Aug 20, 2010 - Chapter 3: Exciton-Plasmon Interaction in a System of Au ..... 20 μl of SA from each buffer solution was added in 3 ml PBS buffer solution (pH 7.2) ..... The optical properties of nanowires make them very attractive for many optical ...... Now we have to find a suitable random potential U(x) based on the Atomic ...
Optical Properties of Nanoparticles and Nanowires: Exciton-Plasmon Interaction and Photo-Thermal Effectcs
A dissertation presented to the faculty of the College of Arts and Sciences of Ohio University
In partial fulfillment of the requirements for the degree Doctor of Philosophy
Pedro Ludwig Hernández-Martínez August 2010 © 2010 Pedro Ludwig Hernández-Martínez. All Rights Reserved.
2 This dissertation titled Optical Properties of Nanoparticles and Nanowires: Exciton-Plasmon Interaction and Photo-Thermal Effectcs
by PEDRO LUDWIG HERNANDEZ-MARTINEZ
has been approved for the Department of Physics and Astronomy and the College of Arts and Sciences by
Alexander O. Govorov Associate Professor of Physics and Astronomy
Benjamin M. Ogles Dean, College of Arts and Sciences
3 ABSTRACT HERNANDEZ-MARTINEZ, PEDRO LUDWIG, Ph.D., August 2010, Physics and Astronomy. Optical Properties of Nanoparticles and Nanowires: Exciton-Plasmon Interaction and Photo-Thermal Effectcs (149 pp.) Director of Dissertation: Alexander O. Govorov The present studies concern optical properties of nanostructures, in particular, those for nanoparticles and nanowires. In our first study, we develop a theoretical formalism for the exciton-plasmon interaction for the case of CdTe nanowire with mobile excitons and Au nanoparticles with localized plasmon. In hybrid CdTe-Au nanowirenanoparticle structure, the excitons and plasmons are coupled via the dipole-dipole interaction akin to the Förster mechanism. The exciton dynamics and the photon emission of the nanowire depend very strongly on the exciton-plasmon distance (separation distance between the nanowire and nanoparticle). We find that the photon emission of the nanowire is blue shifted for smaller separation distances. We also study the processes of exciton transfer in coupled nanoparticles and nanowires, and obtain analytical equation and numerical results for the energy transfer rates. We find that, for large distance, the energy transfer rates are inversely proportional to the fifth power of the nanowirenanoparticle separation distance. We estimate the transfer rates for CdTe nanowire- CdTe nanoparticle, Si nanowire – CdTe nanoparticle, Au nanorod – CdTe nanoparticle. In the case of Si nanowire – CdTe nanoparticles we also study the light harvesting efficiency. Finally, we study the heat generation of photo-excited Au nanoparticles. Here, we focus on a) the heat generation of single nanoparticles embedded in a matrix (water and ice); b)
4 the collective heat generation of an ensemble of Au nanoparticles embedded in a water droplet and in a heat-conductive medium; and c) the temperature distribution on a surface of a crystal due to heat generated by a single Au nanoparticle placed on top of such surface. Approved: _____________________________________________________________ Alexander O. Govorov Associate Professor of Physics and Astronomy
5 ACKNOWLEDGMENTS It is a pleasure to express my gratitude and thanks to my advisor, Dr. Alexander O. Govorov. He provided enlightening and stimulating discussions and guidance throughout my Ph.D. degree. His attitude of supporting and encouraging me with patience were very motivating. It was a privilege to work with him. I enjoyed it thoroughly. I would also like to thank the member of my dissertation committee; Dr. Hugh H. Richardson, Dr. Sergio E. Ulloa, Dr. Eric Stinaff and Dr. P. Greg Van Patten for their kind support. I specially thank Dr. Richardson for helping me to understand some of his experiment where we collaborate. I am grateful to Dr. Ulloa for all his support and hospitality during my time at Ohio University. Additional thanks are extended to the faculty members of the Department of Physics and Astronomy at Ohio University. I am indebted to Dr. Ernesto Cota-Araiza for encouraging me to carry out my Ph.D. at Ohio University. I am also grateful to all of my teachers who have contributed, in one way or another, to shape my academic career; in particularly to Drs. Leonel CotaAraiza, Roberto Romo-Martinez, and Jorge A. Villavicencio-Aguilar. A special thanks to M.C. Miguel A. Martínez-Romero, Director de la Facultad de Ingeniería de la Universidad Autónoma de Baja California, for all his support during my time at Ohio University. I am very grateful to my parents (Pedro and Rosa), brothers (Alimi and Emilio) and sisters (Elibeth and Crisciel) for their deepest love and their support in attaining my goals throughout these many years. God bless you all. I would like to thank all my friends
6 in Athens for their help and support during my graduate studies at OU. I want to acknowledge in a special way Ennice Sweigart for her generous assistance. I am indebted to Facultad de Ingeniería de la Universidad Autónoma de Baja California in México and to the Department of Physics and Astronomy at Ohio University for their financial support. I greatly appreciate it.
M.S. Pedro Ludwig Hernández Martínez
7 TABLE OF CONTENTS Page Abstract ............................................................................................................................... 3 Acknowledgments............................................................................................................... 5 List of Figures ..................................................................................................................... 9 Chapter 1: Introduction ..................................................................................................... 15 Early History of Nanostructures ................................................................................... 17 Applications of Gold Nanoparticles ............................................................................. 19 Gold Nanoparticles for Labeling and Visualizing .................................................... 19 Au Nanoparticles as a Vehicle for Delivery ............................................................. 20 Au Nanoparticle as a Heat Source ............................................................................ 20 Au Nanoparticles as Sensors ..................................................................................... 21 Application of Nanowires ............................................................................................. 22 Magnetic Properties .................................................................................................. 22 Optical Properties...................................................................................................... 22 Chapter 2: Theoretical Background .................................................................................. 25 Bulk Semiconductors .................................................................................................... 25 Band Structure .............................................................................................................. 26 Excitons ........................................................................................................................ 30 Frenkel Excitons ....................................................................................................... 32 Mott-Wannier Excitons ............................................................................................. 35 Quantum Confinement .................................................................................................. 38 Plasmons ....................................................................................................................... 40 Energy Transfer ............................................................................................................ 44 Nonradiative Energy Transfer................................................................................... 44 Theory of Energy Transfer........................................................................................ 45 Chapter 3: Exciton-Plasmon Interaction in a System of Au Nanoparticles and CdTe Nanowires ......................................................................................................................... 51 Introduction ................................................................................................................... 51 Modeling of Exciton Diffusion in Nanowires .............................................................. 53 Chemical Sensor Mechanism Based on Exciton Diffusion in a Nanowire .................. 64
8 Chapter 4: Exciton Energy Transfer between Nanoparticles and Nanowires................... 69 Introduction ................................................................................................................... 69 Nanoparticle-Nanowire Transfer of Excitons ............................................................... 71 Results and Discussion ................................................................................................. 76 Optical Responses of Nanoparticle-Nanowire Complexes and the Light Harvesting Effect ............................................................................................................................. 84 Conclusion .................................................................................................................... 93 Chapter 5: Heat Generation by Optically Excited Gold Nanoparticles ............................ 94 Introduction ................................................................................................................... 94 Models and Calculations ............................................................................................... 95 Heat Generation by a Single Metal NP ..................................................................... 95 Heat Generation in a Collection of NPs .................................................................. 102 Comparison with Experiment on Light Excited NP Solutions ................................... 111 Temperature Problem for a Single NP with Three Media .......................................... 121 Model and Analytical Results ................................................................................. 121 Numerical Results ................................................................................................... 129 Chapter 6: General Conclusions ..................................................................................... 133 References ....................................................................................................................... 136 Appendix A: Orientation Factor κ2 ................................................................................. 148
9 LIST OF FIGURES Page Figure 1.1: a) Optical emission of CdSe quantum dots. After Wikipedia webpage [1]. b) Emission spectra of the quantum dots showing the dependence on size. ..……......... 15 Figure 1.2: Lycurgus Cup. a) The green appearance is produced by reflection. After The British Museum webpage [5]. b) The red and purple are due to transmitted light. After The British Museum webpage [6]. .................................................................................. 18 Figure 1.3: TEM image of cementite nanowires in Damascus Steel. a) and b) Carbon nanotubes in Damascus steel. The dark stripes indicate wires of several hundred nanometers in length. After M. Reibold et al. [12]. .…………………………………... 19 Figure 1.4: Hysteresis loops of Co nanowire arrays with different diameters and roughly the same length with applied magnetic field (H) parallel (||) and perpendicular (⊥)to the wire axes. After Beck et al. 2005 [56]. ..………………………………………………. 23 Figure 1.5: The extinction spectra of Ag nanowire with different diameter. The incident electric field is perpendicular to the wire axis. After Schider et al. 2001 [57]. ..…...…. 24 Figure 2.1: a) The crystal structure of diamond and zinc blende (ZnS). b) The fcc lattice showing a set of primitive lattice vectors. c) The reciprocal lattice of the fcc lattice. Here is shown the first Brillouin zone. Special symmetry points are denoted by Γ, X, and L and high symmetry lines joining some of these points are labeled as Λ and Δ. P. After Y. Yu and M. Cardona, “Fundamentals of Semiconductors: Physics and Materials Properties” [61]. ………………………………………………………………….……. 26 Figure 2.2: Electronic band structure for: a) zinc blende structure (CdTe), and b) Diamond like structure (Si). After M.L. Cohen and T.K. Bergstresser 1966 [63]. ….... 30 Figure 2.3: Exciton levels for a simple band structure at k = 0 . …….………………... 31 Figure 2.4: Energy levels of an exciton created in a direct process. Optical transitions are shown by arrows. ……..……………………………………………………………….. 32 Figure 2.5: a) Schematic illustration of a tightly-bound exciton (Frenkel exciton) localized on one atom in a crystal. ….…………………………………………………. 33 Figure 2.6: Energy for a Frenkel exciton. ….………………………………………….. 35 Figure 2.7: Schematic illustration of a weakly bound exciton (Mott-Wannier-exciton). 36
10 Figure 2.8: Schematic of the degree of freedom and density of state for a) bulk, b) quantum well, c) quantum wire, and d) quantum dot structures [66]. .………………... 40 Figure 2.9: Dispersion relation for transverse electromagnetic waves in a plasma. …... 44 Figure 2.10: Energy transfer efficiency (E) versus distance. R0 is the Förster distance. ………………………………………………………………………………………….. 50 Figure 3.1: (A,B) TEM images of Au bioconjugated CdTe NW in solution B. (A) x 200k, (B) x 500k. Yellow circles indicate the areas where characteristic lattice fringes of 0.23 nm of (111) faces of metallic Au can be seen after bioconjugation. After N.A. Kotov et al. 2004 [75]. .………………………………………………………………………….. 52 Figure 3.2: (a) AFM images of CdTe NWs (2.5 x 2.5 µm, the bar indicate the z-axis). (b,c) TEM images of NP conjugated NWs. Yellow circles in part c indicate CdTe NPs. After N.A. Kotov et al. 2005 [76]. …......……………………………………………… 52 Figure 3.3: Sketch of the systems Nanoparticle-Nanowire. ..…………………………. 54 Figure 3.4: A) AFM profile of a typical NW from corresponding image in B). After N.A. Kotov, A.O. Govorov et al. 2007 [90]. ....………………………………………...…… 57 Figure 3.5: A) NW radius from our model. B) Comparison of the NW diameter from the experiment data and our theoretical model. Experimental data after N.A. Kotov et al. 2007 [90]. …………………………………………………...………………………..... 58 Figure 3.6: A) Exciton potential in the model. B) Exciton density distribution for cm 2 μ = 0.2 and two NW-NP distances. ..…………………………………………... 59 eV ⋅ s Figure 3.7: A) Exact and approximate solution for μ = 0.2 cm 2 eV ⋅ s and large γNP 6 (RNW-NP = 6.2nm, γ NP ∝ 1 R NW − NP ). B) Exact and approximate solution for 6 μ = 0.01cm 2 eV ⋅ s and γNP (RNW-NP = 9.97 nm, γ NP ∝ 1 R NW − NP ). ..………………… 61
Figure 3.8: Exciton density behavior for μ = 0.01
γ NP ∝
1 R
6 NW − NP
cm 2 and γNP (RNW-NP = 9.97 nm, eV ⋅ s
). ..……………………………………………………………………… 62
Figure 3.9: Calculations of emission properties. a) Calculated emission spectrum for two NW-NP distances. b) Calculated maximum peak shift (in nm) as a function of the exciton mobility. After N.A. Kotov, A.O. Govorov et al. 2007 [90]. ....………………………. 64
