Lecture 2: Semiconductor Electronic Structure
MSE 6001, Semiconductor Materials Lectures
Fall 2006
2
2.1
Energy Band Structure of Semiconductors
Electrons in energy bands
Some of the electrons from the atoms in a semiconductor can be freed from their chemical bonds
and can move through the crystal, either by being pulled by an electric field or by diffusion from
a region of high electron concentration, n measured in cm−3 , to low concentration. However,
the wave-like properties of the electrons and their collisions with atoms and other electrons are
important. The possible electron wave states that can exist in a crystal are described by a dispersion
diagram, which depicts how electron states of different momenta are dispersed over energy. In a
crystal, these allowed energies of electron waves lay within bands of energies due to the contructive
2: Semiconductor
Electronic
Structure
and destructive interferencesLecture
of the electron
waves within the periodic
crystal lattice.
MSE 6001, Semiconductor Materials Lectures
Electrons in motion Figure 1 gives a dispersion diagram
an electron moving in free space in
Fallfor
2006
one dimension. For an electron with momentum pe and mass me , the energy is
2 Energy Band Structure
p2e of Semiconductors
E=
(1)
2me
2.1 Electrons in energy bands
Each point on the parabola is an “electron state” that corresponds to the possibility of an electron
of the electrons
from the atoms
semiconductor
can
be freed
their chemical
with that E andSome
pe , moving
at the velocity
ve = inpea/m
gives
the from
curvature
of the bonds
e . The mass
and
can
move
through
the
crystal,
either
by
being
pulled
by
an
electric
field
or
by
diffusion
dispersion curve. If there actually is an electron in a given state, we say that state is “occupied”.
If from
−3
a region of high
electron concentration,
measured
cmthis
, to
lowaconcentration.
we have some concentration
of electrons
n moving at a nvelocity
ve , in
then
gives
current densityHowever,
the
wave-like
properties
of
the
electrons
and
their
collisions
with
atoms
and other electrons are
of
important. The possible electron wave states that can exist in a crystal are described by a dispersion
diagram, which depicts how electron states of different momenta are dispersed over energy. In a
J = −qnv ,
(2)
crystal, these allowed energiesnof electronewaves lay within bands of energies due to the contructive
destructive
interferences
of the×electron
waves within the periodic crystal lattice.
where −q is the and
electron
charge,
and q = 1.602
10−19 coulombs.
Electrons in motion Figure E1 gives a dispersion diagram for an electron moving in free space in
one dimension. For an electron with momentum pe and mass me , the energy is
p2e
(1)
2me
Each point on the parabola is an “electron state” that corresponds to the possibility of an electron
with that E and pe , moving at the velocity ve = pe /me . The mass gives the curvature of the
dispersion curve. If there actually is an electron in a given state, we say that state is “occupied”. If
E=
0
pe
F IGURE 1: Dispersion diagram for an electron in free space.
2-1
F IGURE 2: Band structure of GaAs. The dispersion of electron states is plotted for two directions.
(http://www.ioffe.rssi.ru/SVA/NSM/Semicond/)
Electrons are quantum mechanical objects, with wavelike properties described by a wavevector
ke = pe /h̄ and frequency νe = E/. When considering electrons as quantum mechanical objects,
the dispersion is often plotted as E versus ke .
Band structure In a crystal, electron wave interference dictates which waves fit around the
atoms, and the energies of possible states disperse over wavevectors in complicated ways. Figure 2 gives a dispersion diagram of the electron states in GaAs. The states below E = 0 are
essentially all filled with electrons and correspond to electrons held in chemical bonds between
atoms, and these bands are referred to as the “valence band” of energies. The states that lay at
energies greater than Eg above the valence band maximum are all essentially unoccupied with
electrons, and this higher band is the “conduction band” of energies. No wave states are possible,
because of interference, in the energy range Eg , which is called the semiconductor bandgap.
For most of the semiconductor materials properties needed for devices, the states that make
a contribution to the value of the property, such as α, are those near the top of the valence band
edge EV and near the bottom of the conduction band EC , just below and above the bandgap.
