photonic Devices, Chapter 2.
Wei SHASlide 3/28
Coupled Mode Theory
coupling induces the split of resonant frequency !
energy
conservation
1. Energy Bandgap (1)
11 1 12 2
22 2 21 1
+
+
=
=
daj a jk a
dt
daj a jk a
dt
( )2 2
1 2+ 0=d
a adt
12 21
=k k
1 1 1 12 2
2 2 2 12 1
+
+
=
=
j a j a jk a
j a j a jk a
221 2 1 2
12
+ -=
2 2
+
k
Wei SHASlide 4/28
1. Energy Bandgap (2)
coupling
energy
distance
bandgap
There is no electronic state within bandgap
Wei SHASlide 5/28
Fermi level: If there is a state at the electronic Fermi level, then this state will have a
50% chance of being occupied. Below the Fermi level, the occupation probability
becomes larger and larger.
Metals have a partially filled conduction band. A semimetal has a small overlap between
the bottom of the conduction band and the top of the valence band. In insulators and
semiconductors, the filled valence band is separated from an empty conduction band by
a band gap. For insulators, the magnitude of the band gap is larger.
1. Energy Bandgap (3)
valance band
metal semi-metalsemiconductor
P type N typeintrinsicinsulator
conduction band
Wei SHASlide 6/28
1. Energy Bandgap (4)
1. Intraband transition is the transition
between levels within the conduction
band. It is caused by electron-electron
(electron screening effect) or electron-
phonon interaction, and mainly
responsible for the real part of
permittivity.
2. Interband transition is the transition between conduction and valence
bands. It generates electron-hole pairs, and mainly responsible for the
imaginary part of permittivity (lossy term).
Electronic transitions between valance and conduction bands
3. If electron density is increased or Fermi level is arisen, the interband
transition will be forbidden if excitation (photon) energy is low.
Wei SHASlide 7/28
2. Effective Mass (1)
The effective mass is a quantity that is used to simplify band structures by
modeling the behavior of a free particle with that mass.
At the highest energies of the valence band and the lowest energies of the
conduction band, the band structure E(k) can be locally approximated as
where E(k) is the energy of an electron at wavevector k in that band, E0 is a
constant giving the edge of energy of that band, and m* is a constant (effective
mass). electrons or holes placed in these bands behave as free electrons except
with a different mass.
22
0( )2
= +E Em
k k
22
0 2
0 0
1( ) ( ) ...
2= =
= + + +k k k k
dE d EE k E k k k
dk dk
2
2 2
0
1 1
k k
d E
m dk
=
=
Wei SHASlide 8/28
2. Effective Mass (2)
Band structures of germanium and silicon are complex!
( ) = − + n
dm q q
dt
vE v B
Newton equation of an electron in semiconductor with effective mass
The top of valance band (red) and the bottom of conduction band (blue)
are not aligned. In other words, Si and Ge have indirect bandgap. Thus,
phonon absorption is significant during electronic transition.
Wei SHASlide 9/28
3. Doping (1)
Doping is the intentional introduction of impurities into
an intrinsic semiconductor for the purpose of modulating its
electrical, optical and structural properties.
Small numbers of dopant atoms can change the ability of a
semiconductor to conduct electricity.
When on the order of one dopant atom is added per 100 million
atoms, the doping is said to be low or light.
When many more dopant atoms are added, on the order of one per
ten thousand atoms, the doping is referred to as high or heavy.
This is often shown as n+ for n-type doping or p+ for p-
type doping.
Wei SHASlide 10/28
By doping pure silicon with Group V elements such as phosphorus, extra valence
electrons are added that become unbonded from individual atoms and allow the
compound to be an electrically conductive n-type semiconductor.
Doping with Group III elements, which are missing the fourth valence electron,
creates "broken bonds" (holes) in the silicon lattice that are free to move. The
result is an electrically conductive p-type semiconductor.
In this context, a Group V element is said to behave as an electron donor, and
a group III element as an acceptor.
