Energy bands and charge carriersin semiconductors

In the name of God

- Chapter 3
- Mr. Harriry (Elec. Eng.)
- By Amir Safaei
- 2006

Outlines

- 3-1. Bonding Forces and Energy Bands in Solids
- 3-1-1. Bonding Forces in Solids
- 3-1-2. Energy Bands
- 3-1-3. Metals, Semiconductors Insulators
- 3-1-4. Direct Indirect Semiconductors
- 3-1-5. Variation of Energy Bands with Alloy

Composition

Outlines

- 3-2. Carriers in Semiconductors
- 3-2-1. Electrons and Holes
- 3-2-2. Effective Mass
- 3-2-3. Intrinsic Material
- 3-2-4. Extrinsic Material
- 3-2-5. Electrons and Holes in Quantum Wells

Outlines

- 3-3. Carriers Concentrations
- 3-3-1. The Fermi Level
- 3-3-2. Electron and Hole Concentrations at

Equilibrium

3-1. Bonding Forces Energy Bands in Solids

- In Isolated Atoms
- In Solid Materials

3-1-1. Bonding Forces in Solids

- Na (Z11) Ne3s1
- Cl (Z17) Ne3s1 3p5

- ION
- Bonding

3-1-1. Bonding Forces in Solids

- Metallic
- Bonding

3-1-1. Bonding Forces in Solids

3-1-1. Bonding Forces in Solids

lt100gt

Si

- Covalent
- Bonding

3-1-2. Energy Bands

- Pauli Exclusion Principle

C (Z6) 1s2 2s2 2p2 2 states for 1s level 2

states for 2s level 6 states for 2p level For N

atoms, there will be 2N, 2N, and 6N states of

type 1s, 2s, and 2p, respectively.

3-1-2. Energy Bands

Energy

Diamond lattice spacing

Atomic separation

3-1-3. Metals, Semiconductors Insulators

- For electrons to experience acceleration in an

applied electric field, they must be able to move

into new energy states. This implies there must

be empty states (allowed energy states which are

not already occupied by electrons) available to

the electrons. - The diamond structure is such that the valence

band is completely filled with electrons at 0ºK

and the conduction band is empty. There can be no

charge transport within the valence band, since

no empty states are available into which

electrons can move.

3-1-3. Metals, Semiconductors Insulators

Empty

- The difference bet-ween insulators and

semiconductor mat-erials lies in the size of the

band gap Eg, which is much small-er in

semiconductors than in insulators.

Empty

Eg

Eg

Filled

Filled

Insulator

Semiconductor

3-1-3. Metals, Semiconductors Insulators

- In metals the bands either overlap or are only

partially filled. Thus electrons and empty energy

states

Overlap

Metal

are intermixed with-in the bands so that

electrons can move freely under the infl-uence of

an electric field.

Partially Filled

Filled

Metal

3-1-4. Direct Indirect Semiconductors

- A single electron is assumed to travel through a

perfectly periodic lattice. - The wave function of the electron is assumed to

be in the form of a plane wave moving.

- x Direction of propagation
- k Propagation constant / Wave vector
- ? The space-dependent wave function for the

electron

3-1-4. Direct Indirect Semiconductors

- U(kx,x) The function that modulates the wave

function according to the periodically of the

lattice. - Since the periodicity of most lattice is

different in various directions, the (E,k)

diagram must be plotted for the various crystal

directions, and the full relationship between E

and k is a complex surface which should be

visualized in there dimensions.

3-1-4. Direct Indirect Semiconductors

E

E

Egh?

Eg

Et

k

k

Direct

Indirect

Example 3-1

3-1-4. Direct Indirect Semiconductors

- Example 3-1
- Assuming that U is constant in
- for an essentially free electron, show

that the x-component of the electron momentum in

the crystal is given by

Example 3-2

3-1-4. Direct Indirect Semiconductors

- Answer

The result implies that (E,k) diagrams such as

shown in previous figure can be considered plots

of electron energy vs. momentum, with a scaling

factor .

3-1-4. Direct Indirect Semiconductors

Properties of semiconductor materials

Eg(eV) ?n ?p ?

