ENE 311 Lecture 4

Quantum numbers

• As Pauli exclusion principle stating that no two electrons in an interacting system can have the same set of quantum numbers n, l, m, s .

• Only two electrons can have the same three quantum numbers n, l, m, and those two must have opposite spin. These can be summarized as

n = 1, 2, 3, …

l =0 ,1 ,2 ,…, (n-1)

m = -l , …, -1, 0 ,1 , …, +l

s = 1/2

Quantum numbers

• The quantum states shown in the table are used to indicate the electronic configuration

s for atoms in the lowest energy state.

Quantum numbers

n l m sAllowable states in subshell

Allowable states in complete

shell

1 0 0 1/2 2 2

2

0 0 1/2 2

81

-1 1/2

60 1/2

1 1/2

3

0 0 1/2 2

18

1

-1 1/2

60 1/2

1 1/2

2

-2 1/2

10

-1 1/2

0 1/2

1 1/2

2 1/2

Quantum numbers

Shell(n) K 1 L 2 M 3 N 4

Subshell (l)

0 0 1 0 1 2 0 1 2 3

s s p s p d s p d f

# of electrons

2 2 6 2 6 10 2 6 10 14

2 8 18 32

Quantum numbers

• There is a simple shorthand notation for ele ctronic structures that is the naming of l val

ues expressed as

l = 0, 1, 2, 3, 4

s, p, d, f, g

• These s, p, d, f stand for sharp, principal, diff use, and fundamental. The rest will be writt

en in alphabetical order beyond f.

Quantum numbers

• For example, Si (atomic number = 14): 1s22s22p63s23p2

Bonds

Bonds

where r = interatomic distance

a = attraction constant

b = repulsion constant

m,n = constant of characteristic of each type of bond or structure

• Therefore, and are attraction and

repulsion energy, respectively.

( )m n

a bE r

r r

m

a

r n

b

r

Bonds

We may conclude that

• E 0 at r : Zero energy as the energy i n the absence of interaction.

• At r > r0

, atoms attract each other from r to r r0.

• At r < r0

, atoms repel each other up to the p oint r0.

• At r0

, equilibrium position occurs. It is where the attraction energy and repulsion energy

balance each other.

Types of bonds

• Bonds may be classified into4 types as

1. ionic bond: non-directional

2. metallic bond: non-directional

3. covalent bond: directional

4. van der Waals bond.

Ionic bond

• This happens from electrostatic attraction b etween ions with different charges such as

NaCl or LiF. The cohesive energy, Ec, the en ergy needed to take the crystal apart, may

be written as

where M = Madelung constant

= Coulomb electrostatic

attraction energy between 2 ions.

2

04c n

Me bE

r r

2

04

e

r

Metallic bond

• Metallic bond is similar to the ionic bond as electrostatic forces play big part on it, but this electrostatic forces are everywhere and come from all directions.

• In metals, the negative charges are highly mobile, electrons act like a glue to hold the lattice together.

• The cohesive forces in metals are very stron g and hard to break.

Covalent bond

• This bond happens from the sharing of electrons between two atoms.

• The simplest example of covalent bond is shown by hydrogen atom.

• Hydrogen atom needs another electron to fill its 1s shell.

• It would find that extra electron from another hydrogen atom as they both finally share their electrons.

Covalent bond

• In covalent bond, all electrons pair up and orbit around a pair of atoms, so more of them can wander away to conduct electricity.

• In case of carbon, it acts like an insulator, but this bond in silicon or germanium is weaker.

• Some of electrons in the latter case might b e shaken off and able to conduct electricity, so we call them “semiconductors”.

Covalent bond

(a ) A broken bond at Position A, resulting in a conducti on electron and a hole.

(b ) A broken bond at position B.

The van der Waals bond

• This is like a secondary bond since its force is very weak.

• This bond can be seen in atoms that their outer shell is fully filled.

• Consider atom A has a dipole moment then it will induce an opposite dipole moment on atom B.

• This attraction force is called “van der Waal s bond”.

Energy Bands

• Condiser two identical atoms, when they are far apart, the allowed energy levels for a given principal quantum number (n = 1) consist of one doubly degenerate level (both atoms have exactly the same energy).

• When they are brought closer, the doubly degenerate energy levels will spilt into two levels by the ineraction between the atoms.

Energy Bands

• N isolated atoms are brought together to form a solid, the orbits of the outer electrons of different atoms overlap and interact with each other.

• This causes a shift in the energy levels and N separate closely spaced levels are formed

.

Energy Bands

Energy Bands

• Consider isolated silicon atom, 10 of the 14 electrons occupy energy levels whose orbital radius is much smaller than the interatomic separation in the crystal.

• The four remaining valence electrons are relatively weakly bound and can be involved in chemical reactions.

• Therefore, the valence electrons are the ones that will be considered.

• The two inner shells are completely full and tightly bound to the nucleus.

