Transcript FREE ELECTRON THEORY - West Virginia University

First: Finish tight binding with
an example similar to the HW
Then: Other Methods of
Calculating Energy Bands
Test corrections and homework due Thursday
Learning Objectives for Today
After today’s class you should be able to:
 Discuss the differences, advantage and
disadvantages between various methods
of calculating energy bands
 (If time) Predict general features of
simple structures
Can we simplify this?
Think trig functions.
Preparing you for the homework

The nearest neighbor (nn) positions can help
figure out the energy bands.

In sc, nn’s are at a(1,0,0), a(0,1,0), a(0,0,1)
 

 (k )  Es      eik R
n.n.


Where Es is the atomic s-level orbital energy and
 and  are integrals defined in the book
Putting in the different atom positions:

ik y a
ik y a
ikx a
ikx a
ikz a
ikz a
 (k )  Es     (e  e
e e
e e )

 (k )  Es    2 (cosk x a  cosk y a  cosk z a)
Energy bands in simple cubic

The nearest neighbor (nn) positions can help
figure out the energy bands.

 (k )  Es    2 (cosk x a  cosk y a  cosk z a)
Along X: kx=µ 2/a, 0 µ 1
kxa=2µ
Es    2 (cos 2  1  1)
Along R: kx=ky=kz=µ 2/a, 0 µ 1
Es    6 cos 2
For small k (near ), it’s parabolic!
 (k ~ 0)  Es    6  3k 2 a 2
Give time to discuss homework 9.3 & 10.1

kya
kya
kxa
kxa
kza
kza
 (k )  Es    4 (cos
cos
 cos
cos
 cos
cos
)
2
2
2
2
2
2
 9.3
is the matrix problem (similar
to problem in last class)
You could also discuss test corrections
Tight binding and nearly free
Ch. 10: Tight binding approximation
works well for inner electrons, so most
techniques focus on other bands
Ch. 9: Nearly free electron model can’t
actually be applied to real system,
though more complicated methods
give similar results. Why not?
Why can’t we use the nearly free e- model?
Nearly free electron model assumed the
outer electron wavefunctions could be
made of a small number of plane waves
 Wavefunctions localized (inner electrons)
 Also oscillatory like atoms

The same is true for the valence bands

Since eigenstates of the same Hamiltonian with
different eigenvalues must be orthogonal or
0   dr (r )  (r )
c
k

*
v
k
Then v must have oscillations that carefully
interlace those of c in order for the integral0
 (r )
c
k
 (r )
v
k
Fourier expansion requires many large k terms
The Independent Electron (e-)
Approximation & Sch. Equation
2

2
(
  U (r )) k (r )   (k ) k (r )
2m
A SE for each e- is a huge simplification
 The ind. e- approx. however doesn’t ignore all
 U(r)=periodic potential + periodic interactions
 To know U(r), you need 
 To know , you need U(r)
What to do?
 Guess U(r), use to solve 
Then what?
 Use to get better guess of U(r), repeat same

Generalizations to all Methods
 Except
with the simplest 1D examples,
the S. E. cannot be solved exactly
 All methods require approximations
 And high speed computing!
 Thus the type of approximations
people have tried has been limited by
computing techniques and computing
power
 Focus on higher energy bands as tight
banding pretty good for lower bands
The Cellular Method (1934)
First approach (besides Bloch’s tight binding)
was the cellular method by Wigner and Seitz
(know that name?)
 Since we have periodicity, it is enough to solve
the S.E. within a single primitive cell Co
 The wavefunction in other cell is then

 k (r  R)  e
 k (r )
 ik  R
 is a sum over spherical harmonics, need BCs
 Computationally challenging to solve B.C.s
 Results in potential with discontinuous
derivative at cell boundary

Muffin-tin potential
Solves both complaints of the last method
U(r)=V(|r-R|), when |r-R|  ro (the core region)
=V(ro)=0, when |r-R|  ro (interstitial region)
ro is less than half of the nearest neighbor distance
Two methods use the muffin tin
potential
Augmented Plane-Wave method (APW)
 In the interstitial region k,=eikr
 In the atomic region, k, satisfied S.E.
2

2
(
 k , (r )  V ( r  R )k , (r ))  k , (r )
2m
Only k dependence is in the interstitial region
2 2
 k
 In interstitial region:
Hk , (r ) 
k , (r )
2m


As many as hundreds of APWs can be used
Different starting potentials
How
to
deal
with
a
discontinuous
can give different results
(but continuous )
derivative
Best to use variational principle rather than S.E.
2

2
2
(

(
r
)

U
(
r
))

