Dynamics of Carriers and Photoinjected Currents in Carbon Nanotubes and Graphene

by

Ryan William Newson

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy

Graduate Department of Physics University of Toronto

© Copyright by Ryan Newson 2010

ii

Dynamics of Carriers and Photoinjected Currents

in Carbon Nanotubes and Graphene

Ryan William Newson

Doctor of Philosophy

Graduate Department of Physics University of Toronto

2010

Abstract

This thesis reports results from the investigation of optically-induced carrier dynamics in

graphite and graphitic carbon nanostructures. In this first set of experiments, the dynamics of

photo-excited carriers in exfoliated graphene and thin graphitic films are studied by optical

pump-probe spectroscopy. Samples ranging in thickness from 1 to 260 carbon layers are

deposited onto an oxidized silicon substrate. Time-resolved reflectivity and transmissivity are

measured at 1300 nm, following excitation by 150 fs, 800 nm pump pulses at room temperature.

Two time scales are identified over which the extracted transient dielectric function returns to its

quiescent value. A fast decay time of ~200 fs in graphene is associated with hot phonon

emission and increases to ~300 fs for thicknesses greater than only a few carbon layers. The

slow decay time, associated with hot phonon interaction and/or carrier recombination, increases

more gradually, from ~2.5 to 5 ps over ~30 layers. A simple model suggests the thickness

dependence of the slow decay time is likely a result of thermal diffusion into the substrate.

In the second set of experiments, coherently-controlled two-colour injection photocurrents are

generated via quantum interference of single- and two-photon absorption in bulk graphite and a

variety of single-walled carbon nanotube samples, such as a CVD-grown aligned forest of

iii

nanotubes (tube diameter dt = 2.5 ± 1.5 nm), and both arc discharge (dt = 1.44 ± 0.15 nm) and

HiPco (dt = 0.96 ± 0.14 nm) nanotube films separated by electronic type (metallic vs.

semiconducting). At pump wavelengths of 1500 and 750 nm, the emitted terahertz radiation is

used to estimate a peak current density of ~12 kA cm-2 in graphite and a peak current of ~8 nA

per nanotube. From the dependence of the injected current on pump polarization, the relative

values of the current injection tensor elements are measured, and information is gained on the

alignment and birefringence of the nanotube samples. The dependence of the injected current on

pump wavelength implies that the currents are likely based on band-band electronic transitions

and not on excitonic effects, which govern most linear optical processes.

iv

Acknowledgments

This document would not be possible without the support of many people. I want to thank Henry

van Driel for his guidance through his supervision and discussions, and for the generous use of

his laboratory and equipment. I also thank John Sipe and Dwayne Miller for their helpful input

at my committee meetings, and Ted Sheppard for his service as the Chair of the department.

Thank you to Ken Burch, Joseph Thywissen, and Hoi-Kwong Lo for their participation at my

defenses, and Ted Norris for providing helpful comments and travelling from the University of

Michigan to attend my final defense. Financial support was provided by NSERC, OGSST, and

the Walter C. Sumner Foundation.

A great variety of assistance and advice was provided by Markus Betz. Other graduate students

such as Jesse Dean, Jean-Michel Ménard, and Christian Sames assisted with experiments and

laser system maintenance. I also need to acknowledge the contribution of several undergraduate

and visiting students who assisted me such as Sida Wang, Susanne Scharf, and Ben Schmidt.

Special thanks go to my collaborators in Mark Hersam’s group at Northwestern University,

especially Alex Green, for providing me with the sorted single-wall nanotube samples, and to

Paul Finnie at the Institute for Microstructural Sciences at NRC for providing me with the

aligned forest of nanotubes. Alexandra Haagaard and Richard Loo from Cynthia Goh’s group in

the Chemistry department helped me with Atomic Force Microscopy measurements. Ilya

Gourevich at the Centre for Nanostructure Imaging in the Chemistry department helped me with

Scanning Electron Microscopy measurements.

On a personal note, I’d like to say a special thank you to my parents, Judy and Rick Newson, for

always supporting me in all of my endeavours.

This thesis is first and foremost dedicated to my beloved wife Michele without whom none of

this would have been possible. Her constant love and support through difficult times allowed me

to achieve my dream, and her sacrifices will always be remembered. I love you.

Ryan Newson

Toronto, 2010

v

Table of Contents Abstract ........................................................................................................................................... ii

Acknowledgments.......................................................................................................................... iv

Table of Contents............................................................................................................................ v

List of Tables ................................................................................................................................ vii

List of Figures .............................................................................................................................. viii

Introduction..................................................................................................................................... 1

1.1 Background of graphene and carbon nanotubes ................................................................. 3

1.2 Background of nonlinear optics and coherence control...................................................... 5

1.3 Research objective .............................................................................................................. 8

1.4 Thesis outline .................................................................................................................... 10

Theoretical Background................................................................................................................ 11

2.1 Overview........................................................................................................................... 11

2.2 Carrier dynamics in graphene and graphite ...................................................................... 11

2.2.1 Graphene and graphite .......................................................................................... 12

2.2.2 Silicon ................................................................................................................... 17

2.2.3 Reflection and transmission from a multilayer structure...................................... 18

2.3 Coherent control of photocurrents in carbon nanostructures............................................ 24

2.3.1 Current injection via band-band transitions.......................................................... 24

2.3.2 Excitonic effects in current injection .................................................................... 38

2.3.3 Terahertz radiation emission and collection ......................................................... 43

2.3.4 Electro-optic sampling .......................................................................................... 44

Experimental Techniques and Apparatus ..................................................................................... 47

3.1 Overview........................................................................................................................... 47

3.2 Samples ............................................................................................................................. 47

vi

3.2.1 Exfoliated graphite and graphene ......................................................................... 48

3.2.2 Vertically-aligned carbon nanotube forest............................................................ 52

3.2.3 Unaligned sorted SWNT thick films..................................................................... 55

3.3 Optical sources.................................................................................................................. 61

3.3.1 Optical parametric oscillator................................................................................. 61

3.3.2 Optical parametric amplifier ................................................................................. 61

3.4 Experimental apparatus..................................................................................................... 62

3.4.1 Confocal pump-probe experiment ........................................................................ 62

3.4.2 Coherent control of photocurrents experiment ..................................................... 63

Results and Discussion ................................................................................................................. 68

4.1 Overview........................................................................................................................... 68

4.2 Carrier dynamics in exfoliated thin graphitic films .......................................................... 68

4.2.1 Time-resolved reflection and transmission........................................................... 69

4.2.2 Time-resolved changes in material properties ...................................................... 70

4.2.3 Decay times........................................................................................................... 73

4.3 Coherent control of photocurrents in carbon nanostructures............................................ 77

4.3.1 Graphite................................................................................................................. 78

4.3.2 Aligned SWNT forest ........................................................................................... 83

4.3.3 Unaligned sorted SWNT thick films..................................................................... 91

Conclusions................................................................................................................................. 103

5.1 Summary ......................................................................................................................... 103

5.2 Outlook ........................................................................................................................... 106

References................................................................................................................................... 108

vii

List of Tables Table 2.1 : Selection rules for optically active excitons in SWNTs. ………….………………..40

Table 3.1 : Table of carbon nanotube samples studied. ………………………………………..57

Table 3.2 : Measured thicknesses for each unaligned SWNT film sample. See Table 3.1 for

sample code definitions……………………………………………………...…………………..59

viii

List of Figures Figure 1.1 : Structures of (a) graphene; (b) AB-graphite; and (c) a (9,3) single-walled carbon

nanotube (SWNT). [43] .................................................................................................................. 4

Figure 2.1 : (a) Crystal lattice structure; and (b) reciprocal lattice structure of graphene. .......... 13

Figure 2.2 : 3D plot of (bonding) and * (antibonding) electronic bands in graphene. ........... 14

Figure 2.3 : Electronic band structure for (a) graphene, and (b) graphite [62]............................ 15

Figure 2.4 : Illustration of (a) graphene band structure around K and K’ points; and (b) transient

behaviour of excited carrier population densities Ne and Nh......................................................... 16

Figure 2.5 : Diagram of multilayer structure with forward- and backward-scattering electric field

components shown........................................................................................................................ 19

Figure 2.6 : Approximate luminance and apparent colour of exfoliated graphene and thin

graphitic films with respect to number of layers N, for (a) 0-400 layers; and (b) zoomed in for 0-

15 layers. ....................................................................................................................................... 22

Figure 2.7 : Typical band structure for a direct-gap semiconductor and illustration of coherent

control of photoinjected currents. The sizes of the circles indicate the electron/hole populations

injected at those wavevectors........................................................................................................ 25

Figure 2.8 : Diagram of the wrapping of a single-walled carbon nanotube, with chiral vector

(n,m) = (4,2) [12]. ......................................................................................................................... 31

Figure 2.9 : (a) Diagram of the band structure cutting lines of a (4,2) SWNT in reciprocal space;

and (b) associated band structure taken from the intersection of the cutting lines with the

graphene band structure. ............................................................................................................... 32

Figure 2.10 : (a) Simplified; and (b),(c) more accurate diagram of current injection in a SWNT

for the case when the 2 beam is polarized along the tube axis and the beam is polarized (b)

also along the tube axis; and (c) perpendicular to the tube axis. .................................................. 34

ix

Figure 2.11 : (a) Diagram of single- and two-photon active exciton energies for E11 transition

[75]; and measured (b) two-photon absorption, and (c) single-photon absorption in a (7,5)

nanotube where 011 2( ) 1.190E A eV [77]. ......................................................................................... 41

Figure 2.12 : Collection filter function for the off-axis parabolic mirrors used (N.A. = 0.50) with

100 μm FWHM spot diameter. ..................................................................................................... 44

Figure 2.13 : Magnitude (solid) and phase (dashed) of the EO response filter function for 500

μm thick ZnTe and 800 nm probe wavelength [83]...................................................................... 46

Figure 3.1 : Diagram of the layered structure of the exfoliated graphene and thin graphitic film

samples (not to scale).................................................................................................................... 48

Figure 3.2 : Microscope image of a sample showing areas with 1-layer (B), 2-layer (D), and 3-

layer (C) thickness at 400X magnification. The pixel intensity of the “G” colour component

(RGB) is plotted along the blue line (inset). ................................................................................. 49

Figure 3.3 : Raman spectra of a few of the exfoliated samples studied, with thicknesses of 1

layer (graphene), 2 layers, 3 layers, and hundreds of layers (bulk graphite). Spectra are shifted

vertically for clarity....................................................................................................................... 50

Figure 3.4 : Height profile of one exfoliated graphite sample plotted in 3D, as measured using

Atomic Force Microscopy. ........................................................................................................... 52

Figure 3.5 : Images of the side facet of the vertically-aligned forest of carbon nanotubes, taken at

(a) 20,000X and (b) 200,000X with a Hitachi S-5200 SEM. ....................................................... 53

Figure 3.6 : Histogram of 50 nm long tube segment angles (alignment distribution) measured

from Figure 3.5b (black squares) and Gaussian fit function with a = 19.5o (blue line).............. 54

Figure 3.7 : Normalized absorbance spectra for unaligned (a) arc discharge SWNT sample (dt =

1.44 ± 0.15 nm); and (b) HiPco SWNT sample (dt = 0.96 ± 0.14 nm) in solution after sorting

(courtesy of NWU). ...................................................................................................................... 56

Figure 3.8 : Surface profile contour plot of the unsorted HiPco SWNT film (CO U-X-SWNT)

sample. A plot of surface height along the yellow cutting line is shown in the top panel........... 58

x

Figure 3.9 : Measured absorption coefficient for (a) arc discharge, and (b) HiPco SWNT

films. Semiconducting (red), metallic (green), and unsorted (black) are shown. ........................ 60

Figure 3.10 : Diagram of the pump-probe experimental setup to study carrier dynamics in

graphene and graphite. BS = beamsplitter; PD = biased Ge photodiode; obj = objective lens. .. 63

Figure 3.11 : Diagram of the coherent control of photo-injected currents experimental setup. .. 64

Figure 3.12 : Measured intensity spectrum 2( )S of THz pulse from current injection in InP

(blue dashed) vs. theoretical curve of the product of the two THz filter functions,

|fcoll()*fEO()|² (black solid). ...................................................................................................... 66

Figure 3.13 : (a) Temporal trace ( )S , and (b) intensity spectrum 2( )S of a typical THz pulse

(from current injection in InP) under high humidity, 55.7% R.H. (thin blue), and low humidity, <

0.1% R.H. (thick black) conditions............................................................................................... 67

Figure 4.1 : Time-dependent differential reflectivity R/R and transmissivity T/T of exfoliated

samples with the number of carbon layers indicated. ................................................................... 69

Figure 4.2 : Time-dependent extracted R/R and I/I for the top carbon layer of the

exfoliated samples with number of carbon layers indicated. ........................................................ 72

Figure 4.3 : Extracted fast and slow time constants from time-resolved I/I for the exfoliated

graphene and thin graphitic film samples as a function of the number of layers. ........................ 73

Figure 4.4 : Decay time of a change in a hypothetical material property of graphite affecting

reflection and transmission (averaged over depth) assuming diffusion (D = 4x10-6 cm² s-1),

substrate coupling (S = 104 cm s-1), and decay (D = 5 ps). .......................................................... 76

Figure 4.5 : Measured THz radiation field Š() from graphite for 2 = 750 nm as a function

of time delay and phase parameter between co-polarized pump beams. A single temporal

trace (top panel) and phase parameter plot (right panel) are shown along the yellow cutting lines.

....................................................................................................................................................... 79

xi

Figure 4.6 : Horizontal (blue dots) and vertical (red squares) components of the relative peak

THz amplitude from the graphite sample. Solid lines represent theory curves using equation

(2.35) with /xyyx xxxx –0.19. Data shown for = 190o…350o is repeated from = 10o…170o. 81

Figure 4.7 : Measured THz radiation field Š() from the V-SWNT sample for 2 = 700 nm

as a function of time delay and phase parameter between pump beams co-polarized along

the tube axis. A single temporal trace (top panel) and phase parameter plot (right panel) are

shown along the yellow cutting lines............................................................................................ 83

Figure 4.8 : Normalized relative peak THz amplitudes in V-SWNT sample as a function of

normalized and 2 beam average power with the power of the other beam held constant. The

solid lines indicate linear and square root power laws. ................................................................ 85

Figure 4.9 : Relative peak THz amplitude of V-SWNT sample as incident spot is moved

vertically up the nanotube forest (black squares). Also shown (blue line) is the convolution of

the spatial intensity profile (100 μm FWHM) and a square function representing the 170 μm-high

nanotube forest.............................................................................................................................. 86

Figure 4.10 : Horizontal component of the relative peak THz amplitude (black squares) from the

V-SWNT sample as the sample is rotated from 0 = 0o (horizontal tube alignment) to 0 = 90o

(vertical tube alignment). Both pump beams are polarized along the horizontal direction. The

solid line represents a theory curve using equation (4.13) with / –0.025, / 17.9, and

/t t 1.01. .................................................................................................................................... 87

Figure 4.11 : Horizontal (blue dots) and vertical (red squares) components of the relative peak

THz amplitude from the V-SWNT sample when mean tube direction is aligned (a) horizontally

(0 = 0), and (b) vertically (0 = 90o). Solid lines represent theory curves using equation (4.13)

with / –0.025, / 17.9, and /t t 1.01. Data shown for = 190o…350o is repeated

from = 10o…170o....................................................................................................................... 90

Figure 4.12 : Horizontal (blue dots) and vertical (red squares) components of the relative peak

THz amplitude from the semiconducting arc discharge X-SWNT sample. Solid lines represent

xii

theory curves using equation (2.46) with / –0.068. Data shown for = 190o…350o is

repeated from = 10o…170o. ....................................................................................................... 93

Figure 4.13 : Example of time/phase THz plot on AD S-X-SWNT sample showing some “in

between signal”. ............................................................................................................................ 94

Figure 4.14 : Normalized theoretical peak THz amplitudes (current magnitudes) as a function of

second-harmonic wavelength 2 for InP (solid black) and GaSb (dashed blue)......................... 96

Figure 4.15 : Relative THz field amplitude (injected current magnitude) (black squares) and

corresponding absorbance data from SWNT solutions (red lines) as a function of second-

harmonic wavelength 2 for semiconducting (a) arc discharge; and (b) HiPco SWNT films. ... 97

Figure 4.16 : Injection current absorption coefficient for arc discharge (AD) (thick red) and

HiPco (CO) (thick black) SWNT films, as well as the phase walk-off k for both samples, each

plotted as a function of second-harmonic wavelength 2. .......................................................... 99

Figure 4.17 : Normalized theoretical models of current injection tensor element || (thin black)

and total peak THz amplitude (current magnitude) THzinjE (thick blue), each as a function of second-

harmonic wavelength 2, for semiconducting (a) arc discharge; and (b) HiPco SWNT film

samples. The THzinjE curves are meant to be compared with Figure 4.15...................................... 101

1

Chapter 1

Introduction

The field of nanotechnology is still in its infancy, but before greater strides are made, it is

important to fully understand its building blocks. In the last few years, scientists and engineers

have started to realize the possibilities that carbon nanostructures such as graphene and carbon

nanotubes may bring to this exciting new field. This thesis explores the ultrafast dynamics of

photo-injected carriers and the coherent control of photo-injected currents in graphene, graphite,

and single-walled carbon nanotubes.

Graphene is a single layer of carbon atoms arranged in a hexagonal or “honeycomb” lattice.

This relatively simple material has only recently been the focus of intense theoretical and

experimental investigation due to its interesting physical, electronic, and optical properties.

These exciting properties are typically a result of the unique linear dispersion of the electronic

valence and conduction bands that meet at the so-called Dirac point [1-4]. This linear band

structure gives rise to interesting effects such as the anomalous half-integer quantum Hall effect

[5, 6] and extremely high electron and hole mobilities at room temperature [7, 8]. Graphene is

also intriguing optically as it has been shown to absorb a constant 2.3% of visible and

near-infrared light [9-11], where is the fine structure constant.

A graphene sheet also represents the building block for other important carbon nanostructures.

Carbon nanotubes are long cylinders of graphene with diameters on the order of nanometers.

The band structure of these materials depends on the band structure of graphene and on how the

sheet is rolled into the tube (its chirality) [12]. These materials have also exhibited interesting

electronic properties, such as high electron mobilities and the ability to carry very high electric

current densities [13, 14].

2

Despite research in carbon nanostructures being only in the early stages, development of

electronic and optical devices has already begun. Single-walled carbon nanotubes have been

integrated into field-effect transistor designs [15, 16], while graphene has seen applications as a

photodetector [17] and an ultrafast saturable absorber for passive laser mode-locking [18]. In

order to continue this work it is important to have a thorough understanding of the optical and

electronic properties of such carbon nanostructures, both static and dynamic. An ideal way to

study the dynamics of photo-excited carriers in a material is via ultrafast pump-probe

experiments, whereby the material is excited by an intense pump pulse and the change in

reflectivity and/or transmissivity is measured by a weak time-delayed probe pulse.

Devices that utilize graphene or carbon nanotubes will undoubtedly involve the movement of

charge (a current) through the material. At this time, the most common method for the

generation or detection of an electrical current is through the use of electrodes that physically

contact the material [19-21]; often it is the development of suitable contacts that hinders research

in this area. The use of optical, noncontact methods of generating an electrical current may

ultimately allow for the interfacing of photonics and nanoelectronics [22, 23]. Optical coherence

control is one such method that has been shown to inject significant current densities in three-

dimensional bulk semiconductors and two-dimensional quantum wells [24-29]. In such a

scheme, quantum interference between optical absorption processes (such as single- and two-

photon absorption) breaks the material symmetry, allowing electric currents, and, with suitable

optical polarizations, even spin currents to be created [30, 31]. The photocurrent magnitude and

direction can be controlled by varying the phase difference between the two pump beams as well

as their polarization.

The coherently controlled electric currents are optically-injected over the time scale of the pump

pulse width. When this lies in the picosecond or sub-picosecond regime, the transient nature of

the current density gives rise to electromagnetic radiation emission in the terahertz (THz) range.

Common techniques for measuring THz radiation pulses can therefore be used to detect this

emitted radiation and hence the injected current. In this manner both generation and detection

methods are noncontact.

Before the inception of the experiments discussed in this thesis, neither pump-probe studies on

graphene nor the injection and coherent control of photocurrents in carbon nanostructures had

3

been reported. The experiments discussed herein attempt to fill that gap. But first, a brief

background of carbon nanostructures is presented, along with an overview of coherence control

and its place in the broad field of nonlinear optics.

1.1 Background of graphene and carbon nanotubes Graphite is an extremely common material and is composed of many sheets of graphene bonded

together with van der Waals forces. It almost always appears in the AB Bernal stacking

configuration shown in Figure 1.1b. The single-layer form, graphene (see Figure 1.1a), had been

studied theoretically as early as 1947 [32], but it wasn’t until 2004 that Geim and Novoselov

reported producing electrically-isolated micron-sized samples of graphene on an oxidized Si

substrate by micromechanically cleaving bulk graphite with simple scotch tape [8]. In the past

few years, other types of graphene samples have been fabricated, such as colloidal graphene

suspensions by chemical exfoliation of graphite [33, 34], and epitaxial graphene by the high

temperature graphitization of bulk SiC. Epitaxial graphene consists of many graphene layers

stacked in such a way to apparently eliminate interlayer coupling [35]. The sudden availability

of graphene samples finally allowed experimental investigation of its exciting properties, many

of which were already theoretically predicted.

Prior to the investigations reported in this thesis, electrical transport experiments on graphene

using contacts were common, but optical experiments on graphene were limited. Carrier

dynamics had been studied in graphite by Seibert et al. [36] via an optical pump-probe setup, but

not in graphene. However, in the last few years, several other groups have attempted to study

such dynamics in graphene. Butscher et al. [37] first theoretically calculated the relaxation

dynamics of photo-excited electrons in graphene. Dawlaty et al. [38] and Sun et al. [39]

independently performed optical pump-probe transmission experiments on epitaxial graphene,

while Bruesing et al. [40] conducted similar experiments in free-standing graphite films. Still

none of these investigated exfoliated graphene nor how the ultrafast dynamics evolved in the

transition from graphene to graphite. Nonlinear optical experiments in graphene are even less

common, but a few reports have appeared in recent years [41, 42].

4

Figure 1.1 : Structures of (a) graphene; (b) AB-graphite; and (c) a (9,3) single-walled carbon nanotube (SWNT). [43]

The results from both theoretical calculations on graphene and experimental pump-probe

measurements on graphite and epitaxial graphene give an indication of the various scattering

mechanisms involved in graphene. After intense optical excitation, a non-equilibrium population

of electrons exists in the conduction band with a similar population of holes in the valence band.

These carriers quickly thermalize via carrier-carrier scatting, whereby high temperature Fermi-

Dirac carrier distributions are established. Once thermalized, the carrier distributions cool,

mainly due to intraband optical and acoustic phonon scattering, on the timescale of hundreds of

femtoseconds. Electrons and holes simultaneously begin to recombine with one another,

ultimately restoring the material to its original state before photo-excitation.

In order to achieve the necessary temporal resolution for the study of carrier thermalization,

cooling and recombination via optical pump-probe techniques, ultrafast (sub-picosecond) laser

pulses are required. The development of the all-solid-state self-mode-locking Ti:Sapphire laser

in the 1990’s, which routinely produces pulses with 3-150 fs duration [44, 45], has opened up the

field of ultrafast spectroscopy. These extremely short laser pulses not only represent a tool for

probing dynamics with a very high temporal resolution, but also provide the ultrahigh peak

intensities required for nonlinear optical experiments.

5

The 1990’s was also an exciting time in materials research, following Iijima's discovery of multi-

walled carbon nanotubes formed by arc discharge in 1991 [46], and the prediction by Mintmire,

Dunlap, and White that then-theoretical single-walled carbon nanotubes (see Figure 1.1c) would

exhibit remarkable conducting properties [47]. Since then, these nanometer-thick tubes of

carbon atoms have been studied extensively both theoretically and experimentally to reveal their

unique physical, electronic, and optical properties. In addition to the aforementioned method of

fabrication whereby nanotubes naturally form on graphite electrodes during arc discharge, other

fabrication techniques have been developed. Nanotubes can be made by the laser ablation of

bulk graphite, from the high-pressure flow carbon monoxide over catalytic metallic clusters

(HiPco), or by chemical vapour deposition methods on a substrate.

It is clear from Figure 1.1 that graphene is the basic building block for both graphite and carbon

nanotubes. In fact, the electronic band structures of these 3D allotropes of carbon, which often

determine their electronic and optical properties, are derived from that of 2D graphene. In

graphite, it is the weak interaction between adjacent layers that alters the electron dispersion

from linear to quadratic in momentum space. In a single-walled carbon nanotube, the allowed

electronic states are a subset of those in graphene, depending on how the tube was rolled.

1.2 Background of nonlinear optics and coherence control Since the first experimental demonstration of the ruby laser in 1960 by Maiman [48], the study

of the nonlinear response of materials to high intensity excitation has received considerable

research attention. It has long been known that when a material is exposed to an oscillating

electric field E at frequency , an electric polarization density (dipole moment per unit volume

per unit frequency) is generated that oscillates at the same frequency:

0 1( ) ( ; ) : ( ) P E (1.1)

where 0 is the permittivity of free space and 1 is a matrix that represents the material’s first-

order (linear) optical response. However, under high enough intensities, the material response

becomes nonlinear with respect to the electric field amplitudes involved, and the polarization

density can be expressed as a power series in these fields. The second-order correction involves

6

the mixing of two frequencies 1 and 2:

2 1 2 0 2 1 2 1 2 1 1 2 2( ) ( ( ); , ) : ( ) ( ) P E E (1.2)

where 2 is the second-order nonlinear susceptibility tensor. However, this second-order

correction is zero if the material is centrosymmetric.

The third-order correction to the polarization density is given by:

3 1 2 3 0 3 1 2 3 1 2 3 1 1 2 2 3 3( ) ( ( ); , , ) : ( ) ( ) ( ) P E E E (1.3)

where 3 is the third-order nonlinear susceptibility tensor. For centrosymmetric materials, this

is the first nonlinear term in the expansion of P, and gives rise to a variety of effects such as third

harmonic generation, the Kerr effect, four-wave mixing, and many more. To study these effects,

ultrafast lasers are required in order to provide the high field amplitudes needed for the second-

and third-order polarization terms to become significant.

An intriguing class of ultrafast nonlinear optical effects is referred to as coherence control,

whereby certain physical processes and properties of matter are controlled via coherent

interactions using the phase properties of laser pulses. Examples of this are numerous, and

include the control of light absorption of a material [49], the control of the direction of

photoemission [50, 51], the control of chemical reactions [52], the control of exciton populations

in quantum wells [53], and many more.

In bulk semiconductors, the coherent control of both charge and spin populations and currents

has been demonstrated [25-31]. Of these, the coherent control of injection current is the most

relevant to this thesis [25, 27]. In this configuration, two ultrafast laser pulses of central

frequencies and 2 are incident upon a semiconductor with electronic band gap Eg. When

/ 2g gE E , single-photon absorption (SPA) of the 2 beam occurs simultaneously with

two-photon absorption (TPA) of the beam. A ballistic injection current occurs as a result of

the quantum interference of the transition probabilities that link the same initial (valence) and

final (conduction) states. Just after optical excitation, the carrier distribution in momentum space

is anisotropic and polar, yielding a current density IJ . Thus an electrical current can be

generated in a bulk semiconductor by all-optical means (without the use of a bias voltage). The

injection current can also be controlled by varying the phase difference between the two optical

beams. It can be turned off completely, or reversed in direction.

7

To discuss this and similar processes further, one can think in terms of the nonlinear optical

susceptibility formalism mentioned previously and reported in detail in Ref. [54]. The coherent

injection of a current can be viewed as a special case of a rectification process, which represents

the generation of a DC material polarization. In this case, for both second- and third-order

electric field interactions, the nonlinear susceptibilities are doubly divergent with respect to the

mixing frequency . Focusing on the third-order interactions between electric fields at

frequencies 1, 2, and 3, the third-order polarization at frequency 1 2 3 is given by

equation (1.3), and the third-order nonlinear susceptibility can be expanded in the frequency

domain as:

3 1 2 33 1 2 3 3 1 2 3( ; , , )( ; , , ) ( ; , , )

( )i

3 1 2 32( ; , , )

( )i

(1.4)

These three terms give rise to three different types of current sources in the material, known from

left to right as the rectification current, shift current, and injection current. These three currents

are each represented by their respective tensors, 3 , 3

, and 3 . The rectification current is

based on a non-resonant interaction that represents the displacement of virtually excited carriers.

In other words, it occurs at all excitation frequencies, both above and below the band gap. The

other two currents are both the result of resonant processes, and involve real band-band

transitions. The shift current is associated with a spatial shift of the charge center during

excitation. Finally, the injection current reflects the quantum interference of absorption

pathways linking the same initial and final states in the valence and conduction bands

respectively. Note that one can write an expression similar to equation (1.4) for second-order

currents as well, but for centrosymmetric materials these nonlinear susceptibility tensors will be

zero.

Both the experiments reported in this thesis as well as the previous semiconductor experiments

discussed above concentrate on the third-order injection current, represented by the third-order

injection tensor 3 . It must be noted that since the injected currents are measured by the emitted

electromagnetic radiation, it is of greater interest to concentrate on the temporal derivative of the

8

injected current, IJ . In the frequency domain, the current density rate can be expressed by:

2( ) ( ) ( ) ( ) ( )I I Ii i J J P (1.5)

For the degenerate case of equation (1.4) where 1 1 2 2( ) ( )E E , the injection current density

rate is given by:

0 3 1 1 1 1 2 2( ) : ( ) ( ) ( )I J E E E (1.6)

where the field/frequency index “3” has been renamed “2” and the negative frequency field has

been replaced with its complex conjugate at the positive frequency. This equation will be used

in chapter 2 to describe the properties of the injection current in greater detail.

In 1999 and 2000, E. J. Mele, Petr Král, and David Tománek reported on the theoretical

possibility of injecting photocurrents in carbon nanostructures such as graphene and carbon

nanotubes [55, 56]. Using a long-wavelength theory for the low-energy electronic states and a

density matrix formalism, the authors calculated the third-order transition rate into the

conduction band and found that it had a polar asymmetry in momentum space. This meant that a

third-order injection current was possible in these new materials, but until now there were no

experimental investigations of this exciting phenomenon.

1.3 Research objective The first objective of this work is to use a pump-probe experimental setup to investigate the

dynamics of carriers in exfoliated graphene and thin graphitic films under ultrafast photo-

excitation. This type of study is often conducted after a new material is discovered since it

usually reveals important information about carrier transport and scattering mechanisms. Of

particular interest is how the carrier dynamics of bulk graphite, which have already received

experimental attention [36], change in ultra-thin samples from hundreds of carbon layers down to

a single layer (graphene). In order to determine this, it is necessary to fabricate many ultra-thin

graphitic samples, and this is done using the standard micro-mechanical cleaving method.

Samples are mounted on oxidized silicon substrates for electrical isolation and easy identification

and characterization. On each of these samples, ultrafast pump-probe measurements are

conducted, whereby a weak time-delayed probe pulse samples the material during and after

9

excitation by a stronger pump pulse. Both changes in reflectivity and transmissivity are

measured, and from this information, the changes in the material properties of the graphitic films

are deduced. By analyzing the material properties as they return to their quiescent values,

conclusions are drawn in regards to the dominant scattering mechanisms in graphene and

graphite. Because of the small sizes of the samples (tens of microns in diameter), they must be

imaged in a confocal arrangement to ensure proper targeting, and both beams must be focused to

sufficiently small spot sizes. In order to obtain transmissivity measurements through the thick

(500 μm) silicon substrate, the probe photon energy must be chosen to be less than the indirect

band gap of silicon (1.12 eV). Therefore a 1300 nm probe and an 800 nm pump beam are used.

The results from the pump-probe experiments on carrier dynamics in graphene and graphite have

been published in Optics Express [57].

The second objective of this work is to generate and detect third-order injection currents in

various carbon structures. Initial studies involve bulk graphite, since it is a semi-metal, and it is

unclear whether significant currents can be generated in a system with high absorption of the

beam. Graphene would be the next logical material to study; however, calculations show that the

thickness of such exfoliated samples would not yield a sufficient current density to detect via

THz radiation emission, given the measurements from graphite reported here. In order to

measure injection currents in graphene, one would need a thicker sample, and in fact a recent

study by Norris et al. measures such currents in epitaxial graphene [58]. However, no claims

have been made to differentiate the findings from those of graphite reported in this thesis.

Attention of this work then focuses on current injection in carbon nanotubes. A variety of

single-walled carbon nanotubes are obtained from different sources in order to investigate the

current dependence on several material and experimental parameters. A vertically-aligned

“forest” of carbon nanotubes is used to test the injection current’s dependence on tube alignment

direction. However, the fabrication of this type of sample is not selective enough in regards to

tube diameter or chirality, and therefore other samples are needed to yield more information on

the physics of current injection. Several carbon nanotube samples are then obtained which,

despite the lack of alignment, have high single-wall purity, smaller tube diameters, smaller

diameter distributions, and are sorted by electronic type (semiconducting or metallic). Initial

results reporting the achievement of the coherent control of photocurrents in graphite and the

vertically-aligned carbon nanotube sample were published in Nano Letters [59].

10

For the graphite and carbon nanotube samples, the dependence of the injection current on pump

beam polarization is studied in detail. Based on the theoretical predictions of current injection in

carbon nanotubes [56], there should be two configurations for which such currents are possible,

and by varying the polarization angle between the and 2 beams as well as the sample

orientation, one can determine their relative contributions.

Finally, coherent control experiments are performed on the sorted (unaligned) carbon nanotube

samples at a large range of wavelengths in order to deduce any possible effects excitons may

have on the current injection. It is now known that most of the optical absorption of carbon

nanotubes at room temperature is a result of excitons [60], and it is only natural to question what

effect excitons may have on the current injection. More specifically, it is of interest whether

excitonic effects contribute to or inhibit the traditional injection current initially studied without

attention paid to carrier-carrier effects. Emitted terahertz spectra and time delay/phase parameter

measurements also provide clues to help answer this question.

1.4 Thesis outline In chapter 2, a theoretical framework is presented. The various mechanisms contributing to the

dynamics of photo-excited carriers in graphene and graphite are discussed in section 2.2. The

general formalism of the multilayer transfer matrix method is also given. The theory on the

coherent control of photocurrents is developed in section 2.3, and applied to materials such as

graphene, graphite, and single-walled carbon nanotubes. Chapter 3 begins with a detailed

account of the fabrication and characterization of the samples in section 3.2. The two sections

that follow then outline the optical sources and experimental setups used. The results from these

experiments are shown in chapter 4, including a discussion of carrier relaxation times in the

transition from graphene to graphite in section 4.2, as well as a discussion on the observed

photocurrent generation mechanisms and their material and geometrical dependencies in section

4.3. A summary of the results and a closing discussion appear in chapter 5.

11

Chapter 2

Theoretical Background

2.1 Overview In this chapter, a theoretical formalism is presented on the ultrafast carrier dynamics and

coherent control of photocurrents in carbon nanostructures. In section 2.2, I discuss the theory

related to photo-excited carrier relaxation in graphene and graphite, and how the changes in

material properties affect the total reflection and transmission. In section 2.3, I discuss the

theory related to the coherent injection and control of photocurrents in carbon nanotubes,

graphene and graphite. This theory is compared with experimental measurements in chapter 4.

2.2 Carrier dynamics in graphene and graphite When a photon of energy is absorbed by a semiconductor with an energy band gap

gE , an electron is promoted from the valence band to the conduction band with an excess

energy gE and a nonzero momentum wavevector determined by the band structure. This

electron then loses its excess energy and momentum to other electrons or holes and the crystal

lattice via various carrier-carrier and carrier-phonon scattering mechanisms. Eventually, the

electron recombines with a hole in the valence band. In a metal, the conduction band is already

partially filled with free electrons before excitation. Absorption of a photon then could result in

an intraband excitation of a conduction electron or an interband excitation of a valence electron,

followed by similar subsequent energy and momentum scattering. These scattering and

recombination mechanisms and the timescales over which they occur constitute carrier

dynamics, and can be a rich field of research for any material.

12

Carrier dynamics is often studied experimentally by the influence the excited carriers have on the

material’s optical properties, such as the complex refractive index inn ~ or dielectric

constant (relative permittivity) 2~ni IR . The three most common ways that excited

carriers can change a material’s bulk optical properties (neglecting excitonic effects) are through

band filling (state filling), free carrier absorption, and band gap renormalization (shrinkage).

Band filling occurs because an electron is a fermion and, as a result of the Pauli Exclusion

Principle, cannot be in the same quantum state as another electron. When a large number of

electrons are optically excited at near the same energy, they effectively “block” further electrons

from being excited at that same energy. This typically causes a decrease in the optical

absorption at that photon energy. Free carriers may also absorb photons and move to a higher

energy in the conduction band. This free carrier absorption changes the material’s refractive

index. Finally, band gap renormalization occurs as a result of many-body effects between

carriers and is only apparent at high carrier densities. One must keep in mind that no matter

whether the absorption or refractive index is altered, the other is also changed due to the inherent

co-dependence characterized by the Kramers-Kronig relations.

Carrier dynamics vary greatly depending on the optical and electronic properties of the material,

and whether the material is a semiconductor, dielectric, or metal. Properties of relevant carbon

nanostructures are discussed in the sections to follow. Neither graphene nor graphite has a

positive band gap. Rather graphene is a zero-gap semiconductor, while graphite is a semimetal.

Carbon nanotubes can be either semiconducting or metallic, depending on their chirality. These

important characteristics make the physics of carrier dynamics in carbon nanostructures very

rich.

2.2.1 Graphene and graphite

A single graphene sheet has a two-dimensional hexagonal structure as shown in Figure 2.1a,

which is part of the D6h point group. The primitive cell contains a carbon atom at each of two

sublattice sites labeled A and B, separated by a bond length of d = 1.42 Å. In reciprocal space,

the first Brillouin Zone is hexagonal, as shown in Figure 2.1b, with points of high symmetry

labeled.

13

Figure 2.1 : (a) Crystal lattice structure; and (b) reciprocal lattice structure of graphene.

In graphene (and any carbon material composed of graphene sheets), the electrons involved in

the bonds between atoms (overlapping p orbitals) are responsible for most of the electronic and

optical properties. Using a tight-binding approximation, the dispersion of the bonding ()

valence (– below) and antibonding (*) conduction (+ below) bands is calculated as [12]:

2/1

20 2

1cos421cos

23cos41),(

akakakkkE yyxyx (2.1)

where 0 3 eV is the nearest-neighbour hopping energy and da 3 = 2.46 Å is the lattice

translation vector length. The dispersion relation associated with equation (2.1) is plotted in

Figure 2.2. The wavevector (kx,ky) represents the position relative to the centre of the Brillouin

Zone, known as the point. The most significant symmetry points in the Brillouin Zone of

graphene are the corners, labeled K and Kʹ, since the conduction and valence bands are conical

near this point, and actually meet at the Fermi Energy EF. Cuts through the K and Kʹ points

reveal the linear nature of the band structure in these regions (see Figure 2.3a). This important

property makes the underlying physics of graphene extremely interesting to study.

14

Figure 2.2 : 3D plot of (bonding) and * (antibonding) electronic bands in graphene.

Graphite is formed by layering graphene sheets on top of each other. The layers are weakly

bonded together with van der Waals forces, and have a spacing of c = 3.35 Å. There are three

possible natural stacking configurations for bulk graphite, but the one that is by far most

common is AB Bernal stacking, shown in Figure 1.1. In this arrangement, two neighbouring

sheets are stacked on top of each other, with one rotated 60o relative to the other about the sheet's

normal, through any atom. In any natural stacking configuration, there is always a weak

interaction of electrons between layers, and the band structure changes from that of graphene. In

fact, the linear band structure around the K and Kʹ points becomes parabolic, and the conduction

and valence bands overlap, forming a semimetal. Bulk graphite therefore has a negative band

gap energy of -41 meV [61]. Figure 2.3 compares the band structure of graphene and bulk

graphite [62]. Note the difference around the K point.

15

Figure 2.3 : Electronic band structure for (a) graphene, and (b) graphite [62].

Because of the lack of a positive band gap, graphene and graphite can absorb light at any photon

energy. The absorption of an ultrafast laser pulse of peak frequency generates a non-

equilibrium population of electrons in the conduction band and holes in the valence band. This

is illustrated in Figure 2.4b as a set of population spikes at energies 0 / 2E . Photo-

excitation is followed by thermalization, which takes place in one of two possible scenarios. If

the intra- and interband scattering rates are similar, a hot Fermi-Dirac distribution of electrons

will form with Fermi level 0 . This requires ultrafast carrier recombination, and is followed

by the slower process of carrier cooling, which occurs by optical and acoustic phonon scattering.

However, if interband scattering (recombination) occurs at a much slower rate than intraband

scattering, independent thermalization of electrons and holes takes place, forming two

corresponding Fermi-Dirac distributions with separate non-zero quasi-Fermi levels. This is the

picture developed by the Cornell group [38, 63, 64] and commonly adopted by others [65].

These distributions are also depicted in Figure 2.4b, as carrier densities per unit energy

, , ,( ) ( ) ( )e h e h e hN E D E f E where the subscripts are either “e” or “h”, representing electrons and

holes respectively. Here De,h(E) represent the densities of states, which are linear with respect to

b) a)

16

E for a two-dimensional free electron gas, and fe,h(E) represent the occupation probabilities:

1

( , )1 exp e e ee hB e

E N Tf E f Ek T

(2.2)

where kB is the Boltzmann constant, e hT T is the free electron temperature, Ne is the total free

electron density, and ( , ) ( , )e hN T N T represents a quasi-Fermi level. Recent experiments

have measured the thermalization time to be on the order of only 10 fs [65]. Once thermalized,

the carrier distributions cool (decrease Te), due to carrier-phonon scattering, on timescales of

between hundreds of femtoseconds and several picoseconds [36-40]. At the same time, carriers

begin the process of recombination, which the Cornell group reports to occur on the timescale of

picoseconds in epitaxial graphene [63]. The purpose of the experiments performed in this thesis

is to not only measure relaxation timescales in exfoliated graphene, but to determine how they

evolve as thickness varies in the transition between graphene and graphite.

Figure 2.4 : Illustration of (a) graphene band structure around K and K’ points; and (b) transient behaviour of excited carrier population densities Ne and Nh.

As discussed in the previous section, there are several possible ways that excited carriers can

modify a material’s optical properties. In graphene and graphite, the main modification occurs

through band filling, since the Drude (free carrier absorption) contribution to the dielectric

constant is commonly assumed to be small at visible and near-infrared wavelengths [36, 65].

17

The presence of a large population of excited carriers in graphene effectively blocks additional

carriers from being excited, decreasing absorption and increasing transmission. The change in

the imaginary part of the dielectric constant at optical frequency is given by [66]:

(0) (0)( ) ( ) 2 ( )2 2 2I I e h I e

f f f (2.3)

where )()0( I is the imaginary part of the quiescent dielectric constant at . The main

contribution to the change in the real part of the dielectric constant can be evaluated via the

Kramers-Kronig relations.

2.2.2 Silicon

For experiments on exfoliated graphene and graphite, it is important to understand the carrier

dynamics of silicon, since that is the substrate material. Because the graphitic films are so thin,

excitation of the substrate is inevitable. Silicon is a group IV semiconductor with an indirect

band gap of Eg = 1.12 eV [67]. After excitation to the conduction band from the valence band in

silicon, electrons redistribute their momentum via electron-electron and electron-phonon

scattering. The momentum relaxation time in silicon is 32 fs [68]. The electrons then lose their

excess energy falling to the local minima of the conduction band on the timescale of 260 fs [68].

Finally, electrons experience phonon-assisted recombination to the valence band in a time much

greater than 10 ns. The indirect gap of silicon is significant because an excited electron requires

this interaction with a phonon in order to decay to the top of the valence band, resulting in the

high recombination time value. The complex refractive index of silicon is modified by both

band filling and free carrier absorption, with the latter being more significant [68]. One can

express the change in the real part of the refractive index at optical frequency due to free

carrier absorption by [68]:

2

(0) 2

2( )( ) ( )

e

opt e

Nenn m T

(2.4)

where e is the electron charge and n(0)() is the real part of the quiescent refractive index at .

( )opt em T is the carrier effective optical mass, which depends weakly on temperature and settles

at 0.16opt em m after energy relaxation [68], where me is the free electron mass.

18

2.2.3 Reflection and transmission from a multilayer structure

The fabrication of graphene and thin-film graphite samples is discussed in detail in section 3.2.1.

Atomically thin films of carbon are exfoliated from bulk graphite and applied to an oxidized

silicon substrate. The oxide layer is 300 nm thick, since for this thickness, it is possible to

visually identify graphene using a standard optical microscope. This is because of a Fabry-Perot

interference effect, which makes the sample colour highly dependent on the number of carbon

layers on top of the substrate. This interference effect also has significant implications on pump-

probe experiments that rely on measuring the reflectivity and/or transmissivity of the sample. In

order to properly interpret these measurements, knowledge of the behaviour of light in the

multilayer structure is required to extract optically-induced changes in material properties. In

this section I present the theory behind the interference effect, and explain how it can be used to

evaluate the optical properties of the materials involved.

For light propagation through a multilayer system, a transfer matrix method can be used to

determine the total reflection and transmission, as well as the electric field strengths throughout

the system. Ref. [69] discusses the method for systems containing either coherent layers (thin

layers with smooth interfaces) or incoherent layers (thick layers or layers with rough interfaces),

or a combination of the two. This transfer matrix approach is summarized below, and is not only

used for the analysis of carrier dynamics in exfoliated graphene and thin graphitic films, but also

for the estimation of sample thicknesses.

Consider a multilayer structure of m isotropic homogeneous layers, with plane and parallel

interfaces. Figure 2.5 illustrates this system. Light is described through its electric field at the

left (L) and right (R) sides of each layer, and propagating to the right (+) and left (–) directions.

Incident light is assumed normal to the layers in the structure, and propagates in the positive

direction (from the left hand side). The jth layer has complex refractive index jjj inn ~ and

thickness dj, for j = 0…m+1.

19

Figure 2.5 : Diagram of multilayer structure with forward- and backward-scattering electric field components shown.

The electric field components on the right side of the structure ( LmE )1( ) can be determined from

the electric field components on the left side of the structure ( RE0 ), as long as the total

structure’s transfer matrix T is known, such that:

( 1)0( 1)0

m LR

m LR

EEEE

T (2.5)

where T is a matrix product of matrices representing propagation through each interface ( )1( jjI )

and layer ( jL ):

01 1 12 ( 1)... m m mT I L I L I (2.6)

The interfacial transfer matrix is defined by:

111ij

ij

ijij r

rt

I (2.7)

ji

iij nn

nt ~~~2

ji

jiij nn

nnr ~~

~~

(2.8)

while the layer transfer matrix is defined by:

)exp(00)exp(

j

jj i

i

L (2.9)

jjj

nd ~2 (2.10)

where is the wavelength of incident light. The total reflection and transmission coefficients are

20

therefore respectively defined by:

0 210 11

R

R

E TrE T

(2.11)

( 1)0 11

1m LR

Et

E T

(2.12)

which leads to the total (real) reflectivity and transmissivity:

2rR (2.13)

20

1~

)~Re(t

nn

T m (2.14)

This theory assumes that all the layers are coherent, in the sense that the light retains its temporal

coherence through each interfacial reflection and layer propagation. In many cases, if the

interface is not optically smooth or the layer is thick, the light loses its coherence and the layer is

said to be incoherent. For these cases, the theory can be modified. For a structure where all

layers are incoherent, the light is described in terms of the amplitude of the electric field squared.

Equation (2.5) holds as long as electric field components E are substituted for 2EU and the

incoherent system transfer matrix T is used instead of T:

01 1 12 ( 1)... m m mT I L I L I (2.15)

which is similar to equation (2.6), however the individual transfer matrices are defined by:

222

2

2

11

jiijjiijij

ji

ij

ijrrttr

r

tI (2.16)

2

2

)exp(0

0)exp(

j

jj

i

i

L (2.17)

with equations (2.8) and (2.10) still holding. With the incoherent system transfer matrix T , one

can still compute total reflectivity and transmissivity:

2111

TRT

(2.18)

10 11

Re( ) 1mnTn T

(2.19)

21

For systems composed of a mixture of coherent and incoherent layers, the two algorithms above

can be combined. The entire system should be treated as an incoherent system, with packets of

coherent-layer systems within the incoherent system. For each coherent packet, the coherent

system transfer matrix T is computed. This coherent matrix is then converted into one

incoherent interfacial transfer matrix:

2 211 12

2 22 12 21

21 211

detij

T T

T TT

T

I T (2.20)

and is then combined with the other incoherent layer transfer matrices to form the total

incoherent system transfer matrix T .

For the exfoliated graphene and thin graphitic film samples, the structure consists of N carbon

layers on top of a 300 nm coherent layer of SiO2 )45.1~( n with a 500 μm incoherent layer of Si

underneath. The Si layer is incoherent due to its thickness and unpolished back side. It is

common practice [70] to assume that each carbon layer can be approximated as a 3.35 Å thick

layer of bulk graphite with the same optical properties [71].

With these parameters it is straight forward to compute the linear reflection and transmission

spectra for a sample with any value of N. The reflection spectra can be converted into standard

CIE XYZ tristimulus values [72] to display the apparent colour to the human eye, as follows:

0

)(ˆ)( dxRX (2.21a)

0

)(ˆ)( dyRY (2.21b)

0

)(ˆ)( dzRZ (2.21c)

where )(ˆ),(ˆ),(ˆ zyx are the colour matching functions for the CIE Standard Observer. The Y

value is often regarded as the relative luminance of the colour, while the following two modified

values describe the chromaticity:

ZYX

Xx

(2.22a)

ZYX

Yy

(2.22b)

22

The luminance Y as a function of the number of carbon layers N is plotted below in Figure 2.6,

along with the apparent colour below, as determined by the chromaticity values (x,y). The

bottom graph is the same as the top, except zoomed in for 150 N . It can clearly be seen that

for around 7N , the luminance is roughly linear with respect to N. This helps with sample

characterization as described in section 3.2.1. XYZ tristimulus values can be easily converted to

any other colour model (for example RGB) for display or printing purposes. It should be noted

that all colours are approximate, since they depend not only on the observer and viewing

conditions, but also on the device (monitor, printer, etc).

Figure 2.6 : Approximate luminance and apparent colour of exfoliated graphene and thin graphitic films with respect to number of layers N, for (a) 0-400 layers; and (b) zoomed in for 0-15 layers.

23

The theory described in this section can clearly be used to compute R and T given the structure of

the system and its optical properties. However, it is also possible to reverse this algorithm to

compute the material optical properties from the measured values of R and T. This is the basis of

the analysis of the experiments on carrier dynamics in exfoliated graphene and thin graphitic

films.

Because of the complicated dependence of R and T on the refractive indices of the layered

materials, one cannot obtain a simple analytical expression for the refractive indices as a function

of R and T. However, using a numerical optimization technique, one can computationally vary

the refractive indices to produce values of R and T that match experimental measurements. For

this analysis, it is assumed that excitation from an ultrafast pump pulse changes the refractive

indices n~ of both the carbon layers and the underlying silicon substrate. Changes to both real

and imaginary parts of the refractive index are greatest at the surface of each layer, and decrease

with depth according to Beer's Law:

( ) exp( )n z n z (2.23)

where 2( / )c is the absorption coefficient of the material and z is the depth into the layer.

Therefore both the carbon and silicon layers are partitioned into multiple sublayers and the entire

system is modeled with the transfer matrix model above. The change in refractive index of the

silicon layer is computed beforehand from pump-probe measurements on the bare SiO2/Si

substrate. These material changes are reduced according to the expected absorption from the

upper carbon layers. With these assumptions, only two unknown values remain, the real and

imaginary parts of the refractive index change of the carbon layers.

Section 3.4.1 outlines the pump-probe experiment that measures transient changes in reflectivity

and transmissivity, R/R(t) and T/T(t) respectively. For each of these values, the theory

discussed in this section is used to extract the transient changes in optical properties of the

graphene/graphite. This analysis is presented in section 4.2.

24

2.3 Coherent control of photocurrents in carbon nanostructures From the discussion in section 2.2, it is clear why ultrafast resolution of carrier dynamics is

required to study the novel physics of carbon nanostructures such as graphene. Continuing the

exploration of ultrafast phenomena in carbon nanostructures, I now focus on photoinjected

electric currents. The general theory of the coherent injection and control of photocurrents in

semiconductors is reviewed, followed by specific applications to carbon nanotubes and graphite.

2.3.1 Current injection via band-band transitions

The material polarizations induced via a variety of nonlinear optical effects were introduced in

section 1.2. Third-order nonlinear optical effects occur in all materials, including

centrosymmetric materials. The associated tensor 3 can be expressed as a sum of its

components as in equation (1.4), representing the generation of rectification, shift, and injection

currents respectively. Henceforth this chapter will concentrate on the third-order injection

currents and their optical generation in bulk semiconductors and carbon nanomaterials such as

graphite and single-walled carbon nanotubes.

2.3.1.1 Third-order injection current in semiconductors

A typical direct-gap semiconductor has an electronic band structure similar to that shown in

Figure 2.7 below, with an energy gap Eg between valence and conduction bands. A ballistic

electric current can be optically injected into a semiconductor using two laser beams of

frequencies and 2, as long as 2/gE , and ideally if gE in addition. In this

situation, the 2 beam is absorbed by a single-photon absorption process coinciding with the

promotion of an electron from the valence band to the conduction band, with an excess energy

2 gE . The beam is also absorbed, promoting an electron with the same excess energy;

however the absorption is the nonlinear process of two-photon absorption. An electron then has

two possible pathways to transition to the conduction band, and the two pathways quantum-

25

mechanically interfere with one another. The interference leads to an imbalance of carriers in

momentum space, and a net current develops, known as an injection current.

Figure 2.7 : Typical band structure for a direct-gap semiconductor and illustration of coherent control of photoinjected currents. The sizes of the circles indicate the electron/hole populations injected at those wavevectors.

More specifically, the two pump pulses with central frequencies and 2 and respective

frequency bandwidths and 2 can be expressed through their Gaussian electric field spectra,

given with respect to frequency and depth into the material z:

2

01 12

2 ( )ˆ( , ) exp exp exp ( ) ( )2

eff zz E i k z i

E e (2.24)

2

0 22 2 2 22

2 2

2 ( 2 )ˆ( , ) exp exp exp ( ) ( )2

zz E i k z i

E e (2.25)

where 0,2E represents the (real) peak field strength just under the material surface, ,2ˆ e

represents the (complex) polarization vector of unit length, and 1,2 ( ) represents the phase

spectrum of the field. Absorption of the 2 beam is expressed through the linear absorption

coefficient 2 2 2 / (2 )c . Absorption of the beam is expressed through an effective

absorption coefficient eff which is a combination of nonlinear and linear absorption and is

26

defined by:

0 0 2 / ( )eff I I c (2.26)

where is the two-photon absorption coefficient of the material (assumed constant) and 0I is

the peak intensity of the beam, which is proportional to 20E . Here both single- and two-

photon absorption of the beam are considered, despite the fact that SPA is undesirable. This is

relevant when expanding the discussion to carbon nanomaterials.

The coherent third-order injection current results from a nonlinear optical process governed by

the purely imaginary fourth-rank injection current tensor 3 . The time derivative of the current

density injected by two frequency components is given by equation (1.6). In the case of the

optical pump beams expressed above, any three of the frequency components involved can mix

to yield a current density component at a positive frequency . Integrating over all possible

frequency permutations, equation (1.6) becomes:

1 20 3 1 2 1 2( ; ) 2 ( ; , , 2 )2 2I d dz

J

1 1 1 2 2 1 2: ( , ) ( , ) (2 , ) . .z z z c c E E E (2.27)

where “c.c.” means the complex conjugate of everything that appears before it.

It is shown in section 2.3.4 that the injection current detection system is sensitive to the current

density rate of change. Inserting equations (2.24) and (2.25) for each of the pump field spectra

and performing the integrations,

2

0 0 00 3 2 2 2

2ˆ ˆ ˆ( ; ) 2 : expI z E E E

J e e e

exp sinz k z (2.28)

where 2 2 222 , 1 22eff , and 2 ( ) (2 ) (2 / )( ( ) (2 ))k k k c n n .

is the phase parameter defined by 22 where 1( ) and 2 2 (2 ) .

From this point onward, the 3 tensor is considered real, since a factor of i has already been

incorporated into the phase dependence in equation (2.28).

27

The current detection system used measures the electric field of the THz radiation emitted by the

ultrafast optical injection of the current. The total THz electric field in the far-field is a sum of

the fields generated from the current injected at all depths within the material:

D ITHz

inj dzz0 );()( JE (2.29)

where D is the thickness of the sample. After using the sum angle formula for the sine function

and inserting equation (2.28) into equation (2.29), one finds that:

2

0 0 00 3 2 2 2

2ˆ ˆ ˆ( ) 2 : expTHzinj E E E

E e e e

1 2cos sin (2.30) where 1 and 2 are integrals over depth of exponentially decaying sine and cosine functions

respectively, that can be evaluated to the following form:

1 2 2 1 exp sin cosk D k D k D

k k

(2.31a)

2 2 2 1 exp cos sinkD k D k D

k

(2.31b)

The sum angle formula can be used again to combine the phase parameter terms and simplify

equation (2.30) to:

2

0 0 00 3 2 2 2

2ˆ ˆ ˆ( ) 2 : expTHzinj E E E

E e e e

1/22 2 11 22

sin arctan

(2.32)

If the sample can be approximated as a semi-infinite slab, such that D , equation (2.32)

simplifies to:

2

0 0 00 3 2 2 2

2ˆ ˆ ˆ( ) 2 : expTHzinj E E E

E e e e

1/22 2 sin arctan /k k (2.33)

28

It can be seen that the injection current depends on the intensities of the pump beams, and on the

values of the injection tensor elements (determined by the material properties) and the sample

crystal orientation. The current also depends sinusoidally on the relative phase of the two pump

beams via the phase parameter . This means that by simply delaying one beam relative to the

other, the current can be turned off or even reversed in direction. Equations (2.32) and (2.33)

summarize how one can coherently control the third-order injection current.

In order to determine the dependence of the injection current on a material’s geometry or the

pump beam polarizations, a proper lab frame of reference must be defined. Let the Z axis be

parallel to the sample surface normal, such that the sample surface sits in the XY plane. Let the

X axis be horizontal and the Y axis be vertical. In this case the 2 beam is linearly polarized

along the X axis, while the beam is linearly polarized at an angle to the X axis in the XY

plane. Therefore 0 represents co-polarized pump beams, while 2/ represents cross-

polarized pump beams.

Throughout the experiments on carbon nanostructures, zincblende semiconductors such as InP

and GaSb are used as a reference. For these materials, the generalized current injection tensor

has 21 nonzero elements, only four of which are independent [73]. For (100)-oriented crystals

under normal incidence of two pump beams (, 2), the number of independent tensor elements

reduces to three. In the lab frame, when the principal crystal axes are aligned along the X and Y

axes, the tensor product in the square brackets of equations (2.32) and (2.33) simplifies to:

2sinsincos 22

,

,

xyxy

xyyxxxxxTHz

Yinj

THzXinj

EE

(2.34)

Values for the current injection tensor elements as a function of wavelength for several such

semiconductors are calculated in Ref. [74].

2.3.1.2 Graphene and graphite

Despite the fact that graphite is a semimetal and graphene is a zero-gap semiconductor, third-

order injection currents can still be generated using and 2 pump beams, as discussed in the

previous section. For both graphene and graphite, light can be absorbed at any wavelength, and

29

therefore there will always be single-photon absorption of both pump beams. This may reduce

the beam power available for two-photon absorption, but nevertheless, injection currents

should still be measureable and given by equations (2.32) and (2.33).

Since both graphene and graphite have a hexagonal crystal structure, the generalized injection

tensor has 21 nonzero elements, only 10 of which are independent [73]. The situation is greatly

simplified by limiting ourselves to normal sample incidence, whereby the laser propagation

direction (Z axis) is parallel to the c-axis (perpendicular to the layers), and the degenerate case of

two pump beams (, 2). In this case the tensor product in the square brackets of equations

(2.32) and (2.33) reduces to that in equation (2.34), except for the additional relation

xyyxxxxxxyxy 21 , so that:

2sinsincos

21

22

,

,

xyyxxxxx

xyyxxxxxTHz

Yinj

THzXinj

EE

(2.35)

In fact, this additional symmetry relation implies that the injection tensor has cylindrical

symmetry about the c-axis of these crystal structures. This means that the injected ballistic

current direction does not change when rotating the sample about the propagation direction

(c-axis).

The article by Mele, Král, and Tománek in 1999, Ref. [56], was the first to theoretically study

the coherent control of photocurrents in carbon nanostructures. The authors use an effective

mass theory of low-energy electronic states to develop a formalism for the coherent injection of

currents in an ideal graphene sheet, and then use this to study the same effect in ideal single-

walled carbon nanotubes. Although the theory used is beyond the scope of this thesis, it is

noteworthy that the authors also find that the current direction is invariant to c-axis rotations.

The authors also conclude that if the beam polarization makes an angle of with respect to the

2 beam polarization, the net photocurrent direction makes an angle of 2. In the tensor element

notation, this implies that for an ideal graphene sheet, Ref. [56] predicts that:

xyyxxxxx (2.36)

The article does not develop a theory for bulk graphite, but predicts that the dependencies should

be similar, if not exactly the same, and proposes experiments on graphite to verify this current

direction dependence.

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2.3.1.3 Carbon nanotubes

Carbon nanotubes can be visualized as sheets of graphene rolled up into tubes, with diameters on

the order of nanometers. A tube composed of only one rolled up sheet of graphene is a single-

walled carbon nanotube (SWNT) while a tube composed of multiple rolled sheets (concentric

cylinders) is a multi-walled carbon nanotube (MWNT). In this thesis I concentrate on the study

of the simpler and more fundamental SWNTs. There are many types of SWNTs, depending on

the tube diameter and the way the graphene sheet is rolled up. Figure 2.8 illustrates an example

of a graphene sheet about to be wrapped into a nanotube [12], with the primitive translation

vectors 1â and 2â shown. The chiral vector 21 ˆˆ aaC mnh , also represented with the pair of

indices (n,m), expresses the way the sheet is wrapped, or the chirality. Once wrapped, the length

of the chiral vector becomes the tube circumference, and the tube diameter dt can be calculated

from this circumference as:

22 mnmnadt (2.37)

where a = 2.46 Å from the graphene structure, and by convention mn .

The translational vector T will be along the tube axis, and it is defined by:

)2,2gcd(/ˆ)2(ˆ)2( 21 mnmnmnmn aaT (2.38) where gcd means greatest common divisor. If m = 0, the nanotubes are called “zigzag,” while if

n = m, the nanotubes are called “armchair”. Otherwise, they are called “chiral”.

31

Figure 2.8 : Diagram of the wrapping of a single-walled carbon nanotube, with chiral vector (n,m) = (4,2) [12].

The single-particle electronic band structure of SWNTs can be approximated using a simple

method called band folding or zone folding, with the approximation better for larger diameter

tubes. In this method, the band structure for a SWNT of chirality (n,m) is assumed to be the

same as that for graphene, except that not all electron wave vectors in the azimuthal direction

(around the circumference) are allowed. Instead, periodic boundary conditions must be imposed

in this direction, given by:

Nh 2Ck (2.39)

where N is an integer. This condition leads to a family of straight lines each separated by 2/dt

when plotted in the (kx,ky) plane. Only electrons with these wave vectors are allowed, and their

corresponding energy is still given by equation (2.1). Figure 2.9a shows the unit cell of a (4,2)

nanotube in reciprocal space and the associated cutting lines as determined from equation (2.39).

Figure 2.9b gives the corresponding band structure for this nanotub