Energy Bands

• As the interatomic distance decreases, the 3s and 3p subshell of the N silicon stoms will interact and overlap.

• At the equilibrium interatomic distance, the bands will again split with four quantum states per atom in the lower band (valence band) and four quantum states per atom in the upper band (conduction band ).

Energy Bands

• At absolute zero temperature (T = 0 K), electrons occupy the lowest energy states, so that all states in the lower band will be full and all states in the upper band will be empty.

• The bottom of the conduction band is called Ec, and the top of the valence band is called Ev

.

• The bandgap energy Eg is the width of the forbidden energy level between the bottom of the conduction band and the top of the valence band.

Energy Bands

• The bandgap energy is the energy required to break a bond in the semiconductors to fre

e and electron to the conductgion band and leave a hole in the valence band.

Energy Bands

Energy Bands

- The energy momentum diagram

• The energy E of a free electron is given by

(1)

where p is the momentum

m - 0is the free electron mass

2

02

pE

m

Energy Bands

• In a semiconductor, an electron in the conduction band is similar to a free electron in that it is free to move about inside the crystal as shown in the right figure.

Energy Bands

• However, the above equation for E can not be used due to the periodic potential of the nucleus.

• Anyway, if replacing m0 with an effective m ass, in an equation (1), it yields the energy E

of an electron as

2

*2 e

pE

m

Energy Bands

• The effective mass in a solid is a result of charged particle moving under nucleus of applied electric field in presence of a periodic potential.

• This differs from the mass in free space.

• The electron effective mass depends on the properties of the semiconductor.

Energy Bands

- In quantum mechanic, the velocity of electron is described by its group

1

g

g

dv

dkE h

Ev

k

Energy Bands

The acceleration can be obtained by

(2)

1gdv d Ea

dt dt k

2

2

1 E ka

k t

Energy Bands

• For classical part, it expresses dE as the wor k done by a particle traveling a distance vgd

t under the influence of a force eE . It yields

.

.( )

1.

g

dE F dx

F v dt

EdE F dt

k

Energy Bands

This leads to

(3)

Substituting (3) into (2)

dk F

dt

2

2

2

2 2

1

1

E Fa

k

Ea F

k

Energy Bands

From F = ma, we have

A similar expression can be written for holes with effective mass .

1 12 2*

2 2 2

1e

E Em

k p

*hm

Energy Bands

• -A schematic energy mo mentum diagram for a s

pecial semiconductor

with = 0.25 m0 and

= m0.

• The electron energy is measured upward and hole energy is measured downward.

• This energy-momentum relationship is called “energy-band diagram”.

*hm

*em

Energy Bands

• Energy band diagram may be classified semic

onductors into2 groups as direct semiconducto rs and indirect semicon

ductors. • Energy band structures

of Si and GaAs. Circles ( º) indicate holes in the

valence bands and dots (•) indicate electrons in the conduction bands.

Energy Bands

• Let us consider the figure, GaAs is a direct s/c with a direct bandgap since it does not require a change in momentum for an electron transition from the valence band to the conduction band.

• Unlike in the case of Si, an electron transition from the valence band to the conduction band requires not only an energy change but also momentum change (called indirect s/c).

Energy Bands

• This difference between direct and indirect bandgap is crucial for making the light sour

ces such as LEDs or LASERs.

• These light sources require direct semicond uctors for efficient generation of photons.

Energy Bands

Conduction in Metals, S/C , and Insulators

• The electrical conductivity of metals, semiconductors, and insulators could be explained by their energy bands.

• These can be done by considering the highest two bands, valence and conduction bands, of the materials.

• Electron occupation of the conduction band determines the conductivity of a solid.

Energy Bands

• (left ) a conductor with two possibilities (either the pa rtially filled conduction band shown at the upper port

ion or the overlapping bands shown at the lower portion)

• (middle ) a semiconductor

• (right ) an insulator.

Energy Bands

• Metals: Highest allowed occupied band or conductio n band is partially filled (such as Cu) or overlaps the

valence band (such as Zn or Pb). Therefore, electron s are free to move to the next energy level with only a small applied field.

Energy Bands

• Insulators : The valence electron s form strong bonds between thei r neighboring atoms. These bond s are difficult to break.

• Therefore, valence band is fully fi lled and the conduction band is t

otally empty.

• Also, these two bands are separa ted by a wide bandgap.

• Thermal energy or the energy fro m applied electric field is not eno

ugh to raise the uppermost electr on in the valence band up to the

conduction band.

• Therefore, there is no conductivity.

Energy Bands

• Semiconductor : This is similar to the insulators, but the bandga

p is much smaller than in the cas e of insulators.

• At T =0 K, all electrons are in th e valence band and no electron i n the conduction band.

• Therefore, semiconductors are p oor conductors at low temperatu res. At room temperature, some

electrons are thermally excited f rom the valence band to the con

duction band.

• Also, it needs just small applied electric field to move these elect

rons and that results in conductivity.