(
r
)
dr
 2m
E[ k ] 
2

(
r
)
dr

E[k] is the energy of (k) of the level k
Note similarity to
free bands (right)
Another approach using Muffin tin
The other method is called the Green’s
function approach or the Korringa, Kohn,
and Rostoker (KKR) method
 Formulation seems very different, but it has
been established that the methods yield the
same results using the same potential

 k (r )   G ( k ) (r  r ' )U (r ' ) k (r ' )dr '
iK r  r '
2m e
G (r  r ' )   2
 4 r  r '
Orthogonalized plane wave method (OPW)
Good if don’t want a doctored potential
 Orthogonalized plane waves defined as:

k  e
ik r
  bc (r )
c
k
c
Core levels needed (generally tight binding)
 Constants bc determined by orthogonality

0   dr (r ) k (r )
bc    dr kc (r )*eik r
 This implies
c
k
*
Second term small in interstitial region
 So close to a plane wave in interstitial region

Pseudopotential Method
Began as an extension of OPW
ik r
 If we act H on   e

k

b
c
c
k
(r )
c
In the outer region, this gives ~ free energy
 What goes on in core is largely irrelevant to
the energy, so let’s ignore it

U(r)=0 , when r  Re (the core region)
=-e2/r, when r  Re (interstitial region)
Pseudopotential Method
Calculation of band structure depends only
on the Fourier components of the
pseudopotential at the reciprocal lattice
vectors.
 Usually, only a few values of U are needed
 Constants from models or fits to optical
measurements of reflectance and absorption
 Great predictive value for new compounds
 Often possible to calculate band structures,
cohesive energy, lattice constants and bulk
moduli from first principles

Nearly free e-’s
Tight-binding/LCMO

Large overlap

Wave functions
electrons indep. of
~ plane waves
each other

Assume energy


Linear
combination of
is unchanged
and solve for
Assume some
k·p Theory
• Useful for understanding
interactions between bands
• Critical points of BZ have
specific properties.
• If critical point energies are
Wannier functions
known, treat nearby points
1st order
= unperturbed
as critical energy plus
correction
atomic orbital
perturbation

Pseudopotential Method includes Coulomb repulsion & Pauli exclusion. No exact
way to calculate V(r), guess and iterate. Valence bands->charge density=ѱ*ѱ->V’
Density functional theory (DFT) takes into account Coulomb, exchange and
correlation energies of electrons. Guess and iterate. Gives good bandstructure.
k○p Theory

Most holes (electrons) spend most of their
time near the top (bottom) of the valence
(conduction) band so properties nearby these
points important
Insulators vs Semiconductors (High Temp)
Insulator
A
Semiconductor @ low temp Semiconductor @ high
small fraction of the electrons is thermally excited
into the conduction band. These electrons carry
current just as in metals (holes too)
 The smaller the gap the more electrons in the
conduction band at a given temperature
 Resistivity decreases with temperature due to higher
concentration of electrons in the conduction band
k○p Theory

Most holes (electrons) spend most of their time near
the top (bottom) of the valence (conduction) band so
properties nearby these points important

Based on perturbation theory

V~ is the periodic potential (of the lattice),
and VU is the confinement potential

V0 and x0 are some arbitrary positive
constants. If VU is small, then the solutions to
the S.E. are of the Bloch form:
Essense of k○p Theory
 nk ( x)  unk ( x)eikx
1
2
(
( p  k )  U (r ))uk (r )   k uk (r )
2m
When we plug in the Bloch wavefunction, we
can write the Schrodinger equation in this form.
E’k
Band
Structure
E(k)
Linear H Chain
EF
0
k
•The Fermi energy separates the filled states (E < EF at T = 0 K)
from the empty states (E > EF at T = 0 K).
• Here it splits the band (each band can hold 2 electrons)
•A 1D chain of H atoms is predicted to be metallic because the
Fermi level cuts a band (there is no gap so it takes only an
infinitesimal energy to excite an electron into an empty state).
•The band runs "uphill" (from 0 to /a) because the in phase (at
k=0) combination of orbitals is bonding and the out of phase (at
k=/a) is antibonding.
/a
How Bands “Run”
Yk = S eiknacn (tight binding)
applies in general
It does not, however, say anything about the
energy of the orbitals at the zone center (k=O)
relative to those at the edge (k=n/a).
For a chain of H atoms (s orbitals) it is clear
that E(k = 0) < E(k = n/a).
But consider a chain of p orbitals. Group:
Is bonding or antibonding preferred? Why?
P Orbital Runs Opposite to S
Effect of Orbital Overlap
band width or dispersion=the difference in energy between the
highest and lowest energy levels in the band
If we reduce the lattice
parameter a (bring closer
together) it has the following
effects:
•The spatial overlap of the
orbitals increases
•The band becomes more
bonding (energy reduces) at k=0
•The band becomes more
antibonding (energy up) k=/a.
•The increased antibonding is
larger than the increased
bonding.
•The bandwidth increases.
•The electron mobility increases.
Wide bands  Good orbital
overlap  High carrier mobility
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
/a 0
k
/a 0
EF
k
/a 0
k
Which of the following is the correct band structure for a
linear chain of F atoms (atomic #=9)?
/a