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The Pennsylvania State University

The Graduate School

Department of Physics

STUDY OF CHEMICAL DOPING OF GRAPHENE ON AMORPHOUS SILICA,

ELECTRONIC PROPERTIES OF THE GRAPHENE-FLUORINE SUPER-LATTICE

AND INTERACTIONS OF GRAPHENE WITH ATOMIC FLUORINE

A Dissertation in

Physics

Ning Shen

 2011 Ning Shen

Submitted in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

May 2011

The dissertation of Ning Shen was reviewed and approved* by the following:

Jorge O. Sofo Associate Professor of Physics Associate Professor of Materials Science and Engineering Dissertation Advisor Chair of Committee

Milton W. Cole Distinguished Professor of Physics

Tom Mallouk DuPont Professor of Materials Chemistry and Physics

M. C. Demirel Associate Professor of Department of Engineering Science and Mechanics

Richard W. Robinett Professor of Physics Director of Graduate Studies

*Signatures are on file in the Graduate School

iii

ABSTRACT

Graphene is a novel carbon structure with excellent electronic properties. This dissertation is devoted to the exploration of two graphene related studies: the influence of an amorphous silica substrate on a graphene field effect transistor (FET) and the properties of chemically modified graphene structures with fluorination.

The study of the effect of an amorphous silica substrate on a single graphene layer is interesting since a silicon wafer with thermally grown amorphous silica is commonly used as the substrate for graphene FET devices. Density functional theory (DFT) calculations show that the silica substrate induces a charge transfer to the graphene layer due to surface states of the amorphous silica. The intrinsic n-doping of the graphene is confirmed by the experimental studies of our collaborators. We further propose simple potential profile models to estimate the density of the surface states necessary to explain the magnitude of the observed Dirac Voltages in experiment.

Opening a band gap in graphene is also important to its electronic applications. One possible gap-opening method is through chemical functionalization with fluorine atoms.

Fluorinating certain regions of graphene can transform the carbon atoms in graphene from sp2 to sp3 hybridization, which converts highly conductive graphene into an insulator. We propose the formation of a graphene channel embedded in graphene monofluoride to confine the carriers in graphene. We have studied the electronic structures of two symmetrically orientated graphene channels with armchair or zig-zag boundaries through DFT and tight-binding (TB) calculations.

The armchair channel is found to be metallic or semiconducting depending on the width of the channel. The zigzag channel is found to have dispersive edge bands which are due to the lowering of the site energy of edged carbon compared to the non-edged carbon. We further calculate the local density of states (LDOS) of the edged and non-edged carbon atoms and estimate the charge iv difference between them. Analysis of the LDOS confirms the site energy drop in the edge carbon since it has more electrons than that of the non-edged carbon atom.

Atomic fluorine plasma is found to be an effective method to fluorinate graphene by experimentalists. In order to achieve a better understanding of the fluorination process of graphene and the stability of the generated fluorinated structures, we further explore the interactions between atomic fluorine atoms and graphene through DFT calculations and find the diffusion barriers for isolated fluorine ad-atoms and fluorine atom pairs on graphene. We find that the diffusion barrier of isolated fluorine ad-atom is about 0.3eV while the diffusion barrier of fluorine atom increases to about 1eV when there is another fluorine atom nearby. This indicates that diffusion becomes more difficult when fluorine atoms start assembling together which suggests the stability of the boundary of graphene and graphene monofluoride structures.

TABLE OF CONTENTS

LIST OF FIGURES ...... vi

LIST OF TABLES ……………………………………………………………………...... x

ACKNOWLEDGEMENTS…………………………………………………………………...xi

Chapter 1 Introduction ...... 1

1.1 Interesting Properties of Graphene ...... 1 1.2 Organization of thesis ...... 3

Chapter 2 Overviews of Computational Methods ...... 5

2.1 Tight binding method ...... 5 2.2 Density Functional Theory ...... 10 2.2.1 Basic Equations for Many body Nuclei and Electrons System ...... 10 2.2.2 Hohenberg-Kohn theorem ...... 12 2.2.3 Kohn-Sham Equations ...... 15 2.2.4 Common Approximations in Practical Calculations ...... 17

Chapter 3 Effects of an Amorphous Silica Substrate on a Single Graphene Layer ...... 19

3.1 Introduction ...... 19 3.2 Experimental Details ...... 20 3.2 Ab-initio Results ...... 23 3.3 Potential Profile Model Results ...... 29

Chapter 4 Physics of a Graphene Channel Embedded in Graphite Monofluoride (CF) ...... 36

4.1 Introduction of Opening a Band Gap in Graphene ...... 36 4.2 Electronic Properties of the Armchair Channel ...... 39 4.3 Electronic Properties of Zig-zag Channel ...... 45 4.4 Tight-binding model of the Dispersive Edge band of a Zig-zag Channel ...... 48

Chapter 5 Interaction of Atomic Fluorine with Graphene ...... 55

5.1 Introduction ...... 55 5.2 Transition state theory and the nudged elastic band method ...... 56 5.2 Binding Energy and Diffusion of a Single Fluorine Atom ...... 63 5.3 Binding Energy and Diffusion of Paired Fluorine Atoms ...... 70

Chapter 6 Summary and Future work ...... 77

Bibliography...... 79 vi

LIST OF FIGURES

Figure 2-1. The schematic graph of hexagonal graphene lattice with two sub-lattice carbon atoms: A (black color) and B(grey color).The three nearest neighbors are

represented by vectors ...... 8

Figure 3-1. (a) Schematic graph of a Graphene FET device on a Si substrate with thermally grown SiO2 on top. The source and drain contacts are denoted as S and D while the back-gate is denoted as G. (b) Optical micrograph of the device fabricated with a TEM grid used as a shadow mask. Region I enclosed in a dashed line boundary represents graphene layer. Region II represents the Cr/Au contact. Region III is the Silica dielectric.28[Copyright (2011) by ACS Nano,Reprinted from Link: http://dx.doi.org/10.1021/nn800354m] ...... 21

Figure 3-2. (a) The normalized maximum values of Rds with respect to the initial value of Rds at t=0 (denoted as R0). (b) The time evolution of the Dirac Voltage for the graphene FET. The red dashed line indicates the time evolution of temperature for the FET device. The temperature is about 200ºC for the first 28 hours and suddenly changes to 25 ºC. .28[Figure courtesy of ACS Nano,Reprinted from Link: http://dx.doi.org/10.1021/nn800354m] ...... 22

Figure 3-3. The typical atomic configuration of the unit cell used in our ab-initio calculations. The left panel corresponds to the situation where the graphene sheet is at the minimum energy distance of about 3.6 Å. The right panel is the situation where the graphene layer is at a larger distance. The contours in the figure show the charge transfer and are colored to signify the magnitude of the electron excess when 28 bringing the graphene and SiO2 substrate together. [Copyright (2011) by ACS Nano, Reprinted from the Link : http://dx.doi.org/10.1021/ nn800354m ]...... 25

Figure 3-4. (a) The distance dependence of the amount of charge transferred from the SiO2 substrate to graphene measured in number of electrons Q transferred (left-axis, -3 in unit of 10 e/carbon-atom) and net induced surface charge density n0 (right-axis, in unit of 1013e/cm2). (b) Excess charge per unit length versus distance along z- axis(perpendicular to the graphene layer). The top panel is the situation with the distance between graphene and SiO2 substrate at equilibrium. The bottom panel is the situation with a large separation distance of about 10.8Å. The dashed lines indicate the formal boundary chosen for the SiO2/graphene interface (left) and the outside boundary of the graphene (right).28[ Copyright (2011) by ACS Nano. Reprinted from the link: http://dx.doi.org/10.1021/nn800354m ]...... 27

neglect the interface distance between the thermally grown SiO2 substrate and the silicon substrate. The thickness of the amorphous SiO2 is denoted as dS. ΦS and ΦR correspond to the potential drop between two sides of amorphous SiO2 and between amorphous SiO2 and the graphene layer. The cylinders denote the gauss surfaces in which we apply Gauss’s theorem...... 30

Figure 3-6. The DOS of SiO2-1, SiO2-2 and SiO2-3 amorphous silica substrates respectively. The solid line denotes the position of Fermi Level...... 331

Figure 3-7. The electrostatic potential profile for the graphene layer in a periodic slab of amorphous SiO2 under equilibrium...... 33

Figure 4-1. The schematic graphs of armchair and zig-zag graphene nanoribbons (GNRs). (a) Armchair GNR with N=9. (b) Zig-zag GNR with N=6. The arrows indicate the translational direction of GNRs. The edge carbon atoms are indicated in dark color...... 36

Figure 4-2. The atomic structure of typical relaxed zig-zag (top) and armchair (bottom) configurations. The carbon framework is colored as gray bars and fluorine atoms are light blue balls. The dashed line denotes the counting of row number N for the graphene channel. Similar method is used to count the row number M for the CF barrier. [Copyright (2011) by American Physical Society] ...... 40

Figure 4-3. The dependence of the band gap of armchair channels as the function of channel width N for different barrier width M. The dashed lines correspond to the tight-binding approximation with hopping integral equal to 2.6 eV and a 9% increase of the hopping at the edge. [Copyright (2011) by American Physical Society] ...... 41

Figure 4-4. Typical band structure of a semiconducting armchair channel(armchair(5,13) with a channel width of 15 Å) and corresponding PDOS of the CF barrier and the Graphene channel (top). Band structure of a semi-metallic armchair channel(armchair(6,14) with channel width of 16 Å) and corresponding PDOS of the CF barrier and the Graphene channel(bottom). The G-Y direction in the band structure is the direction along the graphene channel. [Copyright (2011) by American Physical Society] ...... 44

Figure 4-5. Band structure of a zigzag channel (zigzag (6,12)) and PDOS of the CF barrier and the graphene channel. The red continuous line in the band structure is the band of the edge states computed by the tight-binding model discussed in the text. The tight-binding model shows that the dispersive properties of the edge bands are due to the lowering of the site energy of the edge carbon atoms. The X-G direction is the direction along the graphene channel. [Copyright (2011) by American Physical Society] ...... 45

Figure 4-6. The band structure and DOS of zig-zag (N=6) GNRs by Fujita and Nakada 77,78. [Copyright (2011) by American Physical Society. Reprinted from Link: http://link.aps.org/doi/10.1103/PhysRevB.54.17954 ] ...... 47

Figure 4-7. Schematic of a zig-zag graphene channel with periodic boundary conditions in the vertical direction and finite width in the horizontal direction. The first index i viii

th in notation ai,1 or bi,1 is used to denote the i repetitive cell in the vertical direction and the second index 1 is used to denote the row number of the zig-zag graphene channel in the horizontal direction...... 48

Figure 4-8. Comparison of LDOS between DFT and the TB model. The plots are arranges as follows:(a) Edge carbon atom ai,1 at sub-lattice A from the DFT calculation;(b) Edge carbon atom ai,1 at sub-lattice A from the TB model;(c) Bulk carbon atom bi,1 at sub-lattice B from the DFT calculation;(d) Bulk carbon atom bi,1 at sub-lattice B from the TB model ...... 53

Figure 5-1. The one-dimensional energy profile with two local minimum A and B separated by transition state x+.88[Copyright (2011) by John Wiley & Sons, Reprinted from link: http://www.wiley.com/WileyCDA/WileyTitle/productCd- 0470373172.htm ] ...... 57

Figure 5-2. The schematic of the situation where the spring constant K is too small. The solid line denotes the MEP connecting initial and final images with the transition state corresponding to the highest point of the MEP. The grey dots denote the positions of images when K is too small.88[Copyright (2011) by John Wiley & Sons, Reprinted from link: http://www.wiley.com/WileyCDA/WileyTitle/productCd- 0470373172.htm ] ...... 61

Figure 5-3. A schematic of the NEB method.90 The solid contours show equipotential energy surfaces. The images are denoted by the grey dots and reaction paths are the lines joining the images. The inset describes the force diagram of the NEB method. Notice that the force at each image is decomposed into the direction along the path and perpendicular to the path. [Copyright (2011) by American Institute of Physics, Reprinted from link : http://link.aip.org/ link/doi/ 10.1063/1.2841941 ] ...... 62

Figure 5-4. Visualizations of the three high symmetry adsorption sites for a single fluorine atom on graphene and corresponding equilibrium distances between graphene and the fluorine atom. (a) Top, bond and center positions. (b-d) Side views of the top, bond and center site adsorption of the fluorine atom. The vertical distances between the fluorine and graphene layer are 1.56 Ǻ (C-F bond length) for top site, 2.14 Ǻ(vertical distance to the graphene plane) for the bond site adsorption and 2.37 Ǻ(vertical distance to the graphene plane) for the center site adsorption respectively...... 64