11 Figure 3.10: A) Schematics of MSA from Au NPs and CdTe NWs. B) Scanning TEM image of Au-PEG-aB-PEG-NW (the ratio of NP:NW was 5000:1 (NP5K) in the conjugation step). After K.A. Kotov, A.O. Govorov et al. 2007 [90]. ............………... 66 Figure 3.11: A) Schematic illustration of aG-aB reaction in the MSA. Extra amount added will disrupt the aG-aB in NP-PEG-aB-PEG-NW. B) Reversible shift of peak luminescence wavelength in NP5K: a, attachment of NP to NW; b, after adding 20 μL, Streptavidin (SA); c, after adding free aB to the media. After N.A. Kotov, A.O. Govorov et al. 2007 [90]. …...…………………..……………………………………... 67 Figure 3.12: Calibration curve for SA. Averaged wavelength shifts at the hybrids in different analyte concentration for 30~50 min until saturation of wavelength shift. The 20 μl of SA from each buffer solution was added in 3 ml PBS buffer solution (pH 7.2) of an optical cuvette for spectroscopic measurements. Prepared SA buffers the concentrations of SA in optical cuvettes and their numbers of SA. The molarity ratios for the preparation of superstructures were calculated from molarity calculation of nanocolloids. The concentrations of NW and NP solutions were calculated to be 1.0×10-9 M, and 4.5×10-6 M respectively based on the physical dimensions of the particles and synthesis data after N.A. Kotov, A.O. Govorov et al. 2007 [90]. ..……... 68 Figure 4.1: a) Schematics of the coupled NP-NW system. b) Energy diagram of physical processes. Red dashed arrows describe inter-band optical transitions in nanocrystals (exciton generation) and yellow arrows inducts incident photons. A horizontal arrow and two vertical black arrows show the FRET process (Coulomb transfer) between a NP and a NW; in this unidimensional process, an exciton in a NP turns into an excitation in a NW. Blue solid arrows indicate fast intra-band relaxation of excitons and blue dotted arrows are inter-band recombination of excitons. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. …...……………………………………………...…………… 72 Figure 4.2: Rates of exciton transfer from a dye molecule to a NW for different exciton dipole orientations as a function of the molecule-NW distance. Red line shows the asymptotic behavior at large molecule-NW distances. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. ……………………………………...………………... 78 Figure 4.3: Rates of NP-NW transfer of excitons as a function of the CdTe NP-NW separation and available experimental data from reference 76. Green line shows the calculated rate for carbon nanotubes. Inset: FRET rate for the NP-NW complex as a function of the exciton energy of a NP. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. …......…………………………….………………………………………... 82 Figure 4.4: Calculated energy-transfer rate CdTe NP – Au NW complex. The experimental values are taken from reference 96. Red line shows the asymptotical
12 behavior at small Δ = d − RNW and corresponds to NP-surface transfer. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. ..…………………………...…... 84 Figure 4.5: a) and b) Schematics of the complex composed of a NW and NPs. NPs form a shell around a core NW. Similar complexes have been assembled and studied in reference 76. A NW may contain a p-n junction where photogenerated excitons become separated. c) Geometry for generation of photocurrent or photovoltage; a NW with attached NPs bridges two electrical contacts; the normal incidence of light is assumed. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. …....………... 86 Figure 4.6: a) Imaginary part of dielectric constant of Si and CdTe as a function of frequency. b) Energy transfer rate for a CdTe NP- Si NW complex shown in Figure 4.5. Also, Energy transfer rate for a CdTe NP – CdTe NW. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. .…………………………………………………………… 90 Figure 4.7: a) Calculated ratio nexciton , NW / nexciton , NW 0 for two values of angle θ and for unpolarized light for the complex in Figure 4.5; here nexciton , NW is the number of optically-generated excitons inside a NW. b) Calculated number of photo-generated excitons inside a Si NW as a function of the angle for two NP-NW distances. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. ...……………………………..... 92 Figure 4.8: a) Calculated ratio nexciton , NW / nexciton , NW 0 for two values of angle θ and for unpolarized light for the CdTe NPs- Si NW complex; here nexciton , NW is the number of optically generated excitons inside a NW. b) Calculated number of photo-generated excitons inside a CdTe NW as a function of the angle for two NP-NW distances. After P. Hernández-Martínez and A. O. Govorov 2008 [123]. ………………………………… 92 Figure 5.1: System of Au NP and surrounding matrix. Here we are considering 3 media, which are Au, water and Ice. ..………………………………………………………… 97 Figure 5.2: The graph shows the intensity of light required to reach the temperature of 100 oC (vapor). The plot also shows the radius and volume of water that we will expect to have for this temperature. ..………………………………………………… 101 Figure 5.3: System of many NPs surrounded by a medium. The medium can be liquid or gas (dust). ..………………………………………………………………………... 103 Figure 5.4: Nanoparticles in liquid medium. ..……………………………………….. 108 Figure 5.5: A) Temperature trace showing increase in temperature after laser excitation and temperature decay back to the ambient after laser is shut off. B) Plot of the natural log of (T(t)-T0)/(Tmax-T0) as a function of time right after the laser was turned off. The blue dash line in the figure is the fit of the data using the decay constant obtained from
13 this plot. The red dash line in the figure is our model fit to the data with η=1. C) Plot of the residual of the data compared to our model fit. After H.H. Richardson, A.O. Govorov et al. 2009 [170]. …………………………………………….……………………….. 113 Figure 5.6: The temperature trace of the same droplet with different laser intensities. The red dash line is the fit of our model to the data with η=1. The intensity of the laser is 0.28, 0.23 and 0.14 W. After H.H. Richardson, A.O. Govorov et al. 2009 [170]. …... 114 Figure 5.7: A) The temperature trace of a droplet with chopped laser intensity. The fit of our model to the data with η=1 is shown as the red dash line. The difference in the model fit to the data is shown in the upper inset. The temperature limit in the inset is ±1 °C. B) An anisotropic square-wave waveform was applied to the laser intensity (originally at 0.28 W) with the resulting temperature trace shown in A. The laser intensity was on for 2 ms out of 5 ms resulting in a 60% reduction in the overall laser intensity. After H.H. Richardson, A.O. Govorov et al. 2009 [170]. ............................................. 115 Figure 5.8: Image of typical water droplet taken during laser excitation. The thermocouple is embedded in the droplet. The droplet is located at the end of a stainless steel needle. The size (volume) of the droplet is determined by comparison of a grid positioned next to the droplet. After H.H. Richardson, A.O. Govorov et al 2009 [170]. ………………………………………………………………………………………… 117 Figure 5.9: Geometry of the model system with hot NP cylinder; lopt = 0.186 cm and Rbeam = 0.015 cm. ..………………………………………………………………. 118 Figure 5.10: a) Calculated temperature increase on the millimeter scale in the vicinity of the heated cylindrical region. The parameters are chosen similar to those in the experiment. The photon energy is equal to the plasmon peak energy. b) We zoom on a nanoscopic region in the vicinity of a single NP, δT, relative to the local averaged temperature to show a small temperature “bump” of a single NP. ..…………………. 119 Figure 5.11: a) Calculated temperature for the case of a very diluted NP solution and very high laser power. We see very sharp “spikes” in the temperature profile due to optically driven single NPs. Such hot spots can be created and studied using single-NP spectroscopy. b) We zoom on a region in the vicinity of a single NP. ..….. 120 Figure 5.12: System of Au NP and a surrounding matrix. Here, we are considering that, NP is at the bounder of region I and II. NP is illuminated at the plasmon frequency by a laser. ..………………………………………………………………... 122 Figure 5.13: Show temperature as a function of “x”. a) At the positions (x,0nm,-30nm) red line and (x,50nm,-30nm) blue line. The matrix is made of Air, AlGaN and Si. .... 130
14 Figure 5.14: At the position (x,0nm,-50nm) red line and (x,50nm,-50nm) blue line. The matrix is made of Air, AlGaN and Si. ..………………………………………..…….. 130 Figure 5.15: Temperature distribution as a function of (x,y). Here, we plot temperature at position (x,y,-30nm). The surrounding matrix is made of Air-AlGaN-Si. ..………. 131 Figure 5.16: Shows the three dimensional temperature distribution as a function of (x,y) at the plane given by (x,y,-30nm). The surrounding matrix is made of Air, AlGaN and Si. ……………...…………………………………………………………………………. 132 Figure A.1: Coordinate system used to define the orientation factor κ. After JR Lakowicz, “Principles of Fluorescence Spectroscopy” [67]. ……..………………….………….. 149
15 CHAPTER 1: INTRODUCTION The development of nanoscience and nanotechnology provides us with a new generation of miniaturized, highly efficient devices. Nanostructures are utilized in many fields of fundamental and applied sciences, ranging from basic research in chemistry and biology to applications in medicine and electronics. Nanoscale materials exhibit very interesting electronic, optical and catalytic properties. What makes these nanomaterials even more remarkable is the fact that these physical properties can be altered by changing the size and geometry of the nanocrystal. For example, a change in the size of the nanocrystal causes a shift in the photoemission wavelength of the material, as shown in Figure 1.1. The ability to tune their physical properties makes these nanomaterials very attractive for the development of novel devices such as (bio)sensors, (bio)chips, solar cells, junction diodes, transistors, etc.
Figure 1.1. a) Optical emission of CdSe quantum dots. After Wikipedia webpage [1]. b) Emission spectra of the quantum dots showing the dependence on size.
16
The general aims of this dissertation are to concentrate on: (1) the basic physics governing the process of energy transfer in both metal/semiconductor nanoparticles and nanowires; (2) the light-to-heat conversion due to plasmon resonance in metal nanoparticles. The rest of this chapter gives a few historical aspects of nanostructures, as well as some applications which have been reported. A brief overview of the basic concepts and theory needed for this work is given in chapter 2. Chapter 3 discusses the theoretical formalism and its potential application on the exciton-plasmon interaction for CdTe nanowire with mobile excitons and Au nanoparticle with localized plasmons. This chapter includes text and results from Jaebeom Lee, Pedro Hernandez, Jungwoo Lee, Alexander O. Govorov, and Nicholas A. Kotov, “Molecular Spring Assemblies with Emission Controlled by Protein Concentration via Exciton-Plasmon Interaction”, Nature Materials, 6, 291-295, 2007, “Copyright 2007, Nature Publishing Group”. In chapter 4 we obtain analytical equations and numerical results for the transfer rates between CdTe nanoparticle to CdTe, Si, Au nanowire. This chapter includes text and results from P. Hernández-Martínez, A. O. Govorov, “Exciton Energy Transfer Between Nanoparticles and Nanowires”, Phys. Rev. B, B78 035314, 2008, “Copyright 2008 by the American Physical Society”. In chapter 5 we study the heat generation and temperature distribution of optically excited Au nanoparticles embedded in media. This chapter includes text and results from H.H. Richardson, M. T. Carlson, P. J. Tandler, P. Hernández, and A. O. Govorov, “Experimental and Theoretical Studies of Light-to-Heat Conversion and Collective Heating Effects in Metal Nanoparticles Solutions”, Nano Letters, Vol. 9, Issue
17 number 3, 1139-1146, 2009, “Copyright 2009 by the American Chemical Society”. Chapter 6 presents the general conclusion of this work, as well its outlook. Early History of Nanostructures Even without a knowledge of the physical origin of their properties, the metal and semiconductor nanostructures have been in use since the ancient times. Two such examples are the Lycurgus Cup and the Damascus steel which is used to make swords. It is only with the development of new devices to analyze and grow nanomaterials, scientists were able to prove that nanotechnology has been employed by mankind for a long time. The Lycurgus cup is a remarkable piece of Roman glasswork (5th century) with some peculiar optical properties. When it is illuminated from the outside, the cup appears green (see Figure 1.2a). However, when it is illuminated from the inside, it appears crimson red except for the King, who looks purple (see Figure 1.2.b). The dichroism of the cup is due to the presence of nanosized particles of silver (66.2%), gold (31.2%) and copper (2.36%), which are embedded in the glass matrix [2-4]. The majority of nanometals range from 20 to 40 nm in size, but it could go up to 100 nm in size. The red color is due to the absorption of light by gold at approximately 520 nm, while the purple color is due to the absorption by larger nanoparticle. The green color is due to the scattering of light by large silver nanoparticles (greater than 40 nm in size). In addition the composition of the glass, which is 73.5% silica, 14.0% Na2O, 6% lime, and 0.9% K2O, also plays a role in the optical response of the cup.
18
Figure 1.2. Lycurgus Cup. a) The green appearance is produced by reflection. After The British Museum webpage [5]. b) The red and purple are due to transmitted light. After The British Museum webpage [6].
The foundation of Damascus steel is Wootz steel (iron with approximately 1% to 2% carbon) which originated in India and Sri Lanka about 500 AD or earlier [7]. Damascus steel swords became renowned for their extreme strength, sharpness, resilience and beauty of their characteristic surface pattern [8,9]. It was said that Damascus steel blades were able to cut a piece of silk in half as it fells to the ground, as well as being able to chop through normal blades, or even rock, without losing their sharp edge. Scientific studies reveal the presence of cementite nanowires and carbon nanotubes [1012] which give this steel its distinctive properties (see Figure 1.3).
19
Figure 1.3. TEM image of cementite nanowires in Damascus Steel. a) and b) Carbon nanotubes in Damascus steel. The dark stripes indicate carbon nanotubes of several hundred nanometers in length. After M. Reibold et al. [12]
Applications of Gold Nanoparticles In this subsection we discuss the use of Au nanoparticles for: 1) Labeling and Visualizing; 2) Delivery; 3) Heater; and 4) Sensor [13]. The first three applications use the particles as “passive” reporters, i.e. there is no change in the particle properties during the read-out. For the fourth application, the change in the particles properties is used for the read-out. Gold Nanoparticles for Labeling and Visualizing For the purposes of labeling and visualizing, Au nanoparticles are used for: immunostaining; single particle tracking; as contrast agents for X-rays; and also for phagokinetic tracks. For immunostaining, Au nanoparticles, which have been conjugated with ligands against the structure to be labeled, are added to fixed and permeabilized cells. Gold nanoparticles bind to the designated structures due to molecular recognition.
