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MOLECULAR ELECTRONICS
Cuevas, Juan Carlos ; Scheer, Elke
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Últimas novedades
nanotecnología
química general
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This book provides a comprehensive overview of the rapidly developing field of molecular electronics. It focuses on our present understanding of the electrical conduction in single-molecule circuits and provides a thorough introduction to the experimental techniques and theoretical concepts. It will also constitute as the first textbook-like introduction to both the experiment and theory of electronic transport through single atoms and molecules. In this sense, this publication will prove invaluable to both researchers and students interested in the field of nanoelectronics and nanoscience in general.
Molecular Electronics is self-contained and unified in its presentation. It may be used as a textbook on nanoelectronics by graduate students and advanced undergraduates studying physics and chemistry. In addition, included are previously unpublished material that will help researchers gain a deeper understanding into the basic concepts involved in the field of molecular electronics.
Contents:
The Birth of Molecular Electronics
Fabrication of Metallic Atomic-Size Contacts
Contacting Single Molecules: Experimental Techniques
The Scattering Approach to Phase-Coherent Transport in Nanocontacts
Introduction to Green’s Function Techniques for Systems in Equilibrium
Green’s Functions and Feynman Diagrams
Nonequilibrium Green’s Functions Formalism
Formulas of the Electrical Current: Exploiting the Keldysh Formalism
Electronic Structure I: Tight-Binding Approach
Electronic Structure II: Density Functional Theory
The Conductance of a Single Atom
Spin-Dependent Transport in Ferromagnetic Atomic Contacts
Coherent Transport Through Molecular Junctions I: Basic Concepts
Coherent Transport Through Molecular Junctions II: Test-Bed Molecules
Single-Molecule Transistors: Coulomb Blockade and Kondo Physics
Vibrationally-Induced Inelastic Current I: Experiment
Vibrationally-Induced Inelastic Current II: Theory
The Hopping Regime and Transport Through DNA Molecules
Beyond Electrical Conductance: Short Noise and Thermal Transport
Optical Properties of Current-Carrying Molecular Junctions
What is Missing in This Book?
Readership: Advanced undergraduate and graduate students in nanoscience and nanotechnology, physics, chemistry, nanoelectronics and molecular electronics.
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MOLECULAR ELECTRONICS - An Introduction to Theory and Experiment
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Contents
Preface vii
Acknowledgments ix
Brief history of the field and experimental
techniques 1
1. The birth of molecular electronics 3
1.1 Why molecular electronics? . . . . . . . . . . . . . . . . . 5
1.2 A brief history of molecular electronics . . . . . . . . . . . 6
1.3 Scope and structure of the book . . . . . . . . . . . . . . 14
2. Fabrication of metallic atomic-size contacts 19
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Techniques involving the scanning electron microscope
(STM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3 Methods using atomic force microscopes (AFM) . . . . . 21
2.4 Contacts between macroscopic wires . . . . . . . . . . . . 22
2.5 Transmission electron microscope . . . . . . . . . . . . . . 23
2.6 Mechanically controllable break-junctions (MCBJ) . . . . 24
2.7 Electromigration technique . . . . . . . . . . . . . . . . . 31
2.8 Electrochemical methods . . . . . . . . . . . . . . . . . . . 35
2.9 Recent developments . . . . . . . . . . . . . . . . . . . . . 37
2.10 Electronic transport measurements . . . . . . . . . . . . . 38
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
xi
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xii Molecular Electronics: An Introduction to Theory and Experiment
3. Contacting single molecules: Experimental techniques 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2 Molecules for molecular electronics . . . . . . . . . . . . . 46
3.2.1 Hydrocarbons . . . . . . . . . . . . . . . . . . . . 47
3.2.2 All carbon materials . . . . . . . . . . . . . . . . . 50
3.2.3 DNA and DNA derivatives . . . . . . . . . . . . . 51
3.2.4 Metal-molecule contacts: anchoring groups . . . . 52
