Carbon nanomaterials as electrical conductors in electrodes

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Carbon nanomaterials as electrical conductors in electrodes

Delan Shukr

Engineering Physics and Electrical Engineering, master's level 2021

Luleå University of Technology

Department of Computer Science, Electrical and Space Engineering

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Abstract

In this project, different molecules have been investigated with the purpose of creating an ohmic contact between metals and carbon nano materials. In particular, we considered simple molecules connecting a graphene layer and a copper-slab.

In order to determine the capability of such systems, the electronic structure was computed using Density Functional Theory (DFT). Structural relaxation was performed in order to find candidates where the metal and the graphene binds chemically with the hypothesis that the hybridization of the states will induce more states at the Fermi level. Six different molecular chains were tested and three of them were found to chemisorb to the graphene sheet and the copper surface simultaneously. The electronic properties for these systems were then further investigated using the density of states (DOS). An overlap density of states (ODOS) was defined in order to evaluate the respective contribution of the graphene, metal and molecule.

From the DOS analysis, we report that these systems did not form ohmic contacts as the result shows too few states close to the Fermi level. The most interesting case was using a hexanol chain which had some partially overlapping states seen from the ODOS of the graphene- molecule and graphene-Cu at the Fermi level. However, these were only small contributions.

Further research is crucial in order to find a more suitable molecular chain between the graphene and the copper for an ohmic contact.

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1.Introduction

Carbon allotropes are most likely to be a big part in next generation electronics. A broad range of research show its spectacular electronic, thermal- and optical properties which makes it a favorable candidate in electronics and mechanics. In order to use graphene and graphene- like structures in electronics, the creation of ohmic contacts is of great importance. It follows that the way graphene binds to metals are very important for this purpose. Moreover, different types of metals bind in different ways to graphene and thereby affecting its conducting

property.

Graphene has been used to create ohmic contacts together with different types of metals before [1-3]. One example of this was shown by Matsumoto et al. [1], where a graphene sheet was placed on a sheet of silicon dioxide substrate, with a metal on top of the graphene.

Another option to create ohmic contacts are by growing graphene directly on a metal substrate. This was done, experimentally, by Byun et al. [2] with results verified by Raman spectroscopy.

A measure of the effectiveness of a contact is the contact resistance. Moon et al. [3], has shown experimentally, metal-graphene ohmic contacts with electronic contact resistance value of 100 Ω𝜇𝑚. This was accomplished on graphene wafers and measured using the transmission line method. The value should be compared to the electrical contact resistance between pure copper which is approximately 10⁵ Ω𝜇𝑚 [4]. In fact. even lower contact resistance values of 23 Ω𝜇𝑚 have been achieved for the application of high frequency field effect transistors. This was found for a structure constructed by growing graphene using chemical vapor deposition in contact with Au by Luca Anzi et al. [5]. What will also determine the value of the contact resistance is the distance between the metal and the graphene, as the distance depends on how the metal and graphene bind to each other.

Depending on the metal used in these types of systems, either chemisorption or physisorption occurs [1].

Producing graphene in sufficient amounts, using a cheap and simple method, has proven to be a challenge [6]. Thus, a more logical approach to investigate such ohmic contacts is by using simulations. It is important to note that solving the many-body Schrödinger equation,

analytically for most cases is impossible. Hence, one needs to reside to numerical methods. A common method for solving the Schrödinger equation for such system is called Density functional theory (DFT), which will be used in this project.

Rather than adding or growing graphene directly on a substrate we investigate the formation of ohmic contacts using graphene and a metal with a molecule as intermediate. This way, we can avoid physisorption that may occur between the graphene sheet and the metal, and bind chemically via the molecules. In this work the metal of choice is Cu and different types of carbon chains are used as intermediaries. The density of states (DOS) will also be considered in this project to analyze the conductive behavior of the graphene/molecule-chain/Cu systems.

The thesis is structured in the following way that we first tackle the necessary theory needed in order to fully grasp the concept of ohmic contacts and how to solve the Schrödinger

equation for many body systems. This is covered in Chapter 3. In Chapter 4 we will cover the method that will be used in order to solve the many-body Schrödinger equation

computationally. The structures that will be simulated will also be described. Chapter 5 will

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provide the results of the simulations in the form structure relaxation and DOS. In Chapter 6 we analyze and discuss the results from the simulations.

2.Purpose

Implementing graphene in current technology would show great improvement in being more energy efficient and it would be more environmentally friendly. However, much research is needed until the implementation can be done. Also, the production of graphene has proven to be a challenge. The purpose of this research is to investigate the electronic leading capabilities of graphene to be used in electrodes. In theory, because of its impressive characteristics, graphene could be used as ohmic contacts in electric circuits. Nowadays, micro-electronic circuits get smaller and smaller. Since graphene only consist of a single layer of carbon atoms, it is very small (also transparent), so it becomes a promising candidate for such

implementations.

