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3.23 Electrical, Optical, and Magnetic Properties of Materials
Fall 2007
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3.23 Fall 2007 – Lecture 10
TIGHT-BINDING
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3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3.23 Fall 2007 – Lecture 10
TIGHT-BINDING
Images removed due to copyright restrictions. Please see
http://www.helloboston.com/Images/People/5292005Houdini_Harvard_Bridge_2.jpg
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
1
Last time
1.
2.
3.
4.
5.
6.
7.
Explicit solution for the Bloch orbitals
Free ellecttrons
Band structure of free electron vs. silicon
Band edges
Ψnk (r) is not a momentum eigenstate
Group velocity, effective mass
Fermi energy
energy, Fermi surface
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Study
• Chap
Chap. 4 Singleton
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
2
Plane wave expansion
r r
r
r
ψ nkr (r ) = unkr (r ) exp ik ⋅ r
(
)
periodic u is expanded in planewaves, labeled
according to the reciprocal lattice vectors
r
r r
r
G
r
u (r ) = ∑
cnk exp(i G ⋅r )
r
r
nk
G
Explicit solution for the Bloch orbitals
⎛ h 2 ( q − G′ )2
⎞
− E ⎟ Cq −G′ + ∑ VG′′−G′Cq −G′′ = 0
⎜
⎟
⎜
2m
G′′
⎝
⎠
⎛ h2
2
( q − 2G )
⎜
⎜2m
⎜
VG
⎜
⎜
⎜
V2G
⎜
⎜
⎜
V3G
⎜
⎜
⎜
V4G
⎜
⎝
V− G
V−2G
V−3G
h2
2
(q − G)
2m
V−G
V−2G
VG
h2
2
(q)
2m
V− G
V2G
VG
h2
2
(q + G)
2m
V3G
V2G
VG
⎞
⎟
⎟
⎟⎛C
⎞
⎛ Cq − 2 G ⎞
V−3G
⎟ ⎜ q − 2G ⎟
⎟
⎜
⎟ ⎜ Cq − G ⎟
⎜ Cq − G ⎟
⎟⎜ C ⎟ = E ⎜ C ⎟
V−2G
q
⎟⎜ q ⎟
⎜
⎟
⎟ ⎜ Cq + G ⎟
C
⎜ q +G ⎟
⎟⎜
⎟
⎟
⎜C
V−G
⎝ q + 2G ⎠
⎟ ⎝ Cq + 2 G ⎠
⎟
h2
2
( q + 2G ) ⎟⎟
2m
⎠
V−4G
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
3
What choice for a basis ?
• For molecules: often atomic orbitals, or
localized functions as Gaussians
• For solids
solids, periodic functions such as sines
and cosines (plane waves)
The plane waves basis set
• Systematic improvement of
completeness/resolution
• Huge number of basis elements – only possible
because of pseudopotentials
• Allows for easy evaluation of gradients and
Laplacian
• Kinetic energy in reciprocal space, potential in
real space
• Basis set does not depend on atomic positions:
there are no Pulay terms in the forces
4
Hamiltonian in the Bloch representation
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Energy Bands
Α1
-0.939
Ζ3
Q_
Σ1
∆1
-0.539
Γ12
Γ25'
∆2
5
2
Ζ2 1'
1
Ζ3 Ζ1
3
Ζ4
3
2'
1 Ζ
1
∆5
∆2'
2'
Q+
Q_
3
Q+
Q_
Q+
Α1
Σ2
Α3
3 Α
3
Σ3
1
Σ1
K2
K4
K3
K1
K1
Σ1
Α1
∆1
Σ4
Γ1
X
W
L
Γ
Κ
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
5
Brillouin Zone (fcc)
kz
L
Λ
∆
Γ
Σ
K
Q
U
W
Σ'
X
ky
kx
Figure by MIT OpenCourseWare.
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
The Fermi surface
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
Image from the Fermi Surface Database. Used with permission.
Please see: http://www.phys.ufl.edu/fermisurface/jpg/K.jpg,
http://www.phys.ufl.edu/fermisurface/jpg/Cu.jpg.
6
The Fermi surface
http://www phys ufl edu/fermisurface/
http://www.phys.ufl.edu/fermisurface/
3.23 Electronic, Optical and Magnetic Properties of Materials Nicola Marzari (MIT, Fall 2007)
Images from the Fermi Surface Database. Used with permission.
