Materials Theory and Computation

1
Materials Theory and Computation

• S. V. Khare
• Department of Physics and Astronomy
• University of Toledo, Ohio
• 2. Department of Electrical Engineering and
  Computer Science
• University of Toledo, Ohio
• http//astro1.panet.utoledo.edu/khare/
• Funding DARPA, Air Force, NSF, DoE, State of
  Ohio

2
General theme of research
My research involves the application of
appropriate theoretical and computational
techniques to understand condensed matter systems
of significant experimental interest. This work
involves predictions for new phenomena,
explanation of existing data, and collaborations
with experimentalists on their current
experiments. It has involved a variety of thin
film and bulk materials from metals to
semiconductors, crystalline to disordered
materials, and nano- to micro- length scales.
Varied theoretical techniques utilized are
density functional theory based computations,
classical molecular dynamics, Monte Carlo
simulations, and continuum analytical equations.
3
Papers with students I

• Effect of structure, surface passivation, and
  doping on the electronic and optical properties
  of GaAs nanowires A first principles study
• V. Gade, N. Shi, D. Medaboina, S. V. Khare, R.
  Ramprasad (Submitted to journal)
• Structural and Electronic properties of ß-In2X3
  (X O, S, Se, Te) using ab initio calculations
• S. Marsillac, N. S. Mangale, V. Gade, S. V.
  Khare (Submitted to journal)
• Super Hard Cubic Phases of Period VI Transition
  Metal Nitrides A First Principles Investigation
  S. K. R. Patil, N. S. Mangale, S. V. Khare, and
  S. Marsillac Accepted in Thin Solid Films 2008.
• Effect of structure, surface passivation, and
  doping on the electronic properties of Ge
  nanowires A first-principles study D.
  Medaboina, V. Gade, S. K. R. Patil, and S. V.
  Khare Phys. Rev. B 76, 205327 (2007).
• Impact of Structure Relaxation on the Ultimate
  Performance of a Small Diameter, n-Type lt110gt
  Si-Nanowire MOSFET G. Liang, D. Kienle, S. K. R.
  Patil, J. Wang, A. W. Ghosh, and S. V. Khare
  IEEE Trans. Nano. Tech. 6, 225 (2007).

4
Papers with students II

• Mechanical stability of possible structures of
  PtN investigated using first-principles
  calculations S. K. R. Patil, S. V. Khare, B. R.
  Tuttle, J. K. Bording, and S. Kodambaka Phys.
  Rev. B 73, 104118 (2006).
• Ab Initio calculations for Properties of MAX
  phases Ti2TlC, Zr2TlC, and Hf2TlC J. A. Warner,
  S. K. R. Patil, S. V. Khare, and R. S.
  Masiuliniec Appl. Phys. Lett. 88, 101911 (2006).

5
Ab initio computations of structural and
electronic properties of doped and undoped Ge
nanowires

• S. V. Khare1, D. Medaboina2, V. Gade2, and S. K.
  R. Patil3
• Department of Physics and Astronomy
• University of Toledo, Ohio
• 2. Department of Electrical Engineering and
  Computer Science
• University of Toledo, Ohio
• 3. Department of Mechanical and Industrial
  Engineering
• University of Toledo, Ohio
• http//www.physics.utoledo.edu/khare/

6
Outline

• Experimental motivation
• Ab initio methods
• Structural properties
• Band structures of doped and undoped nanowires
• Band gaps of Si and Ge nanowires
• Conclusions

7
Introduction

• Diameter (d) of NWs range from 1 nm 100 nm.
• Length (l) varies from 10nm 1µm
• Different names to NWs in literature
• Nanowires Wires with large aspect ratios (l/d gt
  20)
• Nanorods Wires with small aspect ratios (l/d)
• Nanocontacts Short wires bridged between two
  larger electrodes.

8
Experimental methods for preparing Ge nanowires

• Laser ablation
• Vapor transport
• Low-temperature CVD
• Supercritical fluidliquidsolid synthesis In
  this method thermal evaporation of Ge powder at
  950?C onto silicon wafer and ceramic (alumina)
  substrate using Au catalyst via a
  vapourliquidsolid (VLS) process. Diameters up
  to 30 nm and length tens of micro meters.
  Preferred growth direction for the nanowires is
  111.

