EE 60556: Fundamentals of Semiconductors Lecture Note

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EE 60556 Fundamentals of SemiconductorsLecture
Note 14 (10/14/09)Read a band structure you
know everything about a crystal

• Outline
• Last class post midterm review, effective mass,
  group velocity
• Information from a band structure effective
  mass, group velocity, bandgap, direct or indirect
• The meaning of wave vector k crystal momentum of
  an electron (Kittel p.173)
• Constant energy contours

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Now we can derive a bandstructure (E-k diagram),
what information can we read from it? Effective
mass, group velocity. Let us also revisit the
physical meaning of k the wave vector
Kittel p.173 significance of k 1) the phase
factor in Bloch function, 2) electron crystal
momentum (not its total momentum) and used in the
conservation laws that govern collision processes
in crystals.
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Effective Mass

• Parabolic approximation
• Bigger the curvature, smaller effective mass
• light holes and heavy holes

Electrons in crystal obey the Newtons law when
the external force and its wave vector are used.
Starting from Newtons 2nd law, can you derive
the expression for the effective mass? (30 sec)
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Effective Mass

• Importance the motion of electrons in a crystal
  can be visualized and described in a
  quasi-classical manner!
• electron billiard ball Newtonian mechanics
  because m accounts for the effect of crystal
  forces and quantum mechanical properties!

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Effective mass

• Practice

It is not possible to fit the entire energy band
with a parabolic relation, but near the extreme
points we can generally approximate it with a
parabolic E-k diagram, i.e. find its effective
mass.
The larger the lattice constant, the smaller the
effective mass. The larger the interatomic
interaction (highly delocalized valence
electrons), the smaller the effective mass
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Interesting results from k.p theory

• For details, refer to the handouts but it is not
  required.
• Using perturbation theory and crystal symmetry,
  we can estimate the band structure of
  semiconductors thus effective mass (easy for
  conduction band electrons since C.B. is more
  s-like)

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Indirect direct
Constant energy contours
Using principle axes of the crystal (unit cell)
to reduce m tensor
Figure 310 Realistic bandstructures in
semiconductors (a) Conduction and valence bands
in Si and GaAs along 111 and 100 (b)
ellipsoidal constant energy surface for Si, near
the 6 conduction band minima along the X
directions. (From Chelikowsky and Cohen, Phys.
Rev. B14, 556, 1976).
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Absorption/emission of photon - Vertical
transition
Absorption/emission of phonon - horizontal
transition

• In any transition, K must be conserved as well as
  E.
• A direct gap semiconductor on the left is the
  E-K diagram, and on the right the conventional
  energy band diagram.
• An indirect gap material (so called because
  conduction band minimum and the valence band
  maximum do not occur at the same value of K).
• Figure 3.14

Anderson 2005
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4-fold degenerate at the bottom of Ge conduction
band
6-fold degenerate
Constant energy surface in conduction band
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• Importance
• s-like band is more parabolic and its constant
  energy contour closer to spherical
• p-like band is more ellipsoidal with complicated
  constant energy contours

Orbits, bands and constant energy contour
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Constant energy surface in valence band (Si)
heavy hole band
Split-off hole band
light hole band
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Indirect Bandgap
Direct Bandgap
Spin-orbit split band energy 10 1000 meV,
increases with decreasing Eg.
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Eg m versus T
Strong dependence of Eg on T Weak dependence of
m on T