Transcript My presentation - The University of Toledo

Materials Theory and Computation
S. V. Khare
1. 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
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 nanoto micro- length scales. Varied theoretical techniques utilized are
density functional theory based computations, classical molecular
dynamics, Monte Carlo simulations, and continuum analytical
equations.
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 β-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
<110> 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).
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).
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
1. 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/
Outline
•
•
•
•
Experimental motivation
Ab initio methods
Structural properties
Band structures of doped and undoped
nanowires
• Band gaps of Si and Ge nanowires
• Conclusions
Introduction
• Diameter (d) of NWs range from 1 nm –
100 nm.
ℓ
• Length (ℓ) varies from 10nm – 1µm
• Different names to NWs in literature:
– Nanowires: Wires with large aspect ratios (ℓ/d > 20)
– Nanorods: Wires with small aspect ratios (ℓ/d)
– Nanocontacts: Short wires bridged between two larger
electrodes.
Experimental methods for preparing Ge
nanowires
 Laser ablation
 Vapor transport
 Low-temperature CVD
 Supercritical fluid–liquid–solid synthesis : In this method thermal evaporation of Ge
powder at 950C onto silicon wafer and ceramic (alumina) substrate using Au catalyst
via a vapour–liquid–solid (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.
Orientation of Ge nanowires generated
using SLFS method
[211]
[110]
[111]
[111]
Tip of nanowires generated using supercritical fluid–liquid–solid
(SLFS) method by Hanrath et al*.,
* Hanrath, T.; Korgel, B. A. Small 2005, 1, 7.
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.
Diode made of NWs
n
p
p-n
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.Ŧ
* Duan
Ŧ
et al., Nature 2001, 409, 66, Harvard University, Cambridge.
Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge.
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
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%.
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
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.
Nomenclature used for describing a
nanowire
Number of Ge
atoms in the
nanowire
Number of H atoms
in the nanowire
( Ge - 89, H - 44 )
NW [ 001]
Nanowire
Orientation
of the
nanowire
Diameter of the
nanowire in nm
( 2 . 03 )
Structural Properties of Ge
nanowires
All results in this talk are with DFT-LDA, VASP.
[001]
[110]
(Ge 89, H 44 )
NW[001
]
( Ge89, H  44)
2.03
NW[001
]
[001]
[111]
(Ge 89, H 44 )
NW[001
]
( Ge69, H 32)
2.12
NW[110
]
[110]
( Ge170, H 66)
2.11
NW[111
]
[111]
(Ge 89, H 44 )
NW[001
]
( Ge185, H 60)
3.03
NW[001
]
( Ge133, H  40)
3.3
NW[110
]
( Ge326, H 90)
3.03
NW[111
]
Electronic Properties: Band Structures
of Ge nanowires
[001]
[110]
( Ge89, H  44)
2.03
NW[001
]
[001]
( Ge185, H 60)
[ 001]
NW
( Ge69, H 32)
2.12
NW[110
]
[110]
3.03
( Ge133, H  40)
3.3
NW[110
]
[111]
( Ge170, H 66)
2.11
NW[111
]
[111]
( Ge326, H 90)
3.03
NW[111
]
Band Structures of doped and undoped
Ge nanowires
n-doped
undoped
( Ge88, H  44, P 1)
2.03
NW[001
]
( Ge89, H  44)
2.03
NW[001
]
p-doped
[100]
( Ge88, H  44, B 1)
2.03
NW[001
]
[110]
( Ge68, H 32, P 1)
2.12
NW[110
]
( Ge69, H 32)
2.12
NW[110
]
( Ge68, H 32, B 1)
2.12
NW[110
]
[111]
( Ge169, H 66, P 1)
2.11