Figure 5-5. The MEP diffusion paths and energy barriers. (a) The diffusion path from top site to top site through the bridge site. The energy barrier is about 0.27eV. (b) The diffusion path from top site to center site. The energy barrier is about 0.40eV...... 67

Figure 5-6. The MEP diffusion path and energy barrier for a single hydrogen atom on graphene.98 Site A is the top site while site B and C are bond and center site respectively. [Copyright (2011) by American Physical Society, Reprinted from link: http://prb.aps.org/abstract/PRB/v77/i13 /e134114 ] ...... 68

Figure 5-7. (Top) The different pair configurations of two fluorine atoms on the graphene layer. The first fluorine atom is represented with the blue ball and the ix

position of the second fluorine atom in the pair is numbered from 1 to 6 as shown. (Bottom) The adsorption energies of the pair configurations 1-6. The number in the brackets are notations used in studies by Roman et.al102 on pair configurations of hydrogen atoms. The reference configuration is denoted by the dashed line at - 3.56eV which corresponds to adsorption energy of two isolated fluorine atoms...... 71

Figure 5-8 Studies by Roman et.al102 on pair configurations of hydrogen atoms.(top) pair configurations of two adsorbed hydrogen atoms. (bottom) adsorption energies of different pair configurations. The reference line is at about -1.43eV which is twice the adsorption energy of the isolated hydrogen atom. Note that they are using a different software package, Dacapo103, from Y.Lei98 (VASP)[Copyright(2011) by Elsevier, Reprinted from link: http://dx.doi.org/10.1016 / j.carbon. 2006.09.027 ] ...... 72

Figure 5-9. The MEP diffusion path and energy barrier for a fluorine atom diffusing along the path: site1→site2→site3→site4→site5 on graphene with another fluorine atom is fixed...... 74

Figure 5-10. Energy barriers for atomic hydrogen diffusion and desorption from DFT calculations by L. Horneker107 et al. State A corresponds to pair site1 (equivalent of the configuration p1 in Roman et.al102). The state I corresponds to pair site 2 (equivalent of the configuration p6 in Roman et.al102). The state B corresponds to pair site 3 (equivalent of the configuration p2 in Roman et.al102). [Copyright (2011) by the American Physical Society, Reprinted from link : http://link.aps.org/doi/10.1103/PhysRevLett.96.156104 ] ...... 75

LIST OF TABLES

Table 4-1. Fitting parameters of dispersive edge bands for zig-zag superlattice with different width of graphene channel and CF barrier...... 51

Table 5-1. Binding energy and Carbon-Fluorine atom distance ...... 65

Table 5-2. . Vibrational modes of fluorine atom at the local minimum (top site) and saddle points ( bond and center site ) ...... 69

ACKNOWLEDGEMENTS

I would like to give my deepest acknowledge to my advisor Prof. Jorge O. Sofo for his consistent encouragement and guidance. It was his patience and education that helped me through the difficulties in my scientific career. I also owe the great gratitude to my committee members who gave me valuable advices and help: Prof. Milton W. Cole, Prof. Tom Mallouk and Prof. M.

C. Demirel. There are also many individuals I would like to show my sincere appreciations. They are the faculties who I have the opportunities to learn from and collaborate with: Prof. Peter C.

Eklund, Prof. Vincent H. Crespi and Prof. Jun Zhu. My group members who I have discussed and worked together: Dr Ping Lin, Mr Alejandro M. Suarez, Mr Ivan Iordanov , Mr Nitin Kumar and

Dr. Cuneyt Berkdemir.

Finally, I would like to thank my parents who gave me life, brought me up, and showed me relentless love during all these years. I am indebted to them forever.

Chapter 1

Introduction

Carbon is a fascinating element in the periodic table since it is the most abundant element on earth and has been found to have many different allotropes such as graphite and diamond. The structures of graphite and diamond are three-dimensional while graphite is known to be the stacks of single graphene layers and diamond is known to be the network of tetrahedrons of carbon atoms. People are always interested in discovering new crystalline forms of carbon of lower dimensions. The first kind of fullerene, the buckyball was initially discovered by H. W. Kroto et al.1 in 1985. It is found to be a spherical hollow cluster of sixty carbon atoms with radius about

7Å, which is regarded as quasi zero-dimension carbon allotrope. This discovery is awarded the

Nobel Prize of Chemistry in 1996. Later, the carbon nanotube, a quasi one-dimension crystalline form of carbon was discovered by Sumio Iijima2 in 1991. The structure of the nanotube was found to be single or multiple concentric cylindrical shells capped with half of a fullerene molecule on either end. It is regarded as a quasi-one-dimension structure since it has a remarkable length-to-width ratio with a diameter of a few angstroms and a length of a few micrometers.

Since then, people started attempts to make graphite as thin as possible to achieve few-layer graphite or graphene.

1.1 Interesting Properties of Graphene

In 2004, Novoselov and Geim3 et al. created graphene by micromechanically cleaving a single sheet from highly oriented pyrolytic graphite (HOPG) with a scotch tape method. Since the 2 isolated graphene sheet they made was large enough, they were able to make graphene FET on silicon wafer and measure the electronic properties of the graphene.

They found an ambipolar electric field effect in the graphene FET since the charge carriers in graphene can be tuned between electrons and holes by an external gate voltage. In addition, graphene was found to have astonishing electronic properties. The mobility of graphene exceeds 15000 cm2 V–1 s-1 under ambient conditions3-5. This is much higher than the commonly used materials in electronic devices such as GaAs(~2000 cm2 V–1 s-1 at Room Temperature and

104-107 cm2 V–1 s-1 at low temperature6) and Si-MOSFET(~104 cm2 V–1 s-1 at low temperature6).

These indicate the graphene as a promising candidate for future electronics. Due to their discovery, Novoselov and Geim were awarded the Nobel Prize in Physics in 2010.

Since silicon is widely used in the semiconductor industry, it becomes a natural choice of substrate for graphene FET device and has been commonly used since the first graphene FET experiment by Novoselov and Geim3 et al. Therefore, it is interesting to study the effects of the substrate on the electronic properties of a graphene layer. Also, this thesis will focus on the discussions of the chemical doping effects on the graphene layer with the assumption of ignoring the effects of the substrate on the morphology of graphene.

Although graphene has extraordinary electronic properties, it is a semi-metal with no band gap which becomes problematic in electronic logic applications like FETs. This is because the graphene FET cannot be turned off effectively due to the absence of a band gap. Unlike semiconducting silicon, which has a band gap and can be switched off, graphene will continue to conduct large amount of carriers even in the "off" state of the FET device. When scaled up to a chip made of billions of such transistors, large amounts of energy will be wasted and will make a chip of graphene FETs impractical. Therefore, the ability to open a gap in graphene is crucial to realize its practical electronic applications. Of the different methods to open the bandgap in graphene, such as physical confinement and chemical functionalization, we will focus on 3 chemical functionalization with fluorine atoms. We will study the electronic properties of a graphene channel with armchair or zig-zag boundaries sandwiched between graphene fluorides since this provides a possible way to confine the carriers in graphene.

As atomic fluorine plasma becomes an effective experimental method to fluorinate graphene, it would be interesting to understand the fluorination process. Therefore, we will discuss the possible chemisorption sites of single or pair fluorine atoms and the diffusion barrier of the single atomic fluorine on graphene. We will also study the dependence of the environment, such as existence of a nearby chemisorbed fluorine atom, on the diffusion path.

1.2 Organization of thesis

The organization of the thesis is as follows: In chapter II, I will give an introduction of the techniques we apply in the thesis: Density Functional theory (DFT) and tight-binding (TB) methods. In chapter III, we will study the effects of an amorphous silica substrate on graphene.

The DFT calculations show the intrinsic n-doping of graphene due to the surface states of the amorphous silica substrate. This result explains the experimental observation of a transition in the

Dirac voltage of a graphene FET from positive in ambient conditions to negative under vacuum annealing. We further estimate the amount of surface states needed to explain the large magnitude of the observed Dirac voltage (~50V) in the experiments of our collaborators. In chapter IV, the electronic properties of a graphene channel embedded in graphite monofluoride (CF) are studied with DFT and tight binding methods. We focus on studying two highly symmetric channel orientations: armchair and zig-zag boundaries. We find that the electronic properties of the structures depend on the boundary conditions. We further use a TB model to explain the observed dispersive edge bands of zigzag channels in DFT calculations. In chapter V, the diffusion of atomic fluorine on a single graphene layer is explored through DFT calculation with the nudged 4 elastic band (NEB) method. We first find the energetic adsorption sites of atomic fluorine and then study the diffusion barrier for two basic diffusion paths. We then calculate the diffusion of fluorine under the condition that a second fluorine atom is nearby to study the effect of the environment on the diffusion barrier. We find that the diffusion barrier increases when there is another fluorine atom nearby. When we compare the fluorine diffusion barriers to the hydrogen atom diffusion barriers in the same configurations, we find the energy barrier of fluorine diffusion is lower than that of the hydrogen, which indicates the fluorination of graphene is easier than hydrogenation. In chapter VI, I will summarize the work of the thesis and give some outlooks for future work.

Chapter 2

Overviews of Computational Methods

2.1 Tight binding method

Graphene has an interesting linear band structure in the low energy range near the Fermi

level. Due to its linear band structure following the formula close to the Fermi level

(where is the Fermi velocity of carriers in graphene), the carriers in graphene are thought of as massless Dirac Fermions since they are similar to massless photons for which with c as the speed of light. The tight binding method is widely used to calculate electronic band structure in condensed matter physics and can describe the linear band structure of graphene. Therefore, I will give a detailed description of the tight-binding method in general and its application to calculate the band structure of single layer graphene.

Describing the motion of electrons in a crystal structure is generally a many-body problem. The Hamiltonian for N-electrons can be written as the sum of the kinetic energy of the electrons, electron-electron interactions and electron-ion interactions shown below:

(2.1)

Here N is the number of electrons. P is the number of nuclei and ZJ is the atomic number of the jth nuclei.

If the motion of the electrons is mostly localized around nuclei with a probability to jump to neighboring nuclei, we can apply the tight-binding model to simplify the full periodic crystal.

This new Hamiltonian neglects the electron-electron interactions and approximates the electron- ion interactions as . is the Hamiltonian due to an isolated nuclei located at the lattice site and is the correction of the atomic potential of the nucleus due to the existence 6 of other nuclei in the crystal. Therefore, the crystal Hamiltonian acting on the electrons in the tight-binding model can be written as:

(2.2) Because of the translational symmetry of the unit cell in the crystal, the Hamiltonian is also a periodic function as . So the wave function Ψ of the electrons should satisfy Bloch’s theorem:

(2.3)

th Therefore, the j eigenfunction Ψj can be expressed by a linear combination of the Bloch functions Φj and Φj can be expanded as the summation of the localized atomic wave function as shown below:

(2.4)

(2.5)

th With the Schrödinger equation H|Ψ> = E|Ψ>, the j eigenvalue is given by:

(2.6)

Substituting equation (2.4) into equation (2.6) and changing the subscript yields the following equation:

(2.7)

Here and are the transfer integral matrix element and the overlap integral matrix element respectively. We minimize the energy with respect to coefficients and get: 7

(2.8)

When we substitute equation (2.7) into equation (2.8), we get

(2.9)

Define the column vector of the coefficients and equation (2.9) above can be written as following:

(2.10)

(2.11) The secular equation is defined as the determinant of the coefficient matrix:

(2.12) For single layer graphene, the unit cell has two sub-lattice carbon atoms A and B as shown in figure 2-1 below. Since carbon has four valence electrons and is sp2 hybridized in graphene, it forms three σ bonds with three nearest neighboring carbon atoms and has one free electron forming a pz orbital perpendicular to the graphene plane. Therefore, when we apply the tight binding model to single layer graphene under the nearest neighbor approximation, we are considering the electron in the pz orbital of the carbon atom, which is mostly localized at the lattice site with probabilities of hopping to the nearest neighboring lattice sites. The wave function |Ψ> can be represented as a linear combination of Bloch functions as shown below:

(2.13)

Here and are the Bloch functions of two sub-lattice carbon atoms A and B.

Since we have one pz orbital electron per carbon atom (denoted as or for carbon atom in sub-lattice A or B), we get . 8

Figure 2-1. The schematic graph of hexagonal graphene lattice with two sub-lattice carbon atoms: A (black color) and B(grey color).The three nearest neighbors are represented by vectors .

The transfer matrix H in the secular equation of graphene can be written as

(2.14)

The diagonal matrix elements are

(2.15)

Here the second term is ignored since we consider the nearest neighbor approximation and neglect the small hopping to other non-nearest neighbor carbon atoms.

The off-diagonal matrix elements are

(2.16)

Under the nearest-neighbor approximation, we can simplify equation (2.16) to 9

(2.17)

Here is the hopping integral between the nearest neighbor sub-lattices of A and B and is the summation of phase vectors over the nearest neighboring atoms:

(2.18)

Using the value of shown in figure 2-1, we can simplify

(2.19)

Therefore, the transfer integral matrix H and overlap integral matrix S can be written as

(2.20)

(2.21)

Here S is the overlap integral between the nearest neighbor carbon atoms and is assumed to be zero since we ignore small overlapping between nearest neighbor carbon atoms.

Solving the secular equation (2.12) yields

(2.22)

Here is 10

(2.23)

When we expand around the Dirac points (denoted by wave vector ):

, we simplify as and get the linear band structure as shown below:

(2.24)

2.2 Density Functional Theory

2.2.1 Basic Equations for Many body Nuclei and Electrons System

The complete description of the many-electron system of atomic nuclei and electrons is given by solving the Schrödinger equation,

(2.25) Here the full Hamiltonian H of the system of electrons and nuclei is given as

(2.26)

In practice, it is impossible to solve the above equation directly due to the prohibitive computational costs. We need to have some approximations to simplify the problem. 11

The first approximation applied is the Born-Oppenheimer approximation due to the fact that the masses of nuclei are much larger than those of electrons and the electrons are moving faster than the nuclei. We can use the Born-Oppenheimer approximation by assuming that the electrons can instantaneously adjust their wave functions according to the nuclear wave functions.

Mathematically, we can separate the wave functions of the system into two parts: the part of the electrons in some frozen configurations of the nuclei in quantum treatment and the part of the nuclei in classical treatment.

With the Born-Oppenheimer approximation, the electronic problem is given by solving the electronic Schrödinger equation, where denotes the electron wave function in a frozen configuration of the nuclei denoted as

(2.27)

Here the electronic part of Hamiltonian He is given as

(2.28)

The total energy of the electronic system is the expectation value of the Hamiltonian in equation (2.28) above.

(2.29)

According to the variational principle in quantum mechanics, we can prove that solving the Schrödinger equation is equivalent to minimizing the expectation value of the Hamiltonian He with respect to the wave function Φ. The full minimization of E[Φ] with respect to the electron wave functions will give the true ground state wave function Φ0 and the corresponding ground 12 state energy . The expectation value of the Hamiltonian to any guessed wave

’ function Φ will result in .

The equivalence of the variational principle and solving the Schrödinger equation can be shown as follows.

Suppose the wave function of the Hamiltonian H satisfies the Schrödinger equation

with the normalization condition .

The expectation value of H to Φ is . Applying the variantional principle with the constraint , we have , where λ is the

Lagrange Multiplier. We further simplify it and giving equation (2.30),

(2.30)

Because is arbitrary, we yield . We can see that the Lagrange

Multiplier λ is the eigen-energy of the Hamiltonian.

The proof of is shown as follows.

Suppose we have a guessed wave function which may not be the ground state wave

function. We can expand it as the normalized eigenstates of H, . Then the expectation value of the Hamiltonian becomes

(2.31)

2.2.2 Hohenberg-Kohn theorem

Solving the electronic Schrödinger equation (2.27) directly is still a formidable question.

However, Hohenberg and Kohn7 showed in their seminal paper that an interacting many-electron system can be uniquely determined by the ground state electron density. The ground state energy 13 of the system can be determined by minimizing the energy as the functional of the electron density. The Hohenberg-Kohn theorems are stated below, which serve as the foundations of

Density Functional Theory (DFT).

 For any system of interacting particles in an external potential Vext, the energy is uniquely

determined by the ground state density .

 There exists a universal energy functional E[n] in terms of the density for any

external potential Vext. For any trial charge density that satisfies the condition

, the exact ground state energy of the system E0 is the global minimum of

the functional E[n] and the ground state density is the density that minimizes the

functional: .

The first theorem can be proved by contradiction. Suppose we have two different external potentials Vext1 and Vext2 which differ by more than a constant but share the same ground state density . These two external potentials lead to two different Hamiltonians H1 and H2, which have different ground state wave functions Φ1 and Φ2. Since Φ1 is not the ground state of H2, it follows that

(2.32) The strict inequity is valid if we assume that the ground state is not degenerate. This restriction is proven later by Levy 8 to be not a fatal flaw but a dull technicality. Therefore, the theorem is applicable for either degenerate or non-degenerate ground states. The last term of the above equation can be written as

(2.33) So we have

(2.34)

Similarly, we can have 14

(2.35)

Adding the above two equations gives equation (2.36), which is a contradiction.

(2.36)

Therefore, the ground state charge density uniquely determines the external potential Vext. In addition to the fact that the charge density, gives the knowledge of the total number of electrons in the system, it also determines all the properties of the ground state.

Thus we can write the ground state energy E[ ] as the functional of the density ,

(2.37)

Here denotes the kinetic energy of the electron gas and Vee denotes the electron- electron energy in terms of the electron density. The term includes all the internal energies of the interacting electron system which must be universal by construction since the kinetic energy Te and interacting energy Vee are functionals of the electron density only.

The original proof of the second Hohenberg-Kohn theorem is restricted to the densities

that are the ground state densities of electron Hamiltonian with some external potentials Vext

, which are called “V-representable”. This defines a possible space of densities within which we can use to construct the functional.

The proof is shown as follows. Consider a ground state density corresponding to the external potential Vext1, Hamiltonian H1 and wave function Φ1. The first theorem has established that the total energy functional is

(2.38)

For another different density , this necessarily corresponds to a different wave function Φ2 with the energy 15

(2.39) The above equation shows that the total energy evaluated with the ground state density

is lower than the energy with any other density . Therefore, if we know the universal functional , we minimize the energy functional in equation (2.37) with respect to the

density as and get the eigen-energy

The Hohenberg-Kohn theorems were later generalized by Levy8. As stated above,

Hohenberg-Kohn theorems only apply to the V-representable trial charge density and the conditions for a density to be V-representable are not known. However, Levy showed that the density-functional theory can be formulated in a way that only requires the density to be N- representable. The density is N-representable if it can be obtained by some anti-symmetric wave functions. The conditions for N-representable electron density are

2.2.3 Kohn-Sham Equations

The Hohenberg-Kohn theorems have established the electron density as the basic variable to describe the ground state of a many-electron system. They also show that the minimization of a density functional with respect to the density offers an approach to find the ground state energy. However, the functional in equation (2.37) is not known. In order to have a practical way to find the ground state density, Kohn and Sham proposed in their paper that9 the ground-state density of an interacting system can be represented as the ground-state density of a non-interacting system with a local external potential VKS. This is an unproven 16 assumption; however, the Hohenberg-Kohn theorems guarantee that if the potential is unique if it exists.

Kohn and Sham showed that the ground state density can be obtained by minimizing the energy functional , where is the kinetic energy of the non-interacting system.

The minimization of with respect to charge density under the constraint

with the Lagrange multiplier gives that

(2.40)

On the other hand, we can decompose F[ ] as follows:

(2.41)

Here is the classical Hartree energy and is the non-classical exchange-correlation energy.

Applying the variational principle again gives

(2.42)

By comparing equation (2.40) with equation (2.42), we obtain the Kohn-Sham potential

V as , where is the classical electrostatic Hartree KS

potential, is the exchange-correlation potential and is the external potential.

Since we know that minimizing is equivalent to solving the Schrödinger

equation, all we need to do now is solve the Kohn-Sham equation,

, where the ground state density is given as . 17

The Kohn-Sham equation can be solved interactively as follows: we first start with an input charge density, which gives us the Kohn-Sham potential as

. Then we solve the Kohn-Sham equation with

the Kohn-Sham potential above and get a new charge density. We then update the Kohn-Sham potential with the new charge density and solve the Kohn-Sham equation again. The iterative scheme will stop when the output charge density equals the input charge density within a certain numerical tolerance.

2.2.4 Common Approximations in Practical Calculations

The Kohn-Sham equations are in principle exact if we know the exact form of the exchange-correlation energy . However, the exact form of is unknown since it includes all the non-classical portions of the electron-electron interaction. Therefore, we need to approximate it with different exchange-correlation functionals to put the theory into practical usages.

The simplest approximation is the Local Density Approximation (LDA) which was proposed by Kohn and Sham9 as

(2.43)

Here is the exchange-correlation energy per electron of a homogeneous electron gas of density n.

In practice, the LDA in equation (2.43) is calculated in two parts. The exchange energy is exactly given by the work of Dirac10 as

(2.44)

The correlation energy part is approximated by different forms. The most accurate results are based on the work of Ceperley and Alder 11 using quantum Monte Carlo techniques.

The next levels of approximations are the generalized gradient approximations (GGA),

(2.45)

Here depends on both the electron density and its local gradient . The commonly used GGA functionals in VASP are PW91 by Perdew et al12,and

PBE by the Perdew-Burke-Ernzerhof13.

In addition to the LDA and GGA approximations to the non-classical electron-electron interactions, approximation techniques are also used to calculate the ion-electron interactions to reduce the computational costs, especially in plane-wave methods where the basis sets for expansions of the wave functions are plane waves.

For example, it is known that the physical properties of solids are commonly determined by the bonding of the valence electrons in atoms while the core electrons close to nuclei are too low in energy and thus chemically inert. Pseudopotential techniques have been introduced to avoid the explicit treatment of the core electrons; however, the construction of accurate, transferrable and efficient pseudopotentials is far from straightforward. P. E. Blöchl14 originally proposed the projector-augmented wave (PAW) method , which achieves both computational efficiency of pseudopotential methods and accuracy of full-potential linearized augmented-plane wave (FLAPW) method15, which is commonly used as the benchmark for DFT calculations on solids. More details of the PAW method and its applications in computational tools like Vienna

Ab-Initio Simulation Package (VASP)16-19 can be found in the excellent review papers by J.

Hafner 20, 21.

Chapter 3 Effects of an Amorphous Silica Substrate on a Single Graphene Layer

3.1 Introduction

The motivation for studying the effects of the substrate on graphene is because graphene is a single layer of carbon and generally needs to be deposited upon a substrate to form application devices. Studies show that the substrate can influence various properties of graphene.

For example, Xu Du22 et al. found that the mobility of carriers in suspended graphene can be as high as 200,000 cm2 V–1 s-1, which is ten times larger than that of graphene supported

23 24 on a silicon wafer. Theoretical studies by Nomura et al. and S. Adam et al. show that the carrier mobility in graphene is primarily determined by scattering from charged impurities on the substrate’s surface.

In addition, a substrate can influence the morphology of the graphene layer on top of it.

STM measurements by M. Ishigami, et al. show that graphene primarily follows the underlying

25 26 morphology of the SiO2 substrate . From the 1TPa Young’s modulus measured by Bunch et al , they estimated that the energy stored due to 1% deformation of graphene is about 1meV/Å2. The high Young’s modulus and large surface area make graphene resonators well suited for applications such as mass or force sensors.

The substrate can also dope the graphene. Currently, there are two commonly used substrates. One is the Silicon Carbide (SiC) substrate. A. Mattausch et.al27 have studied the effects of SiC substrates on graphene layers grown on top of them. They find that the first carbon layer is covalently bonded to the SiC substrate and does not show graphene-type behavior. They also found that the work function of the Si-terminated surface is lower than that of free standing graphene while the C-terminated surface is of similar magnitude. Therefore, the charge will flow 20 from the graphene layer to the Si-terminated surface, giving a metallic behavior to the interface.

Conversely, the interface is semiconducting for the C-terminated surface.

Another popular substrate for graphene devices is a silicon wafer with thermally grown amorphous silica on top. In this chapter, I will focus on studying the chemical doping effects of amorphous silica on graphene under the assumption of ignoring the morphology deformation of the graphene layer. In section 3.2, the descriptions of the experimental work by our collaborators will be discussed in detail. In section 3.3, I will explain the details of the DFT calculations which explain the experimental observations of intrinsic n-doping of graphene. In section 3.4, potential profile models will be used to estimate the density of surface states in amorphous silica, which is used to explain the large magnitudes of the Dirac voltages observed in the experiments of our collaborators.

3.2 Experimental Details

The setup for FET graphene devices is shown in figure 3-1. The graphene layer, created by micromechanical cleavage of highly oriented pyrolytic graphite (HOPG), is transferred onto degenerately doped Si(100) substrates with resistivity of . About 300nm of amorphous SiO2 is thermally grown on top of the Si(100) wafer.

The electric quantities measured in the experiments are source-drain resistance Rds, Dirac

Voltage VDirac and their dependences on the environment such as temperature and surrounding gas molecules. The Rds measurement involves monitoring the source-drain current Ids as a function of gate voltage Vg while a small source-drain voltage Vds =1mV is kept constant. The Dirac Voltage

VDirac is the gate voltage Vg corresponding to the maximum Rds. It is named the Dirac Voltage because sweeping the gate voltage induces a transition between electron and hole conduction in 21 the graphene layer and the transition point corresponds to the Dirac point where the graphene is almost neutral.

The measurement is initially conducted on an FET device exposed to ambient conditions.

The Dirac peak in Rds is found at a significantly positive Vg (i.e., +50 V

30 of graphene with NO2 was studied by T. O. Wehling et.al with ab-initio calculations , which shows that the single open shell NO2 molecule dopes graphene as a strong acceptor. H. E.

Romero et al. also suggested that O2 molecules may be weakly adsorbed to graphene and act as acceptors since carbon nanotube films31 are p-doped by them. 22

The p-doping can be reversed to n-doping by vacuum-annealing the FET device at a temperature of about 200ºC and a pressure of about 5×10-7 Torr. This suggests that the graphene layer is intrinsically n-doped by the amorphous silica substrate when gas molecules are removed after vacuum annealing. The time evolutions of the normalized maximum values of Rds and Dirac

Voltage VDirac are shown in figure 3-2.

The ratio Rds/ R0 drops to a minimum of about 0.65 during the first two hours and then recovers to the equilibrium value of about 0.70 during the next 6 hours. When the temperature of the device is suddenly lowered to 25 ºC, the ratio recovers back to 1 over the next 5 hours. The decrease of the ratio at 200ºC compared to 25 ºC is anticipated since this suggests the decrease in scattering in graphene due to degassing at higher temperature, which is due to the removal of gas molecules from the graphene layer. However, we do not have a clear understanding of the undershoot effect of ratio Rds/ R0 to 0.65 in the earlier stage. This effect is also observed for a similar device degassed in Ar or H2 atmospheres by our collaborators. The change of the Dirac

Voltage during vacuum annealing is shown in figure 3-2(b). Notice the overall trend of dropping of the Dirac Voltage from +80V to the range between -48V and -27V. The large negative Dirac 23

Voltage observed shows the intrinsic n-doping of graphene by the amorphous silica substrate, which will be explained by our DFT calculations and simple potential profile models. The interesting reduction of the Dirac Voltage at 28 hours when decreasing the temperature from

200ºC to 25ºC indicates the reduction of n-doping with temperature, which is not discussed in this work since the explanation goes beyond the scope of our current work.

3.2 Ab-initio Results

The experimental observation of the negative Dirac Voltage of a degassed graphene FET device indicates the intrinsic n-type doping of graphene supported on Si/SiO2. We argue that the n-type doping is due to the lower work function of amorphous SiO2 relative to the graphene layer.

In order to support this argument, we generate amorphous SiO2 structures with classical molecular dynamics to try to represent the realistic atomic configuration of amorphous SiO2. We then use the generated structures as examples of the supporting substrate to study the electronic

properties of graphene on top of the silicon wafer since amorphous SiO2 layer is normally grown on top of the silicon wafer.

Our density functional calculations are carried out in a periodic supercell that contains the graphene sheet and the amorphous SiO2 substrate. The number of atoms in the cell is limited to a number close to 100 in order to be able to generate results in a reasonable time and with a good level of accuracy. We use a rectangular supercell of graphene containing 32 carbon atoms with dimensions 8.52 Å × 9.84 Å. These dimensions reflect the optimized cell for isolated graphene

(i.e., lattice constant a = 2.46 Å). The SiO2 substrate is generated with the procedures described in

32 detail by Leed et al . Briefly, we start by generating a periodic cell of amorphous SiO2 with the experimental density of 2.2 g/cm3 by randomly filling the cell with silicon and oxygen atoms with the stoichiometry 1:2 and avoiding atoms to be too close to each other. The x and y dimensions of 24 the cell are chosen to match the dimensions of the graphene supercell of 32 carbon atoms. The z dimension is adjusted to determine the final thickness of the SiO2 slab containing 78 silicon and oxygen atoms. This procedure provides a cell with the proper dimensions and density but wrong co-ordinations of silicon and oxygen atoms. We correct the co-ordinations by running a classical molecular dynamics simulation to anneal the initial structure of SiO2 at a high temperature (2500

K) for 110 ps with time step of 0.001 ps. The Feuston-Garofalini potential33 is used to describe

34 the atomic interactions in SiO2 as implemented in the General Utility Lattice Program (GULP) .

This potential has been tested before and was found to be able to produce amorphous SiO2 surfaces in good agreement with experimental information, i.e., density of defects at surfaces,

32,35 materials density, and coordination. We then create the surface from the bulk SiO2 cell by removing the PBC in the z direction only. The open surface slab was further annealed with high temperature to equilibrate the amorphous SiO2. The computational run takes a longer time (220 ps) to try to cure the dangling bonds created during the removal of the PBC. Finally, as expected from SiO2 surfaces exposed to ambient conditions, we terminate all remaining dangling bonds of oxygen with hydrogen and silicon dangling bond with hydroxyls. One example of a periodic cell of amorphous SiO2 and a flat graphene sheet in proximity to the substrate surface atoms is shown in figure 3-3. The distance between the substrate and the graphene layer shown is defined as the vertical distance between top of the atom in SiO2 and the graphene layer, which is an underestimation of the average distance that would be typically measured. We relax the atomic positions of the amorphous SiO2 but not those of the graphene sheet. Since we focus on the chemical doping effects, we ignore the corrugation of the graphene for simplicity. This will inevitably introduce quantitative errors in our study. However, it will not change our major results qualitatively since we are mostly interested in studying the chemical doping effects, which are due to the relative differences of work functions between the graphene layer and amorphous SiO2.

The electronic properties of the graphene layer will not be qualitatively changed due to the 25 corrugations since corrugated graphene preserves the electronic properties of the suspended graphene. Therefore, we can ignore the observed corrugations in graphene that follow the rough

25, 36 SiO2 surface since we only consider chemical charge transfer effects here.

Figure 3-3. The typical atomic configuration of the unit cell used in our ab-initio calculations. The left panel corresponds to the situation where the graphene sheet is at the minimum energy distance of about 3.6 Å. The right panel is the situation where the graphene layer is at a larger distance. The contours in the figure show the charge transfer and are colored to signify the magnitude of the electron excess when bringing the graphene and SiO2 substrate together.28 [Copyright (2011) by ACS Nano, Reprinted from the Link : http://dx.doi.org/10.1021/ nn800354m ]

The ab-initio calculations are carried out with the Vienna Ab-Initio Simulation Package

(VASP).16-19 The core electrons are treated with the frozen Projector Augmented Wave method14,

37. The exchange and correlation potential is treated with the GGA using the Perdew-Burke-

Ernzerhof (PBE) functional13, 38. The plane-wave energy cutoff determining the basis set size is set to 282.8eV. The Brillouin zone is sampled with a Monkhorst-Pack39 grid of 4×4×1 for the self-consistent (SCF) calculation and 12×12×1 for density of states calculation. Self-interaction 26 correction schemes which describe with greater accuracy the electron affinity are computationally very demanding and cannot be done with such a large unit cell.

The adsorption energy, which is defined as the energy gained when putting graphene on the substrate, is evaluated as the difference between the energy of the combined graphene/substrate system and the sum of the energies of the isolated parts. In all three SiO2 substrate configurations we considered, we find an average adsorption energy of 1meV/Å2 with an equilibrium distance of about 3.6 Å.

Experimentally, C. Lee et al.40 measured the elastic property of a free standing graphene layer by nano-indentation with an atomic force microscope. They extracted the Young’s modulus of graphene to be about 1TPa within the nonlinear elastic stress-strain framework. M. Ishigami et

25 al. measured the height difference between the graphene layer and SiO2 substrate with AFM and found the average thickness to be 4.2 Å in ultra-high vacuum (UHV). They also find the corrugations of a maximum local strain about 1% and estimate the corresponding energy needed to cause the deformation to be about 1meV/Å2. Since our DFT calculation does not include van der Waals (vdW) interactions, the adsorption energy we calculate is an under-estimate. N. G.

Chopra 41 et al. estimated the interlayer vdW interactions of graphite to be 20 meV/Å2 with the interlayer distance of 3.4 Å. J. F. Dobson et al.42 estimate the asymptotic scaling of the vdW energy between parallel planes to be D-3 where D is the separation distance. Thus, we can estimate the vdW energy at the experimental separation of 4.2 Å to be about 10.6 meV/Å2, which is large enough to make the graphene layer corrugate to follow the morphology of amorphous

SiO2. Further study with vdW interactions included will be needed to explore the adsorption energy of graphene on amorphous SiO2.

The amount of the charge transfer between the graphene sheet and the substrate is quantified as the number of electrons Q transferred from SiO2 to graphene. Figure 3-4a shows the dependence of Q on the distance to the substrate d in all three substrate cases (SiO2_1, SiO2_2, 27 and SiO2_3). We observe strong intrinsic n-doping of the graphene sheets when graphene is close to the surface. In figure 3-4b, we show the calculated profile of the charge density redistribution produced after bringing together the graphene sheet and the SiO2 substrate. At every position in the z direction of the cell (perpendicular to the graphene layer), we integrate the difference of the charge densities between the combined system and the sum of the isolated components in the plane. Positive values indicate that there are more electrons in that region when the sheet and the substrate are combined. In order to determine the total charge transfer to graphene per unit area, we integrate the profile in the region corresponding to the graphene sheet which is marked with the vertical dotted line. From figure 3-4(a), it can be seen that Q does not decay to zero when d is large. This can be understood with the periodic boundary conditions (PBC) in our DFT calculation set-up since the distance d is the distance between the graphene and upper surface of amorphous SiO2. As the distance d increases, the graphene layer is moving away from the upper surface of silica while moving closer to the lower surface of silica substrate.

Figure 3-4. (a) The distance dependence of the amount of charge transferred from the - SiO2 substrate to graphene measured in number of electrons Q transferred (left-axis, in unit of 10 3 13 2 e/carbon-atom) and net induced surface charge density n0 (right-axis, in unit of 10 e/cm ). (b) Excess charge per unit length versus distance along z-axis(perpendicular to the graphene layer). The top panel is the situation with the distance between graphene and SiO2 substrate at equilibrium. The bottom panel is the situation with a large separation distance of about 10.8Å. The dashed lines indicate the formal boundary chosen for the SiO2/graphene interface (left) and 28 the outside boundary of the graphene (right).28[ Copyright (2011) by ACS Nano. Reprinted from the link: http://dx.doi.org/10.1021/nn800354m ]

Another interesting effect we observe in all our calculations is that extra charge density gained by graphene is always accompanied by a charge redistribution that promotes electrons from the bonding sp2 bands to the π bands. This is seen as a negative dip in figure 3-4b at the center of the graphene sheet region. This effect is also found by others in the study of adsorptions of metal adatom on graphene.43 Calculations made in an isolated graphene with extra electrons added to the cell show the same charge redistribution effect. This is possibly due to the change in screening with additional charges in the graphene layer.

The charge transfer from the SiO2 substrate to the graphene layer can be understood by the relative level of the work function of the substrate with respect to the graphene layer. The work function of a substance is the energy necessary to remove an electron from the Fermi level and is estimated as the energy difference between the Fermi level and the electrostatic potential in the vacuum region within density functional theory. In this way, we determined the work function of graphene and that of the three substrates considered in this work. Silica is a semiconductor and perhaps we should refer to the ionization potential or the electron affinities. Nevertheless, it is simplest to discuss this problem in terms of the work function. For the substrates studied here, we find small differences in the SiO2 work function: SiO2-1 (W~3.03 eV), SiO2-2 (W~ 3.36 eV), and

SiO2-3 (W ~3.41 eV). In any case, the variations observed between the three substrates are meant to be representative of the variations expected due to the random nature of the amorphous SiO2 structure. However, all of them have a significantly smaller work function than that of graphene

(W~4.23eV, compared with the result with LDA of 4.6eV by Shan Bin et al44) , which explains the charge transfer to the graphene layer when it is in contact with SiO2. The origin of the smaller work function of substrate SiO2-1 is due to the presence of a particular configuration with a low 29 binding energy surface state. In addition, we observe from figure 3-4 (a) that substrate SiO2-1 has much larger amount of charge transfer than the others. This can be partly due to the relative smaller work function of the SiO2-1 than the other two silica substrates. Furthermore, we suggest that this is also due to the spatial distribution of electrons in the states close to the Fermi level since the electrons closer to the surface are more readily transferred to the graphene layer.

In addition to the charge transfer mentioned above, there is also a charge rearrangement effect (shown in the contours of the left panel of Figure3-3) when the graphene layer is at the equilibrium distance of about 3.6 Å. These are charge puddles, which are produced by the inhomogeneity of the charge density of the ionic SiO2 substrate and was observed experimentally by Martin et al. 45 using a single-electron transistor.

In the next sections, we will set up a simple potential profile model to explain the reason of large magnitude of the Dirac voltage observed experimentally.

3.3 Potential Profile Model Results

Let’s first briefly review the experimental operations of the FET device. The schematic figure of the experimental setup of the graphene FET is shown in figure 3-1(a). The source-drain voltage is applied through the metallic electrodes deposited on graphene, which make metallic contact with graphene and pin the chemical potential of graphene to that of the metal. The back- gate voltage is applied on the highly doped silicon substrate where amorphous SiO2 is grown on top. The back-gate voltage is swept during the experiment to induce a transition between electron and hole conduction in the graphene layer. The critical point where the amount of carriers in the graphene layer is at the minimum (close to zero) is called the Dirac Point. The corresponding back-gate voltage is called the Dirac Voltage. Our collaborators have shown that the Dirac

Voltages of graphene samples under the vacuum condition are negative within the range of -30V 30 to -50V. In order to understand the origin of the Dirac Voltage in this range, we set up a simple model of the FET device under equilibrium. Figure 3-5 describes the schematic potential diagram of the FET device under equilibrium conditions.

Figure 3-5. The electrostatic potential profile diagram for the FET device under equilibrium conditions. WM , WS, WG are the work functions of highly doped Silicon, the amorphous SiO2 surface and the graphene sheet. The changes of the Fermi level of the surface states on the left and right are EL and ER respectively. The change of the Fermi level of graphene is denoted as EG. The distance between graphene and the SiO2 surface is characterized with an average distance dR while we neglect the interface distance between the thermally grown SiO2 substrate and the silicon substrate. The thickness of the amorphous SiO2 is denoted as dS. ΦS and ΦR correspond to the potential drop between two sides of amorphous SiO2 and between amorphous SiO2 and the graphene layer. The cylinders denote the gauss surfaces in which we apply Gauss’s theorem.

Experiments have shown that abundant surface states exist on amorphous SiO2 with densities of up to 1014 eV-1×cm-2 measured by W. H. Brattain 46 et al. Our ab-initio calculations of the DOS of the amorphous SiO2 substrates show the surface states near the conduction band edge as shown in figure 3-6. 31

Figure 3-6. The DOS of SiO2-1, SiO2-2 and SiO2-3 amorphous silica substrates respectively. The solid line denotes the position of Fermi Level.

The surface charge densities of the highly doped silicon substrate, the left & right side of thermally grown amorphous SiO2 and the graphene sheet are denoted as , , and respectively. The quantitative explanation of the magnitude of the negative Dirac voltage can be shown by solving the potential diagram of the FET device in figure 3-5.

The energy conservation requires

(3.1) 32

(3.2)

(3.3) The application of Gauss’s theorem at each surface charge region and the use of charge conservation gives

(3.4)

(3.5)

(3.6)

(3.7)

(3.8) If we assume that the density of surface states is constant, the charge densities at the surface of SiO2 and in the graphene layer are given by

(3.9)

(3.10)

(3.11) Combining all these equations, we obtain an expression that provides the Fermi level in graphene, as a function of device parameters. This expression is written as

(3.12)

As defined above, the Dirac Voltage VD is the gate voltage for which the charge in graphene is zero (i.e., EG =0),

(3.13)

As shown by equation (3.13), if the surface state density of amorphous SiO2 β is zero, the

Dirac Voltage will be the difference of , which is on the order of a few eV. Therefore, 33 the observed 30~50V negative Dirac Voltage will need the surface states density β to be on the order of 1012 eV-1 cm-2. In order to show that the magnitude of the surface states density is comparable to the one we derive from the ab-initio calculation, we set up the schematic potential profile model of the graphene sheet in a periodic slab of amorphous SiO2 as shown in figure 3-7.

Figure 3-7. The electrostatic potential profile for the graphene layer in a periodic slab of amorphous SiO2 under equilibrium.

Conservation of potential energy across the left and right gap requires

(3.14)

(3.15)

The same conservation across the amorphous SiO2 gives

(3.16)

Applying Gauss’s law around the left surface of the amorphous SiO2, the graphene sheet and right surface leads to

(3.17)

(3.18)

(3.19)

The charge neutrality of the cell is shown by

(3.20)

From the equations above, we yield solutions for EL , ER and EG consisting of

(3.21)

(3.22)

(3.23)

Here and are given as

(3.24)

(3.25)

(3.26)

(3.27)

From equation (3.23), we observe that EG depends on the parameters: dL ,dR ,dS ,WG ,WS,

L,R. From the ab-initio calculation, we know the work functions for graphene (WG) and the SiO2 substrate (WS). From the atomic configuration of the SiO2 substrate and graphene, we estimate the value of parameters such as the thickness of SiO2 (dS) and the distance between graphene and the SiO2 surfaces (dL ,dR). For simplicity, we assume the left and right surface of amorphous SiO2 have the same surface states density: L= R= . Therefore, we can make EG depend only on the distance between amorphous SiO2 and graphene (dL ) and the density of surface states of amorphous SiO2 ( ), thus eliminating the parameters dR,dS,WG,and WS. 35

On the other hand, we can yield the charge transfer σG versus distance between graphene and SiO2 dL by integrating the charge density from the DFT calculation. We then extract the surface states density parameter of SiO2 by least-squared fitting the function of σG on dL from the DFT calculation. The surface states density is found to be on the order of 1012~1013 eV−1

−2 cm for three SiO2 substrates which agrees with the order of surface state density needed to produce the observed large Dirac Voltage.

In summary, our collaborators have shown experimentally that a graphene FET on a silicon wafer shows intrinsic n-type doping behavior with a large negative Dirac Voltage (|Vg|

~10-50 V) under vacuum annealing. This behavior was observed in six SiO2 samples used in their experiments. We show that the n-type doping of graphene is due to the charge transfer from the amorphous SiO2 to the graphene layer. These results are achieved with ab initio calculations under the assumptions of ignoring the corrugations of the graphene layer. The relative lower work function of SiO2 than that of the graphene layer is due to the surface states of amorphous SiO2 which pin the Fermi level of the SiO2 just below the conduction band. From a potential profile model of the FET device, we show that the surface states density is on the order of 1012 eV−1 cm−2 to produce the large negative Dirac Voltage. This is found to be on the same order as the results estimated from the periodic potential profile model of the ab-initio atomic structure.

Chapter 4 Physics of a Graphene Channel Embedded in Graphite Monofluoride (CF)

4.1 Introduction of Opening a Band Gap in Graphene

Although graphene is a promising candidate for future field effect transistors, the ability to confine the carriers and open a band gap in graphene is crucial to realize practical applications.

One method is to cut graphene into narrow strips known as graphene nanoribbons (GNRs).

There are two basic edge shapes for GNRs, namely the armchair and zigzag edges as shown in figure 4-1 below.

1 2 3 N (b) (a)

Two experimental approaches have been attempted to produce GNRs. One method consists of physically cutting the graphene ribbons with either E-beam 47, 48 or STM49 lithography.

It is found that the energy gap scales inversely with the ribbon width. The reported GNRs created 37 by E-beam lithography are too wide (15-100nm) and correspondingly exhibit a small band gap

(10~100meV). Although STM lithography can produce ribbons with smaller width (2.5~10nm) and large gaps (0.18~0.5eV), it requires more time and dexterity. Another method is to break graphene into pieces with solution-dispersion and sonication50, 51. This method can produce narrow ribbons (on the order of sub 10nm) with varying widths along their lengths. Although it has the advantage of producing much narrower GNRs than physical cutting, it has the disadvantage of much less control over the width of the produced ribbons and the type of edge terminations.

In addition to opening the band gap of graphene with physical methods, it is also possible to functionalize graphene chemically, which transforms the carbon atoms from sp2 to sp3 hybridization.52 It is predicted that full hydrogenation can change the highly conductive graphene into insulating53 graphane with a direct band gap of 3.7eV. The chair conformation structure of hydrogenated graphene, which named “graphane”, was previously predicted as stable by Sluiter and Kawazoe with cluster expansion method.54

Since the Kohn-Sham equations of DFT are derived from non-interacting single particle pictures, they only give solutions of single particle states, which can describe the ground state properties very well but intrinsically fail to describe the excited states’ properties accurately. The band gap measurement is generally achieved with photoelectron spectroscopy, which involves the excitations of electrons out of the system. The process of removing the electron annihilates an ensemble of bare electron and the oppositely charged Coulomb hole. This ensemble is called

“quasi-particle” since it behaves like a single particle with properties such as having a life-time.

The energy difference between the non-interacting electrons and the quasi-particle is called self energy which can be approximately calculated with the GW method55 where G stands for electron Green function and W stands for screened Coulomb interaction. With the corrections of self energy calculated in the GW approximation, it is found that we can improve the 38 underestimated band gap from the DFT calculation to be closer to the band gap measured experimentally. Recent calculation shows that the bandgap can be tuned from 0~6.4eV under GW approximation when graphene is chemically bonded with Group IA and Group VIIA elements and forms a mixed sp2/sp3 hybridization56.

D. C. Elias et al. 57 initially showed the possibility of reversible hydrogenation of graphene when the graphene sample is exposed to cold hydrogen plasma. The dehydrogenation of hydrogenated graphene can be achieved by annealing the sample. Further in-situ “grow-and- pattern” processes of graphane were carried out by Y. Wang58 et al. by incorporating a plasma beam source with a laser micro-structuring system in a single chamber. The pattern, with tunable width and length, can be achieved by moving the computer-controlled sample stage in a programmable step with respect to the focused laser beam.

An alternative approach to hydrogenation of graphene, fluorination of graphite, has been studied for many years.59-61 Experimental studies by Parry et al. 62 and Mahajan et al. 63 have measured the structure and electronic properties of layers of carbon monofluoride (CF)n and found that the lattice constants of the hexagonal unit cell are about 2.53 Ǻ in the plane and about

64 5.7 Ǻ in the c-axis. They also showed (CF)n to be an insulator. Charlier et al. calculated the electronic structure of (CF)n and found a direct band gap of 3.5 eV at Γ. GW calculations of graphene with fluorine by M. Klintenberg 65 et al. show a larger bandgap of 7.4 eV. Some experimental results have shown possible routes to remove fluorine atoms from carbon monofluoride and reduce it back to a graphene layer. For instance, the reduction of (CF)n and

66-70 71, 72 (C2F)n have been attained by hydrogen gas or a NaOH–KOH solution . Recent studies by

R. R. Nair et al.73 and J. T. Robinson, et al.74 showed that the fluorinated graphene can be readily patterned with atomic fluorine plasma and has better insulating properties than hydrogenated graphene. For example, the resistivity of fully hydrogenated graphene is found to be 104Ω while the fully fluorinated graphene has a resistivity on the order of 1GΩ. 39

4.2 Electronic Properties of the Armchair Channel

Our DFT calculations are carried out with VASP.16-19. The plane-wave energy cutoff is set to 400eV. The Monkhorst-Pack39 k-point sampling in the Brillouin zone is 8×1×1 (for zigzag) and 1×8×1 (for armchair). For density of states calculations, we apply a k-point sampling grid of

64×16×1(for zigzag) and 16×64×1(for armchair). A vacuum of 12 Å is added in the direction normal to the graphene (GR)/carbon monofluoride (CF) super-lattice plane to avoid artificial interactions of the images. For the relaxed configurations in this work, the converged atomic forces were smaller than 0.01 eV/Å.

We have tested the above calculation setup for graphene fluoride, which shows a band gap of 3.1 eV at the Γ point and lattice constant of the hexagonal unit cell at 2.61 Å(larger than

2.46 Å of graphene). The lattice constant is in good agreement with the neutron scattering value of 2.61 Å obtained by Y. Sato60. The lattice constant of the super-lattice is approximated by a linear interpolation between the lattice constant of graphene and CF. Figure 4-2 shows the atomic configurations of armchair and zigzag superlattice structures with M rows of CF(the barrier) and

N rows of graphene (the channel). The lattice constant is calculated as

(4.1)

Here aGF= 2.61 Å and aGR= 2.46 Å. All the structures have been optimized with respect to the atomic positions. 40

Similar to GNRs, the electronic properties of the sandwich structure depend on the boundary geometry between the graphene channel and the CF barrier. The properties of the armchair boundary are shown in this section and the properties of the zigzag boundary will be discussed in the next section.

The dependence of the band gap at the Fermi level as a function of the channel width for different widths of the barrier is shown in figure 4-3. 41

The figure shows three features of the band gaps of armchair boundary channel. First, the band gaps are almost independent of the width of the CF barrier but have strong dependence on the width of the graphene channel. Second, as the channel width N increases, the band gap oscillates in a cycle of period three with two semiconducting and one almost semi-metallic case.

More specifically, if we use Eg(M,N) to represent the band gap obtained from DFT calculations for an armchair superlattice consisting of M CF barrier rows and N graphene channel rows , the band gap can be classified into three groups: N=3p, 3p+1 and 3p+2 for any natural number p. The

Eg(M,N) follows the hierarchy of while 3p+2 group is almost semi-metallic with bandgap of few meV and 3p,3p+1 groups are semiconductor.

Son, Cohen, and Louie75 have theoretically modeled the bandgap hierarchy of armchair graphene ribbon terminated with hydrogen and found a similar hierarchy. Instead of using a uniform hopping integral between nearest neighboring carbon atoms, they introduce different 42 hopping integrals for the edge carbon atoms and achieve the following bandgaps to the first order of hopping integral difference δ:

(4.2)

(4.3)

(4.4)

They find they can fit the tight binding model to their DFT result with the hopping integral parameter and . This denotes that the hopping integral of edge carbon atoms increase by 12%. Although there are some variations dependent on the width of the

CF barrier, we can produce the behavior of band gap scaling of our DFT results with parameters

and using their formulas above.

We further use the above equations to achieve the extrapolation of the band gap to larger distance, which is beyond the scope of DFT calculations. Defining the width of the channel as

, we can find the coefficient of the relation by expanding the above

equations while keeping the leading term and yielding

(4.5)

(4.6)

(4.7) The scaling of the band gap of graphene nanoribbons has been studied previously.

Melinda Y. Han et al.47 has also measured the scaling of the band gap of GNRs experimentally,

they fit their measurements with the inverse relation and found the parameters and . The introduction of the fitting parameter W* to the experimental results is anticipated since it corresponds to the minimum width of GNRs they can 43 achieve experimentally. Veronica Barone et al.76 have studied the scaling with DFT calculations and found the scaling coefficients ranging between 0.3 and 1.5 .

In addition to the above features of band gap dependence on the graphene channel width and CF barrier width, we study the band structure and partial density of states (PDOS) as shown in Figure 4-3 below. From the PDOS of the Barrier region, we can see that the band gap of the CF barrier is roughly 3eV which indicates the CF barriers are a good insulator to confine the carriers in the graphene channel.

From the band structure part of figure 4-4, we see that the semiconducting armchair channel (top) has a direct the band gap of about 0.8eV while the semi-metallic armchair channel

(bottom) has an almost nonexistent(~1.6meV) band gap and one dimensional linear band dispersion along the direction of the channel. From the PDOS plots, we see that the CF barrier part displays the band gap of about 3eV in both situations. For the semiconducting armchair channel, the top of the valence band and bottom of the conduction band are well localized within the center of the barrier gap. For the semi-metallic armchair channel, the linear bands correspond to a constant density of states near the Fermi level since the DOS is proportional to inverse of the gradient of the band structure.

From the above analysis, we see that the CF barrier in the armchair channel structure serves as a good insulating barrier. The electronic structures of the graphene channel are similar to GNRs in terms of the bandgap oscillating with the width of the armchair channel. These are not surprising since the armchair channel and GNRs do not have edge states, all the states close to

Fermi Level are confined in the channel, and the corresponding wavefunctions have very low weight at the edge states. The situation is different for the zigzag channel sandwiched by a CF barrier as we will discuss in the next section. 45

4.3 Electronic Properties of Zig-zag Channel

Unlike the armchair channel which has the alternating semiconducting and semi-metallic channels, the zigzag channel has similar band structures and PDOS which is independent on channel width. A typical band structure and PDOS is shown in figure 4-5.

The general features of the band structures for the localized edge states in zig-zag GNRs have been studied by Fujita and Nakada77,78 with the nearest neighbor tight-binding method using the pz orbitals of the carbon atom. For zig-zag GNRs with N rows of carbon atoms, there will be

2N bands in total from the tight-binding calculations with N bands symmetrically below and above the Fermi level. Two manifolds of N-1 each with one below and another one above the

Fermi level are almost degenerate at the X point at energy approximately equal to the first neighbors hopping intergrals, t, below and above the Fermi level. The two bands closest to the 46

Fermi level are of great interest, since they are always degenerate at the X point and stay almost flat and sit steady at the Fermi level up to an intermediate point 1/3 of the distance from the X point along X-G direction for a wider ribbon. This intermediate point is the folded location of the

K point in the graphene band structure. These two bands disperse beyond the intermediate point, with one going up while another goes down and joins the manifolds at the G point. From the analysis of the charge density distribution, Fujita and Nakada et al. found that the electronic states in these two almost flat bands correspond to the state localized on the zig-zag edge. The flat localized bands introduce a large peak of DOS at the Fermi level which causes spontaneous large magnetic moments to emerge on the edge carbon atoms even with infinitesimally small on-site electron-electron Coulomb repulsion U. The formation of the magnetic moment on the edge atoms of the ribbons can be understood with the heuristic argument from the Stoner’s criteria for itinerant ferromagnetism.79,80 The electrons in graphene are delocalized which can be thought of as an itinerant material. The Stoner’s criteria states that the condition for stabilized ferromagnetism is when where U is the on-site Coulomb interaction and

DOS(EF) is the DOS at the Fermi level. With the increasing value of , the susceptibility increases and eventually diverges, which leads to unstable paramagnetic states and a change of the system into a ferromagnetic state. The zig-zag GNRs have a large value of

which satisfies the Stoner’s criteria with small Coulomb repulsion U. The electronic properties of zig-zag GNRs calculated by Fujita and Nakada 77,78 are shown in figure 4-6. 47

Compared to the zig-zag GNRs, the zig-zag graphene channel sandwiched between CF barriers show distinctly different edge states. Unlike the situation where two states remain pinned to the Fermi level until they separate at about the intermediate point 1/3 of the distance from the

X point, these two states disperse together quadratically up to the folded K-point before separating to join the manifolds. Therefore, the effective mass of the carriers close to the Fermi level in the zig-zag graphene channel sandwiched between CF barriers will have lighter effective mass than that of the zig-zag GNRs or graphene sandwiched between graphane. This will induce a change in mobility81, 82 and on the magnetic properties of the channels77. In order to understand the origin of the dispersive edge bands, we construct a tight binding model to reproduce the band structure and local density of states (LDOS) in next section.

4.4 Tight-binding model of the Dispersive Edge band of a Zig-zag Channel

The schematic of the zig-zag lattice used in our tight-binding model is shown in figure

4-7.

Figure 4-7. Schematic of a zig-zag graphene channel with periodic boundary conditions in the vertical direction and finite width in the horizontal direction. The first index i in notation th ai,1 or bi,1 is used to denote the i repetitive cell in the vertical direction and the second index 1 is used to denote the row number of the zig-zag graphene channel in the horizontal direction.

The tight binding Hamiltonian with the nearest-neighbor approximation is

(4.8)

Here denotes the hopping between atom indexed as (i,1) on sub-lattice

A and as (i,1) on sub-lattice B . The parameters and denote the site energy of the edge and bulk carbon atoms respectively. The parameter t denotes the hopping integral between the nearest-neighbor carbon atoms. 49

In order to diagonalize the above Hamiltonian, we apply a Fourier transformation in the vertical direction of the ribbon, giving

(4.9)

This then transform the Hamiltonian into

(4.10)

In matrix format, the Hamiltonian can be written as

(4.11)

Here the parameter .

The solution to this Hamiltonian is of the form

(4.12)

For every k in the Brillouin zone, we get the matrix as shown in equation (4.13), whose dimension is limited by the width of the graphene channel N(number of zig-zag row): 50

(4.13)

We solve the above matrix equation for a graphene channel of width N and fit the dispersive bands from the DFT calculation with two parameters: nearest neighbor hopping integrals between carbon atoms (t) and site energy differences between the edge carbon and bulk carbon atoms ( ) through the least squared method.

We find that the dispersive bands found in DFT calculations are due to the lowering of the site-energy of the edge carbon atoms relative to the bulk carbon atoms. This is due to the fact that the edge carbon atoms are closer to the fluorine atoms in the CF barrier which can be understood as follows. According to Coulson83 and John Lowe84, the site energy of the atom at site m can be regarded as its electron-attracting ability; higher electron-attracting ability will result in lower site energy. The site energy is also dependent on the environment of the atom on the site m. The fluorine atom in the barrier will attract electrons from the bonded carbon atom in the barrier and make the bonded carbon more electron deficient. Therefore, the edge carbon atom in the channel will be more attractive to the π electrons compared to the carbon atom away from the edge since the screening of the nuclei is less. Thus the edge carbon atom has lower site energy than the carbon atom in the bulk.

In order to test whether the fitting parameters to reproduce the dispersive bands depend on the widths of the zig-zag graphene channel or the carbon-monofluoride barrier, we fit the 51 surface bands of the graphene channel with different widths and summarize the results in the table 4-1 below.

Table 4-1. Fitting parameters of dispersive edge bands for zig-zag superlattices with different widths of the graphene channel and CF barrier. Type t(eV) Δ(eV)

Zig68 2.3 0.5

Zig59 2.3 0.5

Zig610 2.2 0.5

Zig511 2.3 0.5

Zig612 2.2 0.5

Table 4-1 shows that the fitting parameters do not depend on the width of the graphene channel nor the CF barrier. This implies that the lowering of the site-energy of the edge carbon atom is a local effect due to the fluorine atoms in the CF barrier.

Due to the lower site energy of the edge carbon atoms, they will have more electrons than the non-edge carbon atoms in the channel. In order to evaluate the charge differences between the edge carbon and the carbon in the center, we calculate the LDOS from the TB model and compare it with the LDOS from DFT.

The local charge at site m can be calculated as

(4.14)

Here the LDOS at the site m is defined as

(4.15)

In order to derive the LDOS from the TB model, we introduce the delta function δ(x).

The delta function can be defined as the limit of Lorentzian function 52

(4.16)

Using the equality

(4.17)

we can define the delta function as

(4.18)

Combining equation (3.18) with equation (3.15), we are left with

(4.19)

The Green’s function is defined as

(4.20)

Here , with h being small and positive, is a complex energy parameter.

Using the definition of the Green’s function, the LDOS at site m can be written as

(4.21)

Here the matrix element Gmn(z) between state m and state m is defined as

(4.22) The total DOS can then be calculated as

(4.23)

For the zig-zag graphene structure (6, 12), figure 4-8 shows the LDOS of carbon atoms from DFT and the TB model. It can be seen that the TB model can reproduce the LDOS features seen in DFT well. From the LDOS of the TB model, we estimate the local charges of edge carbon ai,1 and bulk carbon atom bi,1 to be 1.17 and 0.95 electrons respectively. The same trend is found when we calculate the local charges by integrating the LDOS from the DFT calculation, where 53 we find the local charges for edge carbon ai,1 and bulk carbon atom bi,1 are 1.04 and 0.93 electrons respectively. Both show about 5% more electrons for the edge carbon atoms than the bulk.

In summary, we perform DFT calculations to yield the electronic structure of zigzag and armchair graphene channels sandwiched between CF barriers. The armchair channels show band gap oscillations dependent on the channel width with the period of three. From the work by Son,

Cohen, and Louie75, we fit the scaling and relative values of band gaps for different widths of the channels and find that the interface is defined by a 9% increase of the hopping integral in carbon atoms at the edge over those in the bulk. We further extrapolate the band gap to larger distances, which are beyond the scope of the DFT calculations. 54

The zig-zag channels show dispersive edge bands, which are independent of the channel width. These results are analyzed with the nearest neighbor tight-binding model of the pz orbitals of the carbon atoms in the channel. We find that the site energies of the edge carbon atoms have about 0.5eV lower energy than those of the carbon atoms in the bulk. The quadratic dispersion band structures indicate a lower effective mass of carriers. This observation suggests a method to control the effective mass of carriers in the channel by modifying the electrostatic potential around the edge. The change in the effective mass of the carriers close to the Fermi level will induce a change on the mobility81 and on the magnetic properties of the channels 82. This can be done by a localized knife shaped gate potential85 or by the selective absorption of ferroelectric polymers such as polyvinylfluoride 86 which are chemically compatible with graphene monofluoride.

Chapter 5

Interaction of Atomic Fluorine with Graphene

5.1 Introduction

Fluorine chemistry is an effective way to functionalize carbon materials like graphite and nanotubes.59, 61 Attempts have been made to fluorinate graphite through a reaction with fluorine gas under high temperature and separate the resulting graphite fluoride to get fluorinated graphene.70 Two difficulties are encountered in this method. First of all, achieving stoichiometrically fluorinated graphite with a carbon to fluoride ratio of 1:1 is difficult, since fluorine atoms do not easily diffuse between graphite layers. Furthermore, it is found to be very difficult to mechanically exfoliate fluorinated graphite into good quality single layers of large crystalline size. Recent experimental attempts74,87 to fluorinate graphene with atomic fluorine plasma provided a better solution. With this method, graphene can be fully fluorinated; producing a sample which is inert and stable up to 400ºC even in air. In order to understand the fluorination process of graphene microscopically, we will study the basic interactions of graphene with isolated atomic fluorine or fluorine atom pairs. The interactions are explored in two steps: we first study the lowest energy configurations of one or two fluorine atoms on graphene. Then we use the nudged elastic band (NEB) method implemented in VASP to study the energy barrier of fluorine atom diffusion on graphene. Within the framework of harmonic transition state theory

(HTST), we then estimate the diffusion coefficient of a fluorine atom on graphene. 56

5.2 Transition state theory and the nudged elastic band method

There are many physical and chemical processes where a system makes transitions between different states by traversing a barrier. The initial and final states ( denoted as state A and state B) are generally the local minima in the overall energy surface defined by a set of N atoms, ). The hopping from state A to state B can be generally classified as quantum tunnel transitions at low temperature and classical Arrhenius-type transitions at high temperature. The qualitative analysis of the temperature criteria where the tunneling is predominant can be simply estimated as follows.87

For the tunneling effect to be strong, the system needs to be of a “quantum” nature where the quantum size of the system λ must be greater than the classical size d (characterized as the width of the energy barrier). On the other hand, for the classical effects to be small, the ratio of

must be small. Therefore, we can assume the condition

(5.1)

Here m is the mass of the particle and E is the barrier height. A stricter consideration introduces additional factor π, so we have the condition

(5.2)

Supposing E is about 1eV and d is about 3 Ǻ, the critical temperatures for proton or electron hopping are about 50K and 1600K respectively. Therefore, under the experimental temperature, the transitions involving the diffusion of fluorine atom on graphene are predominantly classical and Arrhenius-type.

The general transition state theory is multi-dimensional. For heuristic purposes, we study the simple one dimensional transition state theory and show that the hopping rate from one local 57 minimum A to the other local minimum B through the transition state X+ follows the relation

shown in figure 5-1.

ΔEbar

The hopping rate from state A to state B from transition state theory is88 :

(5.3)

The probability of observing the atom at x=x+ when the material is at thermal equilibrium with temperature T is given as

(5.4)

Here the integration is taken over all possible positions around the minimum associated with x=A.

The velocity distribution of atoms at temperature T follows the classical Maxwell-

Boltzmann (MB) distribution. The average thermal speed under the MB distribution is given as equation (5.5):

(5.5)

If we use the harmonic approximation of the energy around state A

, where ω is the angular vibrational frequency in the potential minimum, we can

approximate the probability as

(5.6)

Here the energy barrier is .

When we combine equations (5.3), (5.5) and (5.6), we find the transition rate from the harmonic transition state theory:

(5.7)

The multi-dimensional transition state theory under the harmonic approximation is further generalized by Vineyard89 as:

(5.8)

Here υi is the vibrational frequency associated with the minimum and is the real vibrational frequency associated with the transition state. Note that there is one less real vibrational frequency in the transition state since it is the point in the potential energy surface that is a minimum in all directions except one. Therefore, it has N-1 real frequencies and 1 imaginary frequency.

From equation (5.8), we see that the energy barrier is very important in determining the transition rate from harmonic transition state theory since the rate has an exponential dependence on it. One method to search for the local minimum and transition states is the nudged elastic band

(NEB) method 90-92, implemented in VASP by G. Henkelman et al., which will be discussed briefly below and applied in the following calculations to determine the energy barrier.

The NEB method is an improved method of the “chain-of-states” method developed by L.

R. Pratt 93 and R. Elber94. It is useful to define the common terminology used in the NEB method from the beginning. The geometric configurations of the system are generally called images where the initial configuration (reactant) is numbered image zero and final configuration (product) is numbered image p. The images are connected by springs with force constant K to ensure that they are roughly evenly distributed along the reaction path.

The main idea of the method is to find the minimum energy path (MEP) between two local minima. The MEP has the characteristics that any point on the path has the minimum energy in all directions perpendicular to the path. It is proved that the MEP will pass through at least one

90 th first order saddle point. Since the force on the i atom can be defined as: ), the force on the atoms at the local minimum is zero while the force is non-zero elsewhere. Since any point on the MEP has the minimum energy in all directions perpendicular to the path, the force of any image on the MEP is oriented along the path direction only. 60

The NEB method can be understood in two steps: the first step is the elastic method which is used to find the MEP. The second step is to improve the elastic band by updating the nudged force differently than the elastic band method.

The elastic band method defines an object function in terms of a series of p images denoted as :

(5.9)

th Here is the total energy of the i image configuration. The object function does not include the energy of the initial image(image 0) and final image (image p) since these two images are held fixed at the local minimum energy configurations. K is the spring constant that defines the stiffness of the harmonic springs connecting the adjacent images (that is, the “elastic bands”).

The elastic band method establishes that finding the MEP corresponds to the minimization of the above object function with respect to .

The difficulty of the elastic band method comes from the proper choice of the spring constant K. For example, if the penalty of stretching one or more springs is “too low” which you can think of the connecting spring is too soft, the images tend to slide downward in energy, away from the transition state which corresponds to the highest energy image. This is shown in figure

5-2 below. 61

th In the NEB method, the path direction of the i image is denoted as , which follows the

th direction of the line between adjacent images and . The force on the i image is decomposed into two directions: one is parallel with and the other is perpendicular to .

In the elastic band method, the ith image is updated according to the force on it:

(5.10)

Here is expressed as

)+ (5.11)

Note that we use the whole spring force and its perpendicular component to pull the image away from the MEP. 62

As we discussed above, the image points on the MEP have the property that the force perpendicular to the path direction is zero. Therefore, we are only interested in the components of the force perpendicular to the image path. In the nudged elastic band method, we update the force on the ith image as follows:

(5.12)

Here is expressed as

(5.13) The idea of the NEB method discussed above is illustrated in figure 5-3.

A typical NEB calculation starts with an initial path created by interpolating the image positions between the fixed initial and final image configurations. Linear interpolation is generally a good enough starting point, though an interpolation in internal coordinates is also a common choice if the reaction involves rotational motion.95 Different force-based optimizers such as conjugate gradient (CG) or Steepest Descent 96, 97 methods are used in the NEB method until the magnitude of the force in equation (5.12) drops below a specified convergence criterion.

5.2 Binding Energy and Diffusion of a Single Fluorine Atom

We first study the local minimum configuration of a single fluorine atom on graphene using VASP16-19 with the frozen PAW method for the core electrons14, 37 and the Perdew-Burke-

Ernzerhof (PBE) functional for exchange and correlation.13, 38 The plane-wave energy cutoff is set to 400eV and the k-point sampling is 8×8×2 for the SCF calculations. Convergence is obtained when energy differences between iterations fall below 10-5 eV and all forces fall below 10-2 eV/Å.

After identifying the local minimum, we first set the initial transition path to be the linear interpolation between two local minimum configurations and use the VTST implementation of the NEB method90-92 to find the diffusion energy barrier between the initial and final configurations.

As discussed above, the energy barrier is important in determining the transition rate of the chemical process which is calculated as the energy difference between the initial state and transition state. 64

For the adsorption of a single fluorine atom on graphene, there are three possible adsorption sites at high symmetric points: the Top site (over carbon atom), the Bond site (over a

C-C bond) and the Center site (over the center of the hexagonal carbon ring). Figure 5-4 shows the three high symmetry adsorption sites and the shortest distance between the fluorine atom and the graphene plane.

(a) (b)

Figure 5-4. Visualizations of the three high symmetry adsorption sites for a single fluorine atom on graphene and corresponding equilibrium distances between graphene and the fluorine atom. (a) Top, bond and center positions. (b-d) Side views of the top, bond and center site adsorption of the fluorine atom. The vertical distances between the fluorine and graphene layer are 1.56 Ǻ (C-F 65 bond length) for top site, 2.14 Ǻ(vertical distance to the graphene plane) for the bond site adsorption and 2.37 Ǻ(vertical distance to the graphene plane) for the center site adsorption respectively.

The adsorption energy of single fluorine atom on the graphene layer is calculated as

(5.14)

Here is the total energy of the relaxed graphene/fluorine atom system. and are the total energy of the isolated graphene and fluorine gas molecule respectively. A more negative value indicates stronger binding between the fluorine atom and graphene layer. The details of the adsorption energy and carbon fluorine distance are shown in table 5-1 below.

Table 5-1. Binding energy and Carbon-Fluorine atom distance Position Relative to lowest energy (eV) Adsorption energy(eV) C-F distance (Ǻ) Top 0 -1.78 1.56 Bond 0.27 -1.51 2.25 Center 0.40 -1.38 2.76

From figure 5-4 and table 5-1, we find two results. First, the adsorptions of the fluorine atom on all three sites of the graphene layer have no energy barriers. Also, the top site configuration has the largest adsorption energy of -1.78eV; 0.27eV and 0.4eV larger than that of the bond and center site respectively. For the adsorption of a single hydrogen atom on graphene98,

99, the adsorption energy of the top site is -1.3~-1.40eV while the bond site and center site are about -0.1eV and 0.5eV respectively. Therefore, the initial fluorination of graphene is much easier than hydrogenation, since there are many more adsorption sites that are more energetically favorable. Secondly, the bonding interactions between the fluorine atom and graphene are different for different adsorption sites. It can be seen from figure 5-4 that the graphene layer is puckered with the carbon atom bonded to fluorine sticking about 0.4 Ǻ out of the plane. This indicates that the carbon atom changes from sp2 to sp3 hybridization. However, the bonding 66 between the carbon and fluorine atom is not completely covalent. In carbon monofluoride (CF)n , the C-F bond length about 1.38 Ǻ. The C-F bond in this case is regarded as completely covalent, since the atomic radii of the carbon and fluorine atoms are about 0.7 Ǻ and 0.5 Ǻ respectively.100

From table 5-1, the distance between the carbon and fluorine is about 1.56 Ǻ, which indicates the bonding between carbon and fluorine is not completely covalent, but semi-ionic. NMR spectroscopy can be used to test the nature of the C-F bond. Compared with the top site, the graphene layer is not puckered in the bond and center site situations. The closest distances between the carbon and fluorine atom in the bond and center sites are 2.25 Ǻ and 2.76 Ǻ respectively. This indicates that the carbon atoms are not sp3 hybridized and that the adsorption energy is mainly due to coulomb interactions when the charge transfers from the graphene plane to the fluorine atom due to its high electro-negativity.

From the above calculations, we find the top site is the local minimum. We want to explore the diffusion barrier between these local minimums with the NEB method. The energy barrier and corresponding MEP paths are shown in figure 5-5. We can observe that the lowest energy diffusion path from a top site to a nearby top site is via the bond site with energy barrier about 0.27eV. The energy barrier is about 0.4eV via an alternative path through the center site.

For atomic hydrogen diffusion on graphene98 shown in figure 5-6, the energy barriers are 1.19eV

( diffusion through bond site) and 1.84 eV ( diffusion through center site) respectively. Therefore, we find the diffusion barriers of fluorine on graphene are much smaller than that of hydrogen along same paths.

(a)

(b)

The smaller diffusion barriers of the fluorine atom compared to those of the hydrogen atom are expected. The fluorine atom is semi-ionically bonded to the top site and ionically bonded to the bond and center sites. For the hydrogen atom, the hydrogen atom is covalently bonded to the top site and not bonded to the bond and center site adsorptions. Therefore, the diffusion barrier is much smaller for the fluorine atom than the hydrogen atom, since there is less energy cost for bond breaking during the diffusion process for the fluorine atom than the hydrogen atom.

The lower diffusion barrier indicates that a fluorine atom can diffuse more freely on graphene than a hydrogen atom. As shown in equation (5.8), the transition rate can be calculated from the energy barrier and pre-factors in the front. The pre-factors are the so-called attempt rate, which can be calculated from the vibrational modes of the local minimum and saddle points under the harmonic approximation83. The vineyard formalism is implemented as the DYNMAT 69 code by Henkelman101. The calculated vibrational modes of a fluorine atom of the top, bond and center sites are shown in table 5-2:

Table 5-2. Vibrational modes for a fluorine atom at the local minimum (top site) and saddle points (bond and center site )

-1 position Mode 1 (cm-1) Mode 2(cm-1) Mode 3(cm ) Top 222.432929 223.847406 394.488755 Bond -63.207800 94.831561 177.338884 Center -98.954177 -91.511559 138.138948

Since the vibrational frequency can be thought of as the second derivative of the energy with respect to the displacement, we know that the local minimum (top site) has all positive vibrational frequencies since it is at minimum in all displacement directions. The saddle points

(bond and center sites) have negative vibrational frequencies since it is at maximum in some displacement directions. For example, the bond site has one negative frequency since it has the maximum energy when the displacement direction is along the carbon-carbon bond direction. The center site has two negative frequencies since the position is a minimum in energy only in the vertical displacement direction and maximum in energy in the displacements in the x-y plane.

From the above vibrational frequencies, we find the attempt rate to be about 35 THz

( 1/s). The relation between diffusion coefficient D and the transition rate k is given in equation (5.15):

(5.15)

Here d is the jump length (taken to be about 1.42 Ǻ, the carbon-carbon bond distance).

We estimate that the barrier of 0.3eV implies a diffusion coefficient of nm2/s at

300K. Note the above diffusion coefficient estimation is only for a single fluorine atom on the graphene layer. Therefore, the high diffusion coefficient only exists under the initial fluorination 70 process since it depends on the low diffusion barrier of the single fluorine atom on the graphene layer. Therefore, we would like to know how the diffusion barrier of the fluorine atom is affected when there are other fluorine atoms neighboring it since this is going to happen as more and more fluorine atoms are adsorbed onto the graphene layer. This is also interesting for the situation of graphene channel embedded in CF since the electronic properties of the channel depend on the edge configurations. Therefore, the diffusion energy barrier of the fluorine atom gives us some idea about how stable the boundary is. This is going to be discussed in the section below.

5.3 Binding Energy and Diffusion of Paired Fluorine Atoms

In this section, we will study the diffusion barrier of a fluorine atom when there is another fluorine atom nearby which is the simplest example of the environmental dependence of the diffusion barrier of atomic fluorine.

As we showed before, the study of the diffusion barrier with the NEB method involves identifying the local minimum configurations in the beginning. Due to the computation cost limit of the DFT calculation, we limit the size of the cell and the number of possible pair configurations in the search of the local minimum configurations. The pair configurations I searched and corresponding relative energies are shown in figure 4-6 below.

The first fluorine atom shown in figure 5-6 is denoted as the blue ball while the possible positions of the second fluorine atom is numbered from 1-6. The dashed line is the reference energy of -3.56eV (twice the adsorption energy of single fluorine atom on graphene), which stands for the adsorption energy of two isolated fluorine atoms on the graphene layer when they are separated far from each other. The adsorption energies relative to the reference line gives the information about energetic preference of the pair configurations with respect to the isolated situations. If the energy of the pair configuration is above the reference line, it means that 71 bringing the two fluorine atoms from far away into that configuration will cost additional energy and will be less energetically favorable than those configurations below the reference line.

Figure 5-7. (Top) The different pair configurations of two fluorine atoms on the graphene layer. The first fluorine atom is represented with the blue ball and the position of the second fluorine atom in the pair is numbered from 1 to 6 as shown. (Bottom) The adsorption energies of the pair configurations 1-6. The number in the brackets are notations used in studies by Roman et.al102 on 72 pair configurations of hydrogen atoms. The reference configuration is denoted by the dashed line at -3.56eV which corresponds to adsorption energy of two isolated fluorine atoms.

From figure 5-7 above, we see that site 1,3,5, and 6 pair configurations are below the reference energy line and more energetically preferable than those sites 2 and 4. A similar pattern is found by Roman et al.102 from studying the pair configurations of hydrogen atoms on graphene as shown in figure 5-8.

In both situations, the energetic preference is given to those pairs with adsorption sites in different carbon sub-lattices than those on the same sub-lattice. The DFT calculation also shows that the configurations of two fluorine atoms adsorbed on different sub-lattices have zero magnetic moments while non-zero magnetic moments arise when two fluorine atoms are adsorbed on the same sub-lattice. This agrees with the theoretical suggestion that when there are defects present with an equal probability in both sub-lattices in nanographite materials, the overall correlation of the magnetic moments is expected to be antiferromagnetic.104 The antiferromagnetic ordering is also observed in carbon nanohones105, 106. STM images from the work of L. Horneker107 et.al have also shown that pairs of site1( configuration p1 in Roman et.al102) and site3(configuration p2 in Roman et.al102) are the dominant hydrogen dimers on the

Graphite (0001) Surface.

Based on the pair configurations discussed above, I have used nudged elastic band (NEB) method to explore the energy barriers when one fluorine atom diffuses along the path of site1→site2→site3→site4→site5 while the other fluorine atom is held fixed. The results are shown in figure 5-9 below.

Figure 5-9. The MEP diffusion path and energy barrier for a fluorine atom diffusing along the path: site1→site2→site3→site4→site5 on graphene with another fluorine atom is fixed.

From figure 5-9, we observe the following features. Although the diffusion of the fluorine atom is always along the top site to the top site path, the energy barriers are quite different from each other. The energy barrier for diffusing from site1 to site2 is about 1.09eV, much larger than the barrier from site2 to site3 (0.2eV). A similar pattern is found when diffusing from site3 to site4 (~0.94eV) and from site4 to site5 (~0.11eV). Therefore, the energy barrier of the fluorine atom to diffuse away from the other fluorine atom is dominated by the first transition barrier. Once the fluorine atom overcomes the first barrier of the transition, it will overcome the diffusion barrier of the second transition much easier. Therefore, the sites 1, 3 and 5 are at the local minimums while the sites 2 and 4 are at saddle points. Once the initial configuration of the paired fluorine atoms is in either site 1, 3 or 5, it will cost much more energies (0.7~1eV) for the paired fluorine atoms to be separated through diffusion. 75

L. Horneker107 et al. also study the thermal stability of a hydrogen dimer on the Graphite

(0001) Surface. The temperature programmed desorption (TPD) measurements show that pairs at site3 disappear at 445K (with estimated energy barrier to be a 1.25eV desorption energy) while pairs at site1 are more stable and disappear at 560K (with an estimated energy barrier of 1.58eV).

This was understood with their energy barriers from DFT calculations shown in figure 5-10

From figure 5-10, we can see that states A and B have nearly identical adsorption energies while state I is 1.17eV higher in energy. The energy cost of direct recombination of hydrogen from the state A dimer into molecular hydrogen is about 2.49eV while the barrier for hydrogen dimer to recombine at the state B dimer is about 1.4eV. Thus recombination at state B explains the lower temperature TPD peak. The second TPD peak at a barrier of 1.6eV can only be explained through two step processes: one hydrogen atom in the state A dimer diffuses from site

1 to site 3 through site 2 and forms a state B dimer with an energy barrier of 1.63eV for the first 76 transition. Secondly, the state B dimer recombines into molecular hydrogen. This work shows the diffusion from site1→site2→site3 does occur for the case of atomic hydrogen under high temperature about 560K. Therefore, we anticipate that similar diffusion phenomena can be observed for the fluorine atom.

Recent work by Z.M.Ao 108 et al. further shows that the diffusion barrier of a hydrogen atom on the zigzag and armchair configurations of the graphene/graphane interface is almost 10 times larger than the diffusion of hydrogen on graphene. These results provide further evidence that the diffusion of the fluorine atoms will become much more difficult when the atoms are assembled together, even though the diffusion barrier of single fluorine atom is small.

In summary, we first introduce the harmonic transition state theory (HTST) and the nudge elastic band (NEB) method in this chapter. We then find the local minimum configurations of a single fluorine atom or a fluorine atom dimer on graphene layer. NEB calculations are applied to calculate the diffusion barriers between the local minimum configurations. For the single fluorine atom, it is found that the diffusion barrier is about 0.3eV, much smaller than that of the hydrogen atom (about 1.2eV). This is because the binding interaction between the fluorine atom and the graphene layer is ionic, unlike the covalent bond of the hydrogen atom. However, the situation is different for the fluorine dimer. First, a lower energy configuration is found when two fluorine atoms are occupying different sub-lattices compared to when they are on the same sub-lattice. Second, the diffusion barrier (about 1eV) for one fluorine atom in the dimer is much larger than single fluorine atom (about 0.3eV). The corresponding value for the hydrogen dimer is about 1.6eV. Therefore, it is anticipated that once the adsorbed fluorine atoms are clustered together, it is much harder for them to separate away through diffusion.

Chapter 6 Summary and Future work

The general findings of this dissertation are summarized as follows. In the third chapter, we study the chemical doping effects of amorphous SiO2 on single layer graphene. The experimental studies of our collaborators show that a graphene FET transforms from p-doping under ambient conditions to n-doping after vacuum annealing. Through our theoretical study with density functional theory, we find that the intrinsic n-doping effects are due to the surface states of the amorphous SiO2 which pin the Fermi level of the SiO2 to the conduction band edge and cause SiO2 to have a lower work function than that of the graphene layer. We further set up a simple potential profile model to estimate the surface states density which is needed to explain the observed Dirac voltage of about -50V. However, it should be noted that we neglect the fact that that the graphene layer will corrugate and follow the rough topology of the amorphous SiO2.

More sophisticated calculations can be carried out to include the corrugation effects and vdW interactions.

In the fourth chapter, we study the electronic properties of a graphene channel embedded in carbon monofluoride. The DFT calculations show that the electronic properties of the superlattice structure depend on the edge geometry; the armchair channel shows band gap oscillations and the zig-zag channel shows dispersive edge bands. For the armchair channel, we extrapolate

the scaling coefficient of the width dependence relation of the energy gap to be about 0.17~1.33 . For the zig-zag channel, we find that the dispersive edge bands are due to the lowering of the site energy of edge carbon atoms by about 0.5eV with respect to the non- edged carbon atoms. This suggests that it is possible to control the effective mass of the carriers in the channel by modifying the local electrostatic potential. Experimental work has suggested that the local potential can be applied through a knife shaped gate potential85 or physisorption of a 78 ferroelectric material, or poly(vinylidene fluoride) (PVDF)86. This will induce dispersive bands which will have smaller effective mass and larger carrier mobility as suggested by the work of

Meng-Qiu Long et al.81 In the current work, we only consider the two simplest edge geometries with atomic sharpness. Since it is very likely to have graphene channels along other chiral angles or with rough edges, it would be interesting to explore the electronic properties of the graphene channels under these situations.

In the fifth chapter, we study the local minimum configurations of a single fluorine atom and a fluorine atom dimer on graphene. We find that the lowest energy adsorption site for a single fluorine atom is the top site where the fluorine atom forms the semi-ionic bond with the carbon atom. For the fluorine pair, the lowest energy is achieved when two fluorine atoms are bonded to carbon atoms from different sub-lattices. We further calculate the diffusion barriers of a fluorine atom with and without another fluorine atom nearby using the NEB method. We find that the diffusion barrier is affected by the surrounding environment of the diffusing fluorine atom while the barrier in the fluorine dimer situation is much higher (about ~1eV) than the barrier of the single fluorine atom (about ~0.3eV). Techniques such as Green Function methods109 are useful to further understand the interactions of the adsorbed atom interactions and magnetic properties of the fluorinated graphene layer. In addition, we can perform larger scale simulations of the fluorination process with reactive force tools such asReaxFF110 with the force parameters extracted from ab-initio calculations.This can further approximate the dynamic process of graphene fluorination.

In summary, as a fascinating 2D form of carbon, graphene provides a new platform to explore fascinating applications. We have summarized our contributions to the realization of graphene applications and proposed a few outlooks for further studies in this thesis. Much more work is still needed to pave the way for realistic graphene applications in the future.

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Vita

Ning Shen

Education

2011 Ph.D. Department of Physics, The Pennsylvania State University, University Park, PA 2001 B.S. Department of Physics, Wuhan University

Publications

 "n-Type Behavior of Graphene Supported on Si/SiO2 Substrates." ACS Nano, 2008, 2 (10). pp 2037-2044. H. E. Romero, N. Shen, P. Joshi, H. Rodriguez Gutierrez, S. Tadigadapa, J. O. Sofo and P. C. Eklund  "Reversible fluorination of graphene: Evidence of a two-dimensional wide band-gap semiconductor." Phys. Rev. B 81, 205435 (2010). [Subsequently selected to appear in Virtual Journal of Nanoscale Science & Technology, 21, no. 23, 2010],. S. Cheng, K. Zou, F. Okino, H. Rodriguez Gutierrez, A. Gupta, N. Shen, P. C. Eklund, J. O. Sofo and J. Zhu  "Dispersion of Edge States and Quantum Confinement of Electrons in Graphene Channels Drawn on Graphene Fluoride." PRB(Accepted, in press).2011 N. Shen and J. O. Sofo

Workshops and Conferences

 First-Principles Study of Graphene Channel on Graphite Monofluoride - APS March Meeting. Portland, Oregon, 2010

 The effect of SiO2 surface states on the electronic characteristics of graphene FET devices - APS March Meeting 2009. Pittsburgh,PA, 2009  Ab-initio calculation of bonding, charge redistribution and transfer graphene on amorphous silica - APS March Meeting. New Orleans,LA, 2008  WIEN2K Workshop University Park 2007,2009

Awards and Scholarships

 David Duncan Graduate Fellowship Eberly College of Science Pennsylvania State University 2008,2010  Braddock Fellowship Physics Department Pennsylvania State University 2004-2005  RenMin Scholarship Wuhan University 1997-2001