3. Doping (2)
Wei SHASlide 11/28
Electrons in solid crystals Photons in photonic crystals
Wave-like behavior ψ(r, t) in periodic
potential of lattice
Field H(r, t) in periodic dielectric
functions
Electronic band structure Photonic band structure
Electronic Doping/Impurity Photonic Defect
Analogy between solid crystals and photonic crystals (optional)
3. Doping (3)
defect state in photonic crystals
Wei SHASlide 12/28
4. Mobility, Scattering, and Relaxation (1)
Scattering is complex
Wei SHASlide 13/28
4. Mobility, Scattering, and Relaxation (2)
Scattering probability P
( )( )= −
dN tPN t
dt 0( ) −= PtN t N e
initial number of electrons
Number of scattered electrons between t ~ t+dt
0
−PtPN e dt
Mean free time or relaxation time : time between successive scattering
0 0
0
1 1= = −
PtPN e tdt
N P
the number of free electrons at t
Wei SHASlide 14/28
4. Mobility, Scattering, and Relaxation (3)
Newton’s law
,
= − =pn
n p
n p
m q m qvv
E E
Conductivity
,
= = = =p pn n
n p
n p
v qv q
m mE E
Mobility
= − = − =n n n n nqn qnJ v v E
ത𝐯: mean drift velocity
= = =p p p p pqp qpJ v v E
, = =n n p pqn qp
Wei SHASlide 15/28
5. Boltzmann Statistics (1)
Boltzmann law of exponential atmosphere
https://www.feynmanlectures.caltech.edu/I_40.html
If the temperature is the same at all heights, and N is the total number of molecules
in a volume V of gas at pressure P, then we know PV=NkBT. (Density n=N/V)
The force from below must exceed the force from above by
the weight of gas in the section between h and h+dh
( ) ( )+ − = = = −BP h dh P h dP k Tdn mgndh
−=
B
dn mgn
dh k T
0 0exp exp − −
= =
E
B B
mgh Pn n n
k T k T
If the energies of the set of molecular states are called, E0, E1, E2, …, then in thermal
equilibrium, the probability of finding a molecule in the particular state of having
energy Ei is proportional to exp(−Ei/kBT).
Wei SHASlide 16/28
5. Boltzmann Statistics (2)
thermal equilibrium
Two physical systems are in thermal equilibrium if there is no net
flow of thermal energy between them when they are connected by a
path permeable to heat. A system is said to be in thermal equilibrium
with itself if the temperature within the system is spatially uniform
and temporally constant.
thermal equilibrium in semiconductor
Electrons and holes in semiconductor have a unified Fermi level. The
generation and recombination processes are statistically balanced.
External bias and light illumination will break the thermal
equilibrium and thus electrons and holes have different local Fermi
levels under the nonequilibrium state.
Wei SHASlide 17/28
5. Boltzmann Statistics (3)
Carrier concentration in thermal equilibrium
0 exp −
= −
c Fc
B
E En N
k T0 exp
−=
v Fv
B
E Ep N
k T
2
0 0 exp
= − =
g
c v i
B
En p N N n
k T
EF: unified Fermi level for both electrons and holes
Nc: effective density of states for conduction band
Nv: effective density of states for valance band
n0: electron density in conduction band
p0: hole density in valance band
ni: intrinsic carrier density (without doping and defects)
3/2
22
2
n Bc
m k TN
=
3/2
22
2
p B
v
m k TN
=
Wei SHASlide 18/28
5. Boltzmann Statistics (4)
Carrier concentration in thermal non-equilibrium
exp −
= −
c Fnc
B
E En N
k Texp
− =
v Fp
v
B
E Ep N
k T
2
0 0 exp exp− −
= =
Fn Fp Fn Fp
i
B B
E E E Enp n p n
k T k T
EFn: quasi-Fermi level of electrons
EFp: quasi-Fermi level of holes
1. Fast thermal induced transitions at conduction band or valance band (but not in
between) result in the local equilibrium of carrier concentration at each band.
2. Larger EFn-EFp, which suggests larger external bias voltage, more
nonequilibrium carriers (Δn=n-n0, Δp= p-p0) are generated.
Wei SHASlide 19/28
6. Recombination (1)
Band to band (bimolecular) recombination
0 0( )= −R B np n p
The net recombination rate
At thermal equilibrium, R=0, generation (absorption) and recombination (emission)
are statically balanced. It is the essential physics of black-body radiation.
At thermal nonequilibrium, 0 0, , n n p p R Bnp
B: bimolecular recombination coefficient
Radiative recombination is the band-to-band recombination.