Nanowires developed by Nguyen et al., grown
along 110 on heavily doped Si.
Nanowires developed by Kamanev et al., of 40 nm
diameter along 111 growth direction grown on
silicon substrate.
Nguyen, P. Ng, H. T. Meyyappan, M. Adv.
Mater. 2005, 17, 5. Kamanev, B. V. Sharma,
V. Tsybeskov, L. Kamins, T. I. Phys. Stat. Sol.
(a) 2005, 202, 2753.
9
Orientation of Ge nanowires generated using SLFS
method
Tip of nanowires generated using supercritical
fluidliquidsolid (SLFS) method by Hanrath et
al.,
Hanrath, T. Korgel, B. A. Small 2005, 1, 7.
10
Faceting of Ge nanowires
Fourier transform of image representing the 110
pole axis of the wire 110
Tapered end of nanowire showing the facets
HRTEM image of nanowire along 110
growth direction showing the length of nanowire.
Crystallographic model of nanowire showing the
facets of nanowire.
HRTEM image of 110 growth direction developed
by Hanrath et al., representing the faceted cap
structure of nanowire.
Hanrath, T. Korgel, B. A. Small 2005, 1, 7.
11
Diode made of NWs
1 µm

• A SEM image of a p-n diode. Diode obtained by
  simply crossing p- and n-type NW.

FET made of NWs
Schematics illustrating the crossed NW-FET
concept.T
Duan et al., Nature 2001, 409, 66, Harvard
University, Cambridge. T Huang et al., Pure Appl.
Chem. 2004, 76, 2051, Harvard University,
Cambridge.
12
Ab initio method

• Powerful predictive tool to calculate properties
  of materials
• Fully first principles ?
• (1) no fitting parameters, use only fundamental
  constants (e, h, me, c) as input
• (2) Fully quantum mechanical for electrons
• Thousands of materials properties calculated to
  date
• Used by biochemists, drug designers, geologists,
  materials scientists, and even astrophysicists!
• Evolved into different varieties for ease of
  applications
• Awarded chemistry Nobel Prize to W. Kohn and H.
  Pople 1998

13
Pros and Cons of ab initio method

• Pros
• Very good at predicting structural properties
• (1) Lattice constant good to 0-3.
• (2) Bulk modulus good to 1-10.
• (3) Very robust relative energy ordering between
  structures.
• (4) Good pressure induced phase changes.
• Good band structures, electronic properties.
• Used to study the properties of materials at
  unstable conditions.
• Cons
• Computationally intensive.
• Excited electronic states difficult to
  compute.
• Band gaps are under estimated by 50.

14
Ab initio method details

• LDA, Ceperley-Alder exchange-correlation
  functional as parameterized by Perdew and Zunger
• Generalized ultra-soft Vanderbilt
  pseudo-potentials and plane wave basis set
• Supercell approach with periodic boundary
  conditions in all three dimensions
• Wires are infinite along their axis

15
Theoretical and experimental comparison of
lattice constant and bulk modulus of Ge
Lattice constant (nm) Bulk mudulus (GPa)
Theoretical calculations 0.5638 72.57
Experimental calculations 0.5658 75.00
Kittel, C. Introduction to Solid State Physics,
2nd ed., (John Wiley Sons, Inc., New York,
1976), p. 40.
16
Nomenclature used for describing a nanowire
-
)
44
,
(
-
H
Ge
89
NW
17
Structural Properties of Ge nanowires
All results in this talk are with DFT-LDA, VASP.
18
Electronic Properties Band Structures of Ge
nanowires
19
Band Structures of doped and undoped Ge nanowires
n-doped undoped p-doped
100
110
111
20
Plot of Energy gap (eV) versus Diameter (nm)
21
Comparison of band gap of Ge and Si nanowires
along different diameter and axes
Ge nanowires
0.5 1.0 1.5 2.0 2.5 3.0
001 D D I I I I
110 D D D D D D
111 I I I I I I
Si nanowires
Axis 0.5 1.0 1.5 2.0 2.5 3.0
001 I I I I I I
110 D D D D D D
111 D D D D I I
Dia (nm)
D Direct band gap, I Indirect band gap
Zhao, X. Wei, C. M. Yang, L. Chou, M.Y. PRL
2004, 92, 23.
22
Conclusions of work on Ge nanowires

1. Study of structural, energetic, and electronic
   properties of hydrogen-passivated doped and
   undoped germanium nanowires along 001, 110,
   and 111 directions with diameter d up to 3 nm,
   using ab initio methods.
2. The electronic band structure shows a significant
   response to changes in surface passivation with
   hydrogen.
3. Doping of wires with n and p type atoms produced
   a response in the band structure similar to that
   in a doped bulk crystal.
4. Quantum confinement has a substantial effect on
   the electronic band structure and hence the band
   gap, which increases with decreasing diameter.
5. Wires oriented along 110 are found to have a
   direct band gap while the wires along 111 are
   found to have an indirect band gap. Wires along
   001 show a crossover from a direct to an
   indirect band gap as diameter increases above the
   critical diameter for the transition being 1.3
   nm.