Consequently, a simplified dispersion diagram is used, with the dispersion of the band edge states
fit with parabolas near these energies, as depicted in Fig. 3. An electron moving in the conduction
band occupies one of the states on the parabola. The curvature of the parabola is different from that
for an electron in free space, corresponding to a different mass, the electron effective mass mn .
Optical processes A semiconductor generally only absorbs light with photon energy hν ≥ Eg ,
exciting an electron from a valence band state to a conduction band state. An electron from the
conduction band can move to an empty valence band state and emit a photon with an energy given
by the energy difference of these states. These processes are at the heart of solar cells and optical
detectors. An electron can also make this downwards transition by losing energy without emitting
a photon, for example by giving energy to a crystal lattice vibration (a phonon). Photon emission
is used for LEDs and lasers.
2-2
E
Conduction
Band
empty
states
EC
Eg = EC - EV
EV
filled
states
Valence
Band
ke
F IGURE 3: GaAs dispersion diagram near the band edges.
E
Conduction Band
empty
states
Valence Band
filled
states
EC
EV
x
F IGURE 4: Band diagram for a semiconductor, plotting E versus x.
2.2
Band diagrams
The principle way a semiconductor material’s electronic structure is represented in a device is with
a band diagram, which plots the band edge energies EC and EV and other energy levels versus
position x (Fig. 4). Light absorption, with photon energy hν ≥ Eg , can be depicted as a valence
band electron making a transition to a conduction band state. Light with photon energy hν ≤ Eg
is not absorbed and passes through the material.
Electrons and holes Electrons tend to move to the lowest energy state available, which corresponds to moving downwards in the band diagram. In the valence band, this means that the empty
states tend to bubble up. Because the negative charges of the valence band electrons are balanced
by the positive charges on the crystal atoms, the valence band empty state has a positive charge.
These empty states, which have a positive charge and can move around bubble-like, are called
“holes”, and they have an effective mass mp given by the valence band curvature. The concentration of holes in a material is p. A hole moves to a lower energy state by moving upwards in a band
diagram.
2-3
2.3
Electrons motion in bands: Current density
An electric field E appears in a band diagram as a local slope in the band edge energy levels EC
and EV . (Fig. 5) The electric field is given by
E=
1 dEC
.
q dx
(3)
Electrons and holes are accelerated in opposite directions in an electric field, though because they
have opposite charge, their current densities are in the same direction, where the total current
density for electrons and holes drifting in an electron field of
J = Jn + Jp = qµn nE + qµp pE,
(4)
with µn and µp the electron and hole mobilities.
2.4
Doping
The concentration of electrons and holes may be controlled through doping the crystal, introducing
trace amounts of chemical impurities. For example, replacing a silicon crystal atom (with 4 valence
electrons for bonding) with an arsenic atom (with 5 valence electrons), leaves an extra electron not
needed to bond to nearby silicon atoms. This extra electron is only loosely held by the arsenic
atom and easily breaks away to be donated to the conduction band. The concentration of donor
atoms is ND , and if only donors are present, the chemical impurities are generally chosen such
that n = ND . A crystal can also be doped with an atom with one-fewer electron than is needed
for bonding, which can accept an electron from the valence band, leaving a free hole behind. The
concentration of acceptor atoms is NA , and if only acceptors are present, the chemical impurities
are generally chosen such that p = NA .
2.3
Electrons motion in bands: Current density
An electric field E appears in a band diagram as a local slope in the band edge energy leve
and EV . (Fig. 5) The electric field is given by
EC
E=
1 dEC
.
q dx
Electrons and holes are accelerated in opposite directions in an electric field, though becaus
have opposite charge, their current densities are in the same direction, where the total c
density for electrons and holes drifting in an electron field of
EV
J = Jn + Jp = qµn nE + qµp pE,
with µn and µp the electron and hole mobilities.
2.4 Doping
F IGURE 5: Band diagram with an electric field.
The concentration of electrons and holes2-4may be controlled through doping the crystal, introd
trace amounts of chemical impurities. For example, replacing a silicon crystal atom (with 4 v
electrons for bonding) with an arsenic atom (with 5 valence electrons), leaves an extra electr