Energy of a collection of atoms
Hˆ = Tˆe + Vˆe−e +Vˆe− N +VN − N
1
Tˆe =− ∑ ∇ i2
2 i
r r⎤
⎡
Vˆe− N = ∑ ⎢∑ V RI − ri ⎥
i ⎣ I
⎦
(
)
1
Vˆe−e = ∑∑ r r
i j>i | ri − r j |
• Te: quantum kinetic energy of the electrons
• Ve-ee: electron-electron interactions
• Ve-N: electrostatic electron-nucleus attraction
(electrons in the field of all the nuclei)
• VN-N: electrostatic nucleus-nucleus repulsion
3.23 Fall 2006
7
Molecules and Solids:
Electrons and Nuclei
r
r
r
r r
r
r r
ˆ
Hψ ((r1 ,..., rn , R1 ,..., RN ) = Etotψ ((r1 ,..., rn , R1 ,..., RN )
• We treat only the electrons as quantum particles, in the
field of the fixed (or slowly varying) nuclei
• This is generically called the adiabatic or BornOppenheimer approximation
• “Adiabatic” means that there is no coupling between
different electronic surfaces; “B-O” implies there is no
influence of the ionic motion on one electronic surface
3.23 Fall 2006
Complexity of the many-body Ψ
“…Some form of approximation is essential, and this would
mean the construction of tables. The tabulation function of one
variable requires a page, of two variables a volume and of three
variables a library; but the full specification of a single wave
function of neutral iron is a function of 78 variables. It would be
rather crude to restrict to 10 the number of values of each
variable at which to tabulate this function, but even so, full
tabulation would require 1078 entries.”
3.23 Fall 2006
8
Mean-field approach
• Independent particle model (Hartree): each
electron moves in an effective potential,
representing the attraction of the nuclei
and the average effect of the repulsive
interactions of the other electrons
• This average repulsion is the electrostatic repulsion of the average charge density of all other electrons 3.23 Fall 2006
Hartree Equations
The Hartree equations can be obtained directly from the variational
principle, once the search is restricted to the many-body
wavefunctions that are written – as above – as the product of single
orbitals (i.e. we are working with independent electrons)
r
r
r
r
r
ψ (r1 ,..., rn ) = ϕ1 (r1 ) ϕ 2 (r2 )Lϕ n (rn )
r r
⎡
1 2
r 2
r⎤ r
r
1
V
(R
−
∇
+
⎢
r
drj ⎥ϕ i (ri ) = εϕ
i (ri )
∑
i
I −
ri )
+
∑ ∫
|
ϕ
j ( r j ) |
r
I
|
rj −
ri |
⎥⎦
j≠i
⎢⎣
2
3.23 Fall 2006
9
The self-consistent field
• The single-particle Hartree operator is selfconsistent ! It depends on the orbitals that
are the solution of all other Hartree equations
• We have n simultaneous integro-differential
equations for the n orbitals
• Solution is achieved iterativelyy
r r
⎡ 1 2
1
r 2
r⎤ r
r
⎢ − ∇i + ∑ I V ( RI − ri ) + ∑ ∫ | ϕ j ( rj ) | r r drj ⎥ ϕi (ri ) = εϕi (ri )
| rj − ri | ⎦⎥
j ≠i
⎣⎢ 2
3.23 Fall 2006
Iterations to self-consistency
• Initial guess at the orbitals
• Construction of all the operators
• Solution of the single-particle pseudo-
Schrodinger equations
• With this new set of orbitals, construct the
Hartree operators again
• Iterate the procedure until it (hopefully) converges
3.23 Fall 2006
10
What’s missing
• It does not include correlation
• The wavefunction is not antisymmetric
• It does remove nl accidental degeneracy of
the hydrogenoid atoms
3.23 Fall 2006
Spin-Statistics
• All elementary particles are either fermions
(half-integer spins) or bosons (integer)
• A set of identical (indistinguishable) fermions has a wavefunction that is antisymmetric by exchange
r r
r
r r
r
r
r
r
r
ψ (r1 , r2 ,..., rj ,..., rk ,..., rn ) = −ψ (r1 , r2 ,..., rk ,..., rj ,..., rn )
• For bosons it is symmetric
3.23 Fall 2006
11
Slater determinant
• An antisymmetric wavefunction is constructed via a
Slater determinant of the individual orbitals (instead
of just a product, as in the Hartree approach)
r
r r
r
ψ (r1 , r2 ,..., rn ) =
r
r
ϕα (r1 ) ϕ β (r1 ) L ϕν (r1 )
r
r
r
1 ϕα (r2 ) ϕ β (r2 ) L ϕν (r2 )
n!!
