1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and...
Transcript of 1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and...
1.021, 3.021, 10.333, 22.00 : Introduction to Modeling and Simulation : Spring 2011
Part II – Quantum Mechanical Methods : Lecture 4
From atoms to solids
Jeffrey C. Grossman
Department of Materials Science and Engineering Massachusetts Institute of Technology
4. From Atoms to Solids
Part II Outlinetheory & practice example applications
1. It’s A Quantum World:The Theory of Quantum Mechanics
2. Quantum Mechanics: Practice Makes Perfect
3. From Many-Body to Single-Particle; Quantum Modeling of Molecules
7. Nanotechnology
8. Solar Photovoltaics: Converting Photons into Electrons
9. Thermoelectrics: Converting Heat into Electricity
10.Solar Fuels: Pushing Electrons up
5. Quantum Modeling of Solids: Basic
a Hill
11.Hydrogen Storage: the Strength of Weak Interactions
Properties
6. Advanced Prop. of Materials:What 12. Review else can we do?
4. From Atoms to Solids
r1
Review: Next? Helium
Hψ = Eψ
e- �H1 + H2 + W
�ψ(�r1, �r2) = Eψ(�r1, �r2)
�T1 + V1 + T2 + V2 + W
�ψ(�r1, �r2) = Eψ(�r1, �r2)
+
e -
r2
r12
� �22
2 �22
2e e−
2m ∇1 −
4π�0r1 −
2m ∇2 −
4π�0r2 +
e2
4π�0r12
�ψ(�r1, �r2) = Eψ(�r1, �r2)
cannot be solved analytically problem!
!
!
Review:The Multi-Electron Hamiltonian
�2 e2Rememberthe good old days of the
1-electron H-atom??
They’re over!
Hψ = Eψ �
− 2m
∇2 − 4π�0r
�ψ(�r) = Eψ(�r)
N N N n N n n n
H = "% h2
#2 Ri +
1 %% ZiZ je
2
" h2
%#2 ri "%% Zie
2
+ 1 %% e2
i=1 2Mi 2 i=1 j=1 Ri " R j 2m i=1 i=1 j=1 Ri " rj 2 i=1 j=1 ri " rji$ j i$ j
kinetic energy of ions kinetic energy of electrons electron-electron interaction
potential energy of ions electron-ion interaction
H R1,...,RN ;r1,...,rn( ) " R1,...,RN ;r1,...,rn( ) = E " R1,...,RN ;r1,...,rn( ) Multi-Atom-Multi-Electron Schrödinger Equation
!
Born-Oppenheimer Approximation
Electrons and nucleias “separate” systems
H = " 2hm
2
$ n
#r2 i "$
N
$ n
RZ
i
i
"
e2
rj + 21 $
n
$ n
ri
e"
2
rji=1 i=1 j=1 i=1 j=1 i% j
!
Born-Oppenheimer Approximation
Electrons and nucleias “separate” systems
H = " h2
$ n
#2 ri "$
N
$ n Zie
2
+ 1 $
n
$ n e2
2m i=1 i=1 j=1 Ri " rj 2 i=1 ij% =1 j ri " rj
... but this is anapproximation!
• electrical resistivity
• superconductivity
• ....
Review: Solutions
Moller-Plessetperturbation theory
MP2
coupled clustertheory CCSD(T)
quantum chemistrydensity
functionaltheory
density functional
theory
Review:Why DFT?100,000
10,000
1000
100
10
2003 2007 2011 20151
Year
Num
ber o
f Ato
ms
Linear
scalin
g DFT
DFT
QMC
CCSD(T)
Exact treatment
MP2
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wave function:
complicated!
electron densi easy!
ty:
Review: DFT
ψ = ψ(�r1, �r2, . . . , �rN )
Walter KohnDFT, 1964
n = n(�r)
All aspects of the electronic structure of a systemof interacting electrons, in the ground state, inan “external” potential, are determined by n(r)
Review: DFT
ion
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electron density The ground-state energy is a functional
of the electron density.