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3.Theory

3.1 Graphene

Graphene is the name of a single layer of carbon atoms bonded together in a net like shape.

This material is known for its impressive mechanical, electrical, and thermal properties for applications in different areas of science. The structure is commonly described as a

honeycomb lattice, which can be seen in Figure 1. Each carbon in graphene is bonded to three neighboring carbons, which leaves one free electron. The 2𝑠, 2𝑝_𝑥 and 2𝑝_𝑦 form a hybrid 𝑠𝑝²- orbital which form a hybridized bond with the nearest carbons. The free electron comes from the 2𝑝_𝑧-orbitals, which form so-called 𝜋-bonds and this delocalized electron is what gives graphene its impressive electronic properties [7]. Graphene is a semiconductor without a bandgap such that the valence band and conduction band meet at specific points in

momentum space. These points, where the bandgap and density of states are zero, are known as the Dirac points [7].

Figure 1. Single sheet of graphene. The figure was created using VESTA [8].

3.2. Graphene-metal contact

When a semiconductor is placed together with a conductor, for instance, it will form a metal- semiconductor junction. The metal-semiconductor junction will be either an ohmic contact or what is known as a Schottky junction [9]. As they come in contact their Fermi level will align and a potential barrier will be created. The barrier originates from the so-called depletion layer that is formed when electrons going from high energy states to lower energy states. The barrier height determines whether it is an ohmic contact or a Schottky junction. In the former the barrier height is small leading to low contact resistance and in the latter, it is larger giving it larger resistance and rectifying properties [10]. Metal-graphene contacts, like metal-metal contacts, will generally not form a depletion layer due to the absence of band gap and will thus not form conventional Schottky contacts. They are however very limited by the DOS at the Fermi level [1,9].

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A band diagram model over the graphene-metal contact is shown in Figure 2. The figure shows a metal and graphene when separated (left) and in contact (right) with work functions 𝑞𝜙_𝑀 and 𝑞𝜙_𝐺 respectively [12,13].

Figure 2. An energy band diagram of graphene and metal contact. The figure is taken from [9]. This figure shows what happens when a metal and graphene are in contact with each other and their Fermi energy (𝐸_𝐹) aligns. The left side shows before they get closer while the right side shows when the Fermi energy aligns, and a charge transfer can occur. The graphene band structure is drawn as an “X”, where the bottom “triangle” corresponds to the valence band and the top the conduction band, representing the Dirac cone and how it will approach the Fermi energy, seen in the right side of the figure.

At the charge transfer region in Figure 2, the Fermi energy 𝐸_𝐹 is below the Dirac point which suggests that graphene in this figure is p-type semiconductor [1]. In this case the work

function of the metal is larger than the work function of graphene. This will result in the formation of an ohmic contact [12, 13].

The conductive performance will depend on the difference between the metal and graphene together with the DOS close to the graphene Fermi level. The chemistry of the graphene- metal will interface will also influence since binding via chemisorption will perturb the electronic structure of graphene unlike in physisorption.

In this project we consider the metal Cu which under normal circumstances bind to graphene via physisorption. However, by binding graphene carefully with Cu via a molecule chain, the molecules together with the Cu should start interacting with each other and a charge transfer will occur as the Fermi level shifts and aligns with each other.

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3.3 The many-body Schrödinger equation

A theoretical description of a stationary quantum mechanical system is given by the time- independent Schrödinger equation,

𝐻̂Ψ = 𝐸Ψ, (1)

where the wavefunction is denoted as Ψ, and 𝐸 is the energy of the system and 𝐻̂ is the Hamiltonian. For a many-body system consisting of 𝑛 electrons and 𝑁 nuclei the Hamiltonian is given as,

𝐻̂ = − ℏ²

2𝑚_𝑒∑ 𝛻_𝑖²

𝑖

− ∑ 𝑍_𝐼𝑒²

|𝒓_𝑖 − 𝑹_𝐼|

𝑖,𝐼

+1

2∑ 𝑒²

|𝒓_𝑖 − 𝒓_𝑗|

𝑖≠𝑗

− ∑ ℏ² 2𝑀_𝐼𝛻_𝐼²

𝐼

+1

2∑ 𝑍_𝐼𝑍_𝐽𝑒²

|𝑹_𝐼− 𝑹_𝐽|

𝐼≠𝐽

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where the electrons in the system are denoted with lower-case subscripts, with mass 𝑚_𝑒 and charge −𝑒 (𝑒 is the elementary charge). The upper-case subscripts denote the nuclei of the system, with mass 𝑀_𝐼 and charge 𝑍_𝐼𝑒. The first and fourth term in the equation represents the kinetic energy of the electrons, and the nuclei, respectively. The second term expresses the interaction between the nuclei and electron, the third term expresses the electron-electron interaction, and the fifth term expresses the nuclei-nuclei interaction in the system [17]. The wavefunction will depend on both the spatial coordinates of the electron 𝒓 and the coordinates of the nuclei, 𝑹,