M
M
O
M
r
r
r
ϕα (rn ) ϕ β (rn ) L ϕν (rn )
3.23 Fall 2006
Pauli principle
• If two states are identical,
identical the determinant
vanishes (i.e. we can’t have two electrons
in the same quantum state)
3.23 Fall 2006
12
Hartree-Fock Equations
The Hartree-Fock equations are, again, obtained from the variational principle: we
look for the minimum of the many-electron Schroedinger equation in the class of all
wavefunctions that are written as a single Slater determinant
r
r
ψ (r1 ,..., rn ) = Slater
r r ⎤
r
⎡ 1 2
(
V
R
−
∇
+
∑
I −ri ) ⎥ ϕ λ ( ri ) +
⎢ 2 i
I
⎣
⎦
⎡
1
r r⎤
r
* r
⎢ ∑ ∫ ϕ µ (rj ) r r ϕ µ (rj ) drj ⎥ ϕλ (ri ) −
| rj − ri |
⎢⎣ µ
⎥⎦
⎡ * r
1
r r⎤
r
r
r
ϕ
ϕ
(
r
)
(
⎢
∑µ ∫ µ j | rr − rr | λ j )drj ⎥ ϕµ (ri ) = εϕλ (ri )
⎢⎣
⎥⎦
j
i
3.23 Fall 2006
Koopmans’ Theorems
• Total energy is invariant under unitary
unitary
transformations
• It is not the sum of the canonical MO orbital energies
• Ionization energy,
gy, electron affinityy are
given by the eigenvalue of the respective
MO, in the frozen orbitals approximation
3.23 Fall 2006
13
What is missing
• Correlations (by definition !)
– Dynamical correlations: the electrons get too
close to each other in H.-F.
– Static correlations: a single determinant
variational class in not good enough
• Spin contamination: even if the energy is
correct (variational, quadratic) other
properties might not (e.g. the UHF spin is
an equal mixture of singlet and triplet)
3.23 Fall 2006
Linear Combination of Atomic Orbitals
• Most common approach to find out the ground-state solution – it
allows a meaningful definition of “hybridization”, “bonding” and
“anti-bonding” orbitals.
• Also knows as LCAO,
•
LCAO LCAO-MO
LCAO-MO (for molecular orbitals)
orbitals), or tighttightbinding (for solids)
• Trial wavefunction is a linear combination of atomic orbitals – the
variational parameters are the coefficients:
Ψ trial =
∑
r r
c(Inlm ) Ψ (Inlm) r − RI
(
I ,( nlm )
ELCAO = min
)
Ψ trial Ĥ Ψ trial
Ψ trial
Ψ trial
3.23 Fall 2006
14
Hückel approach
• Hückel: Planar / quasi
quasi-planar
planar systems with
delocalized π bonding: two parameters
– α: matrix element between same orbital
– β: matrix element between neighboring orbitals
– Hamiltonian between further neighbors is 0
3.23 Fall 2006
Example: Benzene (C6H6)
H
H
C
H
C
C
C
C
H
H
C
H
σ bond framework
3.23 Fall 2006
Figure by MIT OpenCourseWare.
15
π
π
π
H
H
C
C
H
π
H
C
C
C
C
H
H
π
π
Benzene – energy levels
π bonding
Figure by MIT OpenCourseWare.
β
0
0
0
β ⎞
⎛α − E
⎜
⎟
β
α
−
E
β
0
0
0
⎜
⎟
⎜ 0
β
α −E
β
0
0 ⎟
det ⎜
⎟=0
0
β
α −E
β
0 ⎟
⎜ 0
⎜ 0
0
0
β
α −E
β ⎟
⎜⎜
⎟
0
0
0
β
α − E ⎟⎠
⎝ β
3.23 Fall 2006
Benzene – molecular orbitals
a - 2b
p6
p4
p5
π6
a-b
π5
π4
Energy
p2
p3
p1
π2
a+b
π3
π1
a + 2b
Figure by MIT OpenCourseWare.
Figure by MIT OpenCourseWare.
3.23 Fall 2006
16
Tight-binding (LCAO for solids)
r
• Hamiltonian Hˆ = Hˆ at + ∆U
Û ( r )
3.23 Electronic, Optical and Magnetic Properties of Materials - Nicola Marzari (MIT, Fall 2007)
17