E[n] = T [n] + Vii + Vie[n] + Vee[n] kinetic ion-electron
ion-ion electron-electron
The functional is minimal at the exactground-state electron density n(r)
The functional exists... but it is unknown!
Review: DFT
E[n] = T [n] + Vii + Vie[n] + Vee[n] kinetic ion-ion ion-electron electron-electron
2electron density n(�r) = �
|φi(�r)|i
Eground state = min E[n]φ
Find the wave functions that minimize theenergy using a functional derivative.
Review: DFT
Finding the minimum leads toKohn-Sham equations
ion potential Hartree potential exchange-correlation potential
equations for non-interacting electrons
Review: DFT
Only one problem: vxc not known!approximations necessary
local density general gradient approximation approximation
LDA GGA
3.021j Introduction to Modeling and Simulation N. Marzari (MIT, May 2003)
Iterations to selfconsistency
Review: Self-consistent cycleKohn-Sham equations
-
n(�r) = �
i
|φi(�r)|2
scf loop
Review: DFT calculationssc
f loo
p total energy = -84.80957141 Ry total energy = -84.80938034 Ry total energy = -84.81157880 Ry total energy = -84.81278531 Ry total energy = -84.81312816 Ry exiting loop; total energy = -84.81322862 Ry result precise enough total energy = -84.81323129 Ry
At the end we get: 1) electronic charge density 2) total energy
Review: DFT calculationstotal energy = -84.80957141 Ry total energy = -84.80938034 Ry total energy = -84.81157880 Ry total energy = -84.81278531 Ry total energy = -84.81312816 Ry exiting loop; total energy = -84.81322862 Ry result precise enough total energy = -84.81323129 Ry sc
f loo
p
At the end we get: 1) electronic charge density 2) total energy
Structure Elastic Vibrational ... constants properties
Review: Basis functionsMatrix eigenvalue equation: ψ =
� ciφi
i
expansion in Hψ = Eψ orthonormalized basis
functions H
� ciφi = E
� ciφi
i i � d�r φ∗
j H �
i
ciφi = E �
d�r φ∗ j
�
i
ciφi
�
i
Hjici = Ecj
H�c = E�c
Review: Plane waves asbasis functions
plane wave expansion: ψ(�r) = �
cj e Gj ·�ri �
j plane wave
Cutoff for a maximum G is necessary and results in a finite basis set.
Plane waves are periodic, thus the wave function is periodic!
periodic crystals: atoms, molecules: Perfect!!! (next lecture) be careful!!!
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From atoms to solids
The ground state electron configuration of a system is constructed by putting the available electrons, two at a time (Pauli principle), into the states of lowest energy
Solid Energy Atom Molecule
Antibonding p
p Antibonding s
Bonding p s
Bonding s
Conduction band from antibonding p orbitals
Conduction band from antibonding s orbitals
Valence band from p bonding orbitals
Valence band from s bonding orbitals
k
Image by MIT OpenCourseWare.
Ener
gy
Energy bands
Metal Insulator Semiconductor
occupied
empty
energy gap
NB: boxes = allowed energy regions
Crystal symmetries
A crystal is built upof a unit cell and
periodic replicas thereof.
lattice unit cellImage of M. C. Escher's "Mobius with Birds" removed due to copyright restrictions.
Crystal symmetries
1023 particlesper cm3
Since a crystal is periodic, maybe we can get away with modeling only the unit cell?
Crystal symmetries
BravaisLattice
Triclinic
Monoclinic
Orthorhombic
Tetragonal
Trigonal
Cubic
Hexagonal
Parameters Simple(P)
VolumeCentered (I)
BaseCentered (C)
FaceCentered (F)
a1 = a2 = a3α12 = α23 = α31
a1 = a2 = a3α23 = α31 = 900
α12 = 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = 1200
α23= α31= 900
a1 = a2 = a3α12 = α23 = α31 = 900
a1 = a2 = a3α12 = α23 = α31 < 1200
a3
a1
a2
4 Lattice Types
7 C
ryst
al C
lass
es
Image by MIT OpenCourseWare.
