Transcript FREE ELECTRON THEORY - Web Hosting (community.wvu.edu

A return to density of states
& how to calculate energy
bands
Making HW and test corrections due March 19
Learning Objectives for Today
After today’s class you should be able to:
 Relate
DOS to energy bands
 Compare two main methods for
calculating energy bands
 Be able to use tight binding model to
calculate energy bands
 Understand basics of Dirac notation
The nearly-free-electron model
1 electron per atom:
When EF is well away
from a gap, dispersion is
similar to free-electron
case, but with slight
change in curvature
(electrons in metals act
free)
As we increase the
number of electrons per
atom (or per unit cell), EF
moves up the dispersion
relation.
E
EF
dispersion relation
k
The nearly-free-electron model
When EF is close
to or within a
gap, major
changes in the
material
properties
occur...
E
Conduction Band
(LUMO)
EF
Valence Band (HOMO)
HOMO: Highest Occupied Molecular Orbital
LUMO: Lowest Unoccupied Molecular Orbital
k
4
How DOS g(E) relates to Dispersion
The density-ofstates curve
counts levels.
DOS curves plot the
distribution of
electrons in energy
There are more states in a given energy interval at
the top and bottom of this band.
In general, DOS(E) is proportional to the
inverse of the slope of E(k) vs. k
The flatter the band, the greater the density of
states at that energy.
DOS g(E) is the number of electron states per unit volume
per unit energy at energy E
To find the energy density of states g(E), we need:
the density of states in k-space g(k) and the energy bands.
Not quite so simple
Silicon
critical points of the Brillouin zone
Van Hove points are discontinuities of
the first derivative of g(E)
How would you experimentally
determine the density of states?
X-ray Photoemission Spectroscopy
Like a fancy
photoelectric effect
How would you theoretically
determine the energy bands?
Methods to calculate bandstructures

Solve the Schrodinger Equation and apply the Bloch theorem
 Simplify
the complicated crystal potential to something
solvable. E.g. Kronig-Penney model.
 Treat
the complicated crystal potential as a sum of a simpler
potential (solvable Schrodinger Eqn) and potential
perturbation. E.g. near free-electron model (plane waves +
perturbation), k.p perturbation theory and tight-binding
model (atomic orbitals + perturbation).
 Numerical
methods, e.g. density function theory and
quantum Monte Carlo
For more information, see the following websites for instance,
 http://en.wikipedia.org/wiki/Density_functional_theory
 http://en.wikipedia.org/wiki/Electron_configuration
The main property of solids that determines their electrical
properties is the distribution of their electrons.
Two main models for electron distribution:
1) Nearly Free Electron Approximation
Valence electrons are assumed trapped in a box (the
sample) with a periodic potential
2) The Tight Binding Approximation
Valence electrons are assumed to occupy molecular
orbitals delocalized throughout the solid
Summary of Nearly Free Electron
Model (What We’ve Already Done!)
Nearly free electron Model
 Electrons nearly free due to very
large overlap (opposite from TB)
 Wave functions approximated by
plane waves (free electrons)   Aei (kxt )
 Assume energy is unchanged and
k
E
2m
solve for 1st order correction
 Works for upper states even if tightly
bound electrons in lower states
2
2
My Summary of the Two Main
Approaches
Nearly free e-’s +
pseudopotential
 Electrons
Tight-binding or LCMO

nearly free due
independent of each other
(opposite from TB)
 Wavefunctions
 Assume
~ plane waves
 Works
(often true, tight core)

for upper states even if
TB electrons in lower states
Linear combination of atomic
wave functions (Wannier)
energy is unchanged
and correct to first order
Assume some electrons

Each Wannier function is
equal to the unperturbed
atomic orbital (LCAO approx)
Comparing Chemists and Physicists
Calculation theories fall into 2 general categories, which have
their roots in 2 qualitatively very different physical pictures for
e- in solids (earlier):
“Physicist’s View” - Start from an “almost free”
e- & add the periodic potential
 Nearly
free electrons, Pseudopotential
methods
“Chemist’s View” - Start with atomic energy levels
& build up the periodic solid by decreasing distance
between atoms
Now, we’ll focus on the 2nd method.
Method #2 (Qualitative Physical Picture #2)
“The Chemists’ Viewpoint”
 Start with the atomic/molecular picture of a
solid.