3.2.5 Conclusions: molecular functionalities . . . . . . . 52
3.3 Deposition of molecules . . . . . . . . . . . . . . . . . . . 53
3.4 Contacting single molecules . . . . . . . . . . . . . . . . . 55
3.4.1 Electromigration technique . . . . . . . . . . . . . 56
3.4.2 Molecular contacts using the transmission electron
microscope . . . . . . . . . . . . . . . . . . . . . . 58
3.4.3 Gold nanoparticle dumbbells . . . . . . . . . . . . 59
3.4.4 Scanning probe techniques . . . . . . . . . . . . . 60
3.4.5 Mechanically controllable break-junctions (MCBJs) 64
3.5 Contacting molecular ensembles . . . . . . . . . . . . . . . 66
3.5.1 Nanopores . . . . . . . . . . . . . . . . . . . . . . 66
3.5.2 Shadow masks . . . . . . . . . . . . . . . . . . . . 68
3.5.3 Conductive polymer electrodes . . . . . . . . . . . 69
3.5.4 Microtransfer printing . . . . . . . . . . . . . . . . 70
3.5.5 Gold nanoparticle arrays . . . . . . . . . . . . . . 71
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
Theoretical background 75
4. The scattering approach to phase-coherent transport in
nanocontacts 77
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 From mesoscopic conductors to atomic-scale junctions . . 79
4.3 Conductance is transmission: Heuristic derivation of the
Landauer formula . . . . . . . . . . . . . . . . . . . . . . . 81
4.4 Penetration of a potential barrier: Tunnel effect . . . . . . 83
4.5 The scattering matrix . . . . . . . . . . . . . . . . . . . . 88
4.5.1 Definition and properties of the scattering matrix 88
4.5.2 Combining scattering matrices . . . . . . . . . . . 91
4.6 Multichannel Landauer formula . . . . . . . . . . . . . . . 92
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Contents xiii
4.6.1 Conductance quantization in 2DEG: Landauer
formula at work . . . . . . . . . . . . . . . . . . . 97
4.7 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.8 Thermal transport and thermoelectric phenomena . . . . 104
4.9 Limitations of the scattering approach . . . . . . . . . . . 106
4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5. Introduction to Green’s function techniques for systems
in equilibrium 111
5.1 The Schr¨odinger and Heisenberg pictures . . . . . . . . . 112
5.2 Green’s functions of a noninteracting electron system . . . 113
5.3 Application to tight-binding Hamiltonians . . . . . . . . . 118
5.3.1 Example 1: A hydrogen molecule . . . . . . . . . 118
5.3.2 Example 2: Semi-infinite linear chain . . . . . . . 122
5.3.3 Example 3: A single level coupled to electrodes . 124
5.4 Green’s functions in time domain . . . . . . . . . . . . . . 128
5.4.1 The Lehmann representation . . . . . . . . . . . . 131
5.4.2 Relation to observables . . . . . . . . . . . . . . . 134
5.4.3 Equation of motion method . . . . . . . . . . . . 136
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6. Green’s functions and Feynman diagrams 143
6.1 The interaction picture . . . . . . . . . . . . . . . . . . . . 144
6.2 The time-evolution operator . . . . . . . . . . . . . . . . . 146
6.3 Perturbative expansion of causal Green’s functions . . . . 148
6.4 Wick’s theorem . . . . . . . . . . . . . . . . . . . . . . . . 149
6.5 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . 151
6.5.1 Feynman diagrams for the electron-electron interaction
. . . . . . . . . . . . . . . . . . . . . . . . . 152
6.5.2 Feynman diagrams for an external potential . . . 157
6.6 Feynman diagrams in energy space . . . . . . . . . . . . . 158
6.7 Electronic self-energy and Dyson’s equation . . . . . . . . 162
6.8 Self-consistent diagrammatic theory: The Hartree-Fock
approximation . . . . . . . . . . . . . . . . . . . . . . . . 167
6.9 The Anderson model and the Kondo effect . . . . . . . . . 170
6.9.1 Friedel sum rule . . . . . . . . . . . . . . . . . . . 171
6.9.2 Perturbative analysis . . . . . . . . . . . . . . . . 173
6.10 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 175
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xiv Molecular Electronics: An Introduction to Theory and Experiment
6.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7. Nonequilibrium Green’s functions formalism 179
7.1 The Keldysh formalism . . . . . . . . . . . . . . . . . . . 180
7.2 Diagrammatic expansion in the Keldysh formalism . . . . 184
7.3 Basic relations and equations in the Keldysh formalism . 186