The inverse latticeThe real space lattice is described by three basis vectors:
R� = n1�a1 + n2�a2 + n3�a3
The inverse lattice is described by three basis vectors:G� = m1
�b1 + m2�b2 + m3
�b3
i � � Gj ·ψ(�r) = �
cj e i � �r e G·R = 1
j automatically periodic in R!
The inverse latticereal space lattice (BCC) inverse lattice (FCC)
x y
a
z
a1a2
a3
Image by MIT OpenCourseWare.
The Brillouin zone
inverse lattice
The Brillouin zone is a specialunit cell of the inverse lattice.
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Periodic potentialsmetallic sodium
V (�r)
© Dr. Helmut Foll. All rights reserved. This content is
2�
m ∇2 + V (�r)
�ψ = Eψ
excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/fairuse.
� −
�R �R �R
l
Periodic potentials
It becomes much easier if you use theperiodicity of the potential!
V (�r) = V (�r + R�)
attice vector
Bloch’s theorem ψ�k r) = e i�k·�r u�k(�r)(�
NEW quantum number k that lives in the inverse lattice!
u�k(�r) = u�k(�r + R�)
Periodic potentials
Bloch’s theorem
ψ�k(�r) = e i�k·�r u�k(�r)
u�k(�r) = u�k(�r + R�)
λ = 2π/k
ψk (r) = eik.r
k = 0
u(r)
k = π/a
a
Image by MIT OpenCourseWare.
Periodic potentials Results of the Bloch theorem:
ψ�k(�r + R�) = ψ�k(�r)e i�k·�r
2 2 charge density |ψ�k(�r + R�)| = |ψ�k(�r)| is lattice periodic
ψ�k(�r) −→ ψ�k+ �G(�r)
E�k = E�k+ �G
if solution also solution
with
Periodic potentialsSchrödinger certain quantumequation symmetry number
hydrogen spherical ψn,l,m(�r)
atom symmetry [H, L2] = HL
2 − L2H = 0
[H, Lz] = 0
periodic translational ψ n, (�k �r)
solid symmetry [H, T ] = 0
The band structure
Different wave functions can satisfy the Bloch theorem for the same k: eigenfunctions and eigenvalues labelled with k and the index n
energy bands
The band structure
energy levelsin the Brillouin zone
Figure by MIT OpenCourseWare.
L
EC EV
X1
-10
0
6
E (e
V)
S1
X
k
U,K Γ Γ
Γ1
Γ' 25
Γ15
∆ ΣΛ
Silicon
Image by MIT OpenCourseWare.
The band structure
energy levels in the Brillouin zone
Figure by MIT OpenCourseWare.
L
EC EV
X1
-10
0
6
E (e
V)
S1
X k
U,K Γ Γ
Γ1
Γ' 25
Γ15
∆ ΣΛ
Silicon
Image by MIT OpenCourseWare.
occupied
unoccupied
Energy bandssimple example: infinitesimal small potential
solutions: plane waves with quadratic energies
E�k = E�k+ �G
g-1 g1 g2 g3
k
1. BZ
E
Image by MIT OpenCourseWare.
The band structure
folding of the band structure
3� 2� �
�k
kx
a 3� aa
2� aa
� a
�k
kx 3� 2� � a
3� aa
2� aa
� a
Image by MIT OpenCourseWare.
The band structure
real band structure
�k
kx
�k
kx 3� 2� � a
3� aa
2� aa
� a
3� 2� � a
3� aa
2� aa
� a
Image by MIT OpenCourseWare.
Literature
• Charles Kittel, Introduction to Solid State Physics
• Richard M. Martin, Electronic Structure
• wikipedia,“solid state physics”,“condensed matter physics”, ...
• Simple band structure simulations: http:// phet.colorado.edu/simulations/sims.php? sim=Band_Structure
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3.021J / 1.021J / 10.333J / 18.361J / 22.00J Introduction to Modeling and Simulation Spring 2011
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