Download Band structure - Exam March 2007

Band structure - Exam March 2007
The band structure of a GaN (Galium nitride) is shown below. The zero of energy is chosen to be the top of the
valence band.
(a) Is this a direct or an indirect semiconductor? Direct Why? The bottom of CB is right above/directly over the top of
VB at the same –value k= Γ (Gamma) in the k-space. Δk=0
(b) What is the Fermi energy for GaN? Ef is in the middle of
band gap Eg, thus Ef= Eg/2=(Ec-Ev)/2=1.695eV (c) What is
the band gap? Eg= 3,4eV
(d) What are light holes and heavy holes?
Light hole band has higher/greater slope than heavy hole band resp. light hole band is steeper & heavy hole band
flatter and they are located in the valence band and are degenerate bands. As the names say heavy holes are in the
heavy hole band and light holes in the light hole band, meaning the effective mass m* of heavy holes mhh is larger
than of light holes mlh. Thus holes respond with two different speeds to electric field, meaning light holes are/respond
faster than heavy holes. (Heavy/light) holes, positively charge carries, can also be represented as mass of holes in
different (moving) directions in VB.
Explain how you can determine the effective mass of the holes from this diagram.
The effective mass is used to describe the response of –e/holes to external forces like electric field or other atoms in
the crystal. Near the bottom of CB and top of the valence band, the band structure looks like/can be approximated with
a parabola and the same is true for the simplified free –e model for/of a metal. Free –e only contribute/possess kinetic
energy, thus the same is valid for –e/holes. The relationship between E &k is parabolic. 1st derivative is slope & 2nd is
curvature. CB min. is not perfect/good-shaped parabola, which causes a problem when we use free –e model (parabola
model), thus we want this parabola to fit
more to Si band structure and define effective
mass m*. Hence, -e moves in CB with m*.
m* comes from interaction of the waves with
the crystal structure. –e moving in CB min
have same E-k relationship as free –e but a
different mass, hence m*. Applied electric field accelerates –e as if they have m*. m* contains information how –e
interacts as wave with crystal structure. Thus we can think of –e as particle again, which is easier. m* is smaller or
bigger then electron mass me. m* is negative for –e in VB, which is complicated/confusing to calculate, hence we
define hole (positive charge) which has positive m* in VB. mh*= -me*= negative curvature of VB. m* is calculate
from curvature. m* smaller than me means –e responds faster than free –e would. m= me mx*= me* in CB= -me* in VB
resp. m*<0 in VB of –e. In a nearly fille band the empty space left behind by an –e (the hole) moves in opposite
direction as –e. Charge carriers in VB can be considered to be positively charged holes. The number of holes in VB is
no. of missing –e. m*h= effective mass of hole. The free –e density of states (DOS) is modified by m*.
Behavior of –e near bandedges determines most device properties. Near the
bandedges –e can be described by simple effective mass pictures e.g. –e
behave as if they are in free space except their masses
are m*.
(e) When is a semiconductor degenerately doped?
Heavily doped SC are called degenerately doped.
ND>0.1 NC -> EF in CB & NA>0.1 NV -> EF in VB, thus
no bandgap anymore. Comparable to metal in terms of
behavior, conductivity etc. You can think of it as high
resistance metal.