Principles ofsurface-enhanced Raman

spectroscopy

and related plasmonic effects

Eric C. Le Ru and Pablo G. Etchegoin

Preface xiii

Notations, units and other conventions xvii

1 A quick overview of surface-enhanced Raman spectroscopy 1

1.1 What is SERS? – Basic principles 1

1.2 SERS probes and SERS substrates 31.2.1 SERS substrates 31.2.2 SERS probes 61.2.3 Example 8

1.3 Other important aspects of SERS 91.3.1 SERS enhancements 91.3.2 Sample preparation and metal/probe interaction 101.3.3 Main characteristics of the SERS signals 111.3.4 Related techniques 131.3.5 Related areas 14

1.4 Applications of SERS 141.4.1 Raman with improved sensitivity 151.4.2 SERS vs fluorescence spectroscopy 151.4.3 Applications specific to SERS 17

cVOLUME ISBN 978-0-444-52779-0 All rights of reproduction in any form reservedPRINCIPLES OF SERS Copyright � 2009 Elsevier Ltd

ii ERIC C. LE RU, PABLO G. ETCHEGOIN

1.5 The current status of SERS 171.5.1 Brief history of SERS 171.5.2 Where is SERS now? 201.5.3 Current ‘hot topics’ 21

1.6 Overview of the book content 231.6.1 General outline of the book 231.6.2 General ‘spirit’ of the book 241.6.3 Different reading plans 25

2 Raman spectroscopy and related optical techniques 29

2.1 A brief introduction 302.1.1 The discovery of the Raman effect 302.1.2 Some applications of Raman spectroscopy 312.1.3 Raman spectroscopy instrumentation 32

2.2 Optical spectroscopy of molecules 332.2.1 The energy levels of molecules 332.2.2 Spectroscopic units and conversions 362.2.3 Optical absorption 382.2.4 Emission and luminescence 392.2.5 Scattering processes 422.2.6 The concept of cross-section 462.2.7 The Raman cross-sections 502.2.8 Examples of Raman cross-sections 532.2.9 Mechanical analogs 57

2.3 Absorption and fluorescence spectroscopy 592.3.1 Optical absorption and UV/Vis spectroscopy 592.3.2 Fluorescence spectroscopy 632.3.3 Photo-bleaching 67

2.4 Phenomenological approach to Raman scattering 702.4.1 Dipolar emission in vacuum 712.4.2 The concepts of polarizability and induced dipole 732.4.3 The linear optical polarizability 752.4.4 The Raman polarizability 772.4.5 The local field correction 782.4.6 Polarizabilities and scattering cross-sections 802.4.7 Final remarks on the phenomenological description 87

2.5 Vibrations and the Raman tensor 882.5.1 General considerations 88

iiiPRINCIPLES OF SERS

2.5.2 A primer on vibrational analysis 892.5.3 The Raman tensor 912.5.4 Link to the Raman polarizability 932.5.5 Limitations of the classical approach 972.5.6 A brief overview of related Raman scattering processes 98

2.6 Quantum (or semi-classical) approach to Raman scattering 992.6.1 Justification of the classical approach 1002.6.2 The quantization of vibrations 1012.6.3 The full expressions for the Raman cross-section 1022.6.4 The anti-Stokes to Stokes ratio 105

2.7 Advanced aspects of vibrations in molecules 1062.7.1 More on vibrational analysis 1072.7.2 More on symmetries and Raman selection rules 1142.7.3 Modeling of molecular structure and vibrations 117

2.8 Summary 119

3 Introduction to plasmons and plasmonics 121

3.1 Plasmonics and SERS 122

3.2 The optical properties of noble metals 1233.2.1 The Drude model of the optical response 1243.2.2 The optical properties of real metals 1253.2.3 Non-local optical properties 1273.2.4 What makes the metal–light interaction so special? 127

3.3 What are plasmons? 1313.3.1 The plasmon confusion 1313.3.2 Definition and history 1323.3.3 The relation between plasmons and the dielectric function 1353.3.4 Electromagnetic modes in infinite systems 1363.3.5 Electromagnetic modes of a system of material bodies 1413.3.6 Classification of electromagnetic modes 1433.3.7 Other properties of electromagnetic modes 1443.3.8 Summary and discussion 146

3.4 Surface plasmon–polaritons on planar interfaces 1493.4.1 Electromagnetic modes for a planar dielectric/metal interface 1493.4.2 Properties of the SPP modes at planar metal/dielectric interfaces 1553.4.3 Coupling of PSPP modes with light 1593.4.4 PSPP resonances at planar interfaces 164

iv ERIC C. LE RU, PABLO G. ETCHEGOIN

3.4.5 Local field enhancements and SPPs at planar interfaces 1673.4.6 SPP modes on planar interfaces: A brief summary 173