20 This way Au nanoparticles provide a strong contrast for TEM imaging with high lateral resolution [14,15]. Similarly to immunostaining, single particle tracking, also involves the addition of Au nanoparticles, which have been conjugated with specific ligands against the structures, to the cells. However, in this case fewer Au nanoparticles are added, which makes the average distance between gold particles larger than the optical resolution limit [16-21]. Using the same idea as in the previous cases, Au nanoparticles can be used as contrast agents for X-ray [22,23]. The advantage of this method is that it allows us to make contrast images in vivo of whole organs in animals. The phagokinetic tracks are used in imaging the movement of cells adhering to a substrate [24-26]. Here, the surface of the substrate is coated with a layer of Au nanoparticles. Cells adhering to the substrate will ingest the underlying nanoparticle leaving behind an area free of particles. This path is a blueprint of the migration of the cells. Au Nanoparticles as a Vehicle for Delivery In this case, the molecules of interest are absorbed on the surface of the Au particles and the whole conjugate is introduced into the cell. Once inside the cells the molecules will eventually detach themselves from the Au particles. There are two ways to get into the cells [27,28], one is the so call gene guns (by force) and the other one is by ingestion [29-32]. Au Nanoparticle as a Heat Source When Au nanoparticles absorb light the free electrons in the particles are excited. Excitation at the plasmon resonance frequency causes a collective oscillation of the electrons (plasmons). Due to interaction between the electron and the crystal lattice, the
21 electrons relax and transfer their energy to the lattice, creating phonons which heat the particles. Local heating by Au nanoparticles can be used to manipulate the temperature of the surrounding environment [33,34]. Hyperthermia is an anti-cancer therapy which exploits the fact that the cells are very sensitive to any small increases in temperature. The idea is to enrich cancerous tissues or cells with Au nanoparticles. This can be done by conjugating the gold particles with specific ligands that bind to the cancerous cells. When the Au nanoparticles are illuminated, the temperature close to the particle increases and in this way cells in the vicinity of the particle can be selectively killed [35-38]. Moreover, photo-induced heating of Au nanoparticles can be used for the breaking of chemical bonds [39-41]. For example, double stranded DNA melts into two single strands when it is heated. If a double stranded DNA is linked to the surface of Au nanoparticles, then it will melt into two single strands after illumination. Finally, photo-induced heating of Au nanoparticles can also be used for remotely controlled release of cargo molecules from a container. This concept is based on embedding cargo molecules in containers, such as polymer capsules, where the walls of the containers are functionalized with Au nanoparticles. When the capsule is illuminated, the heat from the Au particles disrupts the wall causing the release of the cargo [42-44]. Au Nanoparticles as Sensors In the previous applications Au nanoparticles are used as “passive” reporters, i.e. there is no change of particle properties for the read out. When Au nanoparticles are used for sensor applications, they are “active” reporters for the read out, i.e. there will be a
22 change in the properties of the particles. For example, the surface plasmon can be used as a sensing mechanism because the plasmon frequency is a function of the average distance between Au nanoparticles [45,46]. This plasmon effect can be used for colorimetric detection of analytes [47-50]. Another example of sensor application of Au nanoparticles is that the fluorescence of many fluorophores quenches when they are in close proximity to the Au nanoparticles [51-53]. This effect is known as fluorescence quenching. Application of Nanowires In the previous subsection we discussed the potential application of the optical properties of Au nanoparticles. Now, we discuss some of the physical properties and potential applications of one-dimensional nanocrystals (nanowires) [53]. Magnetic Properties The magnetic properties of magnetic nanowire arrays make them suitable for the development of magnetic information storages devices. Studies have shown that periodic arrays of magnetic nanowires possess the capability of storing 1012 bits/in2 of information [54,55]. The basic principle in this application is that the magnetic properties of the nanowires depend on the geometry of the wire, in particular, 0n the diameter and aspect ratio (length/diameter). Figure 1.4 shows the dependence of the squareness of the hysteresis loop on the diameter and aspect ratio of the wire (as a function of the applied magnetic field parallel and perpendicular to the wire axes). Optical Properties The optical properties of nanowires make them very attractive for many optical applications. Studies have shown that the absorption, emission and extinction depend
23 strongly on the nanowire diameter. Figure 1.5 shows the extinction spectra of silver (Ag) nanowires with different diameters [53,57]. Also, the high efficiency rate in the generation of carriers due to photo-excitation in CdSe nanowires [58] and coaxial silicon nanowires [59] makes them suitable in the development of solar cells.
Figure 1.4. Hysteresis loops of Co nanowire arrays with different diameters and roughly the same length with applied magnetic field (H) parallel (||) and perpendicular (⊥)to the wire axes. After Beck et al. 2005 [56].
24
Figure 1.5 The extinction spectra of Ag nanowire with different diameters. The incident electric field is perpendicular to the wire axis. After Schider et al. 2001 [57].
25 CHAPTER 2: THEORETICAL BACKGROUND Bulk Semiconductors The crystalline structure of bulk material is important to understand the physical properties of the crystals. Diamond and zinc-blende structure are common in semiconductor crystals, and semiconductors which possess these structures are of interest for device applications [60]. These crystalline structures are shown in Figure 2.1 [61,62]. We can visualize the diamond and zinc-blende structure as two identical interpenetrating face center cubic (fcc) sublattices which are displaced from each other by 1/4 of the cube diagonal along the [111] direction (see Figure 2.1a and 2.1.b). In the case of diamond structure both fcc lattices have the same type of atoms. In contrast, in the zinc blende structure, each fcc has different atoms, one fcc lattice has cations (positive ions) and the other has anions (negative ions). Crystalline structures are characterized by the primitive lattice vectors, a1, a2 and a3 with length a. The choice of primitive lattice vectors for a given lattice is not unique. Figure 2.1b shows one possible choice for the fcc lattice. The physical properties of bulk materials are easier to describe in reciprocal space in particular, at the first Brillouin zone. Figure 2.1c shows the first Brillouin zone with its symmetry points for the fcc lattice.
26
Figure 2.1. a) The crystal structure of diamond and zinc blende (ZnS). b) The fcc lattice showing a set of primitive lattice vectors. c) The reciprocal lattice of the fcc lattice. Here is shown the first Brillouin zone. Special symmetry points are denoted by Γ, X, and L and high symmetry lines joining some of these points are labeled as Λ and Δ. After P. Y. Yu and M. Cardona, “Fundamentals of Semiconductors: Physics and Materials Properties” [61].
Band Structure The Hamiltonian describing a perfect crystal can be written as
27 Pj2 Z j Z j'e 2 pi2 1 +∑ + ∑' H =∑ 2 j ', j 4πε 0 R j − R j ' i 2 mi j 2M j Z j e2
1 e2 −∑ + ∑' 2 i ,i ' 4πε 0 ri − ri ' j ,i 4πε 0 ri − R j
(2.1)
where ri is the position of the ith electron, Rj the position of the jth nucleus, Zj is the atomic number of the nucleus, pi and Pj are the momentum operators of the electron and nuclei, e is the electron charge, and Σ′ means that the summation is only over pairs of indices which are not identical. This many particle Hamiltonian (Equation 2.1) is very difficult to solve because we are dealing with a huge number of particles. Therefore, to solve it we need to make a lot of simplifications. First, we use the valence electron approximation. Here, we reduce the number of electrons in the problem by taking into account only the valence electrons, neglecting the core electrons. We take advantage of the fact that core electrons cannot move as freely as the valence electrons because they are tightly bound to the nucleus in the so-called ion core. As a result of this approximation the indices j and j’ will denote the ion cores while the electron indices i and i’ will label only the valence electrons. The next approximation is the Born-Oppenheimer or adiabatic approximation. This approximation relies on the fact that ions are much heavier than electrons so they move more slowly. Typically, the energy scale involved in the ionic motion in solid is of the order of tens of meV (∼1013 Hz), whereas the excitation energies for electrons in semiconductors is of the order of 1 eV (∼1015 Hz). As a result, the electronic frequencies are two orders of magnitude larger than the ionic vibrations; therefore the electrons see
28 the ions essentially as stationary. Under this assumption, we can rewrite the Hamiltonian (Equation 2.1) as the sum of three terms:
H = H ions (R j ) + H e (ri , R j 0 ) + H e −ion (ri , δ R j )
(2.2)
where H ions (R j ) is the Hamiltonian describing the ionic motion under the influence of the ionic potentials plus the time averaged electronic potentials; H e (ri , R j 0 ) is the Hamiltonian for the electrons with ions in their equilibrium positions R j 0 , and H e −ion (ri , δ R j ) describes the change in the electronic energies as a result of the displacements δ R j of the ions from their equilibrium positions. We have already simplified the problem, but in order to solve each part of the Hamiltonian in equation 2.2, we still need to do more approximations. Next, we discuss the general solution for the electronic, H e (ri , R j 0 ), Hamiltonian in equation 2.2. The electronic Hamiltonian H e (ri , R j 0 ) is the one responsible for the electronic excitation spectra in semiconductors. It is given by
Z je2 pi2 1 e2 + ∑' −∑ H =∑ 2 i ,i ' 4πε 0 ri − ri ' i , j 4πε 0 ri − R j 0 i 2 mi
(2.3)
where the first term is the kinetic energy of the electrons, the second is the Coulomb repulsion and the last term is the Coulomb attraction between the electrons and the
29 nucleus in their equilibrium position. Diagonalizing this Hamiltonian is a challenging task. Therefore, we simplify the electronic Hamiltonian by using the mean field approximation. This approximation assumes that every electron experiences the same average potential V (r ) . Thus, the motion of each electron will be identical and the Coulomb terms in equation 2.3 are replaced by V (r ) . So, the resulting Hamiltonian is given by
p2 H 1e = + V (r ) 2m *
(2.4)
where H 1e is the one electron Hamiltonian. In order to find the electron energy we need to determine V (r ) ; one way to do it is by using first principle calculations and/or semiempirical methods. Moreover, we take into account the rotational and translational symmetry of the crystalline structure. Apart from the Coulomb interaction and the symmetry of the crystal, one must take into account other types of interaction like the spin-orbit effect. As a consequence, the electronic band structure becomes much more complicated. To determine the band structure, a variety of methods have been developed. The nearly free electron, pseudopotential, k • p and tight-binding or linear combination of atomic orbital (LCAO) methods are the most commonly used. Figure 2.2 shows the electronic band structure for CdTe zinc blende structure and Si diamond like structure.
30
Figure 2.2. Electronic band structure for: a) zinc blende structure (CdTe), and b) Diamond like structure (Si). After Cohen M. L. and Bergstresser T. K. 1966 [63].
Excitons An exciton is a quasiparticle consisting of a bound state of an electron and a hole in insulator and semiconductor crystals. An exciton can move through the crystal and transport energy; and since an exciton is electrically neutral it does not transport charge. An exciton can be created by external excitation, for example, the absorption of a photon, with E ≥ E g . In this direct process, an electron is excited from the valence band to the conductive band, leaving behind a hole with opposite charge in the valence band, to which the electron will bind due to the attractive Coulomb interaction. Because of the attractive Coulomb interaction between the electron and the hole in an exciton, the internal states are analogous to those of the hydrogen atom, and some of the lower energy states lie below the conduction band by an energy equivalent to the exciton binding energy in that state (see Figure 2.3 and 2.4).
31 An exciton has two quantities, 1) the pseudomomentum of the electron-hole pair, and 2) the relative momentum of the electron and the hole. The pseudomomentum, which is equal to the vector sum of the individual momenta of the electron and the hole, enables an exciton to move throughout a crystal; and the relative momentum determines its internal structure. Excitons are classified into 1) a tightly bound exciton (Frenkel exciton), and 2) a weakly bound exciton (Mott-Wannier exciton).
Figure 2.3. Exciton levels for a simple band structure at k = 0 .
32
Figure 2.4. Energy levels of an exciton created in a direct process. Optical transitions are shown by arrows.
Frenkel Excitons In a tightly bound exciton the excitation is localized on a single atom (see Figure 2.5), i.e. a Frenkel exciton is an excited state of a single atom. A Frenkel exciton is an excitation that can hop from one atom to another via coupling between neighbors. Similarly to all other excitation in a periodic structure, the translational states of Frenkel excitons take the form of propagating waves. Consider a crystal of N atoms on a line or ring. If u j is the ground state of atom j, the ground state of the crystal is [64]
ψ g = u1u 2 Lu j Lu N −1u N
(2.5)
33
If a single atom j is in an excited state v j , the system is described by
ϕ j = u1u 2 Lu j −1v j u j +1 Lu N −1u N
(2.6)
If we consider that the excited atom interacts only with nearby atoms in its ground state, then the excitation will be passed from atom to atom.
Figure 2.5. a) Schematic illustration of a tightly-bound exciton (Frenkel exciton) localized on one atom in a crystal.