23
Institutional Support

• University of Toledo Parallel Computing Cluster
• Ohio Supercomputer Cluster
• National Center for Supercomputing Applications
  (NCSA)

24
Thank you!
25
Ab initio method details

• LDA, Ceperley-Alder exchange-correlation
  functional as parameterized by Perdew and Zunger
• Used the VASP code with generalized ultra-soft
  Vanderbilt pseudo-potentials and plane wave basis
  set
• Supercell approach with periodic boundary
  conditions in all three dimensions
• Energy cut-offs of 150.00 eV for H-terminated Ge
  nanowires, dense k-point meshes
• Forces converged till lt 0.01 eV/ Å
• Used supercomputers of NCSA and OSC

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Structural and Electronic Properties of Doped and
Undoped GaN Nanowires A First Principles
Investigation
Shandeep Voggu (MS Thesis Candidate) Department
of EECS University of Toledo
29
Acknowledgements

• People
• Prof. Sanjay V. Khare (Thesis advisor)
• Prof. Daniel Georgiev (Committee member)
• Prof. Vijay Devabhaktuni (Committee member)
• Varun Gade, Dayasagar Medaboina, Sunil K. R.
  Patil, Nikhil Mangale, Ashok Kolagatla, Kausthuba
  Ippagunta, Abbas Naseem, Krishnakanth Ganguri
  (Prof. Khares group)

• Institutional support
• Ohio Supercomputer Center (OSC)
• National Center for Supercomputing Applications
  (NCSA)

30
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

31
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

32
Introduction

• Diameter (d) of NWs range from 1 nm 100 nm.
• Length (l) varies from 10nm 1µm
• Different names to NWs in literature
• Nanowires Wires with large aspect ratios (l/d gt
  20)
• Nanorods Wires with small aspect ratios (l/d)
• Nanocontacts Short wires bridged between two
  larger electrodes.

33
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

34
Growth of GaN NWs using the Metalorganic Chemical
Vapour Deposition (MOCVD)
Electron microscopy images of synthesized GaN
nanowires. (a)Scanning electron microscopy
(SEM) images of GaN nanowires grown on sapphire
substrate. Scale bar, 3µm. (b)High-resolution
transmission electron microscopy image of GaN
nanowire.Scale bar, 1 nm. (c)SEM image of single
GaN wire after dispersing onto sapphire
substrate. Scale bar, 5µm.
5 nm
50 nm
J. C. Johnson et al., Nature Materials 1,
106110 (2002), University of California,
Berkeley.
35
Growth of GaN NWs using the Metalorganic Chemical
Vapour Deposition (MOCVD)
TEM images of the GaN nanowires. ac,Wires grown
on (100) ?-LiAlO2.The inset in a is an
electron-diffraction pattern recorded along 001
axis. df,Wires grown on (111) MgO
substrates.The insets in d show the hexagonal
cross-section of the wire and an
electron-diffraction pattern recorded along the
100 axis. c and f show space-filling
structural models for the nanowires with
triangular and hexagonal cross-sections.
Kuykendall et al., Nature Materials 3, 524528
(2004), University of California, Berkeley.
36
Advantages of NWs

• NW devices can be assembled in a rational and
  predictable way because
• NWs can be precisely controlled for structure and
  chemical composition during synthesis.
• NW building blocks can be combined in ways not
  possible in conventional electronics.
• Series of electronic devices are being assembled
  using semiconductor NWs
• Crossed NW p-n diodes,
• Crossed NW-FETs,
• Nanoscale logic gates,
• Optoelectronic devices

37
Diode made of NWs
1 µm

• A SEM image of a p-n diode. Diode obtained by
  simply crossing p- and n-type NW.