NW[111
]
( Ge170, H 66)
2.11
NW[111
]
( Ge169, H 66, B 1)
2.11
NW[111
]
Plot of Energy gap (eV) versus
Diameter (nm)
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
Si nanowires*
I
I
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
Axis
Axis
D = Direct band gap, I = Indirect band gap
* Zhao, X.; Wei, C. M.; Yang, L.; Chou, M.Y. PRL 2004, 92, 23.
Conclusions of work on Ge nanowires
1. Study of structural, energetic, and electronic properties of hydrogenpassivated 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.
Institutional Support
• University of Toledo Parallel Computing Cluster
• Ohio Supercomputer Cluster
• National Center for Supercomputing Applications (NCSA)
Thank you!
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 < 0.01 eV/ Å
• Used supercomputers of NCSA and OSC
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
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. Khare’s group)
Institutional support
• Ohio Supercomputer Center (OSC)
• National Center for Supercomputing Applications (NCSA)
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
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
Introduction
• Diameter (d) of NWs range from 1 nm –
100 nm.
ℓ
• Length (ℓ) varies from 10nm – 1µm
• Different names to NWs in literature:
– Nanowires: Wires with large aspect ratios (ℓ/d > 20)
– Nanorods: Wires with small aspect ratios (ℓ/d)
– Nanocontacts: Short wires bridged between two larger
electrodes.
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
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.
50 nm
* J.
(c)SEM image of single
GaN
5 nm
wire after dispersing onto
sapphire substrate.
Scale bar, 5μm.
C. Johnson et al., Nature Materials 1, 106–110 (2002), University of California, Berkeley.
Growth of GaN NWs using the Metalorganic
Chemical Vapour Deposition (MOCVD)
TEM images of the GaN
nanowires.
a–c,Wires grown on (100) γLiAlO2.The inset in a is an
electron-diffraction pattern
recorded along [001] axis.
d–f,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 crosssections.
* Kuykendall
et al., Nature Materials 3, 524–528 (2004), University of California, Berkeley.
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
Diode made of NWs
n
p
p-n
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.Ŧ
* Duan
Ŧ
et al., Nature 2001, 409, 66, Harvard University, Cambridge.
Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge.
GaN nanowire laser
Far-field image of a single
GaN nanolaser*
1 μm
GaN Nanowire Transistor: n-type
(a) SEM image of a GaN nanowire connected with two electrodes
for the transport study. The inset is an illustration of the GaN
transistor layout.
measurement at different gating voltages for the
(b) Current-voltage
GaN nanowire. Ŧ
*J. C. Johnson et al., Nature Materials 1, 106–110 (2002). Ŧ Kuykendall et al., Nano. Lett. 3, 1063, 2003.
University of California, Berkeley.
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
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
a
Basis atomic
position:
(0.0, 0.0, 0.0)
y
Bases atomic
positions:
z
x
y
(0.0, 0.0, 0.0)
z
x
(0.0, 0.5, 0.5)
(0.5, 0.0, 0.5)
(0.5, 0.5, 0.0)
Hexagonal Bravais lattice structures
Hexagonal Bravais
lattice structure
Wurtzite unit cell
Basis Vectors
Lattice Vectors
A1
=
½ a X - ½ 31/2 a Y
A2
=
½aX+½
A3
=
cZ
The wurtzite lattice.
31/2
aY
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)
Wurtzite structure
Ga atoms
Structure representing the wurtzite lattice.
N atoms
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
Objective of making NW structures
•
Periodically repeating unit along arbitrary direction (m n o)
in a crystal.
- For example consider a [001] axis wire
y
x
z
-
 Indicate Ga atoms
-
 Indicate N atoms
Objective of making NW structures
•
Periodically repeating unit along arbitrary direction (m n o)
in a crystal.
- For example consider a [001] axis wire
y
x
z
-
 Indicate Ga atoms
-
 Indicate N atoms
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
y
z
x
-
 Indicate Ga atoms
-
 Indicate N atoms
Generation of nanowires
Three major steps in generation of nanowires:
1)
Generate a large cube of bulk material using lattice and
basis vectors of wurtzite lattice.
2)
Cut a wire of given length and diameter from the bulk
material using a separate algorithm.
3)
Identify the missing neighbors and passivate the dangling
bonds with hydrogen atoms.
Generation of Bulk material
• Position vector of any atom in bulk material is given by
R =  (ni  ai ) +  b j
•
ai represents the lattice vectors
for i = 1, 2, and 3;
bj represent the basis atoms.
• The generated bulk material has
square cross-section.
-
 Indicate Ga atoms
-
 Indicate N atoms
Extracting the nanowire
For each atom in the bulk material:
1. Cross product of its position vector with the normal along
the axis of wire < radius of the wire.
2. 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
Axis of the NW
* Goldstein.
H, Poole. C, Safko. J, Classical Mechanics, 3rd Edition, Addison Wesley.
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
-
 Indicate Ga atoms
ℓ
 Indicate N atoms
Nanowire cut from bulk material.
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.
H passivated GaN NW.
Top view
-
 Indicate Ga atoms
-
 Indicate N atoms
-
 Indicate H atoms
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
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
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%.
Ab initio codes
• Different codes:
• SIESTA
• VASP
• CASTEP
• Abinit
• CRYSTAL
• VASP - Vienna Ab initio Simulation Package
VASP
• Implementing ab
dynamics.
initio
quantum
mechanical
Input files
Output files
POSCAR
POTCAR
KPOINTS
INCAR
OUTCAR
OSZICAR
CONTCAR
CHGCAR
WAVECAR
EIGENVAL
PROCAR
XDATCAR
LOCPOT
DOSCAR
molecular
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
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
0.500000000
0.000000000
0.500000000
0.500000000
2
2
Selective dynamics
Direct
0.33333333 0.66666667
0.000
0.66666667 0.33333333
0.500
0.33333333 0.66666667
0.385
0.66666667 0.33333333
0.885
0.50000000
0.50000000
0.00000000
T
T
T
F
T T
F T
T F
F T
(Å )
r
a1,
r
a2,
r
a3
n atom
n1, n2
The ordering must
be consistent with
the POTCAR
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.
Test of Pseudopotentials
Lattice
constants (nm)
Bulk modulus
(GPa)
Theoretical
calculations
a=0.3118
c=0.5132
183
Experimental
measurement *
a=0.3189
c=0.5185
187
* http://www.phys.ksu.edu/area/GaNgroup/gparametm.html
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
Nomenclature used for describing a
nanowire
“a” number of Ga
atoms in the
nanowire
“b” number of
N atoms in the
nanowire
“c” number of H
atoms in the
nanowire
(Ga – a, N – b, H – c,..)
Nanowire
NW
Orientation
of the
nanowire
[100]
(d)
Diameter (d) of the
nanowire in nm
Structural Properties
All results in this presentation are obtained using ab initio method.
~ 1.0 nm
> 2.0 nm
~ 2.0 nm
[001]
c-axis
48)
Ga12, N 24,H
NW[(001
]
( 1.09
. )

Ga 72, N
NW[(001
]
96.H96)
(1.94
. )
( Ga 96 , N  120, H 96 )
NW[001]
(2.25)
[100]
a-axis

Ga 19 , N
NW[(100
]
27 , H  32 )
(1 .37 )

Ga 44, N
NW[(100
]
 56 , H  48 )
(2.18
. )

Ga 69 ,N
NW[(100
]
84 ,H60 )
(2.80)
Band structures
[001]
[100]
~ 1.0 nm
K (2π/ℓ)
( Ga12, N 24,H48)
[001]
NW
( 1.09
. )
NW
K (2π/ℓ)
( Ga  19 , N 27 , H  32 )
[100 ]
(1 .37 )
K (2π/ℓ)
( Ga 44, N  56 , H  48 )
[100 ]
(2.18
. )
~ 2.0 nm
K (2π/ℓ)
( Ga 72, N 96.H96)
[001]
NW
(1.94
. )
NW
> 2.0 nm
K (2π/ℓ)
( Ga 96 , N  120, H 96 )
[001]
NW
(2.25)
K (2π/ℓ)
( Ga69 ,N 84 ,H60 )
[100]
NW
(2.80)
Band Structures of doped and undoped
GaN nanowires
n-doped
undoped
p-doped
[001]
K (2π/ℓ)
K (2π/ℓ)
NW
(Ga71, N 96, H 96,C 1)
[001]
(2.02)
NW
K (2π/ℓ)
(Ga 72, N 96, H 96)
[001]
(2.02)
(Ga72, N 95, H 96,C 1)
NW [001]
(2.02)
[100]
K (2π/ℓ)
NW
(Ga43, N 56, H 48,C 1)
[110]
(2.02)
NW
K (2π/ℓ)
K (2π/ℓ)
(Ga 44, N 56, H 48)
[100]
(Ga 44, N 55, H 48,C 1)
[100]
(2.02)
NW
(2.02)
Comparison of band gap of GaN and Ge
nanowires
GaN nanowires
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
Axis
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
Axis
D = Direct band gap, I = Indirect band gap
* Phys. Rev. B 76, 205327 (2007).
Band gap, Eg of GaN nanowires
Wire axis
[h k l]
[001]
[100]
d (nm)
0.83
1.08
1.37
1.84
2.20
0.75
1.00
1.30
1.60
2.18
No. of Ga No. of N No. of H*
atoms
atoms
atoms
12
24
36
54
72
07
10
19
30
44
24
36
54
72
96
12
16
27
41
56
48
48
72
72
96
20
24
32
44
48
Eg (eV)
3.08
2.93
2.74
2.68
2.67
3.00
2.90
2.74
2.63
2.59
Plot of band gap (eV) versus
Diameter (nm)
3.2
3.1
[001] axis
Eg (eV)
3
[100] axis
2.9
2.8
2.7
2.6
2.5
0.5
1
1.5
d (nm)
2
2.5
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
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.
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
Future work (preliminary stages)
• Optical properties of GaN nanowires are being
determined.
e = e r + ie i
(Ga 72, N 96, H 96)
Real and Imaginary plots of the dielectric function of NW [001]
(2.02)
Thank you!
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).
Ab initio techniques and approximations
• Techniques:
1. Density functional theory
2. Pseudopotential theory
3. 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.
Supercell geometry for a molecule
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.
Practical Algorithm
Effective Schrodinger equation for non-interactng electrons
   2 2


+ veff [  (r )] n (r ) = e n n (r ),
 2m

N
 (r ) =   n (r ) ,
2
n =1
Implementation:
1. 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
5. Calculate the new charge density
6. Calculate all the contributions to the effective potential and
iterate until the charge density and effective potential are selfconsistent.
N
7. Then calculate total energy.
E[  (r )] =  e n
n =1
Density Functional Theory (DFT)
Synonyms: DFT = Ab initio = First Principles
Hohenberg Kohn Theorems (1964)
(1)The external potential of a quantum many body system is
uniquely determined by the r), so the total energy is a unique
functional of the particle density E = E[r)].
(2) The density that minimizes the energy is the ground state
density and the energy is the ground state energy,
Min{E[r)]} = E0
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)].
  


+ veff [  (r )] n (r ) = e n n (r ),
 2m

2
2
N
 (r ) =   n (r )
2
n =1
  (r )  3
veff [  (r )]  vnuc . (r ) +  
d r  + vxc [  (r )]
 r  r  
Coulomb
potential of
nuclei
Hartree electrostatic
potential
E xc [  (r )]
vxc [  (r )] =
,
 (r )
Exc[  (r )]
Exchange
correlation
potential
is universal!
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 [GaN]x (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)
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