7.3.1 Relations between the Green’s functions . . . . . 186
7.3.2 The triangular representation . . . . . . . . . . . 187
7.3.3 Unperturbed Keldysh-Green’s functions . . . . . . 189
7.3.4 Some comments on the notation . . . . . . . . . . 191
7.4 Application of Keldysh formalism to simple transport
problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
7.4.1 Electrical current through a metallic atomic contact193
7.4.2 Shot noise in an atomic contact . . . . . . . . . . 199
7.4.3 Current through a resonant level . . . . . . . . . . 200
7.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
8. Formulas of the electrical current: Exploiting the Keldysh
formalism 205
8.1 Elastic current: Microscopic derivation of the Landauer
formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
8.1.1 An example: back to the resonant tunneling model 211
8.1.2 Nonorthogonal basis sets . . . . . . . . . . . . . . 212
8.1.3 Spin-dependent elastic transport . . . . . . . . . . 213
8.2 Current through an interacting atomic-scale junction . . . 215
8.2.1 Electron-phonon interaction in the resonant tunneling
model . . . . . . . . . . . . . . . . . . . . . 217
8.2.2 The Meir-Wingreen formula . . . . . . . . . . . . 222
8.3 Time-dependent transport in nanoscale junctions . . . . . 224
8.3.1 Photon-assisted resonant tunneling . . . . . . . . 231
8.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 233
9. Electronic structure I: Tight-binding approach 237
9.1 Basics of the tight-binding approach . . . . . . . . . . . . 237
9.2 The extended H¨uckel method . . . . . . . . . . . . . . . . 241
9.3 Matrix elements in solid state approaches . . . . . . . . . 242
9.3.1 Two-center matrix elements . . . . . . . . . . . . 244
9.4 Slater-Koster two-center approximation . . . . . . . . . . 246
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9.5 Some illustrative examples . . . . . . . . . . . . . . . . . . 247
9.5.1 Example 1: A benzene molecule . . . . . . . . . . 248
9.5.2 Example 2: Energy bands in line, square and cubic
Bravais lattices . . . . . . . . . . . . . . . . . . . . 250
9.5.3 Example 3: Energy bands of graphene . . . . . . 252
9.6 The NRL tight-binding method . . . . . . . . . . . . . . . 253
9.7 The tight-binding approach in molecular electronics . . . 257
9.7.1 Some comments on the practical implementation
of the tight-binding approach . . . . . . . . . . . . 258
9.7.2 Tight-binding simulations of atomic-scale transport
junctions . . . . . . . . . . . . . . . . . . . . 259
9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 260
10. Electronic structure II: Density functional theory 263
10.1 Elementary quantum mechanics . . . . . . . . . . . . . . . 264
10.1.1 The Schr¨odinger equation . . . . . . . . . . . . . . 264
10.1.2 The variational principle for the ground state . . 265
10.1.3 The Hartree-Fock approximation . . . . . . . . . . 266
10.2 Early density functional theories . . . . . . . . . . . . . . 268
10.3 The Hohenberg-Kohn theorems . . . . . . . . . . . . . . . 269
10.4 The Kohn-Sham approach . . . . . . . . . . . . . . . . . . 271
10.5 The exchange-correlation functionals . . . . . . . . . . . . 273
10.5.1 LDA approximation . . . . . . . . . . . . . . . . . 273
10.5.2 The generalized gradient approximation . . . . . . 275
10.5.3 Hybrid functionals . . . . . . . . . . . . . . . . . . 277
10.6 The basic machinery of DFT . . . . . . . . . . . . . . . . 277
10.6.1 The LCAO Ansatz in the Kohn-Sham equations . 278
10.6.2 Basis sets . . . . . . . . . . . . . . . . . . . . . . . 280
10.7 DFT performance . . . . . . . . . . . . . . . . . . . . . . 282
10.8 DFT in molecular electronics . . . . . . . . . . . . . . . . 284
10.8.1 Combining DFT with NEGF techniques . . . . . 285
10.8.2 Pluses and minuses of DFT-NEGF-based methods 291
10.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
Metallic atomic-size contacts 293
11. The conductance of a single atom 295
11.1 Landauer approach to conductance: brief reminder . . . . 296
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11.2 Conductance of atomic-scale contacts . . . . . . . . . . . 297
11.3 Conductance histograms . . . . . . . . . . . . . . . . . . . 300
11.4 Determining the conduction channels . . . . . . . . . . . . 304
11.5 The chemical nature of the conduction channels of oneatom
contacts . . . . . . . . . . . . . . . . . . . . . . . . . 308
11.6 Some further issues . . . . . . