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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
http://folk.uio.no/ravi/CMT2015
Prof.P. Ravindran, Department of Physics, Central University of Tamil
Nadu, India
Evolution of Bands in Solids
1
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
2
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
3Schematic Illustration of Bands in Insulator,
Semiconductor and Metal
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
4
What are the energy levels?
2 2
( )2m
k
kSommerfeld:
Bloch: For a given band index n, has
no simple explicit form. The only general
property is periodicity in the reciprocal space:
( )n k
( ) ( )n n k G k
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
5
Bandgap and Evaluation of Bands in Na
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
6
Band-Structure Measurements
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
7
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Metals and insulators
In full band containing 2N electrons all states within the first B. Z. are occupied.
A partially filled band can carry current, a filled band cannot
Insulators have an even integer numberof electrons per primitive unit cell.
With an even number of electrons perunit cell can still have metallic behaviourdue to band overlap.
Overlap in energy need not occurin the same k direction
EF
Metal due to overlapping
bands
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Full Band
Empty Band
Energy Gap
Full Band
Partially Filled Band
Energy Gap
Part Filled Band
Part Filled Band
EF
INSULATOR METAL METAL
or SEMICONDUCTOR or SEMI-METAL
EF
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Insulator -energy band theory
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Solid state
N~1023 atoms/cm32 atoms 6 atoms
Energy band theory
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Bound States in atoms
r4
qe = )r(V
o
2
Electrons in isolated
atoms occupy discrete
allowed energy levels
E0, E1, E2 etc. .
The potential energy of
an electron a distance r
from a positively charge
nucleus of charge q is
-8 -6 -4 -2 0 2 4 6 8
-5
-4
-3
-2
-1
0
F6
F7
F8
F9
r
V(r)E2
E1
E0
Increasing
Binding
Energy
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Bound and “free” states in solids
-8 -6 -4 -2 0 2 4 6 8
-5
-4
-3
-2
-1
0
F6
F7
F8
F9
r-8 -6 -4 -2 0 2 4 6 8
-5
-4
-3
-2
-1
0
F6
F7
F8
F9
r
V(r)
E2
E1
E0
The 1D potential energy
of an electron due to an
array of nuclei of charge
q separated by a distance
R is
Where n = 0, +/-1, +/-2 etc.
This is shown as the
black line in the figure.
n o
2
nRr4
qe = )r(V
+ + + + +
RNuclear
positions
V(r) lower in solid (work
function).
V(r)
Solid
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Metal – energy band theory
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Each atomic orbital leads to a band of allowed
states in the solid
Band of allowed states
Band of allowed states
Band of allowed states
Gap: no allowed states
Gap: no allowed states
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band Theory of Solids
What happens in crystalline solids when we bring atoms so close together that
their valence electrons constitute only one electron each?
Band structure of diamond. http://home.att.net/~mopacmanual/node372.html
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Energy-level diagram for an isolated sodium atom.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Energy-level diagram for a hypothetical Nɑ4 molecule. The
four shared, outer orbital electrons are “split” into four slightly
different energy levels, as predicted by the Pauli exclusion
principle.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Energy-level diagram for solid sodium. The discrete 3s energy
level of Fig. has given way to a pseudocontinuous energy band
(half-filled). Again, the splitting of the 3s energy level is
predicted by the Pauli exclusion principle.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band structure
In general the eigenvalues
are in a complicated
function of k:
E(k)=f(k)
Free electron
model:
The dispersion relation between the wave
vector and the energy eigenvalues
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
IV-Column materials
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
The energy levels of the overlapping
electron shells are all slightly altered.
The energy differences are very
small, but enough so that a large
number of electrons can be in close
proximity and still satisfy the Pauli
exclusion principle.
The result is the formation of
energy bands, consisting of many
states close together but slightly
split in energy.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
The energy levels are so close together that for all practical purposes we can consider bands as a continuum of states, rather than discrete energy levels as we have in isolated atoms (and in the core electrons of atoms of metals).
A detailed analysis of energy bands shows that there are as many separate energy levels in each band as there are atoms in a crystal.
Suppose there are N atoms in a crystal. Two electrons can occupy each energy level (spin), so there are 2N possible quantum states in each band.
Let’s consider sodium as an example. Sodium has a single outer 3s electron.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
When you bring two sodium
atoms together, the 3s energy
level splits into two separate
energy levels.
Things to note: 4 quantum
states but only 2 electrons.
You could minimize electron energy by putting both 3s electrons
in the lower energy level, one spin up and the other spin down.
There is an internuclear separation which minimizes electron
energy. If you bring the nuclei closer together, energy increases.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
When you bring five sodium
atoms together, the 3s energy
level splits into five separate
energy levels.
The three new energy levels
fall in between the two for 2
sodiums.
There are now 5 electrons occupying these energy levels.
I’ve suggested one possible minimum-energy configuration. Notice
how the sodium-sodium internuclear distance must increase slightly.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
When you bring N (some big
number) sodium atoms together,
the 3s energy level splits into N
separate energy levels.
The result is an energy band,
containing N very closely-spaced
energy levels.
There are now N electrons occupying this 3s band. They go into the lowest
energy levels they can find.
The shaded area represents available states, not filled states. At the selected
separation i.e. upto EF, these are the available states.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Now let’s take a closer look at the
energy levels in solid sodium.
Remember, the 3s is the outermost
occupied level.
When sodium atoms are brought
within about 1 nm of each other, the 3s
levels in the individual atoms overlap
enough to begin the formation of the 3s
band.
The 3s band broadens as the
separation further decreases.
3s band begins to form
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Because only half the states in the 3s
band are occupied, the electron energy
decreases as the sodium-sodium
separation decreases below 1 nm.
At about 0.36 nm, two things happen:
the 3s energy levels start to go up and
the 2p levels start to form a band.
Further decrease in interatomic
separation results in a net increase of
energy.
3s electron energy is minimized
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
What about the 3p and 4s bands
shown in the figure?
Don’t worry about them—there are no
electrons available to occupy them!
Keep in mind, the bands do exist,
whether or not any electrons are in
them.
What about the 1s and 2s energy
levels, which are not shown in the
figure?
The sodium atoms do not get close
enough for them to form bands—they
remain as atomic states.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Energy levels for an actual crystal structure also
vary with different directions in space.
http://cmt.dur.ac.uk/sjc/thesis/thesis/node39.html, band structure of silicon
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Figure shows energy bands in carbon
(and silicon) as a function of
interatomic separation.
At large separation, there is a filled 2s
band and a 1/3 filled 2p band.
But electron energy can be lowered if the carbon-carbon separation is reduced.
There is a range of carbon-carbon separations for which the 2s and 2p bands overlap
and form a hybrid band containing 8N states.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
But the minimum total electron
energy occurs at this carbon
carbon separation.
At this separation there is a
valence band containing 4N
quantum states and occupied by
4N electrons.
The filled band is separated by a large gap from the empty conduction band.
The gap is 6 eV—remember, kT is about 0.025 eV at room temperature. The
gap is too large for ordinary electric fields to move an electron into the
conduction band. Carbon is an insulator.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Silicon has a similar band
structure. The forbidden gap is
about 1.1 eV.
The probability of a single electron
being excited across the gap is
small, proportional to
exp(-Egap/kT).
However, there are enough 3s+3p electrons in silicon that some of them might
make it into the conduction band. We need to consider such a special case.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids© Copyright 2005
Sharif University of Technology1st
Workshop on Photonic Crystals
Mashad, Iran, September 2005
Brillouin Zones
BZ # 1
a 23 2 3
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Brillouin Zones
BZ #2
a 23 2 3
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Brillouin Zones
Irreducible BZ
a 23 2 3
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Bandstructure: E versus k
The “Extended Zone scheme”
A plot of Ek with no restriction on k
But note! ψk(x) = eikx uk(x) & uk(x) = uk(x+a)
Consider (for integer n): exp[i{k + (2πn/a)}a] exp[ika]
The label k & the label [k + (2πn/a)] give the same ψk(x) (&
the same energy)! In other words, the translational symmetry
in the lattice Translational symmetry in “k space”! So, we
can plot Ek vs. k & restrict k to the range
-(π/a) < k < (π/a)
“First Brillouin Zone” (BZ)
(k outside this range gives redundant information!)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Bandstructure: E versus k
“Reduced Zone scheme”
A plot of Ek with k restricted to the first BZ.
For this 1d model, this is -(π/a) < k < (π/a)
k outside this range is redundant & gives no new information!
Illustration of the Extended & Reduced Zone schemes in 1d with
the free electron energy:
Ek = (ħ2k2)/(2mo)
Note: These are not really bands! We superimpose the 1d lattice
symmetry (period a) onto the free e- parabola.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Free e- “bandstructure” in the 1d extended zone scheme:
Ek = (ħ2k2)/(2mo)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
The free e- “bandstructure” in the 1d reduced zonescheme:
Ek = (ħ2k2)/(2mo)
For k outside the 1st BZ,
take Ek & translate it into
the 1st BZ by adding
(πn/a) to k
Use the translational
symmetry in k-space as
just discussed.
(πn/a)
“Reciprocal Lattice Vector”
![Page 41: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure](https://reader034.vdocuments.net/reader034/viewer/2022050714/5f0827507e708231d4209abd/html5/thumbnails/41.jpg)
P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Reduced Brillouin zone scheme
The only independent values of k are those in the first Brillouin zone.
Discard for
|k| > /a
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
How to Calculate DFT Band structure
I. Solve the Kohn-Sham
equations self-consistently to
determine the effective potential
using an even k-point sampling.
Bouckaert et al., Phys. Rev 50, 58 (1938).
II. Use the effective potential
while solving the Kohn-Sham
equations non self-consistently
along high symmetry lines in the
Brillouin zone
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Example: Band structure of Cu
Cu has FCC structure.
High symmetry points in the Brillouin zone:
G=center of the Brillouin zone
L=mid point on the zone boundary plane in the {111}-directions
W=corner point on the hexagon of the {kikj}-plane
K=mid point on the edge between two hexagons {110}-direction
X= mid point on the zone boundary plane in the {100}-direction
Bouckaert et al., Phys. Rev 50, 58
(1938).
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band structure of Cu
Electronic configuration of Cu: 3d94s2
Bouckaert et al., Phys. Rev 50, 58
(1938).
![Page 46: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure](https://reader034.vdocuments.net/reader034/viewer/2022050714/5f0827507e708231d4209abd/html5/thumbnails/46.jpg)
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
Once SCF has been achieved, we compute the bands along
the high symmetry points in the first-Brillouin zone
Variables to plot the band structure
First-Brillouin zone of a FCC , with the high symmetry points
The band structure is dumped in a file called Al.bands
![Page 47: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure](https://reader034.vdocuments.net/reader034/viewer/2022050714/5f0827507e708231d4209abd/html5/thumbnails/47.jpg)
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
The bands cross the Fermi level
(metallic character)
Bands look like parabollas,
(Al resembles a free electron gas)
Al band structure - analyze your results
![Page 48: Prof.P. Ravindran, - folk.uio.nofolk.uio.no/ravi/cutn/ccmp/4-bandstr-DOS.pdfEnergy band theory P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure](https://reader034.vdocuments.net/reader034/viewer/2022050714/5f0827507e708231d4209abd/html5/thumbnails/48.jpg)
P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
The Fermi energy lies in a gap
insulator
Theo. direct gap = 5.3 eV
Expt. Gap = 7.8 eV
(LDA band gap understimation)
MgO band structure analyze your results
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
Mg: 1s2 2s2 2p6 3s2
Mg loses two electrons
that are gained by O
O: 1s2 2s2 2p6
Mg2+
O2-
One would expect O (one s band and three p bands) bands completely full
and Mg bands completely empty
MgO analyze your results
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
MgO band structure analyze your r- Detail results
O 2s
O 2p
(three bands)
The Fermi energy lies in a gap
insulator
Theo. direct gap = 5.3 eV
Expt. Gap = 7.8 eV
(LDA band gap understimation)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
The Electronic Configuration of a Magnesium Atom
n l ml ms
3 0 0 +1/2
3 0 0 -1/2
Mg: (Ne)3s2
1s
2s
3s2p
3p
Empty 3p
orbitals in Mg
valence shell
How it becomes metal?
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids16a–52
Orbital energy
levels
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
A representation of the energy levels (bands)
in a magnesium crystal
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Nomenclature
For most purposes, it is sufficient to know the En(k) curves - the
dispersion relations - along the major directions of the reciprocal
lattice.
This is exactly what is done when real band diagrams of crystals
are shown. Directions are chosen that lead from the center of
the Wigner-Seitz unit cell - or the Brillouin zones - to special
symmetry points. These points are labeled according to the
following rules:
• Points (and lines) inside the Brillouin zone are denoted with
Greek letters.
• Points on the surface of the Brillouin zone with Roman
letters.
• The center of the Wigner-Seitz cell is always denoted by a G
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
For cubic reciprocal lattices, the points with a high symmetry on the
Wigner-Seitz cell are the intersections of the Wigner Seitz cell with
the low-indexed directions in the cubic elementary cell.
Nomenclature
simple
cubic
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Nomenclature
We use the following nomenclature: (red for fcc, blue for bcc):
The intersection point with the [100] direction is
called X (H)
The line G—X is called D.
The intersection point with the [110] direction is
called K (N)
The line G—K is called S.
The intersection point with the [111] direction is
called L (P)
The line G—L is called L.
Brillouin Zone for fcc is bcc
and vice versa.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
We use the following nomenclature: (red for fcc, blue for bcc):
The intersection point with the [100] direction is
called X (H)
The line G—X is called D.
The intersection point with the [110] direction is
called K (N)
The line G—K is called S.
The intersection point with the [111] direction is
called L (P)
The line G—L is called L.
Nomenclature
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Real crystals are three-dimensional and we must consider
their band structure in three dimensions, too.
Of course, we must consider the reciprocal lattice, and, as
always if we look at electronic properties, use the Wigner-
Seitz cell (identical to the 1st Brillouin zone) as the unit cell.
There is no way to express quantities that change as a
function of three coordinates graphically, so we look at a
two dimensional crystal first (which do exist in
semiconductor and nanoscale physics).
Electron Energy Bands in 3D
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
59
Bands in 3D
In 3D the band structure is
much more complicated than in
1D because crystals do not
have spherical symmetry.
The form of E(k) is dependent
upon the direction as well as the
magnitude of k.
Figure removed to reduce
file size
GermaniumGermanium
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
The lower part (the "cup") is
contained in the 1st Brillouin zone,
the upper part (the "top") comes
from the second BZ, but it is folded
back into the first one. It thus
would carry a different band index.
This could be continued to infinity;
but Brillouin zones with energies
well above the Fermi energy are of
no real interest.
These are tracings along major
directions. Evidently, they contain
most of the relevant information in
condensed form. It is clear that
this structure has no band gap.
Electron Energy Bands in 3D
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band Structure: KCl
We first depict the band structure of an ionic crystal, KCl. The bands are very
narrow, almost like atomic ones. The band gap is large around 9 eV. For alkali
halides they are generally in the range 7-14 eV.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band Structure: silver (fcc)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Band Structure: tungsten (bcc)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Empirical pseudopotential method
Energy band of Si, Ge and Sn
Empirical pseudopotential
method
Si Ge Sn
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
65Band structure of semiconductor
Calculated energy band structure of
Silicon
Calculated energy band structure of
GaAs
- Interband transitions : The excitation or relaxation of electrons between
subbands
- Indirect gap : The bottom of the conduction band and the top of the
valence band do not occur at the same k
- Direct gap : The bottom of the conduction band and the top of the
valence band occur at the same k
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
DOS is Nanomaterials
In bulk (a), layered (b) and wire (c) materials, there are always states populated which do not contribute to gain. These are parasitic states and contribute to inefficiency.
In quantum dot (d) materials, the DOS is a set of discrete states. Theory predicts this type of material is ideal for the gain region of a laser because fewer parasitic states are occupied.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
67
First Brillouin zone E vs. k band
diagram of zincblende semiconductors
One relevant conduction band isformed from s- like atomic orbitals
“unit cell” part of wavefunction isapproximately spherically symmetric.
The three upper valence bands areformed from (three) p- like orbitalsand the spin-orbit interaction splits offlowest, “split-off” hole (i. e., valence)band.
The remaining two hole bandshave the same energy (“degenerate”)at zone center, but their curvature isdifferent, forming a “heavy hole” (hh)band (narrower), and a “light hole” (lh)band (broad)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
SnO2: band structure
VBM
CBM
Density of States
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69
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
70
How to get conduction in Si?
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
71
Doping Silicon with Donors (n-type)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
72
Doping Silicon with Acceptors (p-type)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
73
Atomic Density
for Si
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
74
Summary of n- and p-type Silicon
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
75
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
76
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Introduction to Silicon
6
The appearance of Band Gap, separating CB and VB
The 6 CB minima are not located at the center of 1st
Brillouin zone, INDIRECT GAP
CB VB-H VB-L
1st Brillouin zone of Diamond
lattice
CB
VB
Anisotropy in surface of E
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
78
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
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The Bandgap Problem of DFT
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80
The Bandgap problem [Sham,Schluter, PRL, 51, 1888 (1983).
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81
Baandgap Error in Semiconductors from LDA
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
82Calculated Badgap values of Si from various level of Calculation
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
83GWA calculation of Bandgap of Semiconductors
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84
Graphite band structure (Semi Metal)
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
Fermi level
As Fermions are added to an energy band, they will fill the
available states in an energy band just like water fills a
bucket.
The states with the lowest energy are filled first, followed by
the next higher ones.
At absolute zero temperature (T = 0 K), the energy levels are
all filled up to a maximum energy, which we call the Fermi
level. No states above the Fermi level are filled.
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
22/3
2
( ) 3( )
8F
hc NE
mc V
FERMI ENERGY
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Fermi surface sampling for metallic systems
The determination of the Fermi level might be delicate for metallic systems
Slightly different choices of k-points can lead to bands
entering or exiting the sum, depending if a given
eigenvalue is above or below the Fermi level.
Band structure of bulk Al
For this k-point, three
bands are occupied
For this k-point, two
bands are occupied
For this k-point, one
band is occupied
For a sufficiently dense Brillouin zone sampling, this should not be a problem
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
For the k-points close to the Fermi surface, the
highest occupied bands can enter or exit the sums
from one iterative step to the next, just because the
adjustement of the Fermi energy
Difficulties in the convergence of the self-consistence procedure
with metals: smearing the Fermi surface
Instability of the self-consistent procedure
Solution 1: Use small self-consistent mixing coefficients
Solution 2: Smear the Fermi surface introducing a distribution of occupation number
The occupations are not any longer 1 (if below EF) or 0 (if above EF)
Gaussians
Fermi functions
C. –L. Fu and K. –M. Ho, Phys. Rev. B 28, 5480 (1989)
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Smearing the Fermi surface: the Electronic Temperature
is a broadening energy parameter that is adjusted to avoid instabilities
in the convergence of the self-consistent procedure. It is a technical
issue. Due to its analogy with the Fermi distribution, this parameter is
called the Electronic Temperature
For a finite , the BZ integrals converge faster but to incorrect values. After
self-consistency has been obtained for a relatively large value of Tc , this has to
be reduced until the energy becomes independent of it.
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Comparing energies of structures having
different symmetries: take care of BZ samplings
The BZ sampling of all the structures must be sampled
with the same accuracy
Since for unit cells of different shapes it is not possible to
choose exactly the same k-point sampling, a usual strategy is
to try and maintain the same density of k-points
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
CONTACT POTENTIAL
1 2contactV
e
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
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P.Ravindran, FME-course on Ab initio Modelling of solar cell Materials 03 June 2013 Various Aspects of Energy Issues
BZ integration, “FERMI”-methods Replace the “integral” of the BZ by a finite summation on a mesh of “k-points”
weights wk,n depend on k and bandindex n (occupation)
– for full “bands” the weight is given by “symmetry”
w(G)=1, w(x)=2, w(D)=4, w(k)=8
shifted “Monkhorst-Pack” mesh
– for partially filled bands (metals) one must find the
Fermi-energy (integration up to NE) and determine
the weights for each state Ek,n
linear tetrahedron method (TETRA, eval=999)
linear tetrahedron method + “Bloechl” corrections (TETRA)
“broadening methods”
– gauss-broadening (GAUSS 0.005)
– temperature broadening (TEMP/TEMPS 0.005)
– broadening useful to damp scf oszillations, but dangerous (magnetic moment)
kk
nk
nknknk
EE
n
wkdrFn
*
,
,
3
,
*
,)(
G D X
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Relativistic treatment
Valence states
– Scalar relativistic
mass-velocity
Darwin s-shift
– Spin orbit coupling on demand by second variational treatment
Semi-core states
– Scalar relativistic
– on demand
spin orbit coupling by second variational treatment
Additional local orbital (see Th-6p1/2)
Core states
– Fully relativistic
Dirac equation
For example: Ti
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P.Ravindran, Computational Condensed Matter Physics, Auguest 2015 Evolution of Band structure of Solids
Relativistic semi-core states in fcc Th
additional local orbitals for
6p1/2 orbital in Th
Spin-orbit (2nd variational method)
J.Kuneš, P.Novak, R.Schmid, P.Blaha, K.Schwarz,
Phys.Rev.B. 64, 153102 (2001)