Last Time - West Virginia University

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Transcript Last Time - West Virginia University

Group: Linear Chain of F
F: 1s22s22px22py22pz1
F
F
F
(a)
F
F
F
(b)
F
F
(c)
F
F
(d)
EF
E(k)
EF
EF
0
k
p/a 0
k
p/a 0
EF
k
p/a 0
k
Which of the following is the correct band structure for a
linear chain of F atoms (atomic #=9)?
p/a
Linear F Chain
• There are 4 n=2 orbitals in the unit cell (a
single F atom with 1 2s + 3 2p orbitals)
• There is a lower 1s band (not shown)
EF
E(k)
EF
EF
0
k
p/a 0
k
p/a 0
EF
k
p/a 0
k
p/a
Band Structure: Linear Chain of F
Bonding 2s s
Antibonding 2s s*
A sigma bond shares only one electron
Antibonding 2pz s*
Bonding 2pz s
Which one of these has lower energy?
Linear F Chain
• There are 4 n=2 orbitals in the unit cell (a
single F atom with 1 2s + 3 2p orbitals)
• The fact that the wavefunction corresponding
to a p-orbital changes sign at the nucleus
causes the 2p s band to run downhill
(opposite of the 2s s band).
Band Structure: Linear Chain of F
Bonding 2s s
Antibonding 2s s*
A sigma bond shares only one electron
Antibonding 2pz s*
Bonding 2pz s
Which of these two p orbitals is doubly degenerate
(which will we have two of)?
A pi bond shares two electrons
Bonding 2px/2py p
Antibonding 2px/2py p*
Group: Linear Chain of F
F: 1s22s22px22py22pz1
F
F
F
(a)
F
F
F
(b)
F
F
(c)
F
F
(d)
EF
E(k)
EF
EF
0
k
p/a 0
k
p/a 0
EF
k
p/a 0
k
How do we determine between the final remaining two?
p/a
Linear F Chain
• Do the sigma and pi p orbitals have the same
band width? What affects bandwidth?
• For the same lattice parameter, the reduced
spatial overlap of the p interaction causes the
p bands to be narrower than the s bands.
Band Structure: Linear Chain of F
Antibonding 2pz s*
Doubly degenerate
EF
Antibonding 2px/2py p*
Doubly degenerate
Antibonding 2s s*
Bonding 2px/2py p
Bonding 2pz s
Bonding 2s s
0
k
p/a
Band Structure: Linear Chain of F
A more accurate
treatment of the band
structure would show an
avoided crossing
between the 2pz s and
2s s * interactions at
k=p/a. There would be
mixing between these
two bands (creating sphybrid like states).
Antibonding 2pz s*
EF
Doubly degenerate
Bonding 2px/2py p
Bonding 2s s
0
k
p/a
Band Overlap
Flat band diagrams
• Many materials are metals
due to band overlap
• Often the higher energy
bands become so wide that
they overlap with the lower
bands
Band Hybridization
In some cases the opposite
occurs
– Due to the overlap, electrons
from different shells form
hybrid bands, which can be
separated in energy
– Depending on the
magnitude of the gap, solids
can be insulators (Diamond);
semiconductors (Si, Ge, Sn)
Learning Objectives for Today
After today’s class you should be able to:
• Calculate effective mass and cyclotron
frequency
• Understand the difference between an
electron and an electron quasiparticle
• Distinguish between two types of
excitons
Eph>Eg
Conduction
band
Energy
gap
The ground
state is
sometimes
called the
vacuum state
Electron energy
Flat band structure in semiconductors
-
Forbidden
band
Valence Band = Dirac
Sea of Electrons
If the photon energy is higher than the
energy gap the electron can be excited
Quasiparticles
2
2
 k
E
2m*
• Quasiparticles occur when a solid behaves as if
it contained free particles.
• Example: as an electron travels through a
semiconductor, its scatters with electrons and
nuclei; however it ~behaves like an electron with
a different mass traveling unperturbed through
free space. This "electron" with a m* is called an
"electron quasiparticle". (m* =Effective mass)
• Aggregate motion of electrons in the valence
band of a semiconductor is the same as if the
semiconductor contained positively charged
quasiparticles called holes or positrons.
•
•
•
•
-
Conduction
band
Energy
gap
E
Electron energy
Electron energy band structure in semiconductor
+
Forbidden
band
Valence
band
Quasiparticles: excitations of the ground state (vacuum state),
behave as if particles in free space
Excited electron leaves in the valence band positive hole/positron.
Electron and positron are quasiparticles and charge carriers (equal)
Positively charged hole interacts with negatively charged electron
by Coulomb interaction.
Excitons (quasiparticle) are bound electron-hole states
A free electron and a free hole (empty electronic state in the valence band)
exert Coulomb force on each other:
hydrogen-like bound states possible: excitonic states
n=3
n=2
n=1
e
Coulomb
force
E
Eb is the exciton
binding energy =
h
energy released upon
exciton formation, or
Eb
k
energy required for
exciton breakup
Note: exciton can move through crystal, i.e. not bound to specific atom!
Two Types of Excitons
Wannier – Matt excitons (free exciton): mainly exist in semiconductors,
have a large radius, are delocalized states that can move freely throughout
the crystal, the binding energy ~ 0.01 eV
Frenkel excitons (tight bound excitons): found in insulator and molecular
crystals, bound to specific atoms or molecules and have to move by
hopping from one atom to another, the binding energy ~ 0.1 -1 eV.
The maximum energy of a thermally excited phonon ~ kBT = 0.025 eV (RT)
Wannier – Matt
A phonon
is another
excitons:
quasiparticle
stable
at cryogenic
temperature
Frenkel excitons:
stable at room
temperature
Excitons in most of semiconductors are not
observable at room temperature, because
of the low binding energy
Quasiparticle
in a box
Light Hole and Heavy Hole bands
(holes with different m*)
On the other hand, excitonic
emission is very important for
opto-electronic applications,
as it is narrow and highly
energetic
Confused electron
Let’s focus on
Effective Mass
Confused hole
Comparing free electrons and
the electron quasiparticle
Free electron
2
2
k
E
2me
Electron quasiparticle
2
2
k
E
*
2me
While electrons scatter, we treat electron quasiparticles like
free electrons with m*.
The renormalized mass effectively takes into account all
interactions (with crystal, electrons, phonons)
Thus, electron quasiparticles don’t scatter.
EFFECTIVE MASS
Real metals: electrons still behave like
free particles, but with “renormalized” effective mass m*
In potassium (a metal), assuming m* =1.25m gets the correct
(measured) electronic heat capacity
2
2
k
E
*
2me
Physical intuition: m* > m, due to “cloud” of phonons and other excited electrons
that slow it down (add mass)
Fermi Surface
Interactions with the periodic crystal,
electron-electron interactions and
electron-phonon interactions
renormalize the elementary
excitation to an “electron-like
quasiparticle” of mass m*
Physical Meaning of the Band Effective Mass
2 2
 k
E
*
2m
The effective mass
is inversely
proportional to the
curvature of the
energy band.
Near the bottom of a nearly-free electron band m* is approximately constant,
but it increases dramatically near the inflection point and even becomes
negative(!) near the zone edge. What does that mean for electron near BZE?
Because the energy bands are different
along different directions, the effective
mass depends on which direction in kspace we are “looking”
Effective
Mass
2
Heavy and1light holes
1  dhave
E different
 2  masses
bands so different
  2effective
m*
from
(also different
electrons)
 dk

Group: Find the effective mass
tensor for electrons in a simple
cubic tight-binding band at the
center , face center X, and at
the corner R of the Brillouin zone.
1
1 2E
 2
m *  ki k j
i, j  x, y, z
Besides Getting the Energy
Bands, How Could You
Measure the Effective Mass?
Deflection of Electrons in a
Uniform Magnetic Field (1)
The force F acting on
an electron in a
uniform magnetic
field is given by
F  ev  B
Since the magnetic force F is at a right
angle to the velocity direction, the
electron moves round a circular path.
Deflection of Electrons in a
Uniform Magnetic Field (2)
F  Bev  ma
The centripetal acceleration of the electrons is
Bev
a
m
v 2 Bev
Hence a 

r
m
mv
r
eB
which gives
Cyclotron frequency
Effective Mass
Measure effective mass:
cyclotron resonance v. crystallographic direction
-Measure the absorption of radio frequency energy
v. magnetic field strength.
eB
c 
m*
Put sample in a microwave
resonance cavity at <40 K and
adjust the rf frequency until it
matches the cyclotron frequency.
Will see a resonant peak in the
energy absorption.
Effective Masses in Semiconductors
 v  hh  =
2 2
k
2mhh
m* determined by
cyclotron resonance
k
  lh  =
2m
(rf) at low carrier
k
concentration
  soh  = 
2m
2 2
v
lh
2 2
v
soh
m*  Eg for direct-gap crystals
For
InSb, InAs,
InP
mc
0.015 0.026 0.073

,
,
m Eg 0.23 0.43 1.42
 0.065, 0.060, 0.051
Prep for Homework (due Thursday after break)
Hint:
• How do constants affect the effective mass?
• Is there a way to get around dealing with M-1?
• Any way to simplify matrix given H direction?
• The formula for an ellipse (the orbit) may help. Why an ellipse?
Extra Slides
• Depending on time, I might throw the next
three slides into another lecture as it
shows why a full band doesn’t contribute
to the electrical conduction.
• You are also welcome to look through the
argument yourself.
Dynamics of Bloch Electrons
(electrons in a periodic potential)
Metals have partially-filled upper bands, while semiconductors and insulators
have filled upper bands. What is the qualitative difference between filled and
partially-filled bands?

 


J  ne
v (k )
For a collection of
Current density for j  nev

k
electrons:
single electron:


But the symmetry of the energy bands requires: E (k )  E (k )
Thus we conclude:
 
So for a filled band, which 
J  ne  v (k )  0
has an equal number of
1stBZ
electrons with k positive
and negative,

 
v (k )  v(k )
Filled energy bands carry no
current! We will see that this
is true even when an electric
field is applied.
Note: the electrons in filled bands are not stationary…there are just
the same number moving in each direction, so the net current is zero.
Electron in an Electric Field
An external electric field causes a change in the k vectors of all electrons:





dk
dk  eE
F 
 eE

dt
dt

If the electrons are in a partially filled band, this will
break the symmetry of electron states in the 1st BZ
and produce a net current. But if they are in a filled
band, even though all electrons change k vectors, the
symmetry remains, so J = 0.
E
 pa
v
p
a
kx
When an electron reaches the 1st BZ edge (at k = p/a)
it immediately reappears at the opposite edge (k = p/a) and continues to increase its k value.
kx
As an electron’s k value increases, its velocity
increases, then decreases to zero and then becomes
negative when it re-emerges at k = -p/a!!
Thus, an AC current is predicted to result from a DC
field! (Bloch oscillations)
Do We Observe This?
Not until fairly
recently, due to the
effect of collisions
on electrons in a
periodic but
vibrating lattice.
In both experiments the periodic potential was fabricated in an artificial way
to minimize the effect of collisions and make it possible to observe the Bloch
oscillations of electrons (or atoms!).
Discuss presentation grading
• 10 minutes, no more than 5 PPT slides
• Board work acceptable but visuals encouraged
Graded upon:
• Explains the interest of the topic to the average
taxpayer including possible applications
• Explains at least 2 important findings in the field with
appropriate references
• Relates topic to some portion of the class
• All displayed visuals are at least briefly mentioned
• No more than 20 words per slide
• The main point per slide is stated orally and maybe
briefly in words (recommend as title)
Now we can derive a bandstructure (E-k diagram), what
information can we get? Effective mass, group velocity.
k describes electron’s
response to the external
force (only)
Absorption/emission of
photon - Vertical transition
Absorption/emission of
phonon - horizontal
transition
The onset of indirect transition
should include both Eg-Ephonon,
Eg+Ephonon,
In any transition, K must be conserved as well as E.
(a) A direct gap semiconductor; on the left is the E-K diagram, and on the right the conventional energy band diagram.
(b) An indirect gap material (so called because conduction band minimum and the valence band maximum do not
occur at the same value of K).
Eg & m* versus T
Strong dependence of Eg on T
Weak dependence of m* on T
Indirect
Bandgap
Direct
Bandgap
Spin-orbit split band energy ~ 10 – 1000 meV, increases with decreasing Eg.