Materials Theory and Computation

Materials Theory and Computation
S. V. Khare Department of Physics and Astronomy University of Toledo, Ohio 2. Department of Electrical Engineering and Computer Science 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 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.

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, (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, (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, (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 Department of Physics and Astronomy University of Toledo, Ohio 2. Department of Electrical Engineering and Computer Science 3. Department of Mechanical and Industrial Engineering

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.
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. ℓ Germanium nanowires (GeNWs) have attracted much attention in recent years[1-9] owing to the advanced electrical properties of Ge such as high carrier mobilities[10] and the ability of facile chemical synthesis of single crystal GeNWs at low temperatures well below 400 °C[5]. 7

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]. Excitonic Bohr radius of bulk Ge is much larger (24.3 nm) than that of Si (4.9 nm) and so the quantum size effects will be more prominent in Ge nanowires. Therefore germanium was found to be more advantageous compared to silicon. Famous groups like Harvard University group (Lieber) , UC-Berkeley group (Peidong Yang) and Georgia Tech group (Wang) performed generated Ge nanowires experimentally. Various methods such as laser ablation, vapour transport, low-temperature CVD, and supercritical fluid–liquid–solid synthesis have been developed for the synthesis of Ge NWs. University of Texas, Austin group (Tobias Hanrath and Brian A. Korgel) worked on crystallography and experimental faceting of Ge nanowires. They found that high temperatures are favorable for <111> growth while lower temperatures were favorable for <110> growth direction. New Jersey Institute of Technology and HP laboratories (Kamenev BV et al,.) 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] The fast Fourier transform (FFT) of the TEM image indicates that the nanowire is imaged near the [110] pole with the [110] and 111 directions at 350 and 900 with respect to the [112] growth axis. The figures shows the tip of nanowires for different growth directions. 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. In contrast to the Au/Ge interface, the nascent end of the nanowire exhibits distinctive faceting. The nanowire in figure shows a tapered structure at its end consistent with two {111} facets at 550 relative to the [1bar10] growth axis. Of the low-index facets of diamond cubic Ge, the {111} surface has the lowest surface energy, the tip correspondingly exposes the two lowest-energy ({111} and {100}) surfaces. 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.

* Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge.
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 Ge NWs can serve as active channel material of a FET, they should be assembled directly on insulator layer, electrically decoupled from silicon substrate. The germanium nanowires-on-insulator (GeNOI) technology could offer a nanomaterials-based substrate platform for future chip applications, similar to thin film germanium-on-insulator (GOI) technology that is being actively researched in today’s semiconductor industry for nanoscale CMOS. Using metal-catalyzing heterogeneous growth, we directly assembled Ge nanowires onto thin SiO2 layered Si substrate. Schematics illustrating the crossed NW-FET concept.Ŧ * Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge. Ŧ Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge. 11

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 12

Pros and Cons of ab initio method
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%. 13

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 Our calculations were performed in the framework of density functional theory (DFT), within the local density approximation (LDA) using VASP. The single-particle wave functions were expanded in a plane-wave basis using a 150 eV kinetic energy cutoff which was determined by convergence tests to be sufficient. As a test of the pseudopotentials used, we computed for bulk Si, and Ge. For Ge a lattice constant of nm and bulk modulus of GPa, in excellent agreement with experimental values of nm and 75 GPa respectively. The theoretical lattice constant was used in all the later calculations.

Theoretical calculations Experimental calculations*
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 Diameter of the nanowire in nm Number of H atoms in the nanowire ( Ge - , H - 44 ) ( ) 03 . 2 NW 89 ] 001 [ Nanowire Orientation of the nanowire The compact notation given above represents a NW along an axis of Miller indices [h k l] containing “m” Ge and “n” H atoms, with additional entries for more elements as needed, with a diameter “d” expressed in nm. To represent a doped nanowire we have the third element in the superscript to signify the type and number of dopant atoms.

