Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-1

Chapt 2-1. Semiconductor models

In this chapter, we will study these topics:

The quantization concept

Semiconductor models

Carrier properties

State and carrier distributions

Equilibrium carrier concentrations

Throughout this chapter, we assumes equilibrium

conditions exist within the semiconductor.

a reference

In crystalline Si,

14 electrons per atom and 5☓1022 atoms/cm3

lattice constant of Si: a = 5.43 Å

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-2

Quantization concept

In 1901, Max Planck showed that the energy distribution of the black

body radiation can only be explained by assuming that this radiation

(i.e. electromagnetic waves) is emitted and absorbed as discrete

energy quanta - photons.

The energy of each photon is related to the wavelength of the radiation:

E = h = h c /

where

h = Planck’s constant (h = 6.63 1034 Js)

= frequency (Hz = s1)

c = speed of light (3 108 m/s)

= wavelength (m)

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-3

Example

Our eye is very sensitive to green light. The corresponding

wavelength is 0.555 m or 5550 Å or 555 nm. What is the

energy of each photon?

E = h = = 3.57 10–19 J

These energies are very small and hence are usually

measured using a new energy unit called electron Volts

1 eV = 1.6 1019 CV = 1.6 1019 J

m 610 0.555

m/s 810 3 Js 3410 6.62

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-4

A new unit of energy

Since the energies related to atoms and photons are very small,

(EGREEN LIGHT = 3.57 1019 J), we have defined a new unit of energy

called “electron Volt” or “eV”

One eV is the energy acquired by an electron when accelerated by a 1.0

V potential difference.

+ 1V

Energy acquired by the electron is qV. Since q is 1.6

1019 C, the energy is 1.6 1019 J. Define this as 1 eV.

Therefore, EGREEN LIGHT = 2.23eV

1 eV = 1.6 10–19 J

1 eV = 1 1.610–19 CV = 1.610–19 J

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-5

Niels Bohr in 1913 hypothesized that electrons in hydrogen

was restricted to certain discrete levels. This comes about

because the electron waves can have only certain

wavelengths, i. e. n = 2r, where r is the orbit radius.

Quantization

Based on this, one can show that:

...,,nhn

qm

n

qmE 321for

8)4(2 2220

40

20

40

H

constantsPlanck'and2

where

hh

Quantization concept (continued):

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-6

Bohr’s hydrogen atom model

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-7

For the n = 2 orbit, E2 = 3.4 eV and so on. The number n is called the

principal quantum number, which determines the orbit of the electron.

Since Hydrogen atom is 3-D type, we have other quantum numbers like,

l and ml within each orbit. So, in atoms, each orbit is called a “shell” .

See Appendix A in text for the arrangement of electrons in each shell

and also for various elements in the periodic table.

A numerical example:

orbit1theforeV5.13J107.21

Js1062.61F/m1085.88

C106.1kg1011.9

8

19

234212

41931

222

0

4

0n

n

hn

qmE

Quantum # Designation

n = principal (energy level-shell) K, L, M, N, O (1, 2, 3, etc.)

l = subsidiary (orbitals) s, p, d, f (0, 1, 2, 3,…, n -1)

ml = magnetic 1, 3, 5, 7 (-l to +l)

ms = spin ½ , -½

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-8

Atomic configuration of Si

ex: Si - atomic # = 14

1s

2s 2p

K-shell n = 1

L-shell n = 2

3s 3p M-shell n = 3

3d

4s

4p 4d

Energy

N-shell n = 4

1s2 2s2 2p6 3s2 3p2

valence

electrons

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-9

• Ten of the 14 Si-atom electrons occupy very deep lying energy levels and are tightly bound to the nucleus

• The remaining 4 electrons, called valence electrons are not very strongly bound and occupy 4 of the 8 allowed slots.

• Configuration for Ge is identical to that of Si, except that the core has 28 electrons.

Atomic configuration of Si

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-10

Bond model

• Consider a semiconductor Ge, Si, or C

• Ge, Si, and C have four nearest neighbors, each has 4

electrons in outer shell

• Each atom shares its electrons with its nearest neighbor.