FET made of NWs
Schematics illustrating the crossed NW-FET
concept.T
Duan et al., Nature 2001, 409, 66, Harvard
University, Cambridge. T Huang et al., Pure Appl.
Chem. 2004, 76, 2051, Harvard University,
Cambridge.
38
GaN nanowire laser

• Far-field image of a single GaN nanolaser

1 µm
GaN Nanowire Transistor n-type

1. SEM image of a GaN nanowire connected with two
   electrodes for the transport study. The inset is
   an illustration of the GaN transistor layout.
2. Current-voltage measurement at different gating
   voltages for the GaN nanowire. T

J. C. Johnson et al., Nature Materials 1,
106110 (2002). T Kuykendall et al., Nano. Lett.
3, 1063, 2003. University of California, Berkeley.
39
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

40
Definition of a crystal

• Crystal atomic position Bravais lattice
  position Basis vector
• Bravais lattice is regular arrangement of
  points.
• Vectors determining the position of the atom
  from every Bravais lattice point are called
  basis vectors.
• Basis vector 1 basis atom 4 basis atoms

Bases atomic positions (0.0, 0.0, 0.0) (0.0,
0.5, 0.5) (0.5, 0.0, 0.5) (0.5, 0.5, 0.0)
Basis atomic position (0.0, 0.0, 0.0)
41
Hexagonal Bravais lattice structures
Hexagonal Bravais lattice structure
The wurtzite lattice.
Wurtzite unit cell
Lattice Vectors
Basis Vectors
A1    ½ a X - ½ 31/2 a Y 
A2    ½ a X ½ 31/2 a Y 
A3    c Z  
B1    ½ a X ½ 3-1/2 a Y  (Ga)   (2b)
B2    ½ a X - ½ 3-1/2 Y ½ c Z  (Ga)   (2b)
B3    ½ a X ½ 3-1/2 a Y u c Z  (N)   (2b)
B4    ½ a X - ½ 3-1/2 a Y (½ u) c Z  (N)   (2b)
42
Wurtzite structure
Structure representing the wurtzite lattice.
43
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

44
Objective of making NW structures

• Periodically repeating unit along arbitrary
  direction (m n o) in a crystal.
• - For example consider a 001 axis wire

45
Objective of making NW structures

• Periodically repeating unit along arbitrary
  direction (m n o) in a crystal.
• - For example consider a 001 axis wire

46
Objective of making NW structures

• Periodically repeating unit along arbitrary
  direction (m n o) in a crystal.
• - For example consider a 001 axis wire

• Surfaces should be passivated

47
Generation of nanowires

• Three major steps in generation of nanowires
• Generate a large cube of bulk material using
  lattice and basis vectors of wurtzite lattice.
• Cut a wire of given length and diameter from the
  bulk material using a separate algorithm.
• Identify the missing neighbors and passivate the
  dangling bonds with hydrogen atoms.

48
Generation of Bulk material

• Position vector of any atom in bulk material
  is given by

n
å
å
R
b
a
)
(
j
i
i

• represents the lattice vectors for i 1,
  2, and 3 represent the basis atoms.
• The generated bulk material has square
  cross-section.

bj
49
Extracting the nanowire

• For each atom in the bulk material
• Cross product of its position vector with the
  normal along the axis of wire lt radius of the
  wire.
• Dot product of the position vector of atom and
  normal along the axis of wire lies in the range ?
  -(wire-length)/2 to (wire-length)/2

3. Wire-length determined from the crystal
Goldstein. H, Poole. C, Safko. J, Classical
Mechanics, 3rd Edition, Addison Wesley.
50
Generated nanowire

• The generated nanowires will have dangling bonds
  left on the surface of wire due to the cutting.
• These dangling bonds create states in
  bandstructure.

d
l
Nanowire cut from bulk material.
51
Termination with Hydrogen

• Each atom in the wire is checked to see four
  neighbors. The atoms without four neighbors are
  identified and the missing neighbors are replaced
  with hydrogen atoms.

Top view
52
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

53
Ab initio method

• Powerful predictive tool to calculate properties
  of materials
• Fully first principles ?
• (1) no fitting parameters, use only fundamental
  constants (e, h, me, c) as input
• (2) Fully quantum mechanical for electrons
• Thousands of materials properties calculated to
  date
• Used by biochemists, drug designers, geologists,
  materials scientists, and even astrophysicists!
• Evolved into different varieties for ease of
  applications
• Awarded chemistry Nobel Prize to W. Kohn and H.
  Pople 1998

54
Pros and Cons of ab initio method

• Pros
• Very good at predicting structural properties
• (1) Lattice constant good to 0-3.
• (2) Bulk modulus good to 1-10.
• (3) Very robust relative energy ordering between
  structures.
• (4) Good pressure induced phase changes.
• Good band structures, electronic properties.
• Used to study the properties of materials at
  unstable conditions.
• Cons
• Computationally intensive.
• Excited electronic states difficult to
  compute.
• Band gaps are under estimated by 50.

