Semiconductor Physics

Part 1

Wei E.I. Sha (沙威)

College of Information Science & Electronic Engineering

Zhejiang University, Hangzhou 310027, P. R. China

Email: weisha@zju.edu.cn

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