Density Functional Theory Calculations of Graphene based Humidity and Carbon Dioxide Sensors

Density Functional Theory Calculations of Graphene based Humidity and Carbon

Dioxide Sensors

KARIM ELGAMMAL

Licentiate Thesis in Physics

ii

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TRITA-ICT 2016:02 ISBN 978-91-7595-817-0

KTH School of Information and Communication Technology SE-164 40 KISTA Sweden Akademisk avhandling som med tillstånd av Kungl Tekniska högskolan framlägges till offentlig granskning för avläggande av teknologie licentiatex- amen i fysik fredagen den 19 februari 2016 klockan 10:00 i Sal A, Electrum, Kungl Tekniska Högskolan, Isafjordsgatan 22, Kista.

© Karim Elgammal, February 2016 All rights reserved

Tryck: Universitetsservice US-AB, Stockholm 2016

iii

Abstract

Graphene has many interesting physical properties which makes it useful for plenty of applications. In this work we investigate the possibility of using graphene as a carbon dioxide and humidity sensor.

Carbon dioxide and water adsorbates are modeled on top of the surface of a graphene sheet, which themselves lie on one of two types of silica substrates or sapphire substrate. We evaluate the changes in the elec- tronic and structural properties of the graphene sheet in the presence of the described adsorbates as well as the accompanying substrate.

We perform the study using ab-initio calculations based on density functional theory (DFT), that allows fast, accurate and efficient inves- tigations. In particular, we focus our attention on investigating the effects of defects in the substrate and how it influences the properties of the graphene sheet. The defects of the substrate contribute with impurity bands leading to doping effects on the graphene sheet, which in turn together with the presence of the adsorbates result in changes of the electronic charge distribution in the system. We provide charge density difference plots to visualize these changes and also determine the relaxed minimum distances of the adsorbates from the graphene sheet together with the respective minimum energy configurations. We also include the density of states, Löwdin charges and work functions

v

Sammanfattning

Grafen har många intressanta fysikaliska egenskaper, vilket gör det användbart för många tillämpningar. I detta arbete har vi teo- retiskt undersökt möjligheten att använda grafen som gassensor för koldioxid och fukt. Adsorberade koldioxid- och vattenmolekyler mo- delleras ovanför ytan av ett lager grafen, som i sig ligger ovanpå en av två typer av kiseldioxidsubstrat eller ett aluminiumoxidsubstrat. Vi har utvärderat förändringar i de elektroniska och strukturella egen- skaperna hos grafenlagret i närvaro av de beskrivna molekylerna samt åtföljande substrat. Vi utför studien med ab-initio beräkningar base- rade på täthetsfunktionalteori (DFT), som möjliggör snabba, korrekta och effektiva elektronstruktursberäkningar. Framför allt fokuserar vi på effekten av defekter i underlaget, och hur dessa påverkar egenska- perna hos grafenlagret. Defekter i underlaget bidrar genom att införa elektroniska band som leder till dopningseffekter i grafenlagret, vil- ket i sin tur tillsammans med närvaron av adsorbatmolekylerna leder till förändringar av den elektroniska laddningsfördelningen i systemet.

Vi tillhandahåller s.k. laddningsdensitet-skillnadsfigurer som visualise- rar dessa förändringar. Vi har även beräknat jämviktsavståndet mel- lan adsorbatmolekylerna och grafenlagret tillsammans med respektive minimienergikonfigurationer för molekylerna, Vi åksa tillhandahåller täthet av stater, Löwdin laddningar och arbetsfunktion för fortsatta

Included publications

Paper I:

Anderson David Smith, Karim Elgammal, Frank Niklaus, Anna Delin, Andreas Fischer, Sam Vaziri, Fredrik Forsberg, Mikael Råsander, Håkan W. Hugosson, Lars Bergqvist, Stephan Schröder, Kataria Satender, Mikael Östling and Max Lemme. (2015). Resistive Graphene Humidity Sensors with Rapid and Direct Electrical Readout. Nanoscale.

web link: http://doi.org/10.1039/C5NR06038A

KE carried out the DFT calculations, analyzed the results and contribu- ted to writing the theory part.

Paper II: (in draft phase)

Karim Elgammal, Håkan W. Hugosson, Anderson D. Smith, Mikael Rå-

sander, Lars Bergqvist, Anna Delin. Density functional theory calculations of graphene-based humidity and carbon dioxide sensors: effect of silica and sapphire substrates

KE carried out all the calculations, performed a major literature review,

and wrote important parts of the manuscript.