Structural Properties of Ge nanowires
All results in this talk are with DFT-LDA, VASP. [001] [110] [111] [001] [110] [111] More than 30 wires are constructed along the three axes ([001], [110], and [111]) of different diameters up to 3 nm were fully relaxed. Figure shows the structures of some of these different NWs. Optimized structures were found to be highly symmetric. Above a certain critical diameter dc in the range 2.0 nm < dc < 3.0 nm for all three axes the cross section of wires acquires a faceted shape. Wires along [001] direction appear in Figure to have rectangular bonding geometry rather than the expected square surface arrangement of atoms in a single (001) surface layer because the middle atoms, along the longer side of the rectangle, are nearest neighbors of the top layer atoms and are one layer below the top layer. Cross sections of wires along [110] were found to have cylindrical structure for all the nanowires For higher diameter (d > dc = 2.15 nm) wires along [001], cross-sections were found to be octagonal shaped with facets of the {001} and {110} type which were normal to [001]. For higher diameter (d > dc = 2.11 nm) wires along [111], they were hexagonal shaped with facets of the {110} type which were normal to [111]. We also observe that due to differing atomic densities the wires along [111] surface are found to have larger number of atoms compared to wires of [001] and [110] of same diameter. We also calculated the surface energies of all the wires which were found to be linearly dependent on diamater of wire and are found to be in the ratio 4:5:3 respectively for wires along [001], [110], and [111].

Electronic Properties: Band Structures of Ge nanowires
[001] [110] [111] [001] [110] [111] Electronic band structures of the relaxed wires were calculated from Γ (0, 0, 0) point to X (0, 0, (2*pi)/dr) along the wire axis, where dr is the length of the repeating unit of the wire along its axis. From the band structure calculations we observe that the dispersion of the valence band (VB) for wires with approximately the same diameter is greatest for wires along [110] and least for wires along [111]. We observed the quantum confinement of electrons in the range of diameters studied due to the which absolute value of the VB maximum decreases in energy and the absolute value of the conduction band (CB) minimum increases in energy, as the thickness of the wire decreases. For wires along [111]-axis the VB max occurs at gamma point and CB min occurs at (0.8 * pi )/dr. For wires along [001]-axis the VB max occurs at gamma point and CB min occurs at (0.9 * pi )/dr.

Band Structures of doped and undoped Ge nanowires
n-doped undoped p-doped [100] [110] Georgia Tech (Wang’s) group worked on generation of Ge nanowires using Chemical Vapor deposition process. They also observed that the nanowires with n and p type doping showed band bending towards conduction and valence bands respectively. This motivated us in doing the band structures calculations of doped Ge. Figure shows the band structures of doped Ge NWs for a single wire along each crystallographic direction ([001], [110], and [111]). Highly doped p-type(n-type) wires are obtained by addition of a boron(phosphorus) atom in the interior of the NW. As can be seen from the figure doping was performed for NWs of diameter 2.0 nm along each axis. The concentration of dopants was very high such as one dopant for 89 Ge atoms in unit cell along [001] axis, one dopant for 69 atoms along [011] axis, and one dopant for 170 Ge atoms along [111] direction. These structures were allowed to relax and their band structures studied subsequently High level of doping (0.5 to 1%) obviously has an impact on the electronic structures of Ge NWs. As shown in Figure adding a p-type(n-type) dopant moves the EF towards the VB(CB). Maximum of the VB and minimum of the CB increase(decrease) in energy with addition of p-type (n-type) dopant when measured relative to EF. [111]

Plot of Energy gap (eV) versus Diameter (nm)
From our band structure calculations we observe that for same diameter wires along different axis of rotation, band gap is greatest for wires along [001] and lowest for wires along [110]. Among the band structures we calculated the largest band gap 4.3 eV, within LDA, was found for NW along [001] axis with diameter 0.4 nm.