55
Ab initio codes

• Different codes
• SIESTA
• VASP
• CASTEP
• Abinit
• CRYSTAL
• VASP - Vienna Ab initio Simulation Package

56
VASP

• Implementing ab initio quantum mechanical
  molecular dynamics.

Output files
Input files
OUTCAR OSZICAR CONTCAR CHGCAR WAVECAR EIGENVAL PRO
CAR XDATCAR LOCPOT DOSCAR
POSCAR POTCAR KPOINTS INCAR
57
VASP input files

• POSCAR Positions of ions
• Bravais lattice
• Periodic boundary conditions
• POTCAR Pseudopotentials from VASP
• KPOINTS Would be used for parallelization
• INCAR Different parameters for different
  properties

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POSCAR
Ge Bulk 5.6435 0.00000000 0.50000000
0.50000000 0.50000000 0.00000000 0.50000000
0.50000000 0.50000000 0.00000000 2 Direct
0.000000000000000 0.000000000000000
0.000000000000000 0.250000000000000
0.250000000000000 0.250000000000000

• GaN-bulk
• 5.602
• 0.000000000 0.500000000 0.50000000
• 0.500000000 0.000000000 0.50000000
• 0.500000000 0.500000000 0.00000000
• 2
• Selective dynamics
• Direct
• 0.33333333 0.66666667 0.000 T T T
• 0.66666667 0.33333333 0.500 T F T
• 0.33333333 0.66666667 0.385 T T F
• 0.66666667 0.33333333 0.885 F F
  T

The ordering must be consistent with the POTCAR
59
VASP output files

• OUTCAR Complete information of the simulation
• - Number of irreducible points
• - Final position of ions and forces
• - Time take to complete simulation
• OSZICAR It contains the information about free
  energy (E0) and about convergence speed.
• CONTCAR It contains the positions of ion at
  the final ionic step in relaxations.

60
Test of Pseudopotentials
Lattice constants (nm) Bulk modulus (GPa)
Theoretical calculations a0.3118 c0.5132 183
Experimental measurement a0.3189 c0.5185 187
http//www.phys.ksu.edu/area/GaNgroup/gparametm.
html
61
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

62
Nomenclature used for describing a nanowire
c number of H atoms in the nanowire
b number of N atoms in the nanowire
a number of Ga atoms in the nanowire
Diameter (d) of the nanowire in nm
Nanowire
Orientation of the nanowire
63
Structural Properties
All results in this presentation are obtained
using ab initio method.
gt 2.0 nm
1.0 nm
2.0 nm
001 c-axis

100 a-axis
64
Band structures
001
100
1.0 nm
K (2p/l)
K (2p/l)
2.0 nm

K (2p/l)
K (2p/l)
gt 2.0 nm
K (2p/l)
K (2p/l)
65
Band Structures of doped and undoped GaN nanowires
n-doped undoped p-doped
001
K (2p/l)
K (2p/l)
K (2p/l)
(2.02)
NW
NW
(2.02)
NW
(2.02)


100
K (2p/l)
K (2p/l)
K (2p/l)
NW
(2.02)
NW
NW
(2.02)
(2.02)
66
Comparison of band gap of GaN and Ge nanowires
GaN nanowires
Axis 0.5 1.0 1.5 2.0 2.5 3.0
001 D D D D D D
100 D D D D D D
Dia (nm)
Ge nanowires by Medaboina et al.,
0.5 1.0 1.5 2.0 2.5 3.0
001 D D I I I I
110 D D D D D D
111 I I I I I I
D Direct band gap, I Indirect band gap
Phys. Rev. B 76, 205327 (2007).
67
Band gap, Eg of GaN nanowires
Wire axis d (nm) No. of Ga atoms No. of N atoms No. of H atoms Eg (eV)
h k l d (nm) No. of Ga atoms No. of N atoms No. of H atoms Eg (eV)
001 0.83 12 24 48 3.08
001 1.08 24 36 48 2.93
001 1.37 36 54 72 2.74
001 1.84 54 72 72 2.68
001 2.20 72 96 96 2.67
100 0.75 07 12 20 3.00
100 1.00 10 16 24 2.90
100 1.30 19 27 32 2.74
100 1.60 30 41 44 2.63
100 2.18 44 56 48 2.59
68
Plot of band gap (eV) versus Diameter (nm)
69
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

70
Conclusions of work on GaN nanowires

1. Successfully studied the structural and
   electronic properties of hydrogen-passivated
   doped and undoped GaN nanowires along 001 and
   100 directions with diameter d up to 3 nm,
   using ab initio methods.
2. Doping of wires with n and p type atoms produced
   a response in the band structure similar to that
   in a doped bulk crystal.
3. Quantum confinement has a substantial effect on
   the electronic band structure and hence the band
   gap, which increases with decreasing diameter.
4. All wires studied have direct bandgaps.

71
Outline

• Introduction
• Experimental motivation and applications
• Crystal structures
• Generation of nanowires
• Ab initio methods
• Properties Doped and undoped nanowires
• 1) Structural
• 2) Electronic
• Conclusions
• Future work

72
Future work (preliminary stages)

• Optical properties of GaN nanowires are being
  determined.

(2.02)
Real and Imaginary plots of the dielectric
function of NW
73
Thank you!
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Density Functional Theory (DFT)

• DFT states that the ground state energy of a
  system of particles moving in a potential can be
  consistently expressed as a function of the
  density of the particles, n(r).
• We look for the self-consistent solution to the
  equations that minimize the expression for total
  energy within a unit cell as a function of n(r)
  to find the groundstate n(r).
• We assume that the valence electrons experience
  the effects from nuclei and core electrons as a
  non-interacting pseudopotential.
• The density of electrons in a unit cell is then
  given by the sum of the probability densities
  from a set of orthonormal one-electron orbitals.
• Below solving the Kohn-Sham energy minimization
  equations self-consistently.

Resulting ground state density n(r) substituted
into initial expression for energy gives the
ground state energy for a unit cell.
Formatted Equations taken from Wikipedia.org
Density Functional Theory Content Michael J.
Mehl et al, First Principles Calculations of
Elastic Properties of Metals(1993).
76
Ab initio techniques and approximations

• Techniques
• Density functional theory
• Pseudopotential theory
• Iterative diagonalization method
• Approximations
• Local density approximation
• Generalized gradient approximation
• Different codes like SIESTA, VASP, CASTEP are
  used.
• VASP - Vienna Ab initio Simulation Package

Graph showing the comparison of wave function and
ionic potential in Pseudopotential theory.
77
Supercell geometry for a molecule
78
Evolution of theoretical techniques

• The physical properties of any material are found
  to be related to the total energy or difference
  between total energies.
• Total energy calculation methods which required
  specification of number of ions in the material
  are referred to as ab initio methods.
• Ab initio make use of fundamental properties of
  material. No fitting parameters are involved.

79
Practical Algorithm

• Effective Schrodinger equation for non-interactng
  electrons

• Implementation
• Guess an initial charge density for N electrons
• 2. Calculate all the contributions to the
  effective potential
• 3. Solve the Schrodinger equation and find N
  electron states
• 4. Fill the eigenstates with electrons starting
  from the bottom
• Calculate the new charge density
• Calculate all the contributions to the effective
  potential and iterate until the charge density
  and effective potential are self-consistent.
• Then calculate total energy.

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Density Functional Theory (DFT)
Synonyms DFT Ab initio First Principles

• Hohenberg Kohn Theorems (1964)
• The external potential of a quantum many body
  system is uniquely determined by the r(r), so the
  total energy is a unique functional of the
  particle density E Er(r).
• The density that minimizes the energy is the
  ground state density and the energy is the ground
  state energy,
• MinEr(r) E0

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Kohn Sham Theory (1965)

• The ground state density of the interacting
  system of particles can be calculated as the
  ground state density of non-interacting particles
  moving in an effective potential veff r(r).

Coulomb potential of nuclei
Exchange correlation potential
Hartree electrostatic potential
is universal!
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Self catalitic growth of GaN NWs

• self standing GaN layer
• thinned for TEM ( 300 nm)
• heated at 1050 C in a TEM

Above 850 in high vacuum GaN(s) ? Ga (l) 0.5
N (g) 0.25 N2 (g) GaN(s) ? GaN (g) or GaNx
(g)
in-situ study of the decomposition and resulting
nanostructure evolution
national laboratory for advanced Tecnologies and
nAnoSCience
Stach et al, Nano Lett. 3, 867 (2003)
83

• room temperature analysis
• of the nanostructures
• single crystal GaN NWs
• 0001 oriented
• av diameter 50 nm
• gr rate 300 nm/s
• self catalytic process could be important to
  avoid undesired contamination from foreign metal
  atom (catalyst)

national laboratory for advanced Tecnologies and
nAnoSCience