Acknowledgments

Firstly, I would like to express my sincere gratitude to Anna Delin, my ad- visor for making me part of her research group and the continuous support of my Ph.D study and related research, for her patience, motivation, and immense knowledge. Her guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. She is always keen on delivering the best help possible despite her limited time. She is not only giving her critical comments and motivations for the results and work plan, but also editing the text with me simultaneously with everything we work on. I learnt and still learning much from her.

I do appreciate the continuous help and fruitful inputs form my group seniors: Lars Bergqvist who saved me a lot of my time by giving me hints and assisting all technical hinders from day 1. I believe that it is fun work- ing with Mikael Råsander with much appreciated input and critical thinking with continuous check of my ongoing research and accompanied criticism.

I also appreciate input from Håkan Hugosson for his sharp eyed comments and help in answering many inquiries in my on going research, I do appre- ciate his guidance and his visionary research direction he was pointing to. I am so grateful to have Mohammad Reza as my office mate, who insisted on revising my thesis giving me critical questions on a reviewer level as well as providing technical assistance in many aspects. Despite his preoccupation with his own projects, I always get his comments and suggestions in time.

I do appreciate the time he discuss my research on my desk and trying to figure out the problem.

I thank Amina Mirsakiyeva, Fan Pan, Johan Hellsvik, Pavel Bessarab,

Simone Borlenghi as my fellow group-mates for the support and discussions

x

and for all the fun we have had, I do appreciate the effort done by Pavel helping us run theory study sessions making my hands dirty all the time with from-scratch theory derivations. Also I thank my collaborator Ander- son Smith for the patience and making me part of the community, Andy is always keen on involving me into new ideas, projects and research direction discussions. Additionally, I appreciate the big help from Max Lemme for his constructive comments on our common research projects. I would like to show my deep gratitude to my fellow old friend Loay for his support and critical input once it comes to DFT theories and derivations, he helped me much editing the theory part of this thesis as well with critical proofread- ing. This warm thanks are extended to Hazem and Foda for their review constructive criticism. As usual, my old friend Shady, always gives me the nice English synonyms and review questions once it comes to proofreading, he revised my thesis many times in a similar fashion to what he did with my master thesis 3 years ago. I do appreciate the nice english comments given by Atwa providing a proper final touch on some crucial parts. I would like to give my gratitude to my flatmates Mahmoud and Abubakrelsedik for being supportive making life stress-free. Last but not the least, I would like to thank my family: my parents and to my brother and sister for giving me encoragment to succeed in my life.

I do want to extend my acknowledgment to the Swedish computing cen-

ters where my computations were performed on resources provided by the

Swedish National Infrastructure for Computing (SNIC) at the PDC center

for high-performance computing, KTH as well as NSC Triolith and Abisko

supercomputers.

Contents

Contents xi

1 Introduction 1

2 Theoretical background 5

2.1 The many body problem . . . . 5

2.2 Born-Oppenheimer approximation . . . . 6

2.3 Hohenberg-Kohn theorems . . . . 7

2.4 The Kohn-Sham approach . . . . 8

2.5 Exchange-correlation functionals and van der Waals interactions 8 2.6 Plane waves and pseudopotentials . . . . 9

3 Results and discussion 11

4 Conclusions and future work 13

Bibliography 15

Chapter 1

Introduction

Graphene has demonstrated extraordinary properties such as ultrahigh car- rier mobility and high on-current values which make it an ideal material for sensing purposes [1]. Gas sensors based on solid state devices are promising building blocks for several electronic devices [2–4], opening new horizons in the nano-electronics field. For the last couple of decades, gas sensors have been extensively studied. In details, several materials and structures, such as nanowires [5–8], metal oxides [9], carbon nanotubes [5, 8], graphene oxide, graphene [10–32], transition metal di/tri-chalcogenides (TMDCs/TMTCs) [33]

and other 2D materials [34, 35] have been investigated. Graphene has a low electronic noise, making it an ideal material for detecting adsorbed gas molecules on its surface [1]. Graphene’s dangling π orbitals drive its high conductivity, where each atom’s π orbital interacts with the neighbors’ coun- terparts forming conduction and valence bands [10, 20, 36]. These π orbital electrons cause graphene to be sensitive to the surroundings, giving it the capability of gas sensing [11, 12, 18, 19, 37]. Thus the ability of graphene to sense water molecules as well as carbon dioxide has been extensively studied both theoretically [12, 38–56] and experimentally [13, 14, 17, 43, 46, 48, 52, 57–60]. Graphene-based sensing devices have shown efficiency in detecting other gases, such as carbon monoxide [41, 42, 61, 62], sulfur dioxide [54], nitrogen monoxide/dioxide [41, 42, 62–64], hydrogen sulfide [44, 65] and am- monia [41, 42, 44, 62].