This is called a covalent bonding

• No electrons are available for conduction in this covalent

structure, so the material is and should be an insulator

at 0 K

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-11

2-dimensional semiconductor bonding model

No electrons are available for conduction.

Therefore, Si is an insulator at T = 0 K.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-12

Simplified 2D representation of Si lattice

• How many atom-neighbors has each Si atom in a Si lattice?

• How many electrons are in the outer shell of an isolated Si atom?

• How many electrons are in the outer shell of a Si atom with 4

neighbors?

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-13

(a) Point defect (b) Electron generation

• At higher temperatures (e.g. 300 K), some bonds get

broken, releasing electrons for conduction.

• A broken bond is a deficient electron of a hole.

• At the same time, the broken bond can move about the

crystal by accepting electrons from other bonds thereby

creating a hole.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-14

Energy band model

An isolated atom has its own electronic structure with n = 1,

2, 3 ... shells.

When atoms come together, their shells overlap.

Consider Silicon: Si has 4 electrons in its outermost shell, but

there are 8 possible states. When atoms come together to

form a crystal, these shells overlap and form bands.

We do not consider the inner shell electrons since they are

too tightly coupled to the inner core atom, and do not

participate in anything.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-15

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-16

Energy band model

At T = 0K

No conduction can take place

since there are no carriers in

the conduction band.

Valence band does not

contribute to currents since it is

full.

Both bond model and band

model shows us that

semiconductors behave like

insulators at 0K.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-17

Insulators, semiconductors, and metals

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-18

Chapt 2-2. Carrier properties

Mass-like charge is a very basic property of electrons and

holes. The mass of electrons in a semiconductor may be

different than its mass in vacuum.

Effective mass concept

t

vmqF

d

d0 Ε

t

vmqF *

d

dn Ε

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-19

• Electrons moving inside a semiconductor crystal will collide with

semiconductor atoms, there by causing periodic deceleration of the

carriers

• In addition to applied electric field, the electrons also experience

complex field forces inside the crystals

• The effective mass can have different values along different directions

• The effective mass will be different depending on the property we are

observing. So you can have conductivity effective mass, density of

states effective mass, etc.

Effective mass

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-20

Carrier numbers in intrinsic materials

Intrinsic semiconductor or pure semiconductor has equal numbers of

electrons and holes at a particular temperature.

Number of electrons/cm3 [n] = number of holes/cm3 [p]

Why is n = p?

This is an intrinsic property of the semiconductor and is called intrinsic

carrier concentration, ni

At T = 300 K, ni = n = p 2 106 / cm3 in GaAs

1 1010 / cm3 in Si

2 1013 / cm3 in Ge

How large is this compared to the number of Si atoms/cm3?

What happens to ni at higher temperature? At 0 K?

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-21

Extrinsic semiconductors

Elements in column V of the periodic table have 5 electrons in their outer

shell (one more than Si)! This can be easily released, thus increasing the

net free electrons in the Si crystal.

Elements of column III of the periodic table have only 3 electrons in their

outer shell (one less than Si)! To complete the bond, the atom can

accept an electron by breaking a bond somewhere else, thus creating a

broken bond, or a hole.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-22

Visualization of donors and acceptors

Phosphorus (P) atom Boron (B) atom

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-23

Pseudo-hydrogen atom model for donors

eV0.05

2

0s

0

0

*neV613

) π(42

q*n

20s

4

d

Km

m.

K

mE

eV613)4(2 2

0

40

H .qm

E

Instead of m0, we have to use mn*.

Instead of o, we have to use Ks o.

Ks is the relative dielectric constant

of Si (Ks, Si = 11.8).

This is an approximate value. More accurate values are given next.

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-24

Binding energies for dopants

Questions:

How much energy is required to break a Si-Si bond?

How much energy is required to break the 5th electron from As in

Si?

How much energy is required to break a Si-Si bond when that

bond is adjacent to a B atom?

Does the freeing of an electron from a donor atom create an extra

hole? Chapt.2-Lec3-24

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-25

Temperature effects on donors and acceptors

n-type

p-type

Thermal energy at room temperature (300 K) = kT = 0.026 eV

Prof. Yo-Sep Min Electronic Materials: Semiconductor Physics & Devices Chapt.2-Lec3-26

Announcements

• Next lecture: p. 40~49