Wei SHASlide 20/28
6. Recombination (2)
Nonradiative Shockley-Read-Hall (SRH) recombination
(a) electron capture; (b) electron emission; (c) hole capture; (d) hole emission
electron capture: The recombination rate for electrons is proportional to the density of
electrons n, and the concentration of the traps Nt , multiplied by the probability that the
trap is empty (1-ft), Rn = cn n Nt (1-ft), where cn is the capture coefficient for the electrons.
electron emission: The generation rate of the electrons due to this process is Gn = en Nt
ft, where en is the emission coefficient and Nt ft, is the density of the traps that are
occupied by the electrons.
hole capture and hole emission: Rp = cp p Nt ft and Gp = ep Nt (1-ft)
Wei SHASlide 21/28
6. Recombination (3)
At thermal nonequilibrium, we have stable condition Rn -Gn = Rp - Gp
2
1 1 1 1 0 0, , = = = =pn
i
n p
een p n p n p n
c c
At thermal equilibrium, we have the detailed balancing Gn=Rn and Gp=Rp
2
1 1
=( ) ( )
in n p p
p n
np nR R G R G
n n p p
−= − − =
+ + +
t t
1 1= , = p n
p nc N c N
Wei SHASlide 22/28
6. Recombination (4)
For minority electron, p >> n
n n
n
nR G
−
For minority hole, n >> p
p p
p
pR G
−
SRH recombination can be seen as the monomolecular recombination.
Wei SHASlide 23/28
6. Recombination (5)
Nonradiative Auger recombination
Fig. 1. band-to-band Auger recombination
left: n-type; right: p-type
Band-to-band Auger recombination: An electron in the conduction band recombines
with a hole in the valence band and releases its energy to a nearby electron or hole. The
excited electron or hole will always relax and produce heat (phonon energy).
Impact ionization: one highly-energetic electron (hole) from the conduction (valance)
band transfers its energy to a valance-band electron for generating one e-h pair.
Fig. 2. band-to-band Auger generation
(impact ionization): reverse process of Fig. 1
Wei SHASlide 24/28
6. Recombination (6)
For Auger recombination,
2 2
2 2
0 0 0 0 0 0
,
,
= =
= =
ee e hh h
ee e hh h
R n p R np
R n p R n p
For Auger generation,
0 0 0 0
,
,
= =
= =
ee e hh h
ee e hh h
G g n G g p
G g n G g p
At thermal equilibrium, 0 0 0 0, = =ee ee hh hhG R G R
At thermal nonequilibrium, ee ee
2
( ) ( )
( )( )
= + − +
= + −
hh hh
e h i
R R R G G
n p np n
Wei SHASlide 25/28
1. Recombination can occur in bulk or at surface.
2. Recombination can be direct (through valance band and conduction band) or indirect (thought recombination centers).
3. Energy released from recombination will convert to photon energy, phonon energy, or excite other carriers.
4. Non-radiative recombination is always the dominant electric loss mechanism for state-of-the-art optoelectronic devices.
6. Recombination (7)
More about recombination
Wei SHASlide 26/28
7. Drift-Diffusion Equations (1)
( )exp = exp
− −= −
ni Fni i
B B
E En n n q
k T k T
( )exp = exp
−− = −
pi Fp
i i
B B
E Ep n n q
k T k T
quasi-Fermi
(chemical) potential
thermal
nonequilibrium
= logBn
i
k T n
q n
−
= + log
Bp
i
k T p
q n
electron current
hole current
= +n n n n nJ q n q n qD n = − E
=p p p p pJ q p q p qD p = − − E
, = =B Bn n p p
k T k TD D
q qEinstein relation
= −E
0 =in
Wei SHASlide 27/28
7. Drift-Diffusion Equations (2)
Semiconductor equations
+( ) ( + )D Aq p n N N − = − − −
= − + −
= + −
p
n
dpG R
dt q
dnG R
dt q
J
J
Poisson equation
Current continuity equationsR: net recombination rate
G: generation rate under light illumination
+n n n n
p p p p
J q n qD n
J q p qD p
=
= −
E
EDrift-diffusion equations
Wei SHASlide 28/28
7. Drift-Diffusion Equations (3) — Boundary Conditions (Optional)
Non-electrode boundaries (green line)
0 0, 0ˆ ˆ ˆ
, = = =
n p
n n n
Electrode boundaries (red line)
, a ca a c c
W WV V
q q = − = − Va , Vc: external bias for anode and cathode
Wa ,Wc: work functions for anode and cathode
0 0( - ), ( - )na na pa paJ qS n n J qS p p= =
0 0exp expp g p
v c
B B
Ep N n N
k T k T
− − + = =
,
anode
cathode
0 0exp expg nn
c v
B B
En N p N
k T k T
− + −= =
,
Sna: surface recombination velocity for electrons
Spa: surface recombination velocity for holes
Φp: injection barrier for holes
0 0( - ), ( - )nc nc pc pcJ qS n n J qS p p= =Snc: surface recombination velocity for electrons
Spc: surface recombination velocity for holes
Φn: injection barrier for electrons
normal direction