Recently, extensive studies on the sensing properties of other 2D ma-

terials have been performed. One of those materials is silicene, which is

2 CHAPTER 1. INTRODUCTION

another so-called Dirac material that shows moderate sensitivity towards nitrogen monoxide/dioxide, sulfur dioxide, oxygen and ammonia as demon- strated by Feng et. al. [35] as well as formaldehyde as studied by Wang et. al. [66]. Another example, from the TMDCs is molybdenum disulphide (MoS

), where the adsorbates reside on top of a mono-layer surface of the material. Different gas molecules have been studied on a MoS

mono-layer, including carbon monoxide/dioxide, ammonia, nitrogen monoxide/dioxide, methane, water molecules, nitrogen, oxygen and sulfur dioxide as studied by Yue et. al. [67] and Zhao et. al. [33].

Different stacked 2D materials are promising in terms of the novel prop- erties they posses as demonstrated by Geim et. al. [68]. This leads to many different 2D stacked materials to be studied, a comprehensive survey for these studies was done by Qian et. al. [69]. Graphene bi-layers were also studied by Wang et. al. [70] and Fujimoto et. al. [71]. However, the gas sensing properties for these systems have not yet been as thoroughly studied as is the case for the mono-layered 2D materials.

Introducing defects and dopants in the graphene sheet itself is an in- teresting research topic. Defects such as Stone-Wales defects

have been explored in conjunction with the graphene’s sensing properties. Hajati et.

al. [64] studied nitrogen dioxide gas adsorption on a mono-layer of such de- fected graphene demonstrating a dramatic change in graphene’s electronic properties as compared to pristine graphene. Dutta et. al. [55] examined other defects such as buckled or rippled graphene sites. They found that the binding energies of the graphene with respect to different adsorbates are affected by the buckling and rippling. The underlying substrate surface defects contribute to doping the pristine graphene sheet as confirmed by Wehling et. al. [39]. They studied the electronic properties of a graphene sheet in humid environment on top of a defective substrate surface. Mean- while, doping the graphene sheet itself changes the electronic and sensing properties as shown by Peng et. al. [62] for boron- and nitrogen- doping, Yuan et. al. [44] for p-doping, Deng et. al. [49, 72] for boron doping, Chen et. al. [54] and Denis et. al. [73] for gallium-, germanium-, arsenic- and selenium- doping, Zou et. al. [74] for silicon doping, Dai et. al. [41] and

1A defect where two hexagons (next to each other) in the graphene sheet share ver- tices’s bonds instead of hexagon edges share

3

Sharma et. al. [65] for aluminum doping. Moreover, Nakamura et. al. [32]

showed that silicon substrate coating and conditioning affect the graphene sheet conductivity.

In this work, we report on electronic structure studies of pristine graphene

residing on top of different types of substrates. Specifically, the interplay

between the substrate defects, adsorbed water and carbon dioxide molecules

Chapter 2

Theoretical background

2.1 The many body problem

To solve any quantum mechanical system, the eigenvalues of the Hamiltonian operator shown in equation 2.1 have to be evaluated (HΨ = EΨ). Since the exact solution of such equation for a many body system is analytically impossible, therefore a series of approximations has to be utilized. In this chapter basic theories and approximations, which formulates the density functional theory, will be briefly discussed. The many body Hamiltonian can be written as in equation 2.1:

H = ˆ −

Nuclei K.E.

z }| {

~

2

X

I

∇

M

+

Nucleus-Nucleus Interaction

z }| {

1 2

X

I6=J

Z

Z

e

4π

|R

− R

| −

Electrons K.E.

z }| {

~

2m

X

i

∇

+ 1

2

i6=j

e

4π

− r

| {z }

Electron-Electron Interaction

−

i,I

Z

e

4π

|r

− R

|

| {z }

Electron-Nucleus Interaction

(2.1)