Ψ(𝒓₁, . . . , 𝒓_𝑛, 𝑹₁, . . . , 𝑹_𝑁). (3)

Due to the large mass difference between the nucleus and the electrons orbiting them, less energy is required for the electrons, i.e., the electrons will have a more robust response to energy changes than the nuclei. This creates a case where the problem can be split into two parts. This method of separating the nuclei and electrons into two problems is known as the Born-Oppenheimer approximation. With the Born-Oppenheimer approximation the total wavefunction of the many-body system, can be written as a product,

Ψ(𝒓₁, . . . , 𝒓_𝑛, 𝑹₁, . . . , 𝑹_𝑁) = Ψ_𝑛(𝑹₁, . . . , 𝑹_𝑁) × Ψ_𝑒(𝒓₁, . . . , 𝒓_𝑛). (4)

It is also possible to rewrite the Hamiltonian by considering the electronic problem only, where the electrons are in an external potential by the nuclei,

𝐻̂_𝑒= − ℏ²

2𝑚_𝑒∑ 𝛻_𝑖²

𝑖

− ∑ 𝑍_𝐼𝑒²

|𝒓_𝑖 − 𝑹_𝐼|

𝑖,𝐼

+1

2∑ 𝑒²

|𝒓_𝑖 − 𝒓_𝑗|

𝑁

𝑖≠𝑗

. (2)

For this Hamiltonian, the coordinates of the nuclei are treated as parameters. The parameters that minimize this will thus be the ground state configuration of the system. Starting with the nuclei in fixed positions and solving the electron problem, it is possible to update the

positions of the nuclei and so on in an iterative manner [17, 18].

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3.4 Density functional theory

For large (multi-body) systems, numerical methods are necessary in order to solve the

Schrödinger equation. Arguably the most obvious way to solve the Schrödinger equation is to approximate the total wavefunction in some way. This is what is done in Hartree-Fock and alike methods [19]. These kinds of methods are known as wavefunction methods and are usually suitable for smaller systems since they are computationally expensive. A different approach is to start from the electron density which is the basis of density functional theory (DFT).

3.4.1 Hohenberg-Kohn Theorems

DFT is based on theorems of Hohenberg-Kohn. The first theorem states that the external potential i.e., the potential acting on the electrons due to the nuclei, the second term in

equation (5), is uniquely determined by the ground state density. This means that there is one- to-one mapping between the ground state electron density and the ground state wave function and thus, all properties of the system are determined by the ground state density.

The second theorem states that there exists a universal functional of the density 𝐸[𝑛] and that for any external potential, the exact ground state energy is given as the global minimum and the density that minimize the functional is the ground state density 𝑛₀.

While the Hohenberg-Kohn theorems proofs the existence of such functional, they do not provide any method on how to construct them and no exact functionals are known for more than one-electron system [17]. A solution to this is the Kohn-Sham approach and it is the reason for DFT being one of the most widely used methods for electronic structure calculations.

3.4.2 The Kohn-Sham approach

Kohn and Sham proposed, in the year 1964 [20], that the complicated many-body system could be replaced with a non-interacting system. This approach is known to be a self- consistent method with an interacting density. The non-interacting electrons will have a different wavefunction, but the system will be constructed so that the electron density will be the same as in interacting electron system. In particular, the kinetic energy of the non-

interacting electrons, will be known, so it is not necessary to use approximations for the kinetic energy functionals. The non-interacting electrons will be placed in an external effective potential which will result in the same ground state properties as the system with interacting electrons.

As mentioned above, the kinetic energy in an independent particle system can be expressed explicitly in terms of independent particle wavefunctions, 𝜓_𝑖, as

𝑇_𝑠[𝑛] = −1

2∑ 𝛻_𝑖²𝜓_𝑖

𝑖

, (6)

where atomic Hartree units have been used (ℏ = 𝑒 = 𝑚_𝑒 = 1). With the independent particle kinetic energy 𝑇_𝑠 known, it is possible to rewrite the total energy functional as

𝐸[𝑛] = 𝑇_𝑠[𝑛] + ∫ 𝑉_𝑒𝑥𝑡(𝒓)𝑛(𝒓)𝑑³𝑟 + 𝐸_𝐻[𝑛] + 𝐸_𝐼𝐼+ 𝐸_𝑥𝑐[𝑛] , (7)

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where 𝑉_𝑒𝑥𝑡(𝒓) is the external potential from the nuclei, 𝐸_𝐼𝐼 is the nuclei-nuclei interaction in equation (2), added as a constant and 𝐸_𝐻 it is the so-called Hartree energy,

𝐸_𝐻[𝑛] =𝑒²

2 ∫ ∫𝑛(𝒓)𝑛(𝒓^′)

|𝒓 − 𝒓^′| 𝑑³𝑟𝑑³𝑟^′. (8)

In equation (7), all many-body effects of exchange and correlation between electrons i.e., everything unknown that contributes to the energy are collected into the exchange-correlation functional 𝐸_𝑥𝑐 [17]. While the largest contribution of the kinetic energy is found in equation (6), the rest of the energy is found in 𝐸_𝑥𝑐.

The solution to the Kohn-Sham system for the ground state is given by minimizing equation (8) with respect to the density with the constraint that the orbitals are orthogonal. This leads to a set of Schrödinger-like equations:

𝐻̂_𝐾𝑆𝜓_𝑖 = 𝜀_𝑖𝜓_𝑖, (9)

where 𝜀_𝑖 are the eigenvalues and 𝐻̂_𝐾𝑆 is the effective Kohn-Sham Hamiltonian. The Kohn- Sham Hamiltonian 𝐻̂_𝐾𝑆 can be written as,

𝐻̂_𝐾𝑆 = −1

2𝛻²+ 𝑉_𝐾𝑆(𝒓), (10)

with the effective potential is given by:

𝑉_𝐾𝑆(𝒓) = 𝑉_𝑒𝑥𝑡(𝒓) + 𝑉_𝐻(𝒓) + 𝑉_𝑥𝑐(𝒓). (11) In equation (11), the external potential is 𝑉_𝑒𝑥𝑡(𝒓), the Hartree potential is 𝑉_𝐻(𝒓) and the exchange-correlation potential is 𝑉_𝑥𝑐(𝒓) are given by the functional derivative of their respective functionals,

𝑉_𝐻(𝒓) = 𝛿𝐸_𝐻

𝛿𝑛(𝒓), (12)

and,

𝑉_𝑥𝑐(𝒓) = 𝛿𝐸_𝑥𝑐

𝛿𝑛(𝒓). (13)

The equations (9)-(11) are the so-called Kohn Sham equations with the total energy given by equation (7). The potential must however be found self-consistently [17]. A flow chart of the Kohn-Sham computational process is shown in Figure 3.

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Figure 3. A flow chart for the self-consistent cycle for solving the electronic structure problem in DFT [21].

The computation starts with an initial guess of the electron density 𝑛(𝒓). From the initial guess, the effective potential can be computed. By using the calculated effective potential, the Kohn-Sham equations are solved. The solution of the Kohn-Sham orbitals is then used to calculate the density, the total energy, which is then used to update the effective potential.

Note that this is an iterative computing method. A convergence criterion is used to check for convergence, usually it is done by the following criteria:

|𝐸^(𝑖)− 𝐸^(𝑖−1)| < 𝜂, (14)

where 𝐸^(𝑖) is the current energy and 𝐸^(𝑖−1) is the previous energy and 𝜂 is a tolerance that needs to be specified. If the convergence criterium is achieved, the iterative process breaks.

When the system is converged more information about the system is calculated, such as forces, stresses, eigenvalues etc. [21].

3.4.4 Exchange-Correlation Functional

In order to solve the Kohn-Sham equations, the exchange-correlation functional needs to be identified. Assuming that the exchange-correlation functional can be approximated as local or nearly local, it can be expressed as:

𝐸_𝑥𝑐[𝑛] = ∫ 𝑑³𝑟𝜖_𝑥𝑐𝑛(𝒓) (15)

where 𝜖_𝑥𝑐 is the exchange-correlation energy per electron, at a specific point in space. To a first approximation, this can be approximated by the exchange-correlation energy of the

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uniform gas where the exchange correlation energy at each point is assumed to be the same as in a homogeneous electron gas, resulting in a system with a local electron density [17]. This is also known as the Local density approximation (LDA) and it is likely be a good

approximation if the changes in the density of the system are not too large. This will only give an approximative solution to the Schrödinger equation, because the exchange-correlation is only approximated.

In order to go beyond the local approximation, it is logical to also consider the magnitude of the density gradient. This is commonly known as the Generalized gradient approximation (GGA) and has shown large improvement over LDA for many cases. However, this depends on the type of systems and some exchange-correlation functionals can work better for certain type of systems [22]. GGA is the type of exchange-correlation functional used in this work.

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3.5 Density of states

The density of states (DOS) can be described as the number of states that can be occupied at specific energy levels, per unit volume. This is a useful tool for analyzing conductive

properties of materials [23]. What is often of interest is to investigate the number of states at the Fermi energy. The Fermi energy, denoted as 𝐸_𝐹, can be described as the energy of the highest filled energy level of a system in its ground state [11]. Insulators, unlike transition metals have completely filled orbitals, which makes it difficult for the electrons to move around when affected by an external electric field. For conductors, the Fermi level is found at the conduction band, while for semiconductors, it is found between the valence band and the conduction band [23].

For the case of independent-particle states, where |𝜓_𝑛⟩ are the Kohn-Sham eigenstates the DOS is defined as:

𝜌(𝐸) = ∑〈𝜓_𝑛|𝜓_𝑛〉𝛿(𝜀_𝑛− 𝐸)

𝑛

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where 𝜀_𝑛 is the energy eigenvalue of the eigenstate |𝜓_𝑛⟩. Equation (16) is thus the number of independent-particle states per unit energy. By using the complete orthonormal basis of the spherical harmonics |𝑌_𝑙𝑚^𝑎⟩, and using the property:

∑ |𝑌_𝑙𝑚^𝑎 ⟩⟨𝑌_𝑙𝑚^𝑎 | = 1 (17)

it is possible to rewrite equation (16) in the following way:

𝜌^𝑎(𝐸) = ∑⟨𝜓_𝑛|𝑌_𝑙𝑚^𝑎 ⟩

𝑛

⟨𝑌_𝑙𝑚^𝑎 |𝜓_𝑛⟩𝛿(𝜀_𝑛− 𝐸) (18)

The eigenstates are projected onto the functions 𝑌_𝑙𝑚^𝑎 which are non-zero inside the Wigner- Seitz radius of atom 𝑎, and zero outside. This gives the 𝑙𝑚-resolved DOS projected on site 𝑎.

By summing the contributions from different atoms (and orbital) one can get the structure resolved DOS [17].

Having the structure resolved DOS for two different structures 𝛼 and 𝛽, we define the overlap density of states (ODOS) as the minimum of their densities at a specific energy:

𝜌^𝛼𝛽(𝐸) = 𝑚𝑖𝑛(𝜌^𝛼(𝐸), 𝜌^𝛽(𝐸)) (19)

For this project, in order to investigate if the chosen molecular structures are suitable for ohmic contacts, the DOS will be computed as this will show the number of states at certain energy levels which will give a valid prediction of if the systems chosen will behave as a conductor or not.

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4.Method

4.1 Slab models in DFT calculations on surfaces

This investigation will cover DFT calculations of a metal slab together with a graphene sheet.

In order to describe an extended system, periodic boundary conditions are used, and the states are given as Bloch states. The structure of the system is divided in portions with empty space, called vacuum space, between each layer in order to reduce the influence of periodic images in the normal direction.

The size of the vacuum space plays a vital role in the performed computations of slab models.

In theory, a larger the vacuum space contributes to the electron density that will go to zero.

This entails that the chosen k-point, in the normal direction between each layer will play a vital role in the accuracy of the computation [22]. For the computation to converge properly, a dense k-point mesh is normally required i.e., dense sampling of the Brillouin zone. However, by increasing the number of k-points in the computation will result in an increase in

computation time. But, since the density of points in the reciprocal lattice is inversely

proportional to the density of the real lattice, for large enough systems, often only the Γ-point is needed [24, 25].

4.2 Computational details

The computations were made on the High-Performance Computing Center North, on

Kebnekaise in Umeå. The structural and electronic optimization is performed for each system using density functional theory. The valence electrons are treated using pseudopotentials in the projector-augmented wave method (PAW) as implemented in the Vienna ab-initio package (VASP) [26].

The systems that have been investigated consists of between 185-192 atoms. Due to the size of the systems, and in order to reduce the influence of periodic images in the normal direction, only the Γ-point was used to sample the Brillouin zone. The exchange-correlation functional was treated in the generalized gradient approximation (GGA) in the parameterization by Perdew, Burke and Ernzerhof (PBE) [25]. The convergency condition between consecutive electronic steps in the self-consistent loop was set to 10⁻⁶ eV/atom and the cut-off energy for the plane-wave basis was set to 520 eV. For the structural relaxation, the conjugate-gradient algorithm is used to relax the ions to their instantaneous ground state. The convergence criterion for the relaxation is set to 0.02 eV/Å. The cell shape and the positions are relaxed while keeping the cell volume constant.

In order to describe the van der Waals interactions, as when physisorption occurs between the molecules, resulting from the dynamical correlations between fluctuating charge distributions, a correction to the conventional Kohn-Sham DFT energy was added using DTF-D3 method by Grimme [26]. The Methfessel-Paxton scheme [24] was used to treat the partial

occupancies for each orbital with a smearing of width 𝜎 = 0.05 eV. For the DOS

calculations, Gaussian smearing was used in order to avoid negative partial occupancies as they introduce problems when comparing densities for later analysis (see equation (25)). For the DOS calculations, a large number of grid points (10⁵) was used within an interval from

−5 to 5 eV. In this interval it is possible to study the states close to the Fermi level in more detail and the dense set of grid points makes it possible to resolve all peaks properly.

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The analysis of the DOS and overlap DOS (ODOS) were made with a custom Python script utilizing the Pymatgen library [27]. Using the site projected DOS, the DOS for the

components i.e., the molecule, metal surface and graphene layer could be determined by summing their contribution as described in the previous section. The figures were made using the Matplotlib library [28]. Visualization of the structures was made using VESTA [8].

4.3 Binding positions on graphene and metal

When investigating how a molecule bind to graphene and Cu, we only consider the (111) surface of Cu. The Cu surface has four different adsorption sites and the graphene sheet has three (see Figure 4).

Figure 4. a) Adsorption sites for the Cu surface. The light blue colored atoms represent the first layer, the green color represents the second layer atoms and the dark blue represents the third layer atoms. The different binding sites are shown as: black=bridge, orange = top, yellow = center-fcc and red = center-hcp. b) The adsorption sites of the graphene sheet. In Figure 4a, the light blue colored atoms are the top atomic layer, the green atoms represent the second layer, and the rest of the atoms in the slab are dark blue. In Figure 4b, the C-atoms are shown in brown. The adsorption sites are shown in different colors with the black sites corresponding to bridge- position and orange corresponding to top-position. The yellow and red sites represents the center-positions. In Figure 4a, they represent the fcc and hcp-positions respectively. The fcc binding site is located above a third layer atom and the hcp-site is located above a second layer atom [29]. For the graphene layer in Figure 4b however, there is only one center- position and it is shown in yellow.

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4.4 Binding molecules

For all structures investigated, the chosen atom to bond with the metal is oxygen. The atom that bonds with graphene is either oxygen or nitrogen. Figure 5 shows the structure of the different types of chains used. The red atom represents oxygen, and the grey atom represents nitrogen. Carbon atoms are brown, and the white atoms are hydrogen atoms.

Figure 5. Two alkane chains, two benzene ring chains, a pentylamine chain and a hexanol chain are displayed. These were used to connect the graphene and the metal surface together. Red atom =oxygen, brown = carbon, white = hydrogen, grey = nitrogen.

Each of the chains in Figure 5 are tested in different types of configurations, as presented in Table 1. Following the definitions in Figure 4, we denote a molecule binding to Cu on a

“bridge” site and to the graphene layer on a “bridge” site by “Cu(b)-C(b)”. The same principle goes for all types of chains used in Figure 5. Figures of all systems tested are presented in Appendix A.

Table 1. Bonding configurations between the metal surface (Cu) and the graphene surface (C). The naming scheme for the binding sites used are: letters “b” stands for bridge, “t” stands for top and “c”

means center. For the center binding site on the Cu, it is a fcc binding site.

Type of chain/configuration

Alkane (N-O) Alkane (O-O) Benzene (N-O) Benzene (O-O) Pentylamine

(N-O) Hexanol (O-O) Cu(b)-C(b) Cu(c-fcc)-C(b) Cu(b)-C(b) Cu(b)-C(b) Cu(c-fcc)-C(t) Cu(c-fcc)-C(t) Cu(b)-C(c) Cu(c-fcc)-C(c) Cu(c-fcc)-C(t) Cu(c-fcc)-C(t)

Cu(b)-C(t) Cu(c-fcc)-C(t) Cu(t)-C(c) Cu(t)-C(c) Cu(t)-C(t) Cu(t)-C(t)

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5. Results

This section is divided into two sub-sections. In the first section, the result of the structural relaxation is presented. The DOS calculations of the systems that were found to chemically bind, simultaneously, to the graphene sheet and the Cu surface are presented in the second section.

5.1 Structure relaxation

The only systems that were found to bind simultaneously to graphene and the Cu surface were systems with the hexanol chain, the pentylamine chain and the oxygen-nitrogen terminated benzene chain. These are presented in Figure 6. For the three structures, the molecules bind successfully to graphene only at the “top” position. For the metal part, the oxygen of the molecule chains binds to the Cu surface on the “center-fcc” position (see Figure 4a). The systems where the simple alkane chains and the oxygen-oxygen benzene ring chains were used, were not found to be able to bind either to the Cu surface or to the graphene sheet.

However, for the interested reader, all tested structures are presented in Appendix A.

Figure 6. The following figure shows the structure of the systems that that were found to bind simultaneously to graphene and the Cu-surface. In a) the system with the oxygen-nitrogen benzene chain is shown, b) the system with hexanol chain, c) system with pentylamine chain.

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5.2 Density of states

In Figures 7-11, the different DOS have been plotted for the system with the hexanol chain, the pentylamine chain and the oxygen-nitrogen terminated benzene chain i.e., the systems which were found to bind simultaneously to both graphene and the Cu-surface. The figures also show a Cu-graphene system as a reference in order to evaluate the effect of the molecule chains.

Figure 7 shows the structure projected DOS for the four systems. When the carbon chain molecule is used as the intermediary, the desired outcome is to see an increase in the number of states close to the Fermi level compared to the reference system. These figures are shown as part of a primary analysis and to get a general idea of the distribution of the electron states, before investigating in more detail. The blue lines represent the DOS for the Cu-slab, the orange lines represent the DOS for the graphene sheet, and the green lines for the molecule.

Figure 7d shows the structure resolved DOS for the reference system and Figures 7a-c, for the Hexanol, ON-terminated Benzene and Pentylamine systems, respectively. The distribution is clearly dominated by Cu for all the systems which makes it difficult to draw any conclusions about the graphene and molecular DOS. It is, however, possible to see a decrease of density between −2 and −1eV for systems with a molecule.

Figure 7. The following figures a)-c), shows the structure resolved DOS for the systems with the different molecule chains. Figure d) shows the structure resolved DOS for the reference system. A dark vertical line is drawn across each figure in order to show the Fermi level. The color in the figure represents: Blue=Cu, green line =molecule chain, orange = graphene.

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In order to see the contributions of graphene and the molecules more clearly, the normalized structure resolved DOS has been plotted in Figures 8 and 9 for the relevant systems. They are normalized with respect to the number of atoms, such that they show the DOS per atom. The contribution of graphene and the molecule to the total DOS is seen in Figure 8. In 8a, it can be seen that there is a small graphene contribution at the Fermi level when the hexanol chain is used. This also occurs for the system with the pentylamine chain in 8c. It is also possible to see a small contribution from the molecule chains in the normalized data. However, it is clear that the contribution from the molecules are smaller than for graphene close to the Fermi level.

Figure 8. a-c) shows the normalized structure resolved DOS for the binding systems, d) shows normalized structure resolved DOS for the reference system. A dark vertical line is drawn across each figure to show the Fermi level. The colors in the figure represents: blue=Cu, green =molecule chain and orange = graphene.

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In Figure 9, a Gaussian filter has been applied in order to mimic the smearing that is lacking from only using the Γ-point and using a dense grid. In Figure 9a-c, it can clearly be seen that the number of states from the molecule is still small around the Fermi level. Comparing the results for graphene with a molecule in 9a-c and without in 9d, it is possible to see additional states appearing around −1.2 𝑒𝑉 from the graphene. As mentioned previously it is possible to observe a small number of states from the graphene at the Fermi level in Figure 9a and 9c.

Figure 9. The figure shows the normalized structure resolved DOS with an applied Gaussian filter for a)-c) the binding systems and d) the reference system. The dark vertical line in each figure shows the Fermi level. The colors in the figure represents: blue = Cu, orange = graphene, green = molecule.

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Figure 10 shows the ODOS (defined in equation (19)) i.e., the joint contribution to the density of states between Cu, graphene, and the molecule chains that were used. It is seen that the ODOS is quite low in general for all the systems. The reason for this is of course the low contribution from graphene and the molecule to the total DOS.

The Cu-molecule ODOS (green line) can be observed clearly for all systems with molecule chains, so there is some small Cu-molecule overlap close to the Fermi level. It is also seen that the graphene-molecule ODOS (red line) is generally weaker than the of the Cu-molecule ODOS. However, systems with the hexanol (a) and the pentylamine chain (c), the shows a small contribution close to the Fermi level compared to the benzene system (b).

It is difficult to notice the Cu-graphene ODOS (blue line) in 10a-c but there is actually a small contribution. This is seen in Figure 11 where a Gaussian filter (as for the structure resolved DOS in Figure 9) has been applied.

Figure 10. The following figure shows the overlap DOS (ODOS) for the binding systems in a)-c) and the reference system in d). The black vertical line represents the Fermi level. The colors represent: blue = graphene-Cu, orange = graphene- molecule and green = Cu-molecule.

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The blue line, representing the graphene-Cu ODOS can be seen in Figure 11a and 11c.

However, the value is still very small and has reduced, compared to the reference system in 11d. The graphene-molecule ODOS (orange line) is also more clearly observed here. Again, it is seen that the hexanol chain system has the largest graphene-molecule ODOS close to the Fermi level. As was also found in Figure 10, a small Cu-molecule ODOS (green line) can be seen for all three systems at the Fermi level.

Figure 11. The following figure shows the ODOS with a smearing filter for the binding systems in. a)-c) and the reference system in d). The black vertical line represents the Fermi level. The colors in the figure represents: blue = graphene-Cu, orange = graphene-molecule, green = Cu-molecule.

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6.Discussion and conclusion

Using simulations, it is possible to screen for systems which would otherwise be extremely time consuming and expensive to do experimentally. It also makes it easy to evaluate many different types of properties. With that in mind we systematically investigate the conductive properties of metal-graphene systems using the density of states and the overlap density of states. In particular, we investigate the possibility of creating ohmic contacts using ligand molecules with different functional groups connecting the graphene sheet and the metal (Cu in this case).

In the screening of potential candidates (molecules) we require the molecules to bind

chemically to both the Cu-surface and graphene sheet. In order to find the most stable binding sites, a number of different binding positions and angles were evaluated. For the molecules tested, the “center-fcc” position (shown in yellow in Figure 4a), was proven to be the most stable position for the oxygen to bind to the Cu-surface. For the graphene sheet, the “top”

position (shown in Figure 4b in orange) was most stable for both the oxygen and the nitrogen.

Creating a system to behave as an ohmic contact using graphene and a metal is a difficult task. The structure resolved DOS for the studied systems, did not show an increase in the number of states at the Fermi level when the molecular chains were included. Instead,

comparing them with the Cu-graphene system (the reference system), the molecular chains in this case show that the amount of states actually reduced. From the investigation of the ODOS, the system with the hexanol chain showed better performance than the other systems with some small overlap around the Fermi level. The molecule systems showed overall weaker Cu-graphene overlap compared to the reference system, which is important to note.

As conclusion, these systems will not likely qualify as ohmic contacts. However, the system with the hexanol chain shows more potential than the other molecular chains that were used.

In order to use these kinds of intermediary chains further research is needed. It would be preferred to have more states close to the Fermi level as it would be more similar to a

conducting metal in that sense. Also, by using graphite instead of graphene, suggested by [1], is an option.

7.Future work

In order to create the type of ohmic contacts considered in this project, it will be crucial to bind graphene together with a molecule chain with proper charge transferring properties. An increase of DOS around the Fermi level for graphene and the molecular chain is needed.

Since the organic alcohol chain hexanol performed better than the pentylamine and the benzene chain, other organic alcohols could also be screened. Other candidates can be carboxylic acids or esters. We want to see a larger contribution from the ODOS of the systems. The ODOS of the graphene-molecule were weak for the systems, compared to the ODOS of the Cu-molecule, and the ODOS of the Cu-graphene was very low. Different types of metals should also be investigated. Metals such as Ni, Ag or Au are obviously good candidates as they have been used for this purpose before [1] but also other metals should be considered.

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Acknowledgements

The computations were enabled by resources provided by the Swedish National Infrastructure for Computing (SNIC) at High Performance Computing Center North (HPC2N) in Umeå, partially funded by the Swedish Research Council through grant agreement no. 2018-05973. I would like to thank my family for always supporting me. I would also like to thank my examiner Andreas Larsson and my supervisor Gustav Johansson, especially Gustav for all the time he put into this project and all the help that I got and everything he taught me, thank you so much!

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Appendix A

Figures A1 – A4 shows the systems that did not converge in the VASP simulation. The setup of these structures are similar to each other but with different binding sites used to bind to the graphene sheet and the Cu surface. The first simulations on VASP were started with the systems shown in Figure A1 and A2 before the systems with the benzene rings where used, shown in Figures A3 and A4. Note also that the Figures A1 - A4 only shows how the system was intended to be, before they were run in the simulation, as the structural integrity of these systems did not hold throughout the simulation. The binding sites are noted below for each structure, for a clear description of these binding sites, see Table 1.

Figure A1. Systems with nitrogen-oxygen alkane chains. The grey atoms are nitrogen, the blue atoms are copper, the brown atoms are carbon and the white atoms are hydrogen.

Figure A2. Systems with oxygen-oxygen alkane chains. The grey atoms are nitrogen, the blue atoms are copper, the brown atoms are carbon and the white atoms are hydrogen.

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Figure A3. Systems with nitrogen-oxygen benzene chains. The grey atoms are nitrogen, the blue atoms are copper, the brown atoms are carbon and the white atoms are hydrogen.

Figure A4. Systems with oxygen-oxygen benzene chains. The grey atoms are nitrogen, the blue atoms are copper, the brown atoms are carbon and the white atoms are hydrogen.

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Appendix B

Figure B1. shows the normalized density of states for the systems without Gaussian smearing filter. This was also plotted from the custom Python script. The peak number of states are clearer to see compared to when the smearing filter is used. The blue lines represent the number of states for the Cu, the orange line is showing the states for graphene sheet, and the green line represents the number of states for the molecule chains that were used. Figure d) shows the normalized DOS for the reference system in order to see the effects of a molecular chain being used.

Figure B1. The following figure shows the normalized DOS for the systems. The Fermi energy is marked by a black vertical line. The colors in the figure represents: Blue line = copper, orange line = graphene, green line = molecule.

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