Applying the Hamiltonian of the system on the function ϕ j with the jth atom excited, we obtain the following
34 Hϕ j = εϕ j + T (ϕ j −1 + ϕ j +1 )
(2.7)
where ε is the free atom excitation energy; T is the rate of transfer of the excitation from j to its nearest neighbors, j − 1 and j + 1 . The solutions of equation 2.7 are the waves of the Bloch form:
ψ k = ∑ exp(ijka )ϕ j
(2.8)
j
Operating the Hamiltonian on equation 2.8
[
]
Hψ k = ∑ e ijka Hϕ j = ∑ e ijka εϕ j + T (ϕ j −1 + ϕ j +1 ) j
(2.9)
j
Rearrange the right-hand side of equation 2.9
[
(
)]
Hψ k = ∑ e ijka ε + T e ika + e − ika ϕ j = (ε + 2T cos(ka ))ψ k
(2.10)
j
So that the energy eigenvalues are (see Figure 2.6):
E k = ε + 2T cos(ka )
(2.11)
35 Applying the periodic boundary conditions, the allowed values of the wavevector k are:
k=
2πn ; Na
n = − 12 N ,− 12 N + 1, L, 12 N − 1
(2.12)
Figure 2.6. Energy versus wavevector for a Frenkel exciton.
Mott-Wannier Excitons In a weakly bound exciton the electron-hole distance is larger than the lattice constant of the crystal, meaning that, the exciton is delocalized (see Figure 2.7). The Mott-Wannier exciton is similar to the hydrogen atom problem, in other words, the Mott-
36 Wannier exciton can be treated as a two particle system weakly interacting. In which case, the electron and hole energy (at k = 0 ) is given by [61,65]:
ε c (k ) = ε c (0) +
h2k 2 2 me *
(2.13)
ε v (k ) = ε v (0 ) −
h2k 2 2m h *
(2.14)
and
where me * and − e is the electron mass and electron charge, mh * and + e is the hole mass and hole charge. Here for simplicity, we assume that the crystal has simple valence and conduction bands.
Figure 2.7. Schematic illustration of a weakly bound exciton (Mott-Wannier exciton).
37
The total kinetic energy is
P=
p e2 p h2 + 2me * 2m h *
(2.15)
where p e2 and p 2h are the electron and hole momenta. The effective Hamiltonian for the two particle system when interacting in a dielectric medium of relative dielectric constant
ε , is
H eff
1 h2 h2 e2 2 2 =− ∇e − ∇h − 2me * 2m h * 4πε 0 ε re − rh
(2.16)
The solution for this Hamiltonian is
En = E g −
1
(4πε 0 )2
μ ex e 4 1 h 2 K 2 + 2h 2 ε 2 n 2 2 M *
(2.17)
where E n is the exciton energy, E g = ε c (0) − ε v (0) is the band gap energy,
1
μ ex
=
1 1 + is the reduced exciton mass, and M * = me * + mh * is the effective me * m h *
exciton mass. An useful parameter for an exciton is the exciton Bohr radius (aex). It is obtained from the second term of equation 2.17. Therefore the exciton Bohr is given by
38
h 2ε 2 a ex = 4πε 0 n μ ex e 2
(2.18)
Quantum Confinement The quantum confinement should be taken into account when any dimension of a system is comparable to the particle or quasi-particle wavelength, defined by de Broglie. In the case of an electron with effective mass m* in a semiconductor at temperature T the de Broglie wavelength is
λ=
h h = p 3m * k B T
(2.19)
At this length scale, the particle or quasi-particle must be treated quantum mechanically. Due to the reduction of degree of freedom, the discrete energy levels arise and the continuous energy levels no longer apply. Moreover, the density of states (DOS) changes with the reduction of degree of freedom. The DOS for a bulk semiconductor (3D system) is given by
3
⎛ m* ⎞2 ρ 3 D (E ) = ⎜ 2 2 ⎟ 2 E ⎝π h ⎠
For a 2D system (quantum well), the DOS is
(2.20)
39
⎛ m* ⎞ Θ( E − ε n ) 2 2 ⎟∑ ⎝π h ⎠ n
ρ 2 D (E ) = ⎜
(2.21)
where Θ(E − ε n ) is the Heaviside step function and εn are the discrete energy levels. For a 1D system, the DOS is given by
1
⎛ 2m * ⎞ 2 ρ 1D (E ) = ⎜ 2 2 ⎟ ∑ ⎝ π h ⎠ n,m
1
(E − ε )
Θ(E − ε n ,m )
(2.22)
n,m
And, for the 0D system (quantum dots, nanoparticles) is
ρ 0 D (E ) = 2 ∑ δ (E − ε n ,m ,l )
(2.23)
n , m ,l
where δ (E − ε n ,m,l ) is the Dirac delta function. Figure 2.8 shows the degree of freedom and the density of states for bulk and quantum structures.
40
Figure 2.8. Schematic of the degree of freedom and density of state for a) bulk, b) quantum well, c) quantum wire, and d) quantum dot structures [66].
Plasmons A plasmon is a quantum of a plasma oscillation. Plasmons are collective oscillations of the free electron gas density. A plasmon is a quasiparticle which can be described by the equation of motion for the free electron gas. Plasmons can be understood using classical equations of motion. For simplicity, we consider the one dimensional case:
nm
d 2u n 2 e 2u = − = − neE ε0 dt 2
d 2u + ω p2 u = 0 2 dt
(2.24)
(2.25)
41 where u is the displacement of the electron gas, n is the number of electron per unit volume, m is the electron mass, E = neu ε 0 is the electric field (in SI units) and ω p is the plasma frequency given by
ne 2 ω = mε 0 2 p
(2.26)
In a crystal the plasma oscillation is a collective longitudinal excitation of the conduction electron gas. Such oscillations are explained by the dielectric function. The dielectric function is defined as [64]
ε (ω ) =
D(ω ) P(ω ) = 1+ ε 0 E (ω ) ε 0 E (ω )
(2.27)
where E (ω ) is the external electric field, and P(ω ) is the polarization induced by E (ω ) . For the free electron gas, the polarization P(ω ) is
ne 2 P(ω ) = − nex = − E (ω ) mω 2
(2.28)
where n, e, and m are the same parameter as equation 2.24. From equation 2.27 and 2.28, the dielectric constant of the free electron gas is
42
ε (ω ) = 1 −
ne 2
ε 0ω 2
= 1−
ω p2 ω2
(2.29)
where ω p is the plasma frequency defined in equation 2.26. If the crystal has a background dielectric constant, ε (∞ ) , then
⎛ ω p2 ε (ω ) = ε (∞ )⎜1 − 2 ⎜ ω ⎝
⎞ ⎟ ⎟ ⎠
(2.30)
with
ω p2 =
ne 2 . ε (∞ )ε 0 m
(2.31)
We found an expression for the plasma oscillation in a crystal. Next, we need to find an expression for electromagnetic waves in a crystal. The electromagnetic wave equation for a nonmagnetic isotropic medium is
μ0
∂2D = ∇ 2E ∂t 2
The solution of equation 2.32 has the form
(2.32)
E ∝ exp(− iωt ) exp(iK ⋅ r ) , with
D = ε (ω , K )E . It follows from equation 2.32 that the dispersion relation for
electromagnetic waves is
43
ε (ω , K )ω 2 = c 2 K 2
(2.33)
Combining equation 2.30 and 2.33, the dispersion relation becomes
ε (ω ) = ε (∞ )(ω 2 − ω p2 ) = c 2 K 2
(2.34)
From this equation (Equation 2.34), we find two interesting cases.
Case 1: ω > ω p , in this region, the dispersion relation can be written as
ω 2 = ω p2 +
c2K 2 ε (∞ )
(2.35)
which describes transverse electromagnetic waves in a plasma (see Figure 2.9). Also, the dielectric constant is positive and real, which means, an electron gas in this region is transparent for the electromagnetic wave.
Case 2: ε (ω L ) = 0 , ω = ω L , in this case, the dispersion relation is given by
ε (ω L ) = 1 −
ω p2 ω L2
=0
(2.36)
44 Thus, ω L = ω p . Here the zeros of the dielectric function give the frequencies of the longitudinal oscillation modes of an electron gas at the plasma frequency.
Figure 2.9. Dispersion relation for transverse electromagnetic waves in a plasma.
Energy Transfer Nonradiative Energy Transfer Energy transfer from excited particles to unexcited ones is a common phenomenon that occurs in nature. The excitation processes involved in energy transfer can either be radiative, nonradiative, or both. For radiative energy transfer a (real) photon
45 is emitted by the excited particle and then this photon is absorbed by the unexcited particle. In the case of nonradiative energy transfer, energy is transmitted from the excited particle (donor) to the unexcited one (acceptor) by a process or processes where no (real) photon is emitted by the excited particle. One of the most important examples of nonradiative energy transfer is fluorescence resonance energy transfer (FRET). FRET is an electrodynamic phenomenon and is the result of long-range dipole-dipole interactions between the donor and the acceptor. The rate of energy transfer depends on the extent of spectral overlap of the emission spectrum of the donor with the absorption spectrum of the acceptor, the quantum yield of the donor, the relative orientation of the donor and acceptor transition dipoles, and the distance between the donor and acceptor. FRET is commonly used to measure the distance between the donor and the acceptor when the DA are fixed in space. Also, FRET can be used to measure the conformational changes that move the domains (D-A) closer or further apart. Another measurement which can be done by FRET is the extent of binding between the donor and the acceptor. In this section we discuss some aspects of FRET relevant to our work. Theory of Energy Transfer The process of energy transfer can be described as a transition between two states
k (D*, A) ⎯⎯→ (D, A *) T
where D* (D) is the donor in the excited (unexcited) state, A* (A) is the acceptor in the excited (unexcited) state, and kT is the rate of resonance energy transfer (RET) between
46 the donor and acceptor pair. In this process, the donor absorbs an external photon leaving it in an excited state. Then, the donor transfers its excited energy, via a nonradiative process, to the acceptor leaving it in an excited state. Förster was first to describe this process correctly. Förster derived an expression for the resonance energy transfer. From Förster’s theory, the rate of energy transfer from a donor to an acceptor kT(r) is given by [67]
k T (r ) =
1 ⎛ R0 ⎞ ⎜ ⎟ τD ⎝ r ⎠
6
(2.37)
where τD is the decay time of the donor in absence of acceptor, R0 is the Förster distance, and r is the donor-to-acceptor distance. Hence, the rate of energy transfer depends strongly on distance, and is proportional to r −6 (Equation 2.37). In addition, the rate of transfer is equal to the decay rate of the donor 1 τ D when the D-to-A distance (r) is equal to the Förster distance (R0), and the transfer efficiency is 50%. From the last statement, we define the Förster distance as the distance at which FRET is 50% efficient and its effective range is from 1 nm to 10 nm approximately. At this distance (r = R0 ) the donor emission would be decreased to half its intensity in the absence of acceptors. In a more detailed study of FRET [67,68], the rate of transfer for a single donor and acceptor separated by a distance r can be written as
47 Q κ2 k T (r ) = D 6 τ Dr
⎛ 9000(ln 10 ) ⎞∞ ⎟ F (λ )ε A (λ )λ4 dλ ⎜⎜ 5 4 ⎟∫ D 128 π N n A ⎠0 ⎝
(2.38)
where QD is the quantum yield of the donor in the absence of acceptor, n is the refractive index of the medium, NA is Avogadro’s number, r is the distance between the donor and the acceptor, and τD is the lifetime of the donor in the absence of acceptor. The term κ 2 is the factor describing the relative orientation in space of the transition dipoles of the donor and acceptor. κ 2 is assumed to be equal to 2/3 [See Appendix A for more details], for dynamic random averaging of the donor and acceptor. FD(λ) is the normalized fluorescence intensity of the donor in the wavelength range λ to λ + Δλ with the total intensity (area under the curve) normalized to unity. ε A (λ ) is the extinction coefficient of the acceptor at λ, which is typically in units of M-1cm-1. The overlap integral (J (λ )) expresses the degree of spectral overlap between the donor emission and the acceptor absorption:
∞
J (λ ) = ∫ FD (λ )ε A (λ )λ4 dλ
(2.39)
0
∞
∫ F (λ )ε (λ )λ dλ 4
J (λ ) =
D
A
0
∞
∫ F (λ )dλ D
0
(2.40)
48 FD(λ) is dimensionless. In calculating J (λ ) one should use the corrected emission spectrum with its area normalized to unity (Equation 2.39), or normalize the calculated value of J (λ ) by the area (Equation 2.40). The most common units of J (λ ) are: 1) M −1cm 3 , if ε A (λ ) is expressed in units of M-1cm-1 and λ is in centimeters, and 2) M −1 cm −1 nm 4 , if ε A (λ ) is expressed in units of M-1cm-1 and λ is in nanometers (M =
mol mol = ). L liter For practical reasons it is easier to think in terms of distance rather than transfer
rate. Thus, equation 2.37 is written in terms of the Förster distance R0. From equations 2.37 and 2.38 one obtains:
⎛ 9000(ln 10 )Q D κ 2 R = ⎜⎜ 5 4 ⎝ 128π N A n 6 0
⎞∞ ⎟ ∫ FD (λ )ε A (λ )λ4 dλ ⎟ ⎠0
(2.41)
This expression allows the Förster distance to be calculated from the spectral properties of the donor and acceptor and the donor quantum yield. The efficiency of energy transfer (E) is the fraction of photons absorbed by the donor which are transferred to the acceptor. This fraction is given by
E=
k T (r ) τ + k T (r ) −1 D
(2.42)
49 which is the ratio of the transfer rate to the total decay rate of the donor in the presence of acceptor. From equation 2.42 we can observe: 1) when the transfer rate is much faster than the decay rate, energy transfer is efficient; and 2) when the transfer rate is slower than the decay rate, energy transfer is inefficient because little transfer occurs during the excited state lifetime. The efficiency of energy transfer can be written as a function of distance by substituting equation 2.37 into equation 2.42.