Here the first term



−

I



represents the kinetic energy of all nu- clei, each with mass M

. The second term

0|R_I−R_J|

i

represents

nucleus-nucleus interactions via Coulomb repulsive forces. The second term

can be calculated efficiently using Ewald’s summation method (which de-

termines the electrostatic potential as well as the energy of point charges

in a crystal [75]). The third term

−

∇

represents the kinetic en-

6 CHAPTER 2. THEORETICAL BACKGROUND

ergy of electrons, each with mass m. The fourth term



1 2

P

i6=j e² 4π0

|

|



represents the Coulomb interaction within pairs of electrons (the so called Hartree interaction). The fifth and last term

−

0|r_i−R_I|

i

represents electron-nucleus Coulomb interactions. The 1/2 in the electron-electron and nucleus-nucleus interactions is to correct for the double counting.

The first step to simplify the many body Hamiltonian is to invoke the Born-Oppenheimer approximation, which is described in more detail in the next section.

2.2 Born-Oppenheimer approximation

The Born-Oppenheimer approximation [76] (BO) simplifies the solution of the many body Schrödinger equation (equation 2.1) as it separates the nu- clear and electronic motion. This approximation leads to two wave equa- tions. The first equation describes the electronic motion, which can be solved separately by further approximations to evaluate the electronic wave function and the ground state energy. The second equation provides a de- scription of the motion of the nuclei.

The final result is the simplified Born-Oppenheimer Hamiltonian described in equation 2.2 below:

H ˆ

= − ~

2m

X

i

∇

| {z }

Electrons K.E.

+ 1

2

i6=j

e

4π

− r

| {z }

Electron-Electron Interaction

−

i,I

Z

e

4π

|r

− R

|

| {z }

Electron-Nucleus Interaction

+ 1

2

I6=J

Z

Z

e

4π

|R

− R

|

| {z }

Nucleus-Nucleus Interaction

(2.2)

In atomic units, the Born-Oppenheimer Hamiltonian can be as stated in

1The contents of this section and the upcoming sections closely follows the presentation in standard textbooks in the subject, e.g., ABC of DFT [77] and Density Functional Theory and the family of (L)APW-methods: a step-by-step introduction [78].

2.3. HOHENBERG-KOHN THEOREMS 7

equation 2.3 with ~ = m

= e = 4π

= 1 as:

H ˆ

= − 1 2

X

i

∇

| {z }

Electrons K.E.

+ 1

2

i6=j

1

− r

| {z }

Electron-Electron Interaction

−

i,I

Z

|r

− R

|

| {z }

Electron-Nucleus Interaction

+ 1

2

I6=J

Z

Z

|R

− R

|

| {z }

Nucleus-Nucleus Interaction

(2.3)

2.3 Hohenberg-Kohn theorems

The Born-Oppenheimer approximation simplifies the Hamiltonian of the many body problem. Still, the number of degrees of freedom in the system is prohibitively large. If we could formulate the problem using the charge density instead of the wave function containing all the information about the electrons, the problem would become dramatically simplified. This is the approach taken in the Hohenberg-Kohn (HK) theorems [79]. Those the- orems are considered the foundation of DFT. They are stated below as:

Theorem I: "For any system of interacting particles in an external po-

tential V

(r), the potential V

(r) is determined uniquely, by the ground state particle density n

(r)."

So equation 2.3 will be rewritten as:

H = − ˆ 1 2

X

i

∇

+ 1 2

X

i6=j

1

− r

+

i

V

(r

) (2.4)

Theorem II: "The ground state energy could be expressed in terms of

a universal functional of the electron density E[n(r)] valid for any external potential V

. For any particular V

(r), the exact ground state state en- ergy of the system is the global minimum value of this functional, and the density n(r) that minimizes the functional is the exact ground state density n

(r)."

We can construct a universal function for the energy which contains a

8 CHAPTER 2. THEORETICAL BACKGROUND

depends on the density:

E[n] = F [n] +

d

(r)n(r) (2.5) Where

F [n] = T

[n]

| {z }

K.E.

+ d

n(r) n(r

)

|r − r

|

| {z }

Hartree

+ E

[n(r)]

| {z }

Exchange-Correlation

(2.6)

F [n] is valid for any external potential V

(r). All terms here are solvable apart from the E

[n(r)] term.

2.4 The Kohn-Sham approach

The energy functional above contains a kinetic energy term. There is no known closed expression for this term at present and thus the functional cannot be evaluated as it stands. With the Kohn-Sham approach, one can solve this problem. Achievable through the Kohn-Sham (KS) equations [80]

where the difficult interacting many-body system is replaced with a solvable auxiliary non-interacting system. This formulation is achieved by assuming that the ground state density of the original interacting system is equal to that of some chosen non-interacting system.

Thus, the interacting many-body system in equation 2.3 is represented by Kohn-sham to be in the form in equation 2.7 as:

H ˆ

= − 1

2 ∇

+ V

(r) +

dr

n(r

)

|r − r

| + E

[n(r)] (2.7)

2.5 Exchange-correlation functionals and van der Waals interactions

Next, we need to address the exchange-correlation term E

[n(r)] in the en-

ergy functional. By solving the KS equations, the ground state energy and

the density of the original interacting system are found with an accuracy

limited by approximations utilized in the used exchange-correlation func-

tional.

2.6. PLANE WAVES AND PSEUDOPOTENTIALS 9

The exchange-correlation term can only be calculated approximately.

The local density approximation (LDA) is one of the most well-known ap- proximations [80, 81]. E

is substituted here with the homogeneous electron gas exchange and correlation energies as shown in equation 2.8:

E

[n(r)] =

dr 

[n(r)] n(r) (2.8)

Where 

[n(r)] is the exchange-correlation energy density.

Another way to approximate the E

is the generalized gradient approx- imation (GGA), where the functional includes not only the density but also the density gradient as illustrated in equation 2.9:

E

[n(r)] =

dr f (n(r), ∇n(r)) (2.9) In this work, the parametrized GGA by Perdew, Burke and Ernzer- hof (PBE) [82] was used in conjunction with a semi-empirical scheme for including van der Waals (vdW) interaction dispersive forces developed by Grimme [83]. The corresponding vdW energy is added to the plain KS functional as a correction term. The energy is expressed in equation 2.13 as:

E

= E

+ E

(2.10)

Where

E

= − s

2

i6=j

C

R

f

(R

) (2.11) Here C

is the dispersion coefficient for each pair of atoms i & j

C

=

C

C

(2.12) The damping function f

is given by:

f

(R) = 1

1 + e

(2.13)

2.6 Plane waves and pseudopotentials

Till now, approximations to the many-body Schrödinger equation have been

10 CHAPTER 2. THEORETICAL BACKGROUND

The ground state energy was shown to be a functional of the electronic den- sity which is calculated from the Kohn-Sham eigenstates. But to get the Kohn-Sham eigenstates, the exchange-correlation energy should be known beforehand. This problem is solved by modeling the wave functions in terms of superpositions of numerical functions. The most common in use are Gaus- sian functions and plane waves, used to evaluate the exchange-correlation energy, which allows for solving the Kohn-Sham equation. This solving pro- cess is iterated till a self consistent solution is reached.

In this work plane waves were used together with the so-called pseudopo- tentials, constructed in such a way that as few plane waves as possible are used.

Equation 2.14 expands the eigenstates in terms of an infinite number of plane waves with corresponding coefficients. c

.

ψ

(r) =

K

c

e

(2.14)

It is impossible to numerically evaluate an infinite number of coefficients for the basis set. Therefore the solution is limited for K by specifying a limiting value K

, which is the radius of a sphere in the reciprocal space whose center is the origin. Thus, the limiting factor for all the K is set to K ≤ K

. The corresponding free electron energy is called the cut-off energy which is expressed in equation 2.15 as:

E

= ~

K

2m

(2.15)

Here, m

is the electron mass.

At this point another problem appears. Due to the fact that electronic

wave functions are very steep in the close neighbourhood of the nucleus,

the limitation introduced by choosing a plane wave cutoff will cause high

inaccuracy. This problem is resolved by replacing the potential in the close

neighborhood of the nucleus with a pseudopotential that models the elec-

tronic wave functions properly in the interstitial region. Since most of the

chemical bonding appears away from the nucleus in the inter-atomic region,

the result of using pseudopotentials has a much lower computational cost

for the same accuracy.

Chapter 3

Results and discussion

The results are presented in full in the attached papers. Here, we provide a summary of these papers and how they connect.

