Modeling Molecular Electronics: Applications to Chemical...

Modeling Molecular Electronics:Applications to Chemical Sensors

by

Iker Larraza Arocena

A thesispresented to the Universidad del Paıs Vasco

in fulfilment of thethesis requirement for the degree of

Master’s of Sciencein

Fisıca de Materiales

Supervisors: Prof. Angel RubioDr. Duncan J. Mowbray

Donostia-San Sebastian, Guipuzkoa, Espana, September 7, 2011

c© Iker Larraza Arocena 2011

Contents

1 Introduction 1

2 Basic Concepts 52.1 Landauer approach to molecular transport . . . . . . . . . . . . . . . . . . . . . 52.2 Density Functional Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 Non-Equilibrium Green’s Function Methodology . . . . . . . . . . . . . . . . . 9

3 Carbon Nanotube Molecular Electronics 133.1 Carbon Nanotubes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Carbon Nanotube Based Chemical Sensors . . . . . . . . . . . . . . . . . . . . 143.3 Transition Metal Doped (6,6) SWNTs . . . . . . . . . . . . . . . . . . . . . . . 16

4 Chemical Sensor Applications 194.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2 Active Sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2.1 Divacancy II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.2.2 Divacancy I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.2.3 Monovacancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Target and Background Molecules . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Coverage and Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Conclusions 45

A Adsorption Energies 47

i

B Conductance 49

Bibliography 53

ii

List of Figures

2.1 Non-equilibrium Green’s function approach . . . . . . . . . . . . . . . . . . . . 92.2 Intertube and intratube NEGF conductance calculations for a (13,0) SWNT junction 11

3.1 Schematic of a (13,0) SWNT junction with O2 or N2 . . . . . . . . . . . . . . . 143.2 Methodology for modeling nanosensors . . . . . . . . . . . . . . . . . . . . . . 153.3 Band structure of a (6,6) metallic armchair SWNT . . . . . . . . . . . . . . . . . 17

4.1 Schematic of a (6,6) SWNT chemical sensor . . . . . . . . . . . . . . . . . . . . 204.2 Schematics and formation energies of pure transition metal occupied monova-

cancies, divacancies I, and divacancies II . . . . . . . . . . . . . . . . . . . . . . 214.3 DOS of pure transition metal occupied monovacancy . . . . . . . . . . . . . . . 224.4 Transmission of pure transition metal occupied monovacancy . . . . . . . . . . . 234.5 DOS of pure transition metal occupied divacancy I . . . . . . . . . . . . . . . . 244.6 Transmission of pure transition metal occupied divacancy I . . . . . . . . . . . . 254.7 DOS of pure transition metal occupied divacancy II . . . . . . . . . . . . . . . . 264.8 Transmission of pure transition metal occupied divacancy II . . . . . . . . . . . 274.9 Adsorption energies for molecules on transition metal occupied vacancies . . . . 314.10 DOS of transition metal occupied monovacancy with adsorbed N2, O2, H2O, CO,

and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.11 Transmission of transition metal occupied monovacancy with adsorbed N2, O2,

H2O, CO, and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.12 DOS of transition metal occupied divacancy I with adsorbed N2, O2, H2O, CO,

and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

iii

4.13 Transmission of transition metal occupied divacancy I with adsorbed N2, O2,H2O, CO, and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.14 DOS of transition metal occupied divacancy II with adsorbed N2, O2, H2O, CO,and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.15 Transmission of transition metal occupied divacancy II with adsorbed N2, O2,H2O, CO, and NH3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.16 Circuit theory approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.17 Conductance change for TM occupied vacancies upon molecular adsorption . . . 404.18 Fractional coverage of Ni@DVII and Cu@MV . . . . . . . . . . . . . . . . . . 414.19 Change in resistance of Ni@DVII and Cu@MV . . . . . . . . . . . . . . . . . . 42

iv

List of Tables

A.1 Adsorption energies for molecules on a transition metal occupied monovacancy . 47A.2 Adsorption energies for molecules on a transition metal occupied divacancy I . . 48A.3 Adsorption energies for molecules on a transition metal occupied divacancy II . . 48

B.1 Conductance for molecules on a transition metal occupied monovacancy . . . . . 49B.2 Conductance for molecules on a transition metal occupied divacancy I . . . . . . 50B.3 Conductance for molecules on a transition metal occupied divacancy II . . . . . . 50B.4 Gas phase entropies and equilibrium atmospheric concentrations . . . . . . . . . 51

v

Chapter 1

Introduction

Molecular electronics is often viewed as the concept of single molecules being either activeelements in hybrid electronics or integrating a range of elementary functions necessary in com-putation [1]. This can and should be contrasted with the definition relating the condensed-phaseproperties of a material with the molecular nature of its building blocks, like the case of molecu-lar crystals.

In the year 1974, Ari Aviram and Mark Ratner published a paper entitled Molecular Rectifiers[2]. In this paper, taking into account the reduced amount of experiments that could be made atthe time; they discuss theoretically the possibility to construct a very simple electronic device, arectifier, based on the use of a single organic molecule. The proposal they made would strengthenthe foundations of the field looking forward to electronic applications at a molecular scale. Withthe technology existing nowadays, it is possible to make measurements which allow a regularobservation of what 30 years ago was a challenging theoretical prediction.

The Aviram-Ratner model is probably the first concrete example that involves uni-molecularquantum transport and its applications. Due to this reason, it is closely linked with the electrontransfer theories and experiments that were already being developed at the time. As a short note,the first charge and energy transfer studies in molecules started in the decade of 1940 by RobertMulliken and Albert Szent-Gyorgi with the donor-acceptor systems.

Due to advances and successes on the experiments made, there has been a great progress inmolecular electronics in the last decade. New methods have allowed the use of single moleculeas contacts and the measure of current-voltage characteristics by using molecular functionality

1

2 MODELING MOLECULAR ELECTRONICS

and recognition to assemble an electronic device. Not only this, there has also been some bigadvances in scanning probe microscopy and nano-lithography patterning techniques.

A very good question we must ask ourselves is: why is it necessary to consider molecularelectronics and why is quantum transport so challenging at these scales? The answer to thefirst of these two questions is that the dimensions of conventional silicon based electronics aregetting smaller and smaller due to the technological need to decrease sizes in order to increaseefficiency. These dimensions will soon be so small that quantum effects will begin to influenceand eventually limit the device functionality.

The nomenclature quantum transport is used to describe the phenomenon of electron motiontrough constrictions of transverse dimensions that are smaller than the electron wavelength. Thisis very important in the case of an electron traveling through a single molecule, which is the casein molecular electronics. The studies of charge-transport have lead to the discovery of severalmesoscopic quantum phenomena, like the quantum Hall effect, or the conductance quantization.

All this clearly indicates that the field of molecular electronics has grown so to define imme-diate challenges and new directions [3, 4, 5, 6].

Now that we know a little about the history of molecular electronics, the more than obviousquestion to ask is: what is molecular electronics? Basically, the most fundamental structure in-volved in molecular electronics is a molecular transport junction, which consists of one (ideally)or more molecules located between two electrodes. These junctions combine the fundamentalprocess of intramolecular electron transfer with the mixing of molecular and continuum levels atthe electrodes and the non-equilibrium process of voltage-driven currents. [7]

Molecular electronics is an area of very rapidly growing scientific and applied interest andactivity. It involves the exploration of the electronic level structure, response and transport,together with the development of electronic devices and applications that depend on differentproperties of matter at the molecular scale. The major areas of application are displays, smartmaterials, sensors, logic and memory devices, energy transduction devices, molecular motorsand molecular scale transistors. It is noteworthy that molecules are important not only becausetheir electronic properties, but also because their ability to bind to one another, recognize eachother, assemble into larger structures, and exhibit dynamical stereochemistry.

It is true that all the technological drivers (electrochemical, biological, sensing) are importantbut the use of molecular systems as components of electronic devices raises several fundamental

CHAPTER 1. INTRODUCTION 3

questions. This fundamental issue is the main driver for this science. One of the main differencesbetween traditional quantum chemistry and molecular electronics is that while the first considersmolecules as electronically closed systems, the second analyzes molecular systems that are opento their electronic environment and function in situations far from equilibrium.

Chapter 2

Basic Concepts

2.1 Landauer approach to molecular transport

Electron transfer provides a conceptual and technical basis for molecular electronics. Electronictransfer in supermolecules typically falls into two categories:

1. Bridged species, often consisting f an electron donor (D) and an electron acceptor (A),which are linked by a rigid or non-rigid molecular bridge (B).

2. Biophysical systems such as the photosynthetic reaction centers (RC) of plants and bac-teria, where the photosynthesis process comes from a sequence of highly efficient, direc-tional and specific electron transfer processes between prosthetic groups embedded in theprotein medium.

The control of electron transfer in DA or DBA systems can be accomplished by:

1. Structural control: Molecular engineering of the D, B, and A subunits determines themolecular energetic and the direct (DA) or superexchange (DBA) electronic coupling.

2. Intramolecular dynamic control of the nuclear equilibrium configurational changes thataccompany electron transfer.

3. Medium control of the conventional electron transfer in either a solvent or a cluster. Tounderstand the way this works, we need to know the functions of the medium, which are:

5


the energetic stabilization of the electronic states and the coupling of the electronic stateswith the medium nuclear motion.

4. Dynamic medium electron transfer control, in which the medium acts as a heat bath, veryslow solvent relaxation, solvent controlled electron transfer, or specific dynamic control ofpathways by solvent motion.

The control of electron transfer makes possible the design of molecular systems where the elec-tron transfer can be:

• Ultrafast (1 ps 100 fs), overwhelming any energy waste process.

• Highly efficient, eliminating back reactions.

• Stable with respect to the variation of molecular and medium properties.

• Practically invariant with respect to temperature changes.

In the donor-acceptor charge transfer theory, the non-adiabatic electron transfer rate kD→A isrepresented in the basic form:

kD→A =

(2π

)V2F , (2.1)

in which V is the effective electronic coupling andF is the thermally averaged nuclear vibrationalFranck-Condon factor. Different factors are taken into account in this description:

• Electron transfer is taken as a radiationless transition.

• The Born-Oppenheimer approximation that considers nuclear and electronic motion sepa-rately.

• Incoherent charge transfer between donor-acceptor sites, which involves dephasing at eachelectron transfer step. This is analogous to Holsteins small polaron [8, 9] in the incoherentlimit.

• The electronic coupling is supposed to be weak enough to warrant the description of theradiationless transition in the non-adiabatic limit.

CHAPTER 2. BASIC CONCEPTS 7

• There is a clear difference of time scales for nuclear and electronic processes.

The electronic coupling in the DBA is a sum of the direct D-A exchange contribution VDA be-tween the DA and D+A−, and a superexchange off-resonance interaction [10] Vsuper. Both ofthem show an exponential spatial dependence [11, 12, 13]

V = α exp (−βRDA) , (2.2)

in which RDA is the D-A distance.For the simplest energy representation of a molecular junction the two electrodes are assumed

to be made of the same material, considered as a simple metal with a Fermi level denoted as εF.Under conditions where these metals are thermally equilibrated, the Landauer formula for thecoherent conductance of the junction is given by:

g(Φ) =eπ

δ

δΦ

∫ ∞

0dεT (ε) ( f (ε) − f (ε + eΦ)) , (2.3)

where g,Φ,T, f , ε are the conductance, the voltage, the transmission through the molecular junc-tion, the Fermi functions describing the populations on the two metal leads and the energy vari-able respectively. When the derivative of the right hand side of (2.3) is removed, we obtain aformula for the current, I, in terms of the populations and the elastically-scattered transmission.This equation is the simplest useful result for transport, and has the same role here that (2.1)has in the electron transfer processes. Actually, it is reminiscent of the electron transfer result of(2.1): the rate (electron transfer) or the current (molecular junctions) are given by the product ofa population term and a scattering probability [1].

2.2 Density Functional Theory

DFT is a quantum mechanical method for calculating the ground state energy for many-bodysystems. The reason why DFT is used is that it simplifies a problem in terms of the many electronwave function Ψ(r1, · · · , rN) into another problem, which is only in terms of the electron densityρ(r1). This is an important advantage because it reduces the coordinates from 3N to just 3 (x, y, z).

Ψ(r1, . . . , rN) → 3N coordinates,

ε0[ρ]→ ρ(r1) → 3 coordinates.


This is based on the Hohenberg Kohn theorem which states that for a given external potential theground state energy is a unique functional of the electron density.

ε0[ρ] = F[ρ] +

∫drρ(r)vext(r), (2.4)

where

F[ρ] = T [ρ] + Vee[ρ], (2.5)

where T [ρ] is the kinetic energy, Vee[ρ] describes the electron-electron interaction and vext is theexternal potential. However, an exact form for F[ρ] in terms of the density is unknown, so wewrite it this way:

F[ρ] = T [ρ] +12

"drdr′

ρ(r)ρ(r′)|r − r′|

−

∫drρ(r)vxc[ρ](r). (2.6)

Here vxc is the exchange correlation potential that gives the correct electron-electron interactionVee. vxc has corrections to the Coulomb potential in order to fulfill the Pauli exclusion principle,self interactions corrections, and electron-electron correlations.

The most common implementation of DFT uses the Kohn-Sham method. The procedure forthis method is to assume a form for vxc, and then solve the non-interacting Schrodinger equation

i2∇2Ψi + v(r)Ψi = εiΨi, (2.7)

with the effective potential v given by

v[ρ](r) =

∫dr′

ρ(r′)|r − r′|

+ vxc[ρ] + vext(r). (2.8)

This potential depends on the density

ρ =

N/2∑i=0

Ψ∗i Ψi, (2.9)

in a self consistent manner, where N is the number of electrons.


X[ +TM@VC] (ε)T

X[ +TM@VC] (ε)DOS

NEGF

Figure 2.1: The Hamiltonian H and overlap S matrices obtained from density functional theory(DFT) are used within the non-equilibrium Green’s function (NEGF) methodology to obtainthe density of states (DOS) and transmission probability T for each species X on an active site(X+TM@VC), as a function of energy ε, from Ref. [14].

2.3 Non-Equilibrium Green’s Function Methodology

The non-equilibrium Green’s function methodology (NEGF) is based on the Green’s function,which is dependent on two space-time coordinates. The NEGF can be applied to both extendedand infinite systems and it can handle strong external fields without being perturbed. The approx-imations made in the NEGF can be chosen in order to satisfy macroscopic conservation laws. Aninteresting feature of the NEGF is that memory effects in transport that occur due to electron-electron interactions can be analyzed. As shown in Fig. 2.1, from the NEGF methodology weobtain both the density of states (DOS) and transmission probability T(ε) of an electron beingtransmitted through a scattering region, as depicted schematically for intratube and intertubetransmission in Fig. 2.2.

All transport calculations for the optimized structures have been performed using the NEGF[15] with an electronic Hamiltonian obtained from the siesta code [16] in a double zeta polarized(DZP) basis set.

A large supercell was necessary for the Hamiltonian of each of the four SWNT layers adjacentto the boundaries H prin

C , to be within 0.1 eV of the Hamiltonian for the respective leads Hα, so


that max |H prinC − Hα| < 0.1 eV. In this way the electronic structure at the edges of the central

region was ensured to be converged to that in the leads.The Landauer-Butticker conductance for a multi-terminal system can be calculated from the

Green’s function of the central region, GC, according to the formula [17, 18, 15]

G = G0Tr[GCΓinG†

CΓout]∣∣∣ε=εF

, (2.10)

where the trace runs over all localized basis functions in the central region. To describe theconductance at small bias for semiconducting systems, the Fermi energy εF should be taken asthe energy of the valence-band maximum εVB or conduction-band minimum εCB for p-type andn-type semiconductors, respectively. The central region Green’s function is calculated from

GC(ε) =

zSC − HC −∑α

Σα(ε)

−1

, (2.11)

where z = ε + i0+, SC and HC are the overlap matrix and Kohn-Sham Hamiltonian matrix of thecentral region in the localized basis, Σα is the self-energy of lead α,

Σα(ε) = [zSCα − HCα] [zSα − Hα]−1 [zS†Cα − H †

Cα], (2.12)

and the coupling elements between the central region and lead α for the overlap and Kohn-ShamHamiltonian are SCα and HCα respectively.

The coupling strengths of the input and output leads are then given by Γin/out = i(Σin/out −

Σ†

in/out).To better understand the procedure explained above, we can see in figure 2.2 the supercell,

which is large enough so that the electronic structure at the edges is converged to that of theleads.


O2

FrozenLayers

FrozenLayers

Inte

rtub

eT

rans

port

Tra

nspo

rtIn

trat

ube

Leads

Leads Central Region

Figure 2.2: Schematic of (13,0) SWNT junction consisting of four SWNT leads (light gray) cou-pled to the central scattering region via four SWNT primitive unit cells or layers (gray) “frozen”at their relative positions in the isolated SWNT. The atomic positions of the remaining eightSWNT layers (dark gray) and the physisorbed O2 molecule have been relaxed. A SWNT separa-tion consistent with experiments of approximately 3.4 Å has been used. The intratube transportthrough the SWNT and the intertube transport between the SWNTs has been shown schemati-cally in [19].

Chapter 3

Carbon Nanotube Molecular Electronics

3.1 Carbon Nanotubes

A very good way to understand what a carbon nanotube (CNT), especially a single-walled carbonnanotube (SWNT) is to imagine a sheet of graphene wrapped in a diminutive cylinder. Theproperties of the CNT strongly depend on the way this graphene sheet is wrapped. To representthe way the graphene sheet is wrapped, two indices are used: (n,m), where n and m are integers.These two indices represent the number of unit cell vectors along two directions in the crystallattice of graphene, a1 and a2.

For a (n,m) SWNT, the circumference C is given by

C = ‖na1 + ma2‖. (3.1)

We have a metallic nanotube if n − m = 3p, where p is an integer. If n = m the tube is anarmchair SWNT, if m = 0 the tube is a zigzag SWNT, and otherwise the tube is called chiral.The diameter of a SWNT may be approximated using the following formula:

d∅ ≈aπ

√n2 + nm + m2 (3.2)

where a ≈ 2.46 Å.

13


3.2 Carbon Nanotube Based Chemical Sensors

Single-walled carbon nanotubes (SWNTs) [20] work remarkably well as detectors for small gasmolecules, as demonstrated for both individual SWNTs [21, 22, 23, 24, 25, 26] and SWNT net-works [27, 28]. Previous studies have shown that SWNTs are highly sensitive to most moleculesupon functionalization [29, 30, 31, 32, 33, 34, 35, 36, 37]. However, the difficulty is determiningwhich specific molecules are present. In this study we show how changing the functionalizationprovides “another handle” for differentiating the SWNTs response of different gases/molecules.

Recent experimental advances now make the controlled doping of chirality selected SWNTswith metal atoms a possibility. Specifically, these include (1) photoluminesence, Raman andXAS techniques for measuring the fraction of various SWNT chiralities in an enriched sample[38, 39, 40]; (2) the separation of SWNT samples by chirality using DNA wrapping [41, 42, 43,44], chromatographic separation [44, 45] and Density Gradient Ultracentrifugation (DGU)[46];(3) SWNT resonators for measuring individual atoms of a metal vapor which adsorb on a SWNT

(a) Physisorbed O2 (b) Physisorbed N2

Figure 3.1: Schematics of a (13,0) SWNT junction with (a) physisorbed O2 and (b) physisorbedN2, from [19].

CHAPTER 3. CARBON NANOTUBE MOLECULAR ELECTRONICS 15

[47, 48]; and (4) aberration corrected low energy (<50 keV) transmission electron microscopy(TEM) [49]. The latter provides control over the formation of defects in situ by adjusting theenergy of the electron beam above and below the threshold energy for defect formation at aspecific location in the SWNT. These methods provide such a high level of control that it isnow possible for experimentalists to take a specific SWNT chirality and dope the structure withindividual metal atoms at a specified location.

At the same time, theorists are now able to embrace a “bottom up” approach to the design of

[ ]

[TM@VC]( [ ])

[TM@VC]( [ ])

[ ]

DFT Kinetics

Eads [ +TM@VC]X

X[ +TM@VC]H

X[ +TM@VC] (ε)T

X[ +TM@VC] (ε)DOS

0C X

X[ +TM@VC]S

H [NT] S [NT]

sR [ +TM@VC]X

C X

C X

[TM@VC]formE

gasS X

NEGF Circuits

R∆

Θ

Figure 3.2: Schematic of the overall procedure for modeling a nanosensor, fitting together likefour pieces of a puzzle, density functional theory (DFT) calculations, kinetic modeling, non-equilibrium Green’s function (NEGF) calculations, and circuit theory from Ref. [14].


nanosensors, harnessing the thermodynamics of self-assembly to find useful sensing systems insilico. With recent advances in both computational power and methodologies, theorists can nowefficiently and accurately screen hundreds of candidate sensor designs using a combination ofdensity functional theory (DFT) for energetics of adsorption and stability, and non-equilibriumGreen’s function (NEGF) methodologies for the electrical response [50], as outlined in Fig 3.2.

3.3 Transition Metal Doped (6,6) SWNTs

In this study, nine of the ten 3d transition metals (Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn) wereconsidered as candidates for substitutional doping of a (6,6) SWNT. There are several reasonswhy we consider a (6,6) SWNT in this study, rather than another type of CNT. First of all, a (6,6)SWNT is small enough so that it is easy to calculate. There are 24 atoms in the minimal unitcell, and it is repeated 6 times to ensure the convergence of the Hamiltonian to its bulk valuesat the boundaries of the cell, so there are 144 atoms in the TM@MV (143 for the TM@DVand TM@DVII). In other words, we make the unit cell large enough so structural and electronicchanges created by the addition of the TM atom are not felt at the unit cell boundary. Just togive some size reference, the radius of this (6,6) SWNT is of about 4.1 Å,while in experimentalsamples of chirality sorted nanotubes, the radius is typically around 7 Å.

Another reason to choose a (6,6) SWNT is that it is a metallic tube. This is clear from theDFT calculated band structure for the (6,6) SWNT, shown in Fig. 3.3. In the -1 eV — 1 eVenergy range of interest, for metallic nanotubes, the DOS is flat (2 eV−1), and has a two openeigenchannels, so that T(ε) = 2 eV. In this way, the (6,6) SWNT results also provide a qualitativeunderstanding of the response of larger metallic tubes, which have the same flat DOS and T(ε)between -1 eV and 1 eV. A third reason for choosing the (6,6) SWNT is that it has a simpleelectronic response. For metallic armchair (n, n) SWNTs both the transmission T(ε) and theDOS are completely flat near the Fermi level. This makes the task of differentiating the effectof the transition metal dopant, and the adsorbed molecule on the transmission and DOS muchsimpler. In other words, it is easier to compare our results for the doped systems with a flat line,making it easy to disentangle contributions due to the dopants from those of the pristine SWNT.

As mentioned before, we consider nine of the ten 3d transition metals as candidates forsubstitutional doping of a (6,6) SWNT. It has already been explained the reasons for choosing

CHAPTER 3. CARBON NANOTUBE MOLECULAR ELECTRONICS 17

0 0.2 0.4 0.6 0.8 1k (Å

-1)

-4

0

4

εn,k

− ε

F (

eV)

(6,6) SWNT Bandstructure

Figure 3.3: Band energies εn,k relative to the Fermi level εF in eV versus momentum k in Å−1 fora pristine (6,6) metallic armchair SWNT [14].

a (6,6) SWNT, now let us explain the reasons for choosing 3d transition metals. First of all,pristine SWNTs are rather inert. If we dope them with TM atoms, the molecules bind there andwe can analyze the changes they produce.

The 3d TMs are smaller for a given electronic configuration than the 4d and 5d TM and areexpected to fit best in the structure of a SWNT, with the least strain. Even so, the transition metalatoms are typically pushed somewhat above the SWNT surface, due to both the strain on thegraphene lattice, and the difference in atomic radii. These radii vary from the largest at 1.76 Åfor Ti, to the smallest at 1.42 Å for Ni. These are all between 2 and 3 times the atomic radius ofC of 0.67 Å. Also, the electronic properties of a transition metal dopant are mostly determinedby the filling of their d-band, and should be quite similar for the 4d and 5d counterparts of the3d transition metals studied herein.


There is another aspect of this study that should be taken into account. In order to put theTM atom into the SWNT structure, we must first remove a number of C atoms. The way weremoved C atoms to create our three vacancies (monovacancy, divacancy I, and divacancy II) isnot random. The main reason is that these are the most stable vacancies. An interesting factor isthat comparing divacancy I and divacancy II results shows the effect of strain on the electronicproperties of the active site, since for the divacancy II the strain is evenly distributed over thefour bonds, while for the divacancy I the strain is mostly on two of the four bonds. In the case ofthe monovacancy, the transition metal dopant is three fold coordinated, while for the divacancies,the transition metal dopant is four-fold coordinated. This explains why in general we find thatmolecules are more strongly bonded to transition metals in a monovacancy compared to eitherof the divacancies.

For a strongly binding target molecule, such as CO, a divacancy provides the most suitableactive site, as the site will not be completely saturated/covered, and we have a significant fractionof empty sites. On the other hand, for a weakly binding molecule, such as NH3, a monovacancyis more suitable, since there will then be a measurable change in coverage with molecule con-centration. This is why we consider a possible combined Cu@MV and Ni@DVII device thatshould be able to both detect and differentiate NH3 and CO.

It has been shown that the conductance of SWNT networks increases when exposed to O2

and N2. The results from such experiments suggest that that this sensitivity to O2 and N2 maybe due to the increment of the tunneling probability when physisorbed at SWNT junctions. Thiscan be seen in Fig. 3.1

Chapter 4

Chemical Sensor Applications

4.1 Introduction

As seen by comparing the DOS in Figs. 4.3, 4.5, 4.7, 4.10, 4.12, and 4.14, with the respectiveT(ε) in Figs. 4.3, 4.5, 4.7, 4.11, 4.13, and 4.15, the DOS determines the transmission, withscattering off states on the molecule/active site introducing dips in the transmission, which areFano anti-resonances. This typically blocks only one of the two eigenchannels, as discussed inRef. [36]. The most strongly hybridized states between the TM and the SWNT, which are seenas very broad peaks in the DOS, provide the strongest Fano anti-resonances in the transmission.

In general terms, our goal is to design a chemical sensor making use of carbon nanotubes.For the design of this device, we must take into account four general components that appear inany nanosensing system, as seen in Fig. 4.1:

1. The target molecule, which we intend to detect,

2. The active site, where the molecules can get adsorbed onto our device,

3. The sensing property which will indicate the adsorption of the target molecule,

4. The background molecules, which will give a background signal.

One important detail is that the active site must be designed in a way that allows us to detect theadsorption of the target molecule in the presence of the background molecules.

19


sensing property

background active sitestarget molecule

Figure 4.1: Schematic of a chemical sensor consisting of active sites (metal dopants in a (6,6)SWNT), a target molecule (CO), a background (atmospheric air), and a sensing property (resis-tance) [51].

Since experimentally performing this study would be very demanding, we use a computa-tional screening study approach based on density functional theory (DFT) calculations to decidewhich elements are good candidates to be used as active sites so they can be experimentallyanalyzed.

In this case the change in resistance was used as a sensing property. To do so the non-equilibrium Green’s function method was used. The mains advantages of using the change ofresistance as the sensing property are that it is a non-intrusive measure, and that even smallchanges in the coverage of target molecules produce significant changes in resistance. In thisstudy, nine 3d transition metals were used to search for the best candidates as active sites. Thesetransition metals are: Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Zn [52].

4.2 Active Sites

In this section, we will analyze the transmission data from the graphs and, to help us understandwhat happens, we will make use of the density of states (DOS) to explain the behavior we see inthe transmission T(ε). We will first analyze all the different metals in the monovacancy, then the

CHAPTER 4. CHEMICAL SENSOR APPLICATIONS 21

Figure 4.2: Structural schematics and formation energies E f orm in eV relative to a pristine (6,6)SWNT and a physisorbed transition metal atom for a 3d transition metal occupied monovacancy(TM@MV, black), divacancy I (TM@DVI, gray), and divacancy II (TM@DVII, white) in a (6,6)SWNT. Vacancy formation energies for an empty monovacancy (- - - - -), divacancy I (– – ––), and divacancy II (– – – –) are shown for comparison [36].

same will be done with both divacancies. Since we will talk mostly about DOS, it is importantto understand the way DOS and transmission T(ε) are related. The peaks found in the DOS


0

1

2

3

Tra

nsm

issio

n (a) Ti@MV (b) V@MV (c) Cr@MV

0

1

2

Tra

nsm

issio

n (d) Mn@MV (e) Fe@MV (f) Co@MV

-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@MV

-1 0 1ε − ε

F (eV)

(h) Cu@MV

-1 0 1ε − ε

F (eV)

(i) Zn@MV

Figure 4.3: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied monovacancy (MV) in a (6,6) metallic armchairSWNT (TM@MV). Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓

2 (——) areshown. The constant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided for reference.

match with dips in transmission, which are Fano anti-resonances. While narrow peaks in theDOS denote localized states, broad peaks denote strong bonding or hybridized states.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@MV

-1 0 1ε − ε

F (eV)

(h) Cu@MV

-1 0 1ε − ε

F (eV)

(i) Zn@MV

Figure 4.4: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d tran-sition metal (TM) occupied monovacancy (MV) in a (6,6) metallic armchair SWNT (TM@MV).Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓2 (——) are shown, from Ref. [14].The constant transmission of 2 for a pristine (6,6) SWNT is provided for reference.


0

1

2

3

Tra

nsm

issio

n (a) Ti@DVI (b) V@DVI (c) Cr@DVI

0

1

2

Tra

nsm

issio

n (d) Mn@DVI (e) Fe@DVI (f) Co@DVI

-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVI

-1 0 1ε − ε

F (eV)

(h) Cu@DVI

-1 0 1ε − ε

F (eV)

(i) Zn@DVI

Figure 4.5: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied divacancy I (DVI) in a (6,6) metallic armchairSWNT (TM@DVI). Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓

2 (——) areshown, from Ref. [14]. The constant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided forreference.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVI

-1 0 1ε − ε

F (eV)

(h) Cu@DVI

-1 0 1ε − ε

F (eV)

(i) Zn@DVI

Figure 4.6: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d tran-sition metal (TM) occupied divacancy I (DVI) in a (6,6) metallic armchair SWNT (TM@DVI).Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓2 (——) are shown, from Ref. [14].The constant transmission of 2 for a pristine (6,6) SWNT is provided for reference.


0

1

2

3

Tra

nsm

issio

n (a) Ti@DVII (b) V@DVII (c) Cr@DVII

0

1

2

Tra

nsm

issio

n (d) Mn@DVII (e) Fe@DVII (f) Co@DVII

-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVII

-1 0 1ε − ε

F (eV)

(h) Cu@DVII

-1 0 1ε − ε

F (eV)

(i) Zn@DVII

Figure 4.7: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied divacancy II (DVII) in a (6,6) metallic armchairSWNT (TM@DVII). Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓

2 (——) areshown, from Ref. [14]. The constant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided forreference.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVII

-1 0 1ε − ε

F (eV)

(h) Cu@DVII

-1 0 1ε − ε

F (eV)

(i) Zn@DVII

Figure 4.8: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d transi-tion metal (TM) occupied divacancy II (DVII) in a (6,6) metallic armchair SWNT (TM@DVII).Results for spin ↑ (· · · ·), spin ↓ (– – –), and spin averaged ↑+↓2 (——) are shown, from Ref. [14].The constant transmission of 2 for a pristine (6,6) SWNT is provided for reference.


4.2.1 Divacancy II

For a transition metal (TM) occupying a divacancy II (DVII) in a (6,6) SWNT (TM@DVII),shown schematically in Fig. 4.2, there is an octahedral bonding of the transition metal atom,similar to the bonding in rutile metal oxides. Here, the TM bonding with carbons is dxz±dyz

√2

. Allthe bonds are equally strained due to the curvature of the nanotube. This means that all TM–Cbonds are equivalent. To put it in other words, there is one type of four-fold degenerate TM–Cbonding state.

In the case of Ti@DVII there is an occupied bonding of four C π—Ti dxz±dyz√

2. They are more

than 1 eV below the Fermi level, so they are not relevant for this study. There are also presentcompletely unoccupied dx2−y2 orbitals and anti-bonding Ti dxz±dyz

√2

orbitals. Only the t2g d orbitalsof the transition metals are important in the -1 eV to 1 eV range. These d orbitals are whatdetermines the binding and transmission for the TM@DVII system.

For the case of V@DVII, there is one extra electron in the dx2−y2 orbital, which is a non-bonding orbital. This extra electron yields a spin splitting in the DOS, as shown in Fig. 4.7. Thedx2−y2 orbital is, by symmetry, non-bonding and it is localized on the V atom.

In the case of Cr@DVII there are two extra electrons in the dx2−y2 orbital, instead of only one.The states get shifted down in energy and there is no state around the Fermi level, as seen inFig. 4.7.

With Mn@DVII, electrons start occupying the t2g anti-bonding orbitals. There are localizedspin states above the Fermi level. These spin unpaired d states, which are localized on thetransition metal atom, move down in energy as they are filled with more d-electrons. This iswhat we see in Fig. 4.7 for Fe@DVII and Co@DVII.

In the Ni@DVII, these localized states become spin-paired, being one of them completelyoccupied, and the other one completely empty, while for Cu@DVII both states are fully occupied,localized on the Cu atom, and spin-paired.

In the case of Zn@DVII, the d-orbitals are completely filled. The DOS closely resemblesthat of Ti@DVII, and the states are spin-paired, as shown in Fig. 4.7.


4.2.2 Divacancy I

For a transition metal (TM) occupying a divacancy I (DVI) in a (6,6) SWNT (TM@DVI), shownschematically in Fig. 4.2, the states are not as localized on the metal atom as for the TM@DVII,and are more strongly hybridized with the C π states. Due to the geometric configuration ofTM@DVI, the angle of two bonds with the axis of the nanotube is much smaller than the angle ofthe other two. We will call these tho different bonds near-axis and off-axis This difference in theorientation produces a difference in the strain the bonds withstand. The bonds that undergo morestress from the strain are the far-axis ones, while the other two undergo much less stress. Thismeans that there are two types of degenerate TM–C bonding states. This degeneration results inbroader peaks, compared to those obtained in TM@DVII, and a larger number of them.

Even with these differences it is still possible to see in Fig. 4.5 the filling of states, andtheir movement to lower energy, exactly like in the case of the TM@DVII. It is very likely thatthe differences in the strain contribute to the splitting of the spin degeneracy for Ti@DVI andNi@DVI.

The Ti d orbitals are half-filled through bonding with the C π orbitals, as is the 4s level. TheCu d orbitals are filled by taking an electron from the 4s level. So, in the case of Cu, bondingoccurs through the half-occupied 4s orbital. All of the Zn d orbitals are completely filled, as isthe 4s level. For these three systems, the DOS shown in Fig. 4.5 is very similar to that shown inFig. 4.7for the TM@DVII case. In the case of the Ti@DVI DOS there is a noticable difference,most probably due to the differences in strain for the two types of C–TM bonds (near-axis andoff-axis).

4.2.3 Monovacancy

For a transition metal (TM) occupying a monovacancy (MV) in a (6,6) SWNT (TM@MV),shown schematically in Fig. 4.2, the TM atom has three bonds with three C atoms, two of thebonds being shorter, stronger, and near-axis, and one longer, weaker, and completely perpendic-ular to the nanotube axis. Not only is this bond weaker, but it is also the one that withstandsthe biggest strain. From Fig. 4.3 we see that for Ti@MV and Ni@MV we have spin splitting,which may be attributable to the strain in the weaker C–TM bond. Fig. 4.3 also shows that forthe monovacancy the DOS shifts down in energy as the d orbitals on the TM atom are filled.


The differences between the bond lengths and strengths of the three C–TM bonds yields severalbroad peaks in the DOS.

As shown in Fig. 4.3, for Ti@MV, V@MV, and Cr@MV, we see a broad spin-unpaired peakcentered near the Fermi level. This peak is shifted down in energy for Mn@MV, which has five delectrons. At this point a more narrow peak, for a d-state more localized on the TM atom entersthe energy window and it is positioned at round 0.2 eV, which broadens for Co where it is partlyoccupied, and nearly spin-paired. For Ni this state goes further down in energy relative to theFermi level and becomes spin paired for Cu and Zn.

4.3 Target and Background Molecules

As mentioned before, since pristine SWNTs are rather inert, we have to dope them with TMatoms so it is possible for molecules to bond strongly with our system. In Fig. 4.9 we see theadsorption energies of the target and background molecules with the chosen nine 3d transitionmetals doping three types of vacancies in a (6,6) SWNT. The actual values are tabulated inAppendix A.

Since there is no clear trend with the adsorbed molecules (unlike the case for the clean tubes)we will describe the results in a more general way. Cu@DVI binds with all molecules in arather weak way, as seen in Fig. 4.9, so we can attribute any changes in the DOS from the cleanCu@DVI structure between -1 eV and 1 eV to states completely localized on the molecule. FromFig. 4.10 we only see two states localized on the O2 molecule between 0.6 and 0.8 eV above theFermi level. For CO in Cu@DVI we see a shifting of the DOS, and hybridization with theCu@DVI states. In particular, the broad peak below εF in the Cu@DVI DOS, which is bindingwith the C π states, is shifted down in energy by about 0.1 eV when CO is adsorbed on theCu@DVI site.

In the case of O2, the ground state has spin 2, with one of the electron occupying a pz—pz

anti-bonding π∗ orbital, and the other electron occupying a py—py anti-bonding π∗ orbital, asdepicted in Fig. (insert ref here). For O2 on Ti@DVI, V@DVI, and Cr@DVI, the pz − −pz anti-bonding π∗ orbital hybridizes with the transition metal atom’s dyz state, leaving the spin downcomponent of the other py—py anti-bonding π∗ state on the molecule unoccupied. This state isvisible as a sharp peak above the Fermi level. By symmetry the py—py anti-bonding O2 π

∗ state


Figure 4.9: Adsorption energies of target and background molecules for the 3d transition metalsin a monovacancy (top), divacancy I (middle), and divacancy II (bottom) in a (6,6) SWNT fromRef. [52].

cannot bond to the TM@DVI system.It should be noted that O2 is the only molecule with molecular states in the -1 eV to +1 eV

range relative to the (6,6) SWNT Fermi level. The O2 py—py anti-bonding π∗ states by symmetry


are forbidden from bonding with the TM@DVI and TM@DVII systems. On the other hand, O2

pz—pz anti-bonding π∗ state overlaps with the TM dyz state. For Cu and Zn, as the d orbitals arefilled, we find the O2 anti-bonding π∗ states are both spin unpaired, and exhibit little hybridizationwith the TM@VC site. For Cu and Zn, as the d orbitals are filled, we find the O2 anti-bondingπ∗ states are both spin unpaired, and make up the lowest unoccupied molecular orbital (LUMO).

Overall, the TM d states are shifted up and down in energy when a molecule is adsorbed,based on both charge transfer, and the symmetry of the molecule’s orbitals. This leads to a richresponse in the DOS of the TM@VC system upon adsorption of a molecule. The point is that theDOS and thus the transmission through the TM@VC systems are quite sensitive to the adsorptionof a molecule, and give a strong response. This makes the resistance through such active sites anexcellent sensing property.

4.4 Coverage and Resistance

For a sensor containing hundreds, if not thousands of sites, the fractional coverage of sites isreasonably described by the fractional coverage in thermodynamic equilibrium Θ at standardtemperature and pressure in terms of the target molecule concentration. Specifically for a suffi-ciently large ensemble of active sites, the fractional coverage of sites by a given adsorbate X in abackground set of molecules B at equilibrium is given by

Θ[X] =K[X]C[X]

1 +∑

Y∈B K[Y]C[Y], (4.1)

where K = k+/k− is the ratio of forward and backward rate constants for the adsorption reaction.This may be expressed as

K[X] = exp[Ebind[X] − TS gas[X]

kBT

], (4.2)

where S gas[X] is the gas phase entropy of species X [53], kB is Boltzmann’s constant and T is thetemperature. The values employed for the gas phase entropies S gas are tabulated in Table B.4,shown in Appendix B.

In Fig. 4.18 we see the coverages of both Ni and Cu active sites as a function of CO and NH3

concentration. Clearly, both Cu and Ni active sites will be sensitive to the adsorption of CO. On


0

5

10

DO

S (

eV

-1)

Ti@MV V@MV Cr@MV

0

5

DO

S (

eV

-1)

Mn@MV Fe@MV Co@MV

-1 0 1ε − ε

F (eV)

0

5

DO

S (

eV

-1)

Ni@MV

-1 0 1ε − ε

F (eV)

Cu@MV

-1 0 1ε − ε

F (eV)

Zn@MV

Figure 4.10: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied monovacancy (MV) in a (6,6) metallic armchairSWNT (TM@MV). Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·),O2 (– – – –), H2O (– – · – –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. Theconstant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided for reference.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@MV

-1 0 1ε − ε

F (eV)

(h) Cu@MV

-1 0 1ε − ε

F (eV)

(i) Zn@MV

Figure 4.11: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d tran-sition metal (TM) occupied monovacancy (MV) in a (6,6) metallic armchair SWNT (TM@MV).Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·), O2 (– – – –), H2O (– – ·– –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. The constant transmission of2 for a pristine (6,6) SWNT is provided for reference.


0

5

DO

S (

eV

-1)

Ti@DVI V@DVI Cr@DVI

0

5

DO

S (

eV

-1)

Mn@DVI Fe@DVI Co@DVI

-1 0 1ε − ε

F (eV)

0

5

DO

S (

eV

-1)

Ni@DVI

-1 0 1ε − ε

F (eV)

Cu@DVI

-1 0 1ε − ε

F (eV)

Zn@DVI

Figure 4.12: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied divacancy I (DVI) in a (6,6) metallic armchairSWNT (TM@DVI). Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·),O2 (– – – –), H2O (– – · – –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. Theconstant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided for reference.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVI

-1 0 1ε − ε

F (eV)

(h) Cu@DVI

-1 0 1ε − ε

F (eV)

(i) Zn@DVI

Figure 4.13: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d tran-sition metal (TM) occupied divacancy I (DVI) in a (6,6) metallic armchair SWNT (TM@DVI).Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·), O2 (– – – –), H2O (– – ·– –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. The constant transmission of2 for a pristine (6,6) SWNT is provided for reference.


0

5

DO

S (

eV

-1)

Ti@DVII V@DVII Cr@DVII

0

5

DO

S (

eV

-1)

Mn@DVII Fe@DVII Co@DVII

-1 0 1ε − ε

F (eV)

0

5

DO

S (

eV

-1)

Ni@DVII

-1 0 1ε − ε

F (eV)

Cu@DVII

-1 0 1ε − ε

F (eV)

Zn@DVII

Figure 4.14: Total density of states (DOS) in eV−1 versus energy ε in eV relative to the Fermilevel εF for a 3d transition metal (TM) occupied divacancy II (DVII) in a (6,6) metallic armchairSWNT (TM@DVII). Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·),O2 (– – – –), H2O (– – · – –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. Theconstant DOS at 2 eV−1 for a pristine (6,6) SWNT is provided for reference.


0

1

2

3

Tra

nsm

issio


0

1

2

Tra

nsm

issio


-1 0 1ε − ε

F (eV)

0

1

2

Tra

nsm

issio

n (g) Ni@DVII

-1 0 1ε − ε

F (eV)

(h) Cu@DVII

-1 0 1ε − ε

F (eV)

(i) Zn@DVII

Figure 4.15: Transmission versus energy ε in eV relative to the Fermi level εF through a 3d tran-sition metal (TM) occupied divacancy II (DVII) in a (6,6) metallic armchair SWNT (TM@DVII).Results for the clean TM@MV (———) and with adsorbed N2 (· · · · · ·), O2 (– – – –), H2O (– – ·– –), CO (– · – · –), and NH3 (– · · – · ·) are shown, from Ref. [14]. The constant transmission of2 for a pristine (6,6) SWNT is provided for reference.


[TM@VC]( [ ])sR [ +TM@VC]X

C X

Circuits

R∆

Figure 4.16: The coverage Θ and transmission probabilities T(ε) are used within simple circuittheory to estimate the scattering resistance Rs for species X on an active site (X+TM@VC),and hence derive the overall change in resistance ∆R for a particular active site (TM@VC) as afunction of the target molecules’ concentration C[X].

the other hand, the coverage of the Cu active sites is sensitive to NH3 concentration, while Niactive sites are not. This suggests that by combining the response of Cu and Ni active sites, wemay obtain a multifunctional sensor for both CO and NH3.

To measure the change in resistance produced by the adsorption of target and backgroundmolecules we first need to consider the change in conductance when a single molecule is ad-sorbed on an active metal site of an otherwise pristine SWNT. In Fig. 4.17 it is shown the cal-culated change in conductance relative to the active site with no adsorbate. It was possible toobserve a trend in the binding energies but there is no clear trend in the conductance. The sen-sitivity of the conductance is perhaps most clearly demonstrated by the absence of correlationbetween different types of vacancies, i.e. between the three panels in Fig. 4.17.

Close to the Fermi level, the conductance of a pristine armchair SWNT equals 2G0. Thepresence of the transition metal dopant yields to several dips in the transmission function knownas Fano anti-resonances [35]. The location and shape of these dips depend on the d-levels ofthe transition metal atom, the character of its bonding to the SWNT, and is further affected bythe presence of the adsorbate molecule. The coupling of all these factors is very complex and


O2

3NH

N2

H S2

H O2

CO

O2

3NH

N2

H S2

H O2

CO

O2

3NH

N2

H S2

H O2

G0

Cr Fe Co Ni Cu ZnMnTi V

Monovacancy

Div

acancy II

Div

acancy I

CO

−1.0

−0.8

−0.6

−0.4

−0.2

0.0

+0.2

+0.4

+0.6

+0.8

+1.0

Change in C

onducta

nce [ ]

Figure 4.17: Change in conductance upon adsorption of target and background molecules onto3d transition metals in a monovacancy (top), divacancy I (middle), and divacancy II (bottom) ina (6,6) SWNT from Ref. [52].

makes it difficult to estimate the value of the conductance. For the spin-polarized cases, we usethe spin-averaged conductance, i.e. G = (G↑ + G↓)/2.

We now estimate the resistance of a SWNT containing several impurities (a specific transition


10-4

10-2

100

Co

ve

rag

e o

f N

i@D

VII

1 10 100CO Concentration (ppm)

10-6

10-4

10-2

100

Co

ve

rag

e o

f C

u@

MV

1 10 100NH

3 Concentration (ppm)

CO

COClean

O2

O2

Clean

(a) (b)

NH3

CO

O2

O2

Clean

Clean

(c) (d)

Figure 4.18: Fractional coverage Θ in thermal equilibrium of (a,b) Ni in a divacancy II(Ni@DVII) and (c,d) Cu in a monovacancy (Cu@MV) versus (a,c) CO concentration and (b,d)NH3 concentration in a background of air at room temperature and 1 bar of pressure [51].


10-2

10-1

100

101

102

103

104

∆R

(Ω

/ A

ctive

Site

)

Ni@DVIICu@MV

1 10 100CO Concentration (ppm)

0

2

4

6

8

10-2

∆R

(m

Ω /

Active

Site

)

1 10 100NH

3 Concentration (ppm)

(a) (b)

(c) (d)

Figure 4.19: Change in resistance ∆R in (a,b) Ω and (c,d) mΩ per active site versus (a,c) COand (b,d) NH3 concentration in ppm for Ni in a divacancy II (Ni@DVII, – – – –) and Cu in amonovacancy (Cu@MV, ———) of a (6,6) carbon nanotube. The reference concentrations ofCO and NH3 are 0.1 and 0.01 ppm, respectively, in a background of air at room temperature and1 bar of pressure.


metal dopant with different molecular adsorbates). Under the assumption that the electron phase-coherence length, lφ, is smaller than the average distance between the dopants, d, we may neglectquantum interference and obtain the total resistance by adding the scattering resistances due toeach impurity separately. The scattering resistance due to a single impurity is given by

Rs[X] =1

G[X]−

12G0

, (4.3)

where G[X] = T [X+TM@VC](εF) is the Landauer conductance of the pristine SWNT with asingle metal dopant occupied by molecule X, i.e. the transmission at the Fermi level εF, 1/(2G0)is the contact resistance of a (6,6) SWNT, and G0 ≡ 2e2/h is the quantum of conductance. Thecalculated conductances are provided in Tables B.1, B.2, and B.3 for a molecule X adsorbed ona 3d transition metal occupied monovacancy, divacancy I, and divacancy II, respectively.

We may now obtain the average change in resistance ∆R of an active site as a function oftarget molecule concentration. As discussed in Ref. [36], this change in resistance is reasonablywell described by

∆R ≈∑

X

Rs[X](Θ[X,C] − Θ[X,C0]), (4.4)

where C is the concentration, C0 is the concentration at standard temperature and pressure.In Fig. 4.19 we show ∆R for a single active site (Ni or Cu) as a function of target molecule

concentration (CO or NH3). Keeping in mind that a change in concentration from 1 ppm to 50ppm amounts to a change from allowable to toxic concentrations for both CO and NH3, we seethat this sensor design should work effectively for both target molecules.

Chapter 5

Conclusions

To summarize the work done in this Master’s Thesis, we have explained some basic concepts ofmolecular electronics, putting special emphasis on the density functional theory (DFT) and thenon-equilibrium Green’s function methodology, since these are the tools we used in this study.We have defined what carbon nanotubes are and explained the reasons for choosing 3d TM doped(6,6) SWNTs as our candidate systems for chemical sensors.

Analyzing the obtained results, it is clear that hybridization between the adsorbed molecularstates and d states on the transition metal dopant determine the changes in transmission. We canalso see that only O2 has states in the -1 eV to 1 eV range, relative to the Fermi level. Sincethe TM d states get shifted up and down in energy upon the adsorption of a molecule, there arestrong changes in transmission, which is the reason for using these devices as chemical sensors.

To conclude we can propose a combined Ni and Cu doped device, which would be effectiveto detect and differentiate between both CO and NH3.

45

Appendix A

Adsorption Energies

Table A.1: Adsorption energies ∆Eads in eV for N2, O2, H2O, CO, NH3, and H2S on a 3d transi-tion metal occupied monovacancy (TM@MV), from Ref. [14].

Ti V Cr Mn Fe Co Ni Cu ZnN2 -0.61 -0.76 -0.57 -0.70 -0.77 -0.60 -0.49 -0.34 -0.51O2 -3.16 -3.39 -2.61 -2.57 -2.17 -1.88 -2.00 -1.02 -0.90H2O -1.09 -1.08 -0.97 -0.89 -0.84 -0.64 -0.54 -0.52 -0.62CO -0.89 -1.21 -1.06 -1.38 -1.54 -1.23 -1.16 -0.97 -1.07NH3 -1.39 -1.46 -1.35 -1.32 -1.31 -1.03 -0.94 -0.89 -1.12H2S -0.78 -0.88 -0.77 -0.78 -0.88 -0.58 -0.69 -0.63 -0.67

47


Table A.2: Adsorption energies ∆Eads in eV for N2, O2, H2O, CO, NH3, and H2S on a 3d transi-tion metal occupied divacancy I (TM@DVI), from Ref. [14].

Ti V Cr Mn Fe Co Ni Cu ZnN2 -0.58 -0.65 -0.65 -0.38 -0.46 -0.63 -0.15 -0.01 -0.01O2 -2.72 -3.41 -2.74 -1.92 -1.67 -1.42 -0.61 -0.06 -0.12H2O -1.21 -1.19 -0.79 -0.49 -0.51 -0.33 -0.19 -0.05 -0.25CO -0.80 -1.11 -1.14 -1.06 -1.62 -1.90 -1.10 -0.12 -0.04NH3 -1.51 -1.58 -1.11 -0.89 -0.91 -0.75 -0.40 -0.20 -0.53H2S -0.87 -0.78 -2.13 -0.50 -0.64 -0.59 -0.31 -0.04 -0.17

Table A.3: Adsorption energies ∆Eads in eV for N2, O2, H2O, CO, NH3, and H2S on a 3d transi-tion metal occupied divacancy II (TM@DVII), from Ref. [14].

Ti V Cr Mn Fe Co Ni Cu ZnN2 -0.50 -0.62 -0.41 -0.38 -0.60 -0.39 -0.07 -0.02 -0.11O2 -2.27 -2.64 -2.35 -2.17 -1.48 -1.31 -0.53 -0.09 -0.18H2O -1.15 -1.20 -0.74 -0.64 -0.49 -0.55 -0.16 -0.05 -0.17CO -0.78 -1.10 -1.05 -0.96 -1.49 -1.30 -0.82 0.02 0.01NH3 -1.45 -1.47 -1.14 -1.09 -0.99 -0.68 -0.23 -0.13 -0.42H2S -0.80 -0.83 -0.62 -0.62 -0.56 -0.50 -0.10 -0.01 -0.10

Appendix B

Conductance

Table B.1: Conductance G[X] in G0 ≡ 2e2/h for a 3d transition metal occupied monovacancy(TM@MV) with N2, O2, H2O, CO, NH3, and H2S adsorbed. The conductance of the cleanTM@MV is provided for comparison, from Ref. [14].

Ti V Cr Mn Fe Co Ni Cu ZnN2 1.44 1.63 1.44 1.72 1.88 1.19 1.40 0.97 1.16O2 1.44 1.78 1.72 1.66 1.56 1.83 1.33 1.13 1.69H2O 1.42 1.31 1.64 1.39 1.50 1.47 1.18 0.97 0.99CO 1.31 1.61 1.61 1.59 1.33 1.09 1.58 0.97 1.79NH3 1.57 1.37 1.68 1.47 1.51 1.50 1.39 1.05 1.01H2S 1.40 1.51 1.64 1.51 1.50 1.43 1.32 0.98 1.13Clean 1.40 1.41 1.38 1.77 1.87 1.08 1.06 1.31 1.02

49


Table B.2: Conductance G[X] in G0 ≡ 2e2/h for a 3d transition metal occupied monovacancy(TM@DVI) with N2, O2, H2O, CO, NH3, and H2S adsorbed. The conductance of the cleanTM@DVI is provided for comparison, from Ref. [14].


Table B.3: Conductance G[X] in G0 ≡ 2e2/h for a 3d transition metal occupied monovacancy(TM@DVII) with N2, O2, H2O, CO, NH3, and H2S adsorbed. The conductance of the cleanTM@DVII is provided for comparison, from Ref. [14].


APPENDIX B. CONDUCTANCE 51

Table B.4: Equilibrium atmospheric concentrations C0[X] at 4% humidity and gas phase en-tropies S gas[X] in meV/K at T = 300 K where X is one of the background molecules N2, O2, andH2O or one of the target molecules CO, NH3, and H2S, from Ref. [14]. Experimental gas phaseentropies are taken from Ref. [53]

X C0[X] S gas[X] X C0[X] S gas[X]N2 74.96% 1.988 meV/K CO 96.00 ppb 2.050 meV/KO2 20.11% 2.128 meV/K NH3 16.32 ppb 2.000 meV/KH2O 4.00% 1.959 meV/K H2S 0.96 ppb 2.136 meV/K

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