Lattice Å

- Si
- Ge
- GaAs
- AlAs
- Gap

1.11 1350 480 2.5E5 D 5.43 0.67 3900

1900 43 D 5.66 1.43 8500 400

4E8 Z 5.65 2.16 180 0.1 Z

5.66 2.26 300 150 1 Z 5.45

3-1-5. Variation of Energy Bands with Alloy

Composition

E

3.0

E

2.8

2.6

2.4

2.2

2.0

1.43eV

2.16eV

1.8

k

1.6

AlxGa1-xAs

AlAs

GaAs

1.4

0

0.2

0.4

0.6

0.8

1

X

3-2. Carriers in Semiconductors

Ec

Eg

0ºK

3ºK

2ºK

4ºK

5ºK

1ºK

6ºK

7ºK

8ºK

10ºK

11ºK

12ºK

13ºK

14ºK

300ºK

15ºK

16ºK

17ºK

18ºK

19ºK

20ºK

9ºK

Ev

Electron Hole Pair

E

P

H

3-2-1. Electrons and Holes

E

kj

-kj

k

j

j

3-2-2. Effective Mass

- The electrons in a crystal are not free, but

instead interact with the periodic potential of

the lattice. - In applying the usual equations of

electrodynamics to charge carriers in a solid, we

must use altered values of particle mass. We

named it Effective Mass.

3-2-2. Effective Mass

- Example 3-2
- Find the (E,k) relationship for a free electron

and relate it to the electron mass.

E

k

3-2-2. Effective Mass

- Answer
- From Example 3-1, the electron momentum is

3-2-2. Effective Mass

- Answer (Continue)
- Most energy bands are close to parabolic at

their minima (for conduction bands) or maxima

(for valence bands).

EC

EV

3-2-2. Effective Mass

- The effective mass of an electron in a band with

a given (E,k) relationship is given by

- Remember that in GaAs

3-2-2. Effective Mass

- At k0, the (E,k) relationship near the minimum

is usually parabolic

- In a parabolic band, is constant. So,

effective mass is constant. - Effective mass is a tensor quantity.

3-2-2. Effective Mass

EV

EC

Table 3-1. Effective mass values for Ge, Si and

GaAs.

Ge Si GaAs

m0 is the free electron rest mass.

3-2-3. Intrinsic Material

- A perfect semiconductor crystal with no

impurities or lattice defects is called an

Intrinsic semiconductor. - In such material there are no charge carriers at

0ºK, since the valence band is filled with

electrons and the conduction band is empty.

3-2-3. Intrinsic Material

e-

Si

h

npni

3-2-3. Intrinsic Material

- If we denote the generation rate of EHPs as

and the recombination rate as

equilibrium requires that

- Each of these rates is temperature depe-ndent.

For example, increases when the

temperature is raised.

3-2-4. Extrinsic Material

- In addition to the intrinsic carriers generated

thermally, it is possible to create carriers in

semiconductors by purposely introducing

impurities into the crystal. This process, called

doping, is the most common technique for varying

the conductivity of semiconductors. - When a crystal is doped such that the equilibrium

carrier concentrations n0 and p0 are different

from the intrinsic carrier concentration ni , the

material is said to be extrinsic.

3-2-4. Extrinsic Material

V

P

As

Sb

Ec

Ed

0ºK

3ºK

2ºK

4ºK

5ºK

1ºK

6ºK

7ºK

8ºK

10ºK

11ºK

12ºK

13ºK

14ºK

50ºK

15ºK

16ºK

17ºK

18ºK

19ºK

20ºK

9ºK

Ev

Donor

3-2-4. Extrinsic Material

?

B

Al

Ga

In

Ec

0ºK

3ºK

2ºK

4ºK

5ºK

1ºK

6ºK

7ºK

8ºK

10ºK

11ºK

12ºK

13ºK

14ºK

50ºK

15ºK

16ºK

17ºK

18ºK

19ºK

20ºK

9ºK

Ea

Ev

Acceptor

3-2-4. Extrinsic Material

e-

Sb

h

Al

Si

3-2-4. Extrinsic Material

- We can calculate the binding energy by using the

Bohr model results, consider-ing the loosely

bound electron as ranging about the tightly bound

core electrons in a hydrogen-like orbit.

3-2-4. Extrinsic Material

- Example 3-3
- Calculate the approximate donor binding energy

for Ge(er16, mn0.12m0).

3-2-4. Extrinsic Material

- Answer

Thus the energy to excite the donor electron from

n1 state to the free state (n8) is 6meV.

3-2-4. Extrinsic Material

- When a ?-V material is doped with Si or Ge, from

column IV, these impurities are called

amphoteric. - In Si, the intrinsic carrier concentration ni is

about 1010cm-3 at room tempera-ture. If we dope

Si with 1015 Sb Atoms/cm3, the conduction

electron concentration changes by five order of

magnitude.

3-2-5. Electrons and Holes in Quantum Wells

- One of most useful applications of MBE or OMVPE

growth of multilayer compou-nd semiconductors is

the fact that a continuous single crystal can be

grown in which adjacent layer have different band

gaps. - A consequence of confining electrons and holes in

a very thin layer is that

3-2-5. Electrons and Holes in Quantum Wells

- these particles behave according to the

particle in a potential well problem.

GaAs

Al0.3Ga0.7As

Al0.3Ga0.7As

50Å

E1

0.28eV

1.43eV

1.85eV

1.43eV

0.14eV

Eh

3-2-5. Electrons and Holes in Quantum Wells

- Instead of having the continuum of states as

described by , modified for

effective mass and finite barrier height. - Similarly, the states in the valence band

available for holes are restricted to discrete

levels in the quantum well.

3-2-5. Electrons and Holes in Quantum Wells

- An electron on one of the discrete condu-ction

band states (E1) can make a transition to an

empty discrete valance band state in the GaAs

quantum well (such as Eh), giving off a photon of

energy EgE1Eh, greater than the GaAs band gap.

3-3. Carriers Concentrations

- In calculating semiconductor electrical

pro-perties and analyzing device behavior, it is

often necessary to know the number of charge

carriers per cm3 in the material. The majority

carrier concentration is usually obvious in

heavily doped material, since one majority

carrier is obtained for each impurity atom (for

the standard doping impurities). - The concentration of minority carriers is not

obvious, however, nor is the temperature

dependence of the carrier concentration.

3-3-1. The Fermi Level

- Electrons in solids obey Fermi-Dirac statistics.
- In the development of this type of statistics
- Indistinguishability of the electrons
- Their wave nature
- Pauli exclusion principle
- must be considered.
- The distribution of electrons over a range of

these statistical arguments is that the

distrib-ution of electrons over a range of

allowed energy levels at thermal equilibrium is

3-3-1. The Fermi Level

- k Boltzmanns constant
- f(E) Fermi-Dirac distribution function
- Ef Fermi level

3-3-1. The Fermi Level

T0ºK

T1gt0ºK

T2gtT1

3-3-1. The Fermi Level

E

f(Ec)

Ec

Ef

Ev

f(E)

Intrinsic

n-type

p-type

0

1

1/2

3-3-2. Electron and Hole Concentrations at

Equilibrium

- The concentration of electrons in the conduction

band is

- N(E)dE is the density of states (cm-3) in the

energy range dE. - The result of the integration is the same as that

obtained if we repres-ent all of the distributed

electron states in the conduction band edge EC.

3-3-2. Electron and Hole Concentrations at

Equilibrium

E

Electrons

N(E)f(E)

EC

EV

N(E)1-f(E)

Holes

Intrinsic

n-type

p-type

3-3-2. Electron and Hole Concentrations at

Equilibrium

3-3-2. Electron and Hole Concentrations at

Equilibrium

3-3-2. Electron and Hole Concentrations at

Equilibrium

3-3-2. Electron and Hole Concentrations at

Equilibrium

- Example 3-4
- A Si sample is doped with 1017 As Atom/cm3. What

is the equilibrium hole concentra-tion p0 at

300K? Where is EF relative to Ei?

3-3-2. Electron and Hole Concentrations at

Equilibrium

- Answer
- Since Ndni, we can approximate n0Nd and

3-3-2. Electron and Hole Concentrations at

Equilibrium

- Answer (Continue)

Ec

EF

0.407eV

Ei

1.1eV

Ev

References

- Solid State Electronic Devices
- Ben G. Streetman, third edition
- Modular Series on Solid State Devices, Volume I

Semiconductor Fundamentals - Robert F. Pierret