3.5 Localized surface plasmon–polaritons3.5.1 Introduction to localized SPPs3.5.2 LSP on planar structures3.5.3 LSP modes of a metallic sphere3.5.4 LSP modes of nano-particles3.5.5 LSP resonances3.5.6 Local field enhancements and LSP3.5.7 Interaction of SPPs – gap SPPs

3.6 Brief survey of plasmonics applications3.6.1 Applications of surface plasmon resonances3.6.2 SPP propagation and SPP optics3.6.3 Local field enhancements

4 SERS enhancement factors and related topics

4.1 Definition of the SERS enhancement factors4.1.1 General considerations4.1.2 The analytical point of view4.1.3 The SERS substrate enhancement factor –

approach4.1.4 The SERS cross-section and single-molecule EF

174174175175177177178179

181181182182

185

186187190

Experimental191192

4.1.5 The SERS substrate enhancement factor – Formal definition 1974.1.6 Discussion and merits of the various definitions 198

4.2 Experimental measurement of SERS enhancement factors 2004.2.1 The importance of the non-SERS cross-section 2024.2.2 Example of AEF measurements 2034.2.3 Link between SSEF definition and experiments 205

4.3 Overview of the main EM effects in SERS 2094.3.1 Analysis of the EM problem of SERS 2094.3.2 Local field enhancement 2124.3.3 Radiation enhancement 2144.3.4 Other EM effects 2164.3.5 The common |E|4-approximation to SERS enhancements 217

4.4 Modified spontaneous emission 2194.4.1 Introduction 2194.4.2 The link between spontaneous emission and dipolar emission 2204.4.3 Modification of dipole emission: definitions of enhancement

factors 224

vPRINCIPLES OF SERS

4.4.4 Spontaneous emission and self-reaction 2294.4.5 The Poynting vector approach 2314.4.6 Spontaneous emission and the optical reciprocity theorem 233

4.5 Formal derivation of SERS EM enhancements 2374.5.1 Definitions, notations, and assumptions 2374.5.2 The SERS EM enhancement: general case 2404.5.3 SERS EM enhancements in the back-scattering configuration 245

4.6 Surface-enhanced fluorescence (SEF) 2484.6.1 Similarities and differences between SEF and SERS 2484.6.2 Modified (enhanced) absorption 2494.6.3 Modified fluorescence quantum yield 2504.6.4 Fluorescence quenching and enhancement 252

4.7 Other EM effects in SERS 2544.7.1 Fluorescence quenching in SERS 2544.7.2 Photo-bleaching under SERS conditions 2554.7.3 Non-radiative effects in SERS 257

4.8 The chemical enhancement 2584.8.1 Introduction 2584.8.2 The charge-transfer mechanism 2594.8.3 Electromagnetic contribution to the chemical enhancement 2614.8.4 The chemical vs electromagnetic enhancement debate 263

4.9 Summary 263

5 Calculations of electromagnetic enhancements 265

5.1 Definition of the problem and approximations 2665.1.1 The EM problem 2665.1.2 Far field and local/near field 2695.1.3 Some key EM indicators 2735.1.4 The electrostatic approximation (ESA) 2795.1.5 Other approximations 285

5.2 Analytical tools and solutions 2865.2.1 Plane surfaces 2875.2.2 The perfect sphere 2875.2.3 Ellipsoids 2895.2.4 Other approaches 289

5.3 Numerical tools 2905.3.1 A brief overview of the EM numerical tools 290

vi ERIC C. LE RU, PABLO G. ETCHEGOIN

5.3.2 A semi-analytical approach: the discrete dipole approximation 2925.3.3 Direct numerical solutions 2945.3.4 Other approaches 297

6 EM enhancements and plasmon resonances: examples anddiscussion 299

6.1 Quenching and enhancement at planar surfaces 3006.1.1 The image dipole approximation for the self-reaction field 3006.1.2 Enhancement and quenching at plane metal surfaces 303

6.2 A simple example in detail: The metallic sphere 3076.2.1 Metallic sphere in the ES approximation 3086.2.2 Localized surface plasmon resonances and far-field properties 3146.2.3 Local field effects 3246.2.4 Distance dependence 3356.2.5 Non-radiative effects – surface-enhanced fluorescence 337

6.3 The effect of shape on the EM enhancements 3426.3.1 Shape effects on localized surface plasmon resonances 3436.3.2 Shape effects on local fields 3466.3.3 Summary of shape effects 353

6.4 Gap effects – junctions between particles 3546.4.1 Coupled localized surface plasmon resonances and SERS 3546.4.2 EF distribution and hot-spot localization 359