E=
R06 R06 + r 6
(3.7)
This equation shows that the transfer efficiency is strongly dependent on distance when the D-A distance is near to R0 (see Figure 2.10). The efficiency quickly increases to 1 as the D-A distance decreases below R0. Conversely, the efficiency quickly decreases to 0 if
r is greater than R0. Note that, when r = 2 R0 the transfer efficiency is 1.54%, and when r = 0.5R0 the transfer efficiency is 98.5%.
50
Figure 2.10. Energy transfer efficiency (E) versus distance. R0 is the Förster distance.
51 CHAPTER 3: EXCITON-PLASMON INTERACTION IN A SYSTEM OF AU NANOPARTICLES AND CDTE NANOWIRES Introduction The development of (bio)sensors and actuators based on metal and semiconductor nanoparticles (NPs) and nanowires (NWs) has had a great deal of interest in recent years. The most successful and dominant technology in optical sensing and imaging are systems which use detection of analytes based on variation of intensity as a transduction mechanism [69-73]. Molecular spring assembly (MSA) made from Au NPs and CdTe NW have been demonstrated to have reversible variations of the intensity of excitonic CdTe luminescence [74-77]. This system combines two types of excitation, excitons and plasmons, which are common in nanomaterials. The optical emission of such superstructures is strongly temperature dependent due to the exciton-plasmon interaction. In paper [75], the authors report the optical effects stemming from collective interactions of metal NPs and semiconductors NWs (see Figure 3.1). They demonstrated the strong enhancement of fluorescence of CdTe-NWs conjugated with Au-NPs. Reference [76] describes the multistep cascade Förster resonance energy transfer (FRET) between NPs and NWs (see Figure 3.2). This effect is important for optoelectronic devices and energy conservation systems.
52
Figure 3.1. (A,B) TEM images of Au bioconjugated CdTe NW in solution B. (A) x 200k, (B) x 500k. Yellow circles indicate the areas where characteristic lattice fringes of 0.23 nm of (111) faces of metallic Au can be seen after bioconjugation. After N. A. Kotov et al. 2004 [75].
Figure 3.2. (a) AFM images of CdTe NWs (2.5 x 2.5 µm, the bar indicate the z-axis). (b,c) TEM images of NP conjugated NWs. Yellow circles in part c indicate CdTe NPs. After N. A. Kotov et al. 2005 [76].
We introduce a theoretical model consisting of a superstructure made from CdTe NWs and Au NPs connected by molecular springs, which utilize the changes in the luminescence emission wavelength (wavelength shift) as a transduction mechanism. We calculate the linear density of the excitons in the CdTe NW and how it is related to the
53 wavelength shift. One estimates the wavelength shift as: 1) function of mobility and 2) function of distance between the Au NP and CdTe NW. Modeling of Exciton Diffusion in Nanowires Exciton diffusion in the CdTe NW can be described by the transport equations. The exciton current j in the NW is given by:
j = μ ⋅ F ( x ) ⋅ n( x ) − D ⋅
dn( x) dx
(3.1)
where the linear density of the excitons is n(x), μ is the mobility coefficient, D is the diffusion coefficient which is related to the mobility by the Einstein relation ( μ =
and F ( x) = −
D ), k BT
dU ( x) = −U ' ( x) is a force acting on the exciton due to the presence of a dx
random potential U(x) [78]. Applying the continuity equation under the steady state condition, one gets:
j ' = −γ tot n( x) + I 0
(3.2)
where I0 is the rate of exciton generation by the incident light, γ tot = γ 0 + γ NP is the exciton recombination rate, τ 0 =
1
γ0
is the intrinsic exciton lifetime in a NW, and
54
τ NP =
1
γ NP
is the energy transfer time of the exciton from the NW to the Au-NP corona
(see Figure 3.3).
Figure 3.3. Sketch of the systems Nanoparticles-Nanowire.
The energy transfer rate (γNP) is very sensitive to the NW-NP distance, RNW-NP. The dependence of the energy transfer rate and the NW-NP distance is given by the Förster formula [76,80-83]. Considering that, the rate of transfer to a single NP decreases rapidly with the distance, and taking into account only the NPs nearest to the exciton, the Förster formula can be written as:
55
γ NP ∝
1
R
6 NW − NP
(3.3)
And the exciton lifetime is given by:
τ exc =
1 γ 0 + γ NP
(3.4)
Combining equation 3.1 and 3.2, the resulting differential equation for the exciton’s diffusion is given by:
D
d 2 n( x ) dn( x) + μU ' ( x) + μU " ( x)n( x) = γ tot n( x) + I 0 2 dx dx
(3.5)
Now we have to find a suitable random potential U(x) based on the Atomic Force Microscope (AFM) data (see Figure 3.4). In order to do so, we consider that the exciton diffusion can be described as a problem of a cylindrical NW; that is, one has to solve the Schrödinger equation in cylindrical coordinate [84], which is:
−
h2 2 ∇ ψ (r ) + V (r )ψ (r ) = Eψ (r ) 2m
(3.6)
56
⎧∞ if ρ > 0 where V (r ) = ⎨ ⎩0 if ρ < 0
ρ 2 = x 2 + y 2 is the potential for cylindrical NW. The
solution for this problem is well known, and the eigenenergies are:
E n ,l = E g0 +
h 2 β n2,l
(3.7)
2 MR 2
where E g0 is the bulk band-gap energy, β n,l is the nth-order zero of the Bessel function J l ( ρ ) , R is the radius of the cylindrical NW and,
1 1 1 = + , me = 0.095m0 M me m h
and mh = 0.5m0 are the effective electron and hole masses, respectively. Assuming that the random potential can be written as the eigenenergies for the cylindrical NW and, considering that we have a periodic modulation of the radius R(x) (see Figure 3.4A), the random potential takes the form of:
h 2α 2 U ( x) = E + 2 MR 2 ( x) 0 g
R ( x) = 3.4nm + 1.25nmCos(qx)
where q =
(3.8)
(3.9)
2π and d = 300nm is the period and α ≅ 2.405 is the first zero of the Bessel d
function J 0 ( x ) . For convenience, we use the diameter instead of the radius:
57
D( x) = 6.8nm + 2.5nmCos(qx)
(3.10)
Figure 3.4. A) AFM profile of a typical NW from corresponding image in B). After N.A. Kotov, A.O. Govorov et al. 2007 [90].
To compare the theoretical model and the experimental data, we compute the NW radius and diameter by using equations 3.9 and 3.10. The results are shown in Figure 3.5. Then, the corresponding driving potential is calculated using equation 3.8 and is shown in Figure 3.6A. Next, to compute the linear density of the excitons, we solve equation 3.5 numerically and the result is shown in Figure 3.6B. As we expect, the linear density of
58 the excitons is more efficient for larger NP-NW distance since excitons have more time to move. The parameters used to compute these results were determined experimentally and are typical for CdTe NWs [74].
A)
B) Experiment
12 6 NW Diameter (nm)
NW Radius
10
4
2
8 6 4
Model
2 0
0 0
200
400
600
800
1000
1200
0
Position (nm)
200
400
600
800
1000
1200
Position (nm)
Figure 3.5. A) NW radius from our model. B) Comparison of the NW diameter from the experiment data and our theoretical model. Experimental data after N.A. Kotov et al. 2007 [90].
To compute the linear density of the excitons for small mobility (μ) and large exciton to NP energy transfer rate (γNP), we introduce an approximate analytical solution by defining two new variables which are: n0 =
I0
γ tot
and ρ ( x) =
n( x ) . So, equation 3.5 n0
can be written as:
D d 2 ρ ( x) μU ' ( x) dρ ( x) μU " ( x) + + ρ ( x) = ρ ( x) + 1 γ tot dx 2 γ tot γ tot dx
(3.11)
59
A)
B)
0.40
R RW-NP =12.2
R RW-NP =11.0
0.35
Exciton Density, n(x) (arb. units)
0
U(x) - Eg (eV)
Exciton Potential,
0.30 0.25 0.20 0.15
2
0.10
Potential Traps
0.05
0
0.00 0
200
400
600
800
1000
1200
0
200
Positon (nm)
400
600
800
1000
1200
1400
Position (nm)
A) Exciton potential from the theoretical model. B) Exciton density cm 2 distribution for μ = 0.2 and two NW-NP distances. eV ⋅ s
Figure 3.6.
To avoid instability of numerical solution at the boundaries of the NW, we propose an analytical solution to equation 3.11. Assuming γ tot → ∞ in that case, ρ(x) can be expanded as:
ρ ( x) = ρ 0 + ρ1 + L + ρ i + ...
where ρ 0 = 1 , ρ1 =
ρi =
μU " ( x) μU " ( x) ρ0 = and, γ tot γ tot
D dρ i −1 ( x) μU ' ( x) dρ i −1 ( x) μU " ( x) + + ρ i −1 ( x) i ≥ 2 . γ tot γ tot γ tot dx dx
(3.12)
60 Selected results are shown in Figures 3.7 and 3.8. Figure 3.7A shows a comparison between the analytical and numerical solutions (Equation 3.12 and 3.11 respectively) for the linear density of excitons. As we can see from Figure 3.7B, the analytical solution is a good fit for the linear density of exciton. The results presented in both the Figures 3.7A and 3.7B are calculated with a large γNP. On the other hand, Figures 3.7C and 3.7D are for the case of a small exciton mobility. Figure 3.7C shows another comparison between the analytical and numerical solution for the linear density of excitons. While Figure 3.7D gives another good fit for the linear density of excitons. An interesting result can be seen Figure 3.8, where we find small minima every 300 nm in the linear density of excitons. These minima may be explained in the following manner: with a small exciton mobility, some of the excitons have enough time to recombine with the plasmon, thus reducing the number of excitons in the NW.
61
B)
A) 1.10
1.10
Analytical Solution Numerical Solution
1.05
Exciton Density, n(x) (arb. units)
Exciton Density, n(x) (arb. units)
1.05
Analytical Solution
1.00
0.95
0.90
1.00
0.95
0.90
0.85
0.85
100
120
140
160
180
200
0
50
100
Position (nm)
150
200
250
300
Position (nm)
C)
D) Analytical Solution 1.05
Analytical Solution Numerical Solution
1.00
1.00
Exciton Density, n(x) (arb. units)
Exciton Density, n(x) (arb. units)
1.05
0.95
0.95
0.90 0.90 0.85 100
120
140
160
Position (nm)
180
200
0
50
100
150
200
250
300
Position (nm)
Figure 3.7. A) Comparison between the analytical and numerical solution for the case of 6 μ = 0.2 cm 2 eV ⋅ s and large γNP (RNW-NP = 6.2nm, γ NP ∝ 1 R NW − NP ). B) Analytical 6 solution for μ = 0.2 cm 2 eV ⋅ s and large γNP (RNW-NP = 6.2nm, γ NP ∝ 1 R NW − NP ). C) Comparison between the analytical and numerical solution for the case of 6 μ = 0.01cm 2 eV ⋅ s and γNP (RNW-NP = 9.97 nm, γ NP ∝ 1 R NW − NP ). D) Analytical solution 6 for μ = 0.01cm 2 eV ⋅ s and γNP (RNW-NP = 9.97 nm, γ NP ∝ 1 R NW − NP ).
62
1.08
Small Minima
1.06 1.04
Exciton Density, n(x) (arb. units)
1.02 1.00 0.98 0.96 0.94 0.92 0.90 0.88 0.86 0
200
400
600
800
1000
Postion (nm)
Figure 3.8. Analytical solution for the linear density of exciton for the case of 1 cm 2 μ = 0.01 and γNP (RNW-NP = 9.97 nm, γ NP ∝ 6 ). eV ⋅ s R NW − NP
The exciton emission for a small fraction of NW is γ rad n( x)δ [hω − E g0 − U ( x)]dx and by integrating the total NW length, we can express the spectrum of emission in terms of an integral, which is:
LNW γ rad h d Γ I (ω ) = n( x)dx ∫ 2 π 0 Γ + hω + E g0 − U ( x) 2 d
[
]
(3.13)
where LNW is the NW length, γrad is the rate of radiative recombination, and the parameter
Γ is introduced to model inhomogeneous broadening.
63
Wavelength shift is given by:
⎛ Δω ⎞ ⎛ Δω ⎞ Δλ = −λ20 ⎜ ⎟ ⎟ = −λ 0 ⎜ ⎝ ω ⎠ ⎝ 2πc ⎠
(3.14)
where Δω is obtained by finding the peak in the exciton emission (Equation 3.13). Now, we compute the wavelength shift as a function of mobility. First, we calculate the shift of the emission peak for various mobilities. The maximum blue shift of the spectrum for fixed mobility and for given variations in RNW-NP is:
Δλ max ( μ ) = λ peak ( R NW − NP = 12.2) − λ peak ( R NW − NP = 9.97)
(3.15)
Then, from equations 3.14 and 3.15, the calculated function Δλmax(μ) has a maximum (see Figure 3.9). This maximum in the wavelength shift might be explained as follows: for large mobility all excitons have enough time to reach the potential minimum and the optical spectrum becomes insensitive to the NW-NP distance, and, for small mobility, excitons do not drift and diffuse at all and the optical spectrum vanishes again.
64
Figure 3.9. Calculations of emission properties. a) Calculated emission spectrum for two NW-NP distances. b) Calculated maximum peak shift (in nm) as a function of the exciton mobility. After N.A. Kotov, A.O. Govorov et al. 2007 [90].
Chemical Sensor Mechanism Based on Exciton Diffusion in a Nanowire Wavelength shift in the NW is due to the change in distance between the NW and the NP. When one decreases the NW-NP distance, the exciton lifetime becomes shorter and not all excitons can reach the potential minimum; thus the emitted photon of the exciton-plasmon interaction becomes blue shifted. Similarly, when we increase the NWNP distance, the exciton lifetime becomes longer and many excitons can reach the potential minimum; thus the emitted photon of the exciton-plasmon interaction becomes red shifted. To estimate the exciton wavelength shift for a change in the NW-NP distance (RNW-NP), we use a molecular spring assembly (MSA) structure made from Au NPs and CdTe NWs (see Figure 3.10) [74,76]. These structures are based on a polyethyleneglycol (PEG) backbone with biological functionality since we are looking for bio-sensing applications. Biological functionalities can be obtained by incorporating an antibody (aB)
65 in the PEG chain as PEG-aB-PEG. These latter chains react to the presence of antigen (aG), which changes the RNW-NP (see Figure 3.11). This change was measured, and it was from 9.97 to 12.2 nm. And by using equation 3.3, the corresponding γNP is in the range of
0.28 ns-1 < γNP < 0.93 ns-1. With these experimental results, we can see that the above wavelength shift is Δλ max,exp ≈ 8 − 10nm . And as expected, the shift depends on the exciton mobility. Experimentally the emission wavelength shift of the superstructure NP5K was blue shifted by 8-10 nm (see Figure 3.11B, a Æ b) during the attachment of NP to NW. Then, when 20 μL of streptavidin (SA) was added, the red shift of the spectra was observed (see Figure 3.11B, b Æ c). This effect is due to the free SA which extends the molecular spring, causing the exciton-plasmon interaction to decrease. This increase in the NW-NP distance allows the exciton to diffuse more along the NW and find the lower energy (the potential minimum), which produced the red shift. To induce the reverse reaction, ~10-7 ng/ml of non-conjugated free aB was added to the media, resulting in the disruption of the SA-aBs immunocomplexes, which recovered the original superstructure (see Figure 3.12). This procedure was repeated several times to show the reversibility of emission wavelength shift. The reversibility was shown until two or three laps of wavelength shifts, but the amplitude of shifts was reduced due to progressive overloading of the solution with proteins.
66
Figure 3.10. A) Right: Schematics of MSA from Au NPs and CdTe NWs. Left: The PEG molecules are expected to cover the entire surface of the NW. B) Scanning TEM image of Au-PEG-aB-PEG-NW (the ratio of NP:NW was 5000:1 (NP5K) in the conjugation step). After N.A. Kotov, A.O. Govorov et al. 2007 [90].
As a final comment to this section, we find that the experimental wavelength shift is about 8 to 10 nm, while our model has a wavelength shift of about 6 nm. This difference is due to the fact that we considered a periodic potential in the model while in the experiment it was not (see Figure 3.4 and 3.5B). Also, we used experimental data which had some error bars in the measurements. So, our model is a semi-quantitative description of the wavelength shift in the system of Au NP and CdTe NW.
67
Figure 3.11. A) Schematic illustration of aG-aB reaction in the MSA. Extra amount added will disrupt the aG-aB in NP-PEG-aB-PEG-NW. B) Reversible shift of peak luminescence wavelength in NP5K: a, attachment of NP to NW; b, after adding 20 μL, Streptavidin (SA); c, after adding free aB to the media. Solid line is a theoretical fit to the experimental data. After N.A. Kotov, A.O. Govorov et al. 2007 [90].
68
Figure 3.12. Calibration curve for SA. Averaged wavelength shifts at the hybrids in different analyte concentration for 30~50 min until saturation of wavelength shift. The 20 μl of SA from each buffer solution was added in 3 ml PBS buffer solution (pH 7.2) of an optical cuvette for spectroscopic measurements. Prepared SA buffers the concentrations of SA in optical cuvettes and their numbers of SA. The molarity ratios for the preparation of superstructures were calculated from molarity calculation of nanocolloids. The concentrations of NW and NP solutions were calculated to be 1.0×10-9 M, and 4.5×10-6 M respectively based on the physical dimensions of the particles and synthesis data. After N.A. Kotov, A.O. Govorov et al. 2007 [90].
69 CHAPTER 4: EXCITON ENERGY TRANSFER BETWEEN NANOPARTICLES AND NANOWIRES Introduction Modern nanotechnology assembles and studies hybrid superstructures composed of nanocrystals and biomolecules as building blocks [92-94]. Individual building blocks of a hybrid superstructure can be made of different materials (metals, semiconductors, and biomolecules) and might have very different physical characteristics. Each building block of a superstructure contributes unique properties. Interactions between building blocks give a superstructure enhanced properties. One important mechanism of internanocrystal interaction is fluorescence resonance energy transfer (FRET) or Förster energy transfer [95]. This transfer mechanism provides very efficient coupling between optically excited nanocrystals. FRET comes from the Coulomb interaction between nanocrystals and does not require tunneling. In a typical scheme for FRET (see Figure 4.1), exciton energy flows from a large band-gap nanocrystal (donor) to a nanocrystal with a smaller band gap (accepter). Here we model Förster energy transfer in a novel class of superstructures composed of nanowires (NWs) and nanoparticles (NPs) [72,96,97]. Importantly, NP-NW complexes combine nanocrystals of different dimensionality, and this can lead to optical properties and functionality.
In a
semiconductor NP-NW complex, zero-dimensional NPs have strong and isotropic optical absorption, whereas one-dimensional NWs can carry electrical currents and deliver optical energy to external circuits. We note that one factor limiting the use of NPs for solar cell application is that NPs are zero dimensional objects and are often not connected
70 electrically. Therefore, it is challenging to extract optical energy from NPs. The NP-NW complexes described here have the ability to extract optical energy from NPs via the FRET mechanism. Similar properties can be expected in the systems composed of carbon nanotubes and NPs/bio-molecules [98,99]. FRET and related effects in the systems of mixed dimensionality have been discussed both theoretically and experimentally in several papers. In particular, various aspects of NP-NP, NP-biomolecule, NP-nanorod, NP-surface, and quantum well-NP transfers were described in references 100-107. The case of NW-NW transfer was theoretically analyzed in reference 108. Experimentally, FRET in the system based on carbon nanotubes was recently reported in reference 109 and 110. A theoretical description of transfer between a localized exciton and a two-dimensional quantum well was given in reference 111. Electrodynamics of a localized exciton (dye molecule) in the vicinity of a one-dimensional carbon nanotube is a subject of the paper 112. Here we study optical properties of coupled NP-NW complexes and focus on the FRET process. We obtain convenient analytical equations for the FRET rates for NP-NW and NP-nanotube systems in the limiting cases of short and long distances. For arbitrary distances, we obtain numerical results. We show that, for realistic parameters, FRET between NPs and NWs is an efficient energy-transport mechanism. Our results are in good agreement with available experimental data [76,96]. Regarding material systems, we explore nanocrystals made of semiconductors (CdTe and Si), metals (Au), and carbon nanotubes. Our calculations show that a NP-NW complex has an enhanced rate of exciton generation in the NW component due to energy channeling from NPs. We model this
71 effect for a Si NW - CdTe NP complex. The CdTe-Si complex takes advantage of peculiar physical properties of the involved materials. In such a superstructure, the Si NW is an indirect-band crystal with long-lived electron-hole pairs whereas the CdTe NPs are the direct-band, light-emitting component, providing efficient FRET. Combination of these properties makes the Si-CdTe structure suitable for photovoltaic applications. We also show that the metal-semiconductor NW-NP complexes have very fast and efficient FRET, and may be used for sensor applications [78,90]. Nanoparticle-Nanowire Transfer of Excitons We now calculate Förster energy transfer from an optically excited NP to NW (see Figure 4.1). The center-to-center distance between NP and NW is denoted as d , and the distance between the NP center and the NW surface is given by Δ . A NW, NP and matrix are described with local dielectric constants denoted as ε NW , ε NP , and ε 0 , respectively. The local dielectric constant approach provides us with reliable description if the transferred exciton energy (band gap of a donor nanocrystal) is not very close to the band-gap of a NW (accepter) [113,114].
72
Figure 4.1. a) Schematics of the coupled NP-NW system. b) Energy diagram of physical processes. Red dashed arrows describe inter-band optical transitions in nanocrystals (exciton generation) and yellow arrows inducts incident photons. A horizontal arrow and two vertical black arrows show the FRET process (Coulomb transfer) between a NP and a NW; in this unidimensional process, an exciton in a NP turns into an excitation in a NW. Blue solid arrows indicate fast intra-band relaxation of excitons and blue dotted arrows are inter-band recombination of excitons. After P. Hernández-Martínez and A.O. Govorov 2008 [123].
An optically excited NP may have three types of excitons with optical dipole moments along the unit vectors xˆ , yˆ , and zˆ . The rate of energy transfer for the exciton with an optical dipole αˆ ( αˆ = xˆ , yˆ , zˆ ) can be calculated using a convenient formalism
73 developed in references 115 and 116. The calculation provides us the probability of the process exc NP → exc NW , where exc NP(NW) are the exciton states in a NP and a NW. In this transfer process, a NP returns to its unexcited state with simultaneous excitation of NW [see Figure 4.1b]. The process is induced by the internanocrystal Coulomb interaction. The α -exciton transfer rate can be calculated as
2
r r* ⎡ ε NW (ωexc ) ⎤ dV E α ⋅ Eα ⎥⎦ ∫ 2π
γ α (ωexc ) = Im ⎢ h ⎣
(4.1)
r where Eα is the electric field induced by the dipole field of α -exciton given by
Φα e
where
ε eff =
− iωt
r r ed exc (r − rNP )αˆ − iωexc t = e ε eff rr − rrNP 3
d exc is the dipole moment of exciton,
(4.2)
ωexc is the exciton frequency,
2ε 0 + ε NP r , and rNP is the spatial position of NP; Im ε NW > 0 . The dielectric 3
constant of NP in the equation for ε eff corresponds to the high-frequency limit,
ω exc >> ωTO , where ωTO is the optical phonon energy; in other words, the effect of lattice polarization can be neglected. For CdTe NPs used here, we take a conventionally used optic dielectric constant ε eff = 7.2 [117]. We now assume that the NP radius, RNP , is relatively small, i.e. RNP RNW . In this case, we expand equation 4.6 in terms of the parameter
R NW d
and obtain convenient equations:
2 2 RNW γ α (ωexc ) = h d5
2
2 ⎞ ⎛ ed exc ⎞ 3π ⎛⎜ ε0 ⎜ ⎟ aα + bα ⎟ Im ε NW ⎜ ε ⎟ 32 ⎜ ⎟ ε NW + ε 0 ⎝ eff ⎠ ⎠ ⎝
(4.8)
where the coefficient aα is 15/16, 0, and 9/16 for α = x, y and z , respectively; the corresponding values for the coefficient bα are: 41/4, 4, and 15/4. We see that the distance dependence of Förster transfer for the dipole-to-nanowire case is γ α ∝ note that in the case of traditional dipole-dipole transfer γ dipole − dipole ∝
1 . We d5
1 [67,95]. Slower d6
spatial decay of the energy transfer rate comes from the one-dimensional character of a NW. Since the exact solution of equation 4.6 is rather complicated, the expansion of equation 4.8 can be very convenient to estimate transfer times in structure with
R NW n z > n x ; the number of y-excitons would be the largest since the rate γ y is the smallest one. In Figure 4.3 we now show the results for the complex composed of CdTe NPs and CdTe NWs. This complex was assembled and optically characterized in reference 76. Experimental values for rates of FRET from NPs to NWs were extracted from the photoluminescence spectra recorded during the assembly process. The experiment [76] was performed with orange and green CdTe NPs: λexc ,orange NP = 582nm ( Rorange NP = 2nm ) and λexc , green NP = 526nm ( Rgreen NP = 1.6nm ). The NW radius RNW = 3.3nm and its emission is at λexc ,NW = 689nm . The NP-NW complex was assembled using the biotin-streptavidin biolinker with a length of 5nm . The resultant NP-NW distances were estimated as: d orange NP = 10.3nm and d green NP = 9.9nm . Exciton lifetimes in NPs were measured as
τ exc = 19 ns and quantum yield as Y ~ 0.2 . The radiative recombination rate of NP
excitons can be estimated as γ rad = γ exc , NP ⋅ Y = 1 / 95 ns −1 , where γ exc ,NP = 1 / τ exc ,NP . The exciton dipole moment can now be evaluated from the equation for the radiative rate [120]:
γ rad =
3 2 8π ε 0 ωexc e 2 d exc 3(ε eff / ε 0 ) 2 h ⋅ c 3
(4.10)
80
& . From the experiment, with ε 0 = 2.2 , ε NP = 7.2 , and λexc ~ 582nm , we obtain: d exc ~ 0.8A it was determined that γ trans ,orange = 1 / 16 ns −1 and γ trans , green = 1 / 12 ns −1 . The corresponding theory −1 theory −1 and γ trans . We see that the theoretical numbers are: γ trans , orange ≈ 1 / 13.1 ns , green ≈ 1 / 9 ns
calculations provide us with reliable estimates for the FRET rates (see Figure 4.3). In Figure 4.3 we also show the dependence γ trans (ωexc , d = 10.3nm ) as an inset. The function
γ trans (ωexc ) reflects the frequency dispersion of the CdTe dielectric constant, ε NW = ε CdTe (ω ) . Among available one-dimensional nanocrystals, there are also metal NWs and carbon nanotubes (CNTs). A metal is a stronger absorber compared to a semiconductor, especially in the regime of the plasmon resonance, ωexc ≈ ω plasmon . In other words, a metal has a larger density of states for the optical absorption or energy transfer. We now consider the CdTe-NP Au-NW complex assembled in reference 96. The parameters are as follows: RNW = 6.5nm , λexc = 515nm , and d = 13nm ; the estimated exciton dipole
& . The time-resolved photoluminescence spectroscopy showed that the FRET d exc ~ 0.7A rate is in the range 1 / 2 ns −1 < γ trans < 1 / 0.3 ns −1 . We can see that the FRET rate for Au NWs is indeed greater than that for the CdTe NWs. This is because the metal NWs can absorb energy more efficiently than the semiconductor ones. In addition, the exciton energy of CdTe NP in our complex was chosen to be close to the plasmon resonance in an Au NP. Figure 4.4 shows calculated FRET rates for the CdTe NP - Au NW complex. Again we can see that our theory is able to reproduce the experimental data [96]. For the matrix dielectric
constant,
we
took
the
value
for
water,
ε 0 = 1.8 .
For
small
81 Δ = d − RNW ( Δ ≥ RNW ), the calculated function γ trans (Δ ) follows the asymptotic behavior
(see Equation 4.7):
γ trans (Δ) =
where β (ω ) =
(ed exc )2 3hε
2 eff
(ed exc )2 3hε
ε 0 Im[
2 eff
ε0 Δ
3
Im[
(ε NW − ε 0 ) ] = β (ω exc ) / Δ3 (ε NW + ε 0 )
(4.11)
(ε NW − ε 0 ) ] . Very fast FRET can be used to design NP (ε NW + ε 0 )
sensors. The optical emission of NPs strongly depends on the energy transfer rate. Basically, a large energy transfer rate means a low intensity of optical emission. Since the FRET rate strongly depends on the linker length Δ , the emission from NPs also acquires a strong Δ dependence. Linkers for bioassembled complexes can be specially designed to be dependent on temperature or chemical environment [78,90]. Therefore, the length of a specially designed biolinker becomes a function of local temperature or chemical content. Then, by looking at the intensity and wavelength of emission from semiconductor nanocrystals, one can obtain information about local temperature or chemical content. In this way, biosensors based on meal-semiconductor complexes were designed and tested in reference 90 and 78.
82
Figure 4.3. Rates of NP-NW transfer of excitons as a function of the CdTe NP-NW separation and available experimental data from reference 76. Green line shows the calculated rate for carbon nanotubes. Inset: FRET rate for the NP-NW complex as a function of the exciton energy of a NP. After P. Hernández-Martínez and A.O. Govorov 2008 [123].
Finally, we also can model FRET from an optical dipole to a CNT. We cannot apply our equation 4.6 for small NP-CNT distances d ≈ RCNT , where RCNT is the CNT radius. The reason is that the equation 4.6 is derived for a solid dielectric cylinder whereas the electrostatics of a hollow cylinder is different. However, at large distances
d >> RCNT , we can use our theory. Namely, we can apply equation 4.8. It is known that
83 CNTs have a strong depolarization effect for the electric field perpendicular to the CNT axis [121]. In other words, in the first approximation, we can neglect the second term in equation 4.8. Then, individual transfer rates take the form:
2 2 RCNT γ α (ω exc ) = aα h d5
2 2 RCNT where γ 0 = h d5
⎛ ed exc ⎜ ⎜ ε ⎝ eff
⎛ ed exc ⎜ ⎜ ε ⎝ eff
2
⎞ 3π ⎟ Im ε CNT = γ 0 aα ⎟ 32 ⎠
(4.12)
2
⎞ 3π ⎟ Im ε CNT , where ε CNT is the “z” component of the ⎟ 32 ⎠
dielectric constant averaged over a CNT volume. For the averaged transfer rate, we have γ trans (ωexc ) = (3 / 2)γ 0 . The important parameter Im ε CNT can be estimated from the
absorption cross section [122]:
σ CNT L
10 −13 cm 2 = , where L is the CNT length. The 380nm
absorption cross section of CNT
σ CNT L
where SCNT
=
S Q = S CNT ω CNT Im ε CNT L ⋅ I0 c
(4.13)
is the cross-sectional area of a CNT. Assuming hω = 1.2eV and
RCNT = 0.37nm , we obtain Im ε CNT ~ 12.2 and SCNT ⋅ Im ε CNT ~ 5.4 ⋅ 10−14 cm 2 . From Figure 4.3, we see that NP-CNT transfer is slower compared to that for the NP-NW system. This is because an effective cross section of CNT is smaller than that of CdTe
84 NW. A relatively small σ CNT corresponds to a relatively small number of atoms in a CNT. However, for small d , FRET can be quite efficient (see Figure 4.3).
Figure 4.4. Calculated energy-transfer rate CdTe NP – Au NW complex. The experimental values are taken from reference 96. Red line shows the asymptotical behavior at small Δ = d − RNW and corresponds to NP-surface transfer. After P. Hernández-Martínez and A. O. Govorov 2008 [123].
Optical Responses of Nanoparticle-Nanowire Complexes and the Light Harvesting Effect We now model processes of optical absorption and energy channeling in a NPNW complex. In a complex composed of a core NW and a NP shell (see Figure 4.5a and
85 4.5b), incident photons generate excitons in both NW and NP components. However, excitons generated in the NPs can flow to the NW due to FRET. In this way, exciton energy becomes harvested in the NWs. If a NW has electrical contacts (see Figure 4.5c) the exciton energy-harvesting effect can be observed as an enhanced photocurrent or photovoltage. The photocurrent response can be observed in a homogenous NW with an applied external bias and is proportional to the number of electron-hole pairs generated in a NW, nexciton , NW . The photovoltage signal can appear without an applied external bias if a NW contains built-in electric field (p-n junction or Schottky barrier) that spatially separates electron-hole pairs (excitons). In the latter case, the signal also depends on the number of optically generated electron-hole pairs, nexciton , NW . We now calculate nexciton , NW for CdTe NP - Si NW complex. This may be an optimal choice of materials for a device with enhanced photo-responses: CdTe has a relatively large absorption (see Figure 4.5a) whereas Si is an indirect-band-gap material widely used for photodetectors and solar cells. In an indirect-band-gap crystal, the radiative recombination rate is very small which guarantees small radiation losses and larger photocurrent/photovoltage responses. In principle, CdTe can also be used a NW materials, like it was realized in reference 76. However, in this case, a CdTe NW as a direct band-gap system may have relatively large radiative losses of photogenerated carriers. But, nevertheless, we calculate nexciton , NW for CdTe NP – CdTe NW as (for) completeness.
86
Figure 4.5. a) and b) Schematics of the complex composed of a NW and NPs. NPs form a shell around a core NW. Similar complexes have been assembled and studied in reference 76. A NW may contain a p-n junction where photogenerated excitons become separated. c) Geometry for generation of photocurrent or photovoltage; a NW with attached NPs bridges two electrical contacts; the normal incidence of light is assumed. After P. Hernández-Martínez and A. O. Govorov 2008 [123].
The dynamics of excitons in a NP-NW complex at room temperature under constant illumination is described with the stationery rate equations:
dnexciton, NP dt
and
= −(γ exc , NP + γ trans )nexciton, NP + I NP = 0
(4.14)
87
dnexciton, NW dt
= −γ exc, NW nexciton, NW + N NP γ trans nexciton, NP + I NW = 0
(4.15)
where I NP is the rate of exciton generation in a single NP, I NW is the exciton generation rate inside a fraction of NW of a width Δ (see Figure 4.5b); the cylindrical fraction of a NW of width Δ corresponds to one shell of attached NPs. In the above rate equations,
γ exc, NW and N NP are the exciton recombination rate of a NW and the number of NPs in a ring, respectively. Note that FRET between NPs is not included since, in our model, it does not create any effect on nexciton , NW . Solving the rate equations, we obtain:
nexciton, NW =
1
γ exc, NW
⎧⎪ γ trans I NP N NP ⎫⎪ + I NW ⎬ ⎨ ⎪⎩ (γ exc , NP + γ trans ) ⎪⎭
(4.16)
The absorption rates can be expressed as:
I exciton =
1 2hωlaser
Re
r r* j ( r ) ⋅ E ( r ) dV ∫
(4.17)
V NC
r ε −1 r E ( r ) is the induced current in where ωlaser is the excitation frequency, j ( r ) = −iω NC 4π r a nanocrystal with dielectric constant ε NC , and E (r ) is the electric field inside a nanocrystal. The integral in the equation for I exciton is taken over a nanocrystal volume,
88 r VNC , and the laser electric field is taken in the form E0 = eˆE0 e − iωlaser t , where the r polarization vector eˆ = (0, sin θ , cosθ ) and θ is the angle between E0 and zˆ . Above we
assumed the geometry shown in Figure 4.5c; incident light is normal to the NW axis and the electric field angle θ is arbitrary. An essential approximation in this paper is that we will neglect the effect of interaction between NWs and NPs in the calculation of the light absorption rate of a complex. Simultaneously, we take into account the FRET coupling between NPs and NWs. Assuming that the wavelength of incident light is much greater than the diameters of NPs and NWs ( λlaser >> RNP ( NW ) ), we can use the near-field r approximation and obtain simple equations for the electric fields, E (r ) , inside nanocrystals. In the case of a NW, we have
E z = E 0 cos θ
(4.18) Ey =
2ε 0 E 0 sin θ ε NW + ε 0
i.e. the electric field parallel to a NW remains unscreened. This brings very strong anisotropy in the absorption spectrum of a NW. Inside a NP, the electric field is r E =
3ε 0 E 0 eˆ ε NP + 2ε 0
The resultant absorption rates become:
(4.19)
89
I NW
2 2 ⎛ ⎞ Δ ⋅ π ⋅ RNW 2 ⋅ ε0 2 ⎜ = I0 Im ε NW cos θ + sin 2 θ ⎟ ⎜ ⎟ ε NW + ε 0 hc ε 0 ⎝ ⎠
I NP = I 0
3 4π ⋅ R NP
3hc ε 0
3ε 0 2ε 0 + ε NP
(4.20)
2
Im ε NP
(4.21)
cE02 ε 0 where I 0 = is the incident light flux. We see from equation 4.21 that the 8π absorption by a NP is isotropic whereas the NW absorption (Equation 4.20) acquires very strong anisotropy due to the dielectric screening in a NW. To calculate the number nexciton , NW , we should first compute numerically the important parameter, γ trans , using equations 4.4, 4.6 and 4.9. Results of numerical calculations for CdTe NP – Si NW and CdTe NPs – CdTe NW are shown in Figure 4.6b. We see that Si NWs have relatively small γ trans (compared with CdTe NWs) in the region λexciton > 400nm ( hω exciton < 3eV ). This is due to a relatively small absorption of Si in this wavelength interval (see Figure 4.6a).
90
Figure 4.6. a) Imaginary part of dielectric constant of Si and CdTe as a function of frequency. b) Energy transfer rate for a CdTe NP- Si NW complex shown in Figure 4.5. Also, Energy transfer rate for a CdTe NP – CdTe NW. After P. Hernández-Martínez and A. O. Govorov 2008 [123].
The results for generation of electron-hole pairs in a CdTe NP- Si NW complex are summarized in Figure 4.7a. Similarly, the results for generation of electron-hole pairs in a CdTe NP - CdTe NW are shown in Figure 4.8a. In figure 4.7a and 4.8a, we show a relative value nexciton , NW / nexciton , NW 0 as a function of the NP-NW distance; here nexciton , NW 0 (θ ) = I NW (θ ) / γ exc , NW is the absorption by a single NW in the absence of NPs and d = RNW + RNP + Δ lin ker , where Δ lin ker is a biolinker length. Note that the function nexciton , NW / nexciton , NW 0 is strongly anisotropic. We can see that the relative rate of carrier generation in a NP-NW complex becomes strongly enhanced compared to that in a single NW. The enhancement effect is especially strong for θ = π / 2 since, in this case, the absorption by a NW alone is very weak while the absorption by the NP shell is isotropic and remains strong for any angle θ .
91 For unpolarized light, the enhancement should be calculated as a ratio of two averaged rates, < nexciton , NW (θ ) >θ / < nexciton , NW 0 (θ ) >θ . Figures 4.7b and 4.8b shows the number of excitons in a NW as a function of the angle θ for different NP-NW distances. We see that the rate nexciton , NW becomes less anisotropic for small NP-NW distances; this is due to the influence of exciton flow (FRET) from the NPs. Therefore, for unpolarized light, a NP-NW complex has a much larger rate of exciton generation in the conducting channel compared to a single NW. In addition, we observe that the enhanced effect is large in the CdTe NP - Si NW than in the CdTe NP - CdTe NW. Regarding optoelectronic applications, such as photodetectors and solar cells, NP-NW complexes have the following advantages: (1) increased absorption cross sections and increased nexciton , NW / nexciton , NW 0 , and (2) efficient absorption of photons with perpendicular polarizations. In a single Si NW, the absorption cross section in the visible range is relatively small (see Figure 4.6a) and highly anisotropic. In fact, a NW practically does not absorb photons with two polarizations of light (x and y). To conclude this section, we also mention that the exciton collection effect in a NP-NW complex resembles the light harvesting in natural photosynthetic systems. NPs play a role of antenna chlorophylls whereas a NW serves as a reaction center where the charge separation takes place.