We carried out DFT calculations, simulating the presence of adsorbates of carbon dioxide (CO

) and water (H

O) molecules on graphene. We found that the interaction between the water molecules adsorbed on the graphene sheet and the defect states coming from the underlying substrates has a doping effect in the graphene sheet. That is explained in Paper I, where the substrate defects together with the H

O molecules change the electronic structure properties represented by charge density difference contours. The phenomenon is further detailed in Paper II where three different substrates were studied. Two of them were silica substrates with different types: one of them was a cristobalite substrate with a "Q

" defect

[84, 85] at the substrate surface (i.e. Si-terminated cristobalite(111) substrate) and the other was an α-quartz silicon dioxide substrate with an under-coordinated silicon atom at the surface (i.e. Si-terminated α-quartz(0001)). The third substrate was a sapphire (Al

O

) substrate

with an under-coordinated aluminum atom at the substrate surface (i.e. Al-terminated sapphire(0001)). Moreover, a comparison with an adsorbate species lacking electric dipole moment, CO

molecule was done in Paper II on all the three different substrates.

As detailed in Paper II, we have relaxed the graphene with respect to

1That is a well known silicon substrate surface defect.

2Which is used as a silicon substrate coating material in many cases.

12 CHAPTER 3. RESULTS AND DISCUSSION

the different substrates in use. The adsorbates were relaxed as well on the relaxed graphene-substrate systems. We have examined several adsorbate configurations, distinguished by the relative (x-y) plane positioning of the adsorbates with respect to the graphene sheet. We have investigated the par- allel configuration with respect to the graphene sheet. Among all the relaxed configurations, adsorbate configurations with the lowest energy (a preferred configuration) are the closest to the graphene sheet among the same sub- strate type. This is expected since a stronger bond (more hybridization) will usually also imply a shorter bonding distance. Water adsorbates always prefer to be above the center of the graphene hexagon regardless of the type of the underlying substrate. On the other hand, carbon dioxide adsorbates preferred configurations differed according to the substrate; being above the midway between two carbon atoms (the so-called bridge position) of the graphene hexagon in the case of cristobalite substrate, while being above the top silicon atom of the Si-terminated α-quartz(0001) substrate, and fi- nally above a carbon atom in the graphene sheet as well as an aluminum atom (inside the substrate and not the substrate’s surface top atom) in the case of Al-terminated sapphire(0001) substrate. Moreover, carbon dioxide adsorbates are apart from the graphene sheet by 3.20 Å in all the preferred configurations across all these cases.

Upon studying those different systems, we can say that the doping ef-

fect differs slightly upon different types within the silica substrates, while it

differs dramatically in comparison with sapphire substrate type or as called,

the alumina coated silicon substrates. Thus, the effect of adsorbates exis-

tence on the graphene sheet depends on the substrate properties, types and

its associated defects.

Chapter 4

Conclusions and future work

In conclusion, we have studied some aspects of graphene based sensors using density functional calculations. In particular, we have simulated the pres- ence of adsorbates of carbon dioxide and water (i.e humidity) on graphene.

We found that the interaction between the adsorbing molecules and the de- fect states coming from the underlying substrates appears to have a doping effect. This doping is sensitively dependent on the type of the substrate. The alumina substrate affects the charge density difference in the graphene layer much more than the silica based substrates. Also, the molecular binding energies are much larger in the alumina case. In comparison, the differences between the cristobalite and α-quartz substrates are very small. Therefore, alumina may be a good choice as a coating material for substrate surfaces, since it enhances the effect of adsorbates on the electronic properties of graphene. Meanwhile, cristobalite based silica substrate can be suggested as a suitable silica based substrate for graphene sensing purposes as it can enhance the graphene’s sensing properties while preserving the graphene’s electronic structure.

The second paper is in the draft phase, and additional calculations will

be performed to find e.g. the global minimum configuration of the adsorbed

species for each case. As future work, we will also continue investigat-

ing graphene sensor properties for a range of other molecules such as car-

bon monoxide, ammonia, nitrogen monoxide and dioxide. Furthermore, we

aim to address the conductance directly by employing the non-equilibrium

Green’s functions scheme [86]. Ultimately, We wish to merge our quantum

14 CHAPTER 4. CONCLUSIONS AND FUTURE WORK

mechanical studies with device and circuit level simulations using a multi-

scale approach [87].

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