The atomic energy levels merge to form molecular
levels, & merge to form bands as periodic interatomic
interaction V turns on.
TIGHTBINDING or
Linear Combination of Atomic Orbitals
(LCAO) method.


This method gives good bands, especially valence
bands!
The valence bands are almost the same as those
from the pseudopotential method! Conduction
bands are not so good because electrons act free!
The Tightbinding Method
Some believe the Tightbinding / LCAO
method gives a clearer physical picture (than
pseudopotential method does) of the causes of
the bands & the gaps.
 In this method, the periodic potential V is
discussed as in terms of an Overlap

Interaction of the electrons on
neighboring atoms.

As we’ll see, we can define these interactions in
terms of a small number of parameters.
Tightbinding/LCAO

Assume the atomic orbitals
~ unchanged
bare atoms
solid
Atomic energy levels merge to form
molecular levels & merge to form
bands as periodic interatomic
interaction V turns on.
Covalent Bonding Revisited


When atoms are covalently bonded
electrons are shared by atoms
Example: the ground state of the
hydrogen atoms forming a molecule


If atoms far apart, little overlap
If atoms are brought together the
wavefunctions overlap and form the
compound wavefunction, ψ1(r)+ψ2(r),
increasing the probability for electrons to
exist between atoms
These two possible
2
 2 d  1, 2 ( x)
combinations represent 2

 U( x)[ 1,2 ( x)]  E 1,2 ( x)
2
2m dx
possible states of two
2
 2 d [ 1 ( x)  2 ( x)]
atoms system with


U
(
x
)[

(
x
)


(
x
)]

E
[

(
x
)


(
x
)]
1
2
1
2
different energies
2m
dx2
LCAO: Electron in Hydrogen Atom
(in Ground State)
 K x1
1 (r)  Ae
Second hydrogen atom
 K x2
 2 (r)  Ae
2.0
1(r)
1.5
1.0
0.5
0.0
-10
-5
0
5
10
r (aB)
Approximation: Only Nearest-Neighbor interactions
H2 Molecule
Chain of 5 H atoms
c0
E
Antibonding
c1
c1
c2
c3
c4
4
c2
3
Nonbonding
2
1
Bonding
0
# of Nodes
Group: For 0, 2 and 4 nodes,
determine wavefunctions
If there are N atoms in the chain c0
there will be N energy levels and
N electronic states (molecular
orbits). The wavefunction for
each electronic state is:
c1
c2
c3
c4
k=p/a
Yk = S eiknacn
k=p/2a
Where:
a
is the lattice constant,
n
identifies the individual atoms
within the chain,
cn represents the atomic
orbitals

k
is a quantum # that identifies
the wavefunction and tells us the
phase of the orbitals.
a
k=0
The larger the absolute value
of k, the more nodes one has
Infinite 1D Chain
of H atoms
c
k = p/a
0
c1
c2
c3
c4
k=p/a
Yp/a = c0+(exp{ip})c1 +(exp{i2p})c2
+(exp{i3p})c3+(exp{i4p})c4+…
Yp/a = c0 - c1 + c2 - c3 + c4 +…
k = p/2a
k=p/2a
Yp/2a = c0+(exp{ip/2})c1 +(exp{ip})c2
+(exp{i3p/2})c3+(exp{i2p})c4+…
Yp/2a = c0 + 0 - c2 + 0 + c4 +…
k = 0
k=0
Y0 = c0+c1 +c2 +c3 +c4 +…
a
k=0  orbital phase does not change when we translate by a
k=p/a  orbital phase reverses when we translate by a
Infinite 1D Chain of H atoms
What would happen if consider k> p/a?
 If not obvious, try in groups k=2 p/a.
What is the wavefunction?

