PDF Device Physics

Device Physics

1. Energy Band

- Atomic bonding and energy band

- Fermi energy and carrier concentration

Atomic Bonding & Energy Band

Atomic Bonding

sp³ hybridized atomic orbitals

Covalent bonding

Si: [Ne]3s

²

3p

²

Crystal Structure

(Si or Ge)

CN = 4

Energy Splitting by Interacting Particles

Energy band splitting and the formation of allowed and forbidden bands

Formation of Energy Bands

Si crystal

4 valence electrons

Insulator, Semiconductor, Metal

Insulator Semiconductor

Metal

3.5 ~ 6.0 eV or larger ~ 1.0 eV

Electron Energy in Solid

Energy Band and Bond Model

T = 0 K

T > 0 K

For an intrinsic silicon,

𝑛 𝑝 𝑛 1.5 10 𝑐𝑚

@ 300 K

Concept of Hole

The movement of a valence electron into the “empty state” is equivalent to the movement of the positively charged “empty state” itself.

The is equivalent to a positive charge (“hole”) moving in the valence band.

Temp. Dependence of Bandgap

The bandgaps of most semiconductors decrease with increasing temperature.

N-Type Doping

A substitutional phosphorous atom (donor) with five valence electrons replaces a silicon atom and a negatively

charged electron is donated to the lattice in the conduction band.

T = 0 K T > 0 K

P-Type Doping

A boron atom (acceptor) with three valence electrons

substitutes for a silicon atom and an additional electron is accepted to form four covalent bonds around the boron

leading to the creation of

positively charged hole in the valence band.

T = 0 K T > 0 K

Fermi Energy & Carrier Concentration

Fermi Energy

Electrons in solids obey Fermi-Dirac statistics.

The distribution of electrons over a range of allowed energy levels at thermal equilibrium is governed by the equation,

𝑓 𝐸 1

1 exp 𝐸 𝐸 𝑘𝑇

𝑓 𝐸 gives the probability that an available energy state at 𝐸 is occupied by an electron at absolute temperature 𝑇.

𝑘 is Boltzmann’s constant (𝑘 8.62 10 𝑒𝑉/𝐾 1.38 10 𝐽/𝐾).

𝐸 is called the Fermi energy.

For an energy state at 𝐸 equal to the Fermi energy level 𝐸 , the occupation probability is 1/2.

Fermi-Dirac Distribution

(𝐸 𝐸 ≫ 𝑘𝑇)

𝑓 𝐸 𝑒𝑥𝑝 𝐸 𝐸 𝑘𝑇

Thermal-Equilibrium Electron Concentration

Number of electrons in the conduction band is given by the total number of states 𝑔 𝐸 multiplied by the occupancy 𝑓 𝐸 , integrated over the conduction band

𝑛 𝑔 𝐸 𝑓 𝐸 𝑑𝐸

𝑓 𝐸 1

1 exp 𝐸 𝐸 𝑔 𝐸 4𝜋 2𝑚^{∗ /} 𝑘𝑇

ℎ 𝐸 𝐸

𝑛 4𝜋 2𝑚

^{∗ /}

ℎ 𝐸 𝐸 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑑𝐸 𝑁 𝑒𝑥𝑝 𝐸 𝐸 𝑘𝑇

𝑁 2 2𝜋𝑚^∗𝑘𝑇 ℎ

/

Boltzmann approximation 𝐸 𝐸 ≫ 𝑘𝑇

Effective density of state function in the conduction band

𝜂

𝜂 ^/ exp 𝜂 𝑑𝜂 1 2 𝜋

Thermal-Equilibrium Hole Concentration

Similarly, number of electrons in the valence band is given by the total number of states 𝑔 𝐸 multiplied by the probability that a state is not occupied by an electron 1 𝑓 𝐸 , integrated over the valence band

𝑝 𝑔 𝐸 1 𝑓 𝐸 𝑑𝐸

𝑝 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇

𝑁 2 2𝜋𝑚^∗𝑘𝑇 ℎ

/

Effective density of state function in the valence band

Distribution of Electron and Holes

Intrinsic Carrier Concentration

Intrinsic Concentration. For intrinsic semiconductors at finite temperatures, thermal agitation occurs which results in continuous excitation of electrons from the valence

band to the conduction band, and leaving an equal number of holes in the valence band.

This process is balanced by recombination of the electrons in the conduction band with holes in the valence band. At steady state, the net result is 𝑛 𝑝 𝑛 , where 𝑛 is the intrinsic carrier density.

𝐸 𝐸 𝐸 𝐸

2

𝑘𝑇

2 𝑙𝑛 𝑁 𝑁

𝑛 𝑁 exp 𝐸 𝐸

𝑘𝑇 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇

𝑁 𝑁 𝑒𝑥𝑝

Extrinsic Semiconductor

n-type semiconductor p-type semiconductor

Fermi Level Position vs. Doping

Mass Action Law

𝑝 𝑛 𝑁 𝑁 𝑒𝑥𝑝 𝐸

𝑘𝑇 𝑛

𝑝 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑛 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑛 𝑁 𝑁 𝑒𝑥𝑝 𝐸

2𝑘𝑇

for nondegenerate semiconductor

Extrinsic Carrier Concentration

𝑛 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑁 𝑒𝑥𝑝 𝐸 𝐸 𝐸 𝐸

𝑘𝑇 𝑛 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇

𝑛 𝑁 exp 𝐸 𝐸 𝑘𝑇

𝑝 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑁 𝑒𝑥𝑝 𝐸 𝐸 𝐸 𝐸

𝑘𝑇 𝑛 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝑛 𝑁 𝑒𝑥𝑝 𝐸 𝐸

𝑘𝑇 𝐸 𝐸 𝑘𝑇 𝑙𝑛 𝑛

𝑛

𝐸 𝐸 𝑘𝑇 𝑙𝑛 𝑝 𝑛

Donor and Acceptor Level

𝑇 300 𝐾

𝑇 0 𝐾

Freeze-out

Impurity Levels

Carrier Conc. vs. Temperature

@ RT, 𝑛 𝑁 𝑁 for nondegenerate semiconductor

Fermi Level Position vs. Temp.

Constancy of Fermi Level

In thermal equilibrium, the Fermi energy level is constant throughout a system !