6.5 Additional effects 3616.5.1 Nano-particles on a supporting substrate 3626.5.2 Surface roughness 364

6.6 Factors affecting the EM enhancements: Summary 364

7 Metallic colloids and other SERS substrates 367

7.1 Metallic colloids for SERS 3687.1.1 Silver vs gold 3687.1.2 Citrate-reduced colloids 3697.1.3 Other types of colloids 3717.1.4 Remarks on colloid fabrication methods 3737.1.5 Dry colloids and other ‘2D planar’ SERS substrates 373

7.2 Characterization of SERS substrates 3757.2.1 Microscopy 376

viiPRINCIPLES OF SERS

7.2.2 Extinction or UV/Vis spectroscopy of SERS substrates 3777.2.3 Other techniques: dynamic light scattering (DLS) for colloidal

solutions 381

7.3 The stability of colloidal solutions 3857.3.1 Introduction 3857.3.2 The van der Waals interaction between metallic particles 3877.3.3 The screened Coulomb potential 3897.3.4 The DLVO interaction potential 3947.3.5 Colloid aggregation within the DLVO theory 396

7.4 SERS with metallic colloids 3997.4.1 Molecular (analyte) adsorption and SERS activity 3997.4.2 Colloid aggregation for SERS 4037.4.3 Focus on the ‘chloride activation’ of SERS signals 4067.4.4 SERS from ‘dried’ colloidal solutions 4087.4.5 SERS signal fluctuations 410

8 Recent developments 415

8.1 Single-molecule SERS 4158.1.1 Introduction 4158.1.2 Early evidence for single-molecule detection 4178.1.3 Langmuir–Blodgett monolayers 4238.1.4 Bi-analyte techniques 4258.1.5 Single-molecule SERS enhancement factors 4338.1.6 Single-molecule SERS: Discussion and outlook 435

8.2 Tip-enhanced Raman spectroscopy (TERS) 4368.2.1 Introduction to TERS 4368.2.2 TERS with an atomic force microscope (AFM) 4378.2.3 TERS with a scanning tunneling microscope (STM) 4388.2.4 Theoretical calculations on tips 4408.2.5 Discussion and outlook 442

8.3 New substrates from nano-technology 4438.3.1 Chemical synthesis of metallic nano-particles 4448.3.2 Self-organization 4478.3.3 Nano-lithography 4488.3.4 Adaptable/Tunable SERS substrates 4518.3.5 Micro-fluidics and SERS 454

8.4 Optical forces 4558.4.1 A simple theory of optical forces 455

viii ERIC C. LE RU, PABLO G. ETCHEGOIN

8.4.2 Radiation pressure in colloidal fluids 4578.4.3 Optical trapping of metallic particles 4588.4.4 Optical forces on molecules 459

8.5 Applications of SERS 4608.5.1 Analyte engineering and surface functionalization 4608.5.2 Substrate reproducibility and SERS commercialization 462

8.6 Epilogue 463

9 Density functional theory (DFT) calculations for Ramanspectroscopy 465

A.1 A brief introduction to DFT 465A.1.1 Computing aspects of DFT 465A.1.2 Principles of DFT 467A.1.3 Important parameters 469

A.2 Applications of DFT to Raman 471A.2.1 Principle 471A.2.2 Geometry optimization using DFT 472A.2.3 Limitations of DFT calculations for Raman 473

A.3 Practical implementation 474A.3.1 Brief overview of the input and output files 474A.3.2 Common units and definitions in Raman calculations from DFT 477A.3.3 Normal mode patterns and Raman tensors 479

A.4 Examples of DFT calculations for SERS applications 485A.4.1 Validation of absolute Raman cross-sections of reference

compounds 485A.4.2 Raman tensor and vibrational pattern visualizations 485A.4.3 Depolarization ratio breakdown under SERS conditions 489

B The bond-polarizability model 491

B.1 Principle and implementation 491B.1.1 Principle 491B.1.2 Calculation of bond polarizabilities 492B.1.3 Practical implementation 495

B.2 A simple example in detail 496B.2.1 Bond-polarizability analysis 496B.2.2 Raman polarizabilities 497B.2.3 A brief comment on the symmetry 498

ixPRINCIPLES OF SERS

C A brief overview of Maxwell’s equations in media 499

C.1 Maxwell’s equations in vacuum 499C.1.1 The equations 499C.1.2 Maxwell’s equations for harmonic fields in vacuum 501C.1.3 Plane wave solutions in free-space 503

C.2 Maxwell’s equations in media 503C.2.1 Microscopic and macroscopic fields 503C.2.2 The electromagnetic response of the medium 504C.2.3 Electric polarization and magnetization 505C.2.4 Constitutive relations 508C.2.5 Boundary conditions between two media 512

C.3 Other aspects relevant to SERS and plasmonics 513C.3.1 The microscopic field 513C.3.2 Plane waves in media 515C.3.3 Electromagnetic problems in SERS 517C.3.4 Link with the static approach 518

D Lorentz model of the atomic/molecular polarizability 523

D.1 The Lorentz oscillator 523D.1.1 Principle 523D.1.2 Multiple transitions (multiple resonances) 525D.1.3 Example: linear optical polarizability of rhodamine 6G 525

D.2 Link with macroscopic properties 526D.2.1 Dielectric function in a dilute medium 526D.2.2 Dielectric function in solids 526D.2.3 The metallic limit 527

D.3 Summary 528

E Dielectric function of gold and silver 529

E.1 Model dielectric function for silver 529E.1.1 Analytical expression 529E.1.2 Comparison to experimental results 530

E.2 Model dielectric function for Au 531E.2.1 Analytical expression 531E.2.2 Comparison to experimental results 532

x ERIC C. LE RU, PABLO G. ETCHEGOIN

E.3 Remarks on the model dielectric functions 533E.3.1 Limitations of the models 533E.3.2 Comparison between Ag and Au 534

F Plane waves and planar interfaces 537

F.1 The plane wave electromagnetic fields 537F.1.1 General expressions 537F.1.2 Propagating plane waves 539F.1.3 Evanescent plane waves 539F.1.4 Inhomogeneous plane waves: hybrid propagating/evanescent

waves 540

F.2 Plane waves at a single planar interface 541F.2.1 Plane wave polarization at an interface 541F.2.2 General solution for plane waves at a planar interface 542F.2.3 Physical waves in a semi-infinite region 546F.2.4 The Fresnel coefficients 550F.2.5 Surface modes 551F.2.6 Incident wave modes 556

F.3 Reflection/Refraction at a planar interface 556F.3.1 Incident, reflected, and transmitted waves 557F.3.2 Snell’s law 558F.3.3 TM or p-polarized waves 558F.3.4 TE or s-polarized waves 561F.3.5 Special cases 563

F.4 Multi-layer interfaces 564F.4.1 Principle 564F.4.2 p-polarized or TM waves 565F.4.3 s-polarized or TE waves 567F.4.4 Particular cases of interest 567F.4.5 Implementation in Matlab 567

F.5 Dipole emission close to a planar interface 570F.5.1 Total decay rates 571F.5.2 Radiative decay rates 571

G Ellipsoids in the electrostatic approximation 573

G.1 General case 573G.1.1 Some definitions 573G.1.2 Ellipsoidal coordinates 574

xiPRINCIPLES OF SERS

G.1.3 The electrostatic solution 575G.1.4 Some important EM indicators for ellipsoids 578G.1.5 Some aspects of the numerical implementation 581

G.2 Oblate spheroid (pumpkin) 583G.2.1 Geometrical factors 583G.2.2 Surface averages 584G.2.3 Limit of large aspect ratio 585

G.3 Prolate spheroid (rugby ball) 585G.3.1 Geometrical factors 585G.3.2 Surface averages 586G.3.3 Limit of large aspect ratio 587

H Mie theory and its implementation 589

H.1 Introduction 589H.1.1 Motivation 589H.1.2 Overview of this appendix 590

H.2 The concepts of Mie theory 590H.2.1 The electromagnetic equations 590H.2.2 The vectorial wave equation in spherical coordinates 591H.2.3 Scattering by a sphere 593H.2.4 Optical resonances of the sphere 596H.2.5 Some aspects of the practical implementation of Mie theory 597

H.3 Basic formulas of Mie theory 598H.3.1 Conventions 599H.3.2 Spherical coordinates: A brief reminder 599H.3.3 Definition and properties of the vector spherical harmonics 600H.3.4 Expressions for the susceptibilities 605H.3.5 More on optical resonances 607H.3.6 Absorption, scattering, and extinction for an incident beam 608H.3.7 Absorption and radiation for a localized source 611H.3.8 Far-field radiation profile 612H.3.9 The local field at the surface 612

H.4 Plane wave excitation of a sphere: The ‘original Mie theory’ 613H.4.1 Expansion of a plane wave in vector spherical harmonics 613H.4.2 Extinction, scattering, and absorption for plane wave excitation 614H.4.3 Average local field at the surface 615H.4.4 Useful expansions for plane wave excitation 615

xii ERIC C. LE RU, PABLO G. ETCHEGOIN

H.5 Extensions of Mie theory 615H.5.1 Emitter close to a sphere 615H.5.2 Coated spheres 621H.5.3 Multiple spheres and generalized Mie theory (GMT) 625

H.6 Example of implementation of Mie theory with Matlab 626H.6.1 Common problems 626H.6.2 Other issues specific to Matlab 627H.6.3 Some aspects of our implementation 627

References 629

Index 655