Comparison of band gap of Ge and Si nanowires along different diameter and axes
Ge nanowires Axis Dia (nm) 0.5 1.0 1.5 2.0 2.5 3.0 [001] D I [110] [111] Si nanowires* Axis 0.5 1.0 1.5 2.0 2.5 3.0 [001] I [110] D [111] Dia (nm) Bulk Band Gap of Ge = 300K Indirect Bulk Band Gap of Si = 300K Indirect The table shows the comparison of band gaps for Ge and Si semiconductor material nanowires along different axis of orientation with different diameters. As shown in the above tables, there are distinct differences in the behavior of band gap for NWs of Ge compared to those of Si, as seen from earlier theoretical computations. For Ge nanowires, NWs along [110] and thin (d1.3 nm) ones along [001] have direct band gaps occurring at the Γ point. Such wires due to their direct gaps would be less suitable for applications in optics. Wires along [001] were found to transit from direct to indirect Eg as the diameter increased above 1.3 nm while all wires along [111] have indirect band gaps. For Si, NWs along [110] and thin (d 2.2 nm) NWs along [111] have direct band gaps occurring at the Γ point while wires along [001] have indirect band gaps. It has also been shown that Si NWs along [111] have a greater band gap than those along [110] with the same diameter. These differences in the variation of the band gaps of NWs, from direct to indirect with orientation and diameter, can be attributed to differences in the location of the conduction band minimums of the two materials. Si has 6 ellipsoidal conduction bands with axes along the 111 directions while Ge has 8 ellipsoidal conduction bands along the 001 axes. 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
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. The electronic band structure shows a significant response to changes in surface passivation with hydrogen. Doping of wires with n and p type atoms produced a response in the band structure similar to that in a doped bulk crystal. Quantum confinement has a substantial effect on the electronic band structure and hence the band gap, which increases with decreasing diameter. 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 eV for H-terminated Ge nanowires, dense k-point meshes Forces converged till < 0.01 eV/ Å Used supercomputers of NCSA and OSC Our calculations were performed in the framework of density functional theory (DFT), within the local density approximation (LDA) using VASP. The single-particle wave functions were expanded in a plane-wave basis using a 150 eV kinetic energy cutoff which was determined by convergence tests to be sufficient. As a test of the pseudopotentials used, we computed for bulk Si, and Ge. For Ge a lattice constant of nm and bulk modulus of GPa, in excellent agreement with experimental values of nm and 75 GPa respectively. The theoretical lattice constant was used in all the later calculations.

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 28

Acknowledgements People Institutional support
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) 29

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 I will give a brief outline of thesis in this slide 30

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 > 20) Nanorods: Wires with small aspect ratios (ℓ/d) Nanocontacts: Short wires bridged between two larger electrodes. ℓ Germanium nanowires (GeNWs) have attracted much attention in recent years[1-9] owing to the advanced electrical properties of Ge such as high carrier mobilities[10] and the ability of facile chemical synthesis of single crystal GeNWs at low temperatures well below 400 °C[5]. 32

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. Due to high intrinsics carrier density , high compatibility with high dielectric constants. Excitonic Bohr radius of bulk Ge is much larger (24.3 nm) than that of Si (4.9 nm) and so the quantum size effects will be more prominent in Ge nanowires. Therefore germanium was found to be more advantageous compared to silicon. Famous groups like Harvard University group (Lieber) , UC-Berkeley group (Peidong Yang) and Georgia Tech group (Wang) performed generated Ge nanowires experimentally. Various methods such as laser ablation, vapour transport, low-temperature CVD, and supercritical fluid–liquid–solid synthesis have been developed for the synthesis of Ge NWs. University of Texas, Austin group (Tobias Hanrath and Brian A. Korgel) worked on crystallography and experimental faceting of Ge nanowires. They found that high temperatures are favorable for <111> growth while lower temperatures were favorable for <110> growth direction. New Jersey Institute of Technology and HP laboratories (Kamenev BV et al,.) 50 nm 5 nm * J. C. Johnson et al., Nature Materials 1, 106–110 (2002), University of California, Berkeley. 34

Growth of GaN NWs using the Metalorganic Chemical Vapour Deposition (MOCVD)
TEM images of the GaN nanowires. a–c,Wires grown on (100) γ-LiAlO2.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 cross-sections. Due to high intrinsics carrier density , high compatibility with high dielectric constants. Excitonic Bohr radius of bulk Ge is much larger (24.3 nm) than that of Si (4.9 nm) and so the quantum size effects will be more prominent in Ge nanowires. Therefore germanium was found to be more advantageous compared to silicon. Famous groups like Harvard University group (Lieber) , UC-Berkeley group (Peidong Yang) and Georgia Tech group (Wang) performed generated Ge nanowires experimentally. Various methods such as laser ablation, vapour transport, low-temperature CVD, and supercritical fluid–liquid–solid synthesis have been developed for the synthesis of Ge NWs. University of Texas, Austin group (Tobias Hanrath and Brian A. Korgel) worked on crystallography and experimental faceting of Ge nanowires. They found that high temperatures are favorable for <111> growth while lower temperatures were favorable for <110> growth direction. New Jersey Institute of Technology and HP laboratories (Kamenev BV et al,.) * Kuykendall et al., Nature Materials 3, 524–528 (2004), University of California, Berkeley. 35

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 Germanium nanowires (GeNWs) have attracted much attention in recent years[1-9] owing to the advanced electrical properties of Ge such as high carrier mobilities[10] and the ability of facile chemical synthesis of single crystal GeNWs at low temperatures well below 400 °C[5]. 36

* Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge.
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 Ge NWs can serve as active channel material of a FET, they should be assembled directly on insulator layer, electrically decoupled from silicon substrate. The germanium nanowires-on-insulator (GeNOI) technology could offer a nanomaterials-based substrate platform for future chip applications, similar to thin film germanium-on-insulator (GOI) technology that is being actively researched in today’s semiconductor industry for nanoscale CMOS. Using metal-catalyzing heterogeneous growth, we directly assembled Ge nanowires onto thin SiO2 layered Si substrate. Schematics illustrating the crossed NW-FET concept.Ŧ * Duan et al., Nature 2001, 409, 66, Harvard University, Cambridge. Ŧ Huang et al., Pure Appl. Chem. 2004, 76, 2051, Harvard University, Cambridge. 37

GaN Nanowire Transistor: n-type
GaN nanowire laser Far-field image of a single GaN nanolaser* 1 μm GaN Nanowire Transistor: n-type Ge NWs can serve as active channel material of a FET, they should be assembled directly on insulator layer, electrically decoupled from silicon substrate. The germanium nanowires-on-insulator (GeNOI) technology could offer a nanomaterials-based substrate platform for future chip applications, similar to thin film germanium-on-insulator (GOI) technology that is being actively researched in today’s semiconductor industry for nanoscale CMOS. Using metal-catalyzing heterogeneous growth, we directly assembled Ge nanowires onto thin SiO2 layered Si substrate. SEM image of a GaN nanowire connected with two electrodes for the transport study. The inset is an illustration of the GaN transistor layout. Current-voltage measurement at different gating voltages for the 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. 38

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 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) x z y x z y Basis atomic position: (0.0, 0.0, 0.0) 40

Hexagonal Bravais lattice structures
Wurtzite unit cell The wurtzite lattice. 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 B3 ½ a X + ½ 3-1/2 a Y + u c Z (N) B4 ½ a X - ½ 3-1/2 a Y + (½ + u) c Z 41

Structure representing the wurtzite lattice.
Wurtzite structure N atoms Ga atoms Structure representing the wurtzite lattice. 42

Objective of making NW structures
Periodically repeating unit along arbitrary direction (m n o) in a crystal. - For example consider a [001] axis wire z x y -  Indicate Ga atoms  Indicate N atoms 44

Periodically repeating unit along arbitrary direction (m n o) in a crystal. - For example consider a [001] axis wire z x y -  Indicate Ga atoms  Indicate N atoms 45

Periodically repeating unit along arbitrary direction (m n o) in a crystal. - For example consider a [001] axis wire Surfaces should be passivated x z y -  Indicate Ga atoms  Indicate N atoms 46

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. 47

Generation of Bulk material
Position vector of any atom in bulk material is given by R = å n a + ( ) å b i i j represents the lattice vectors for i = 1, 2, and 3; represent the basis atoms. The generated bulk material has square cross-section. ai bj -  Indicate Ga atoms  Indicate N atoms 48

Extracting the nanowire
For each atom in the bulk material: Cross product of its position vector with the normal along the axis of wire < 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 Axis of the NW * Goldstein. H, Poole. C, Safko. J, Classical Mechanics, 3rd Edition, Addison Wesley. 49

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  Indicate N atoms ℓ Nanowire cut from bulk material. 50

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. -  Indicate Ga atoms  Indicate N atoms  Indicate H atoms Top view 51

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 53

Pros and Cons of ab initio method
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%. 54

Ab initio codes Different codes: SIESTA VASP CASTEP Abinit CRYSTAL
VASP - Vienna Ab initio Simulation Package 55

VASP Implementing ab initio quantum mechanical molecular dynamics.
Output files Input files POSCAR POTCAR KPOINTS INCAR OUTCAR OSZICAR CONTCAR CHGCAR WAVECAR EIGENVAL PROCAR XDATCAR LOCPOT DOSCAR 56

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 57

POSCAR Ge Bulk 5.6435 2 Direct (Å ) 3 2 1 a , r n atom GaN-bulk 5.602 2 Selective dynamics Direct T T T T F T T T F F F T n1, n2 The ordering must be consistent with the POTCAR 58

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. 59

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 * 60

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 NW (Ga – a, N – b, H – c,..) [100] (d) Diameter (d) of the nanowire in nm Nanowire Orientation of the nanowire The compact notation given above represents a NW along an axis of Miller indices [h k l] containing “m” Ge and “n” H atoms, with additional entries for more elements as needed, with a diameter “d” expressed in nm. To represent a doped nanowire we have the third element in the superscript to signify the type and number of dopant atoms. 62

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 ) . 1.09 ( NW 48 H , 24 N 12 Ga ] 001 [ - ) . 1.94 ( NW 96 H N , 72 Ga ] 001 [ - ) .25 (2 NW 96 H , 120 N Ga ( ] 001 [ - More than 30 wires are constructed along the three axes ([001], [110], and [111]) of different diameters up to 3 nm were fully relaxed. Figure shows the structures of some of these different NWs. Optimized structures were found to be highly symmetric. Above a certain critical diameter dc in the range 2.0 nm < dc < 3.0 nm for all three axes the cross section of wires acquires a faceted shape. Wires along [001] direction appear in Figure to have rectangular bonding geometry rather than the expected square surface arrangement of atoms in a single (001) surface layer because the middle atoms, along the longer side of the rectangle, are nearest neighbors of the top layer atoms and are one layer below the top layer. Cross sections of wires along [110] were found to have cylindrical structure for all the nanowires For higher diameter (d > dc = 2.15 nm) wires along [001], cross-sections were found to be octagonal shaped with facets of the {001} and {110} type which were normal to [001]. For higher diameter (d > dc = 2.11 nm) wires along [111], they were hexagonal shaped with facets of the {110} type which were normal to [111]. We also observe that due to differing atomic densities the wires along [111] surface are found to have larger number of atoms compared to wires of [001] and [110] of same diameter. We also calculated the surface energies of all the wires which were found to be linearly dependent on diamater of wire and are found to be in the ratio 4:5:3 respectively for wires along [001], [110], and [111]. [100] a-axis ) .37 1 ( NW 32 H , 27 N 19 Ga ] 100 [ - ) . 2.18 ( NW 48 H , 56 N 44 Ga ] 100 [ - ) 80 . 2 ( NW 60 H , 84 N 69 Ga ] 100 [ - 63

Band structures [001] [100] ~ 1.0 nm ~ 2.0 nm > 2.0 nm ) . 1.09 (
K (2π/ℓ) K (2π/ℓ) ) . 1.09 ( NW 48 H , 24 N 12 Ga ] 001 [ - ) .37 1 ( NW 32 H , 27 N 19 Ga ] 100 [ - ~ 2.0 nm K (2π/ℓ) K (2π/ℓ) ) . 2.18 ( NW 48 H , 56 N 44 Ga ] 100 [ - ) . 1.94 ( NW 96 H N , 72 Ga ] 001 [ - > 2.0 nm K (2π/ℓ) K (2π/ℓ) ) .25 (2 NW 96 H , 120 N Ga ( ] 001 [ - ) 80 . 2 ( NW 60 H , 84 N 69 Ga ] 100 [ - 64

Band Structures of doped and undoped GaN nanowires
n-doped undoped p-doped [001] K (2π/ℓ) K (2π/ℓ) K (2π/ℓ) NW (2.02) NW (2.02) NW (2.02) Georgia Tech (Wang’s) group worked on generation of Ge nanowires using Chemical Vapor deposition process. They also observed that the nanowires with n and p type doping showed band bending towards conduction and valence bands respectively. This motivated us in doing the band structures calculations of doped Ge. Figure shows the band structures of doped Ge NWs for a single wire along each crystallographic direction ([001], [110], and [111]). Highly doped p-type(n-type) wires are obtained by addition of a boron(phosphorus) atom in the interior of the NW. As can be seen from the figure doping was performed for NWs of diameter 2.0 nm along each axis. The concentration of dopants was very high such as one dopant for 89 Ge atoms in unit cell along [001] axis, one dopant for 69 atoms along [011] axis, and one dopant for 170 Ge atoms along [111] direction. These structures were allowed to relax and their band structures studied subsequently High level of doping (0.5 to 1%) obviously has an impact on the electronic structures of Ge NWs. As shown in Figure adding a p-type(n-type) dopant moves the EF towards the VB(CB). Maximum of the VB and minimum of the CB increase(decrease) in energy with addition of p-type (n-type) dopant when measured relative to EF. [100] K (2π/ℓ) K (2π/ℓ) K (2π/ℓ) NW (2.02) NW (2.02) NW (2.02) 65

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 [100] Dia (nm) Ge nanowires by Medaboina et al.,* 0.5 1.0 1.5 2.0 2.5 3.0 [001] D I [110] [111] Axis Dia (nm) The table shows the comparison of band gaps for Ge and Si semiconductor material nanowires along different axis of orientation with different diameters. As shown in the above tables, there are distinct differences in the behavior of band gap for NWs of Ge compared to those of Si, as seen from earlier theoretical computations. For Ge nanowires, NWs along [110] and thin (d1.3 nm) ones along [001] have direct band gaps occurring at the Γ point. Such wires due to their direct gaps would be less suitable for applications in optics. Wires along [001] were found to transit from direct to indirect Eg as the diameter increased above 1.3 nm while all wires along [111] have indirect band gaps. For Si, NWs along [110] and thin (d 2.2 nm) NWs along [111] have direct band gaps occurring at the Γ point while wires along [001] have indirect band gaps. It has also been shown that Si NWs along [111] have a greater band gap than those along [110] with the same diameter. These differences in the variation of the band gaps of NWs, from direct to indirect with orientation and diameter, can be attributed to differences in the location of the conduction band minimums of the two materials. Si has 6 ellipsoidal conduction bands with axes along the directions while Ge has 8 ellipsoidal conduction bands along the axes. D = Direct band gap, I = Indirect band gap * Phys. Rev. B 76, (2007). 66

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] [001] 0.83 12 24 48 3.08 1.08 36 2.93 1.37 54 72 2.74 1.84 2.68 2.20 96 2.67 [100] 0.75 07 20 3.00 1.00 10 16 2.90 1.30 19 27 32 1.60 30 41 44 2.63 2.18 56 2.59 The table shows the comparison of band gaps for Ge and Si semiconductor material nanowires along different axis of orientation with different diameters. As shown in the above tables, there are distinct differences in the behavior of band gap for NWs of Ge compared to those of Si, as seen from earlier theoretical computations. For Ge nanowires, NWs along [110] and thin (d1.3 nm) ones along [001] have direct band gaps occurring at the Γ point. Such wires due to their direct gaps would be less suitable for applications in optics. Wires along [001] were found to transit from direct to indirect Eg as the diameter increased above 1.3 nm while all wires along [111] have indirect band gaps. For Si, NWs along [110] and thin (d 2.2 nm) NWs along [111] have direct band gaps occurring at the Γ point while wires along [001] have indirect band gaps. It has also been shown that Si NWs along [111] have a greater band gap than those along [110] with the same diameter. These differences in the variation of the band gaps of NWs, from direct to indirect with orientation and diameter, can be attributed to differences in the location of the conduction band minimums of the two materials. Si has 6 ellipsoidal conduction bands with axes along the directions while Ge has 8 ellipsoidal conduction bands along the axes. 67

Plot of band gap (eV) versus Diameter (nm)
From our band structure calculations we observe that for same diameter wires along different axis of rotation, band gap is greatest for wires along [001] and lowest for wires along [110]. Among the band structures we calculated the largest band gap 4.3 eV, within LDA, was found for NW along [001] axis with diameter 0.4 nm. 68

Conclusions of work on GaN nanowires
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. Doping of wires with n and p type atoms produced a response in the band structure similar to that in a doped bulk crystal. Quantum confinement has a substantial effect on the electronic band structure and hence the band gap, which increases with decreasing diameter. All wires studied have direct bandgaps. 70

Future work (preliminary stages)
Optical properties of GaN nanowires are being determined. i r e + = Real and Imaginary plots of the dielectric function of NW (2.02) 72

Thank you! 73

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. If we minimize the energy, the n(r) will be consistent with the orbitals. The n(r) at which the energy is minimized is the self consistent solution to the We said ground state energy written in terms of n(r). From minimization we need to satisfy the Kohn Sham equations. Find the n(r) that produces a potential that gives orbitals that reproduce the potential. 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). 75

Ab initio techniques and approximations
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. 76

Supercell geometry for a molecule
77

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. 78

Effective Schrodinger equation for non-interactng electrons
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. 79

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 = E[r(r)]. The density that minimizes the energy is the ground state density and the energy is the ground state energy, Min{E[r(r)]} = E0 80

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! 81

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) N (g) N2 (g) 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) 82

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 83

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