92
Figure 4.7. a) Calculated ratio nexciton , NW / nexciton , NW 0 for two values of angle θ
and for
unpolarized light for the complex in Figure 4.5; here nexciton , NW is the number of opticallygenerated excitons inside a NW. b) Calculated number of photo-generated excitons inside a Si NW as a function of the angle for two NP-NW distances. After P. HernándezMartínez and A. O. Govorov 2008 [123].
Figure 4.8. a) Calculated ratio nexciton , NW / nexciton , NW 0 for two values of angle θ
and for
unpolarized light for the CdTe NPs- Si NW complex; here nexciton , NW is the number of optically generated excitons inside a NW. b) Calculated number of photo-generated excitons inside a CdTe NW as a function of the angle for two NP-NW distances. After P. Hernández-Martínez and A. O. Govorov 2008 [123].
93 Conclusion Using realistic parameters and conditions, we calculated the rates of energy flow in optically
excited
NP-NW
superstructures
composed
of
different
materials
(semiconductors, metals, and carbon nanotubes). We have shown that FRET is an efficient mechanism for optical energy transport from NPs to NWs/nanotubes. Since nanowires can carry an electrical current, optical energy transferred from NPs can be further transferred to external circuits and utilized. This scheme provides an efficient method of extraction of optically generated energy (excitons) from zero-dimensional nanocrystals.
94 CHAPTER 5: HEAT GENERATION BY OPTICALLY EXCITED GOLD NANOPARTICLES Introduction Solid state nanoparticles (NPs) have great potential in modern bionanotechnology as biosensors and actuators [124,125]. Metal NPs have useful thermal properties. Under optical illumination, metal NPs efficiently create heat [73,74,126-128]. Recently, heat generation by NPs under optical illumination (nanoheater) has attracted much interest [129-146]. The heat generation from these nanoheaters involves not only the absorption of incident photons, but also the conversion of photon energy into heat energy as well as heat transfer from the NP to the surrounding matrix. The heating effect is especially strong for metal NPs since they have many mobile electrons and become strongly enhanced under the plasmon frequency or when the laser frequency hits the collective resonance of a NP. Since metal NPs have very low optical quantum yield (i.e. they are very poor light emitters), the total amount of heat generated can be estimated in a relatively simple way to the total optical absorption rate. Biomedical applications involving nanoheaters rely on a simply mechanism [133,134,137,142,147-158]. First, the NPs become attached to targeted biological centers such as tumor cells using selective bio-molecular linkers. Then, heat is generated remotely by optically-stimulating the NPs. Finally, the heat generation causes an actuation of a biological process. The important physical properties in this process are the temperature at the surface of the NP and the scope and length over which this temperature changes occurs, as well the collective response of NPs. Some applications
95 that involve the optical and heat generation properties of metal NPs include biological imaging and detection of DNA, RNA and proteins [73,124,125,155,159-163], drug delivery [164-166] as well as treatment of diseases (photo-thermal cancer therapy [128,134,137,154,167,168]). Models and Calculations Heat Generation by a Single Metal NP The heat transfer in an isotropic system, consisting of Au NP and a surrounding matrix (see Figure 5.1), can be described by the usual heat transfer equation [85]:
ρ (r )c(r )
∂T (r, t ) = ∇k (r )∇T (r, t ) + Q(r, t ) ∂t
(5.1)
where T(r,t) is the temperature as a function of coordinate r and time t, ρ(r) is the mass density, c(r) is the heat capacity, k(r) is the thermal conductivity and Q(r,t) is the local heat intensity generated from light dissipation in Au NP, which is given by:
1 ⎡ ε (r ) − 1 ~ ~ ⎤ Q(r, t ) = j(r, t ) ⋅ E(r, t ) t = − Re ⎢iω E(r )E * (r )⎥ 2 ⎣ 4π ⎦
[
(5.2)
]
~ Here j(r,t) is the current density, E(r, t ) = Re E(r ) ⋅ e − iωt is the resulting electric field in the system, and ε(r) is the dielectric constant. Assuming that the system is excited with
[
]
~ the external laser field E 0 (t ) = Re E 0 e − iωt at t ≥ 0 , then, the light intensity is:
96
c ⎧ ε m E 02 t ≥ 0 ⎪I 0 = I (t ) = ⎨ 8π ⎪⎩0 t Rb
(5.8)
98 a) T (r → ∞) = T0 b) Twater (r = Rb ) = Tice (r = Rb c) Twater (r = R Au ) = T Au (r = R Au ) d ) k Au
dT Au (r ) dr
e) k water
= k water r = Au
dTwater (r ) dr
dTwater (r ) dr
= k ice r = Rb
(5.9)
dTice (r ) dr
r = Au
r = Rb
where T0 is the equilibrium temperature of the matrix (ice). So, the solution is:
Tice (r ) = T0 +
3 QR Au Twater (r ) = T0 + 3Rb
3 QR Au T Au (r ) = T0 + 3Rb
3 QR Au 1 3k ice r
⎛ 1 1 ⎜⎜ − ⎝ k ice k water
⎛ 1 1 ⎜⎜ − ⎝ k ice k water
r > Rb
3 ⎞ Q R Au ⎟⎟ + ⎠ 3k water r
⎞ 1 1 2 ⎛ ⎟⎟ + QR Au ⎜⎜ + ⎠ ⎝ 3k water 6k Au
(5.10)
R Au < r < Rb
⎞ Q 2 ⎟⎟ − r ⎠ 6k Au
(5.11)
0 < r < R Au
(5.12)
The position of the ice-water boundary (Rb), is given by the condition of Tice ( Rb ) = Ttrans , where Ttrans is the phase transition temperature. Then, we obtain:
99
3 QR Au Rb = 3k ice (Ttrans − T0 )
(5.13)
And melting occurs under the condition of Q > Qmelting, which can be written as Rb > RAu or:
2 QR Au > k ice 3(Ttrans − T0 )
(5.14)
To compute the volume of water, we use the expression:
(
4 3 Vw = π Rb3 − R Au 3
)
(5.15)
Now, we calculate the melting point for different sizes of Au NPs. For comparison we use the parameter given by A. O. Govorov et al. 2006 [171]. For the forward parameter melting occurs at: I 30m > 1.3·104 W/cm2, I 25m > 1.8·104 W/cm2, and I 20m > 2.8·104 W/cm2, for RAu = 30 nm, RAu = 25 nm, and RAu = 20 nm respectively. The
parameters for the ice-water matrix are the following: T0 = -2 oC, Ttrans = 0 oC, kice = 1.6 W/mK, kwater = 0.6 W/mK, and kAu = 318 W/mK. Next, we compute that the temperature
depends on the ice-water matrix for the sizes of Au NPs above. The results are shown in figure 5.2. As we expect, the temperature depends on the size of the NP, and the radius
100 and volume of water as well. The results given in figure 5.2 show the temperature at which the ice-water matrix has a phase change – that is, water becomes vapor. The parameters showing in figure 5.2 are: I 30g = 24.74·104 W/cm2, Rb = 592.65 nm, and Vw = 8.71·108 nm3 for RAu = 30 nm; I 25g = 35.62·104 W/cm2, Rb = 493.80 nm, and Vw = 5.04·108 nm3 for RAu = 25 nm; and I 20g = 55.66·104 W/cm2, Rb = 395.06 nm, and Vw = 2.58·108 nm3
for RAu = 20 nm. The maximum temperature increase occurs at r = RAu, and using equation 5.11, we get:
T ( R Au ) = T0 +
3 QR Au 3Rb
⎛ 1 1 ⎜⎜ − ⎝ k ice k water
⎞ Q 2 ⎟⎟ + R Au 3 k water ⎠
(5.16)
2 . where the main contribution is given by the last term of equation 5.16 – that is, T ∝ R Au
This dependence of temperature as a function of RAu can be understood because heat is transferred between the Au NP to the matrix through the surface of the NP. Temperature (T) is proportional to surface area, and the heat comes from the plasmon resonance of Au NP, which makes possible the melting in the matrix.
101
Figure 5.2. The graph shows the intensity of light required to reach the temperature of 100 oC (vapor). The plot also shows the radius and volume of water that we will expect to have for this temperature.
102 Heat Generation in a Collection of NPs Energy Balance Equation
Now, we consider a system of many particles surrounded by a medium. Such a system is show in figure 5.3. Here, one has many NPs in a matrix, and they are distributed homogenously. Then, the system is illuminated by a laser at the plasmon frequency. At this point, NPs absorb light and convert it into heat. To calculate the amount of energy that the system is absorbing and dissipating, we use the energy balance equation given by:
∑m C i
i
p ,i
dT = QI − Qext dt
(5.17)
where, mi and Cp,i are the mass and heat capacity components of the system, T is the temperature, t is time, QI is the laser-induced energy source and Qext is the energy dissipated by the system. The laser-induced energy source represents heat dissipated by electron-phonon relaxation of plasmons on the metal NP surface at a resonant wavelength λ, and it is given by:
(
)
Q I = P 1 − 10 − Aλ η
(5.18)
where P is incident laser power, η is the efficiency of transduction incident resonant absorbance to thermal energy via plasmon, and Aλ is the absorbance of the NP solution
103 given by Beer-Lambert’s Law, A λ = l opt Cε , here lopt is the optical path, C is the molar concentration, and ε is the molar extinction coefficient.
Figure 5.3. System of many NPs surrounded by a medium. The medium can be liquid or gas (dust).
The rate of energy flowing out of the system is assumed to be proportional to the linear thermal driving force which has a heat transfer coefficient, h, and cross-sectional area perpendicular to conduction, S. This dissipation energy is given by:
Qext = hS (T − T0 )
(5.19)
104 Next, we simplify equation 5.17 by a change variable and reordering terms, and we get,
dT * = A − BT * dt
(5.20)
where T * = T − T0 is the temperature difference from the ambient temperature T0, A (°C/s) is the rate of energy absorption given by equation 5.21, and B (s-1) is the rate constant associated with heat loss given by equation 5.22. Here, A and B have been limited to the dominant component of the surrounding matrix. Those components are mass (m0) and (Cp,0) the heat capacity of the matrix.
A=
(
)
QI P 1 − 10 − Aλ η = m0 C p , 0 ∑ mi C p ,i
(5.21)
i
B=
hS hS = ∑ m i C p ,i m 0 C p , 0
(5.22)
i
The rate constant of heat loss from the system to an external reservoir (B) is determined by following the temperature decay back to the ambient temperature after laser excitation is turned off (A=0). Under this condition, the temperature profile is obtained by integrating equation 5.20 with A=0. So, the solution is
105 T (t ) = T0 + (Tmax − T0 ) exp(− Bt )
(5.23)
where Tmax is the temperature when the laser is tuned off. Similarly, the temperature trace after the laser is turned on (A≠0) is given by:
T (t ) = T0 +
A (1 − exp(− Bt )) B
(5.24)
The steady-state temperature (TSS) is achieved when the rate of energy absorption is equal to the rate of heat loss, in other words,
dT * = A − BT * = 0 dt
(5.25)
Then
ΔT = TSS − T0 =
A I (1 − 10 − Aλ )η = B Bm0 C p ,0
(5.26)
From equation 5.26 we calculate the efficiency of converting absorbed light to heat as,
η=
B(TSS − T0 ) ρ 0 c0V0 I (1 − 10 − Aλ )
(6.27)
106
Microscopic Theory
In this section, we describe the heat generated by NPs in liquid medium (see Figure 5.4). Here, we have to consider the most general case for the heat transfer. In an isotropic system, the heat transfer equation is [169]:
∂ρ (r, t )c(r, t )ΔT (r, t ) ⎛ ∂ρ (r, t )c(r, t )ΔT (r, t ) ⎞ + v(r, t )⎜ ⎟= ∂t ∂r ⎝ ⎠ ∇k (r, t )∇ΔT (r, t ) + Q(r, t )
(5.28)
where T(r,t) is the temperature as a function of coordinate r and time t, ρ(r,t) is the mass density, c(r,t) is the heat capacity, k(r,t) is the thermal conductivity and Q(r,t) is the local heat intensity generated from light dissipation in metal NP, which is given by:
1 ⎡ ε (r ) − 1 ~ ~ ⎤ Q(r, t ) = j(r, t ) ⋅ E(r, t ) t = − Re ⎢iω E(r )E * (r )⎥ 2 ⎣ 4π ⎦
[
(5.29)
]
~ here j(r,t) is the current density, E(r, t ) = Re E(r ) ⋅ e −iωt is the resulting electric field in the system, and ε(r) is the dielectric constant. Assuming that the system is excited with
[
]
~ the external laser field E 0 (t ) = Re E 0 e − iωt at t ≥ 0 , then, the light intensity is:
107 c ⎧ ε m E 02 t ≥ 0 ⎪I 0 = I (t ) = ⎨ 8π ⎪⎩0 t