Yk = S eiknacn
 nk ( x)  u nk ( x)e ikx ,
LCAO +Bloch notation
u nk ( x)  u nk ( x  a )
 & potential U(r)=U(r+Na) obeys the Born-von Karman condition
Where: ck  e
2pni / N
 eika
Combining the Bloch theorem and the above gives that
Let’s simplify by considering just two states. (Dirac notation)
  c1  1  c 2  2
 a1 


The collection of all functions of x constitutes a vector space. a
  2
But to present a possible physical state, the wave
functions

~a 
must be normalized:



2 
   ( x) dx 
1
a

 N
Brief Summary of Dirac Notation
Wave Mechanics (Position Space)
Dirac Notation
Dirac ket:
Linear operator A
|Y(t)
System State
Measurement
ˆ n  a n n
A
A
Eigenvalue Equation
Y(x,t)
wavefunction
differential or
multiplicative operator
ˆ n ( x )  a n n ( x )
A
eigenvalue
 gives possible results of a measurement
eigenstate or eigenket
eigenfunction
 gives probability of measurement result an
Expectation Value (average of many identical measurements)
ˆ Y
A  YA
ˆ Y( x)dx
A   Y* ( x)A
all x
Dirac Notation with LCAO
Approximation
Dirac Notation for 2 atoms:
  c1  1  c 2  2
If we assume little overlap, can expand H   E  to
1 H   1 E1 
and
Or expanding 
c1 1 H 1  c2 1 H  2  c1E
c1  2 H 1  c2  2 H  2  c2 E
 2 H    2 E2 
Eigenvalue problem with
two solutions. E1 and E2
are the unperturbed atom
energies. Cross term is the
overlap.
Solution:
Lower energy result is the bonding state
c1 E1  c 2V12  c1 E
 c1V12 *  c 2 E 2  c1 E
V12 is overlap integral
c1 1 H 1  c2 1 H  2  c1E
c1  2 H 1  c2  2 H  2  c2 E
Example similar to homework
Find the energies at
the H point of BCC.
H
2
h
2
(
(k  K )   )ck  K  U K ' K ck  K '  0
2m
K'
(   )ck  U k ck K '  0
0
k
How do we plot the Empty Lattice Bands?
The limit of a vanishing
potential is called the “empty
lattice”, and the empty-lattice
bands are often plotted for
comparison with the energy
bands of real solids.
Here plotted in the reduced
zone scheme (translations back
into the 1st BZ).
Example: 1D Empty Lattice
• V  0:
2
k2
E (k ) 
,  (k )  eikx
2m
• We assume a periodicity of a. Define the reciprocal
lattice constant G = 2p / a. We can therefore
restrict k within the range of [-G/2, G/2].
Bloch’s theorem
Sorry G=K
 nk (k  nG)  eik nGx  eikxunk ( x), unk ( x)  einGx
implies
Enk ( x) 
2
(k  nG ) 2
2m
Free Electrons in 1D
2
V  0:
Enk (k ) 
 k  nG 
2m
2
,
 nk (k )  ei k nG  x
Where k1 is a

 
The symmetry of the
E(k )  E(k1  G)
wavevector lying in
reciprocal lattice requires:
the 1st BZ.
2



2
 
E (k )  E (k1  G ) 
k1  G
The  sign is redundant.
2m
Empty Lattice Bands for bcc Lattice
For the bcc lattice, let’s plot the empty lattice bands
along the [100] direction in reciprocal space.
General reciprocal lattice translation
vector:
 


Ghkl  ha  kb  lc
Let’s use a simple cubic lattice,
for which the reciprocal lattice
is also simple cubic:
 2p
 2p
 2p
ˆ
ˆ
a
x b
y c
zˆ
a
a
a
And thus the general reciprocal
lattice translation vector is:

