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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.
xjkxk
xexkUx ),()( 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.
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3-1-4. Direct & Indirect Semiconductors
Eg=hνEg Et
k k
EE
Direct IndirectExample 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
xx khP Example 3-2
),()( xkUx xk xjkxe
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3-1-4. Direct & Indirect Semiconductors
x
x
xjkxjk
x
khdxU
dxUkh
dxU
dxexj
heU
P
xx
2
2
2
2
)(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 .
h
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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
khmvp
222
2
22
1
2
1k
m
h
m
pmvE
Answer: From Example 3-1, the electron momentum is:
m
h
dk
Ed 2
2
2
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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
2
2
2*
dkEd
hm
X
L
k
E
1.43eV
) ()( or** LXmm
Remember that in GaAs:
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3-2-2. Effective Mass
• At k=0, the (E,k) relationship near the minimum is usually parabolic:
gEkm
hE 2
*
2
2In a parabolic band, is constant. So, effective mass is constant.
Effective mass is a tensor quantity.
2
2
dk
Ed
2
2
2*
dkEd
hm
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3-2-2. Effective Mass
EV
EC
02
2
dk
Ed
02
2
dk
Ed
0* m
0* m2
2
2*
dkEd
hm
Ge Si GaAs
† m0 is the free electron rest mass.
Table 3-1. Effective mass values for Ge, Si and GaAs.
mn
*
mp
*
055.0 m 01.1 m 0067.0 m
037.0 m 056.0 m 048.0 m
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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
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3-2-5. Electrons and Holes in Quantum Wells
these particles behave according to the particle in a potential well problem.
GaAs Al0.3Ga0.7AsAl0.3Ga0.7As
50Å
E1
Eh
1.43eV1.85eV
0.28eV
0.14eV
1.43eV
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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.
2
222
2mL
hnEn
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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 Eg+E1+Eh, greater than the GaAs band gap.
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