Energy bands and charge carriers in semiconductors

1
Energy bands and charge carriersin semiconductors
In the name of God

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

2
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

3
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

4
Outlines

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

5
3-1. Bonding Forces Energy Bands in Solids

• In Isolated Atoms
• In Solid Materials

6
3-1-1. Bonding Forces in Solids

• Na (Z11) Ne3s1
• Cl (Z17) Ne3s1 3p5

• ION
• Bonding

7
3-1-1. Bonding Forces in Solids

• Metallic
• Bonding

8
3-1-1. Bonding Forces in Solids
9
3-1-1. Bonding Forces in Solids
lt100gt
Si

• Covalent
• Bonding

10
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.
11
3-1-2. Energy Bands
Energy
Diamond lattice spacing
Atomic separation
12
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.

13
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
14
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
15
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

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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.

17
3-1-4. Direct Indirect Semiconductors
E
E
Egh?
Eg
Et
k
k
Direct
Indirect
Example 3-1
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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
19
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 .
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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
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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
22
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
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3-2-1. Electrons and Holes
E
kj
-kj
k
j
j
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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.

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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
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3-2-2. Effective Mass

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

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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
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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

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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.

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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.
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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.

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3-2-3. Intrinsic Material
e-
Si
h
npni
33
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.

34
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.

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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
36
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
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3-2-4. Extrinsic Material
e-
Sb
h
Al
Si
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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.

39
3-2-4. Extrinsic Material

• Example 3-3
• Calculate the approximate donor binding energy
  for Ge(er16, mn0.12m0).

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3-2-4. Extrinsic Material

• Answer

Thus the energy to excite the donor electron from
n1 state to the free state (n8) is 6meV.
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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.

42
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

43
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
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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.

45
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.

46
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.

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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

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3-3-1. The Fermi Level

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

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3-3-1. The Fermi Level
T0ºK
T1gt0ºK
T2gtT1
50
3-3-1. The Fermi Level
E
f(Ec)
Ec
Ef
Ev


f(E)
Intrinsic
n-type
p-type
0
1
1/2
51
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.

52
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
53
3-3-2. Electron and Hole Concentrations at
Equilibrium
54
3-3-2. Electron and Hole Concentrations at
Equilibrium
55
3-3-2. Electron and Hole Concentrations at
Equilibrium
56
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?

57
3-3-2. Electron and Hole Concentrations at
Equilibrium

• Answer
• Since Ndni, we can approximate n0Nd and

58
3-3-2. Electron and Hole Concentrations at
Equilibrium

• Answer (Continue)

Ec
EF
0.407eV
Ei
1.1eV
Ev
59
References

• Solid State Electronic Devices
• Ben G. Streetman, third edition
• Modular Series on Solid State Devices, Volume I
  Semiconductor Fundamentals
• Robert F. Pierret