2p
2p
2p
Ghkl 
h xˆ 
k yˆ 
l zˆ
a
a
a
Energy Bands in BCC
 2p
2p
2p
k1 
x xˆ 
y yˆ 
z zˆ
a
a
a
We write the reciprocal lattice
vectors that lie in the 1st BZ as:
The maximum value(s) of x, y, and z depend on
the reciprocal lattice type and the direction
within the 1st BZ. For example:
[100]
0<x<1
0 < x < ½, 0 < y < ½
[110]

Remember that the
reciprocal lattice for a
bcc direct lattice is fcc!
Here is a top view, from
the + k direction:
2p
a
H
kx
2p
a
H
N
ky
Group: Plot the Empty Lattice Bands for bcc Lattice
Thus the empty lattice energy bands are given by:

2

2   2 2  2p 
2
2
2
E (k ) 
k1  Ghkl 
  x  h    y  k   z  l 
2m
2m  a 

Along [100], we can enumerate the lowest few bands for the y = z = 0
case, using only G vectors that have nonzero structure factors (h + k + l
= even, otherwise S=0):
2
 2  2p  2
2
{G} = {000} E 
  x  E0 x
2m  a 
 
{G} = {110}
 
(110) (1 1 0) (101) (101)
(1 1 0) (1 10) (1 0 1) (1 01)
(011) (011) (0 1 1) (0 1 1)
{G} = {200}
(200)
( 200)
(020) (020) (002) (002)

 

E  E x 1 1   E x 1 1
E  E x 1 1   E x  2
E  E x  2 
E  E x  2 
E  E x  2   E x  4
E  E0 x 1 12  E0 x 1 1
2
2
2
2
2
0
0
2
2
2
2
0
0
2
0
2
0
2
0
2
2
0
Empty Lattice Bands for bcc Lattice: Results

2

2   2 2  2p 
2
2
2
E (k ) 
k1  Ghkl 
  x  h    y  k   z  l 
2m
2m  a 
Thus the lowest energy empty lattice
energy bands along the [100] direction
for the bcc lattice are:



2
E(k )  E0 x  h  k 2  l 2
6
5
4
Series1
E/E0
Series2
Series3
3
Series4
Series6
Series7
Series8
2
1
0
0
0.2
0.4
0.6
x
0.8
1

Summary Band Structures

What is being plotted? Energy vs. k, where k is the
wavevector that gives the phase as well as the wavelength of the
electron wavefunction (crystal momentum).

How many lines are there in a band structure diagram? As
many as there are orbitals in the unit cell.

How do we determine whether a band runs uphill or
downhill? By comparing the orbital overlap at k=0 and k=p/a.

How do we distinguish metals from semiconductors and
insulators? The Fermi level cuts a band in a metal, whereas
there is a gap between the filled and empty states in a
semiconductor.

Why are some bands flat and others steep? This depends
on the degree of orbital overlap between building units.
Wide bands  Large intermolecular overlap  delocalized eNarrow bands  Weak intermolecular overlap  localized e-
How the energies split as we increase N
•Energy levels get closer as N increases.
•Degenerate pairs, except for ground state.
Energy Bands and Fermi Surfaces in 2-D Square Lattice
Reminder: the Brillouin
zones of the reciprocal
lattice can be identified
with a simple construction:
1st
The
BZ is defined as the
set of points reached from
the origin without crossing
any Bragg planes.
3
2
3
3
2
2
1
3
3
A truly free electron
system would have a Fermi
circle to define the locus of
states at the Fermi energy.
3
2
3
3
2 p/a
2 p/a
Group Problem
Show for a square lattice (2D) that the
kinetic energy at the corner of the 1st BZ
is larger than that of an electron at the
midpoint of a side face. By how much?
 What is the corresponding factor in 3D?
 Draw the energy bands from the zone
center to these to points on the boundary.
 What bearing might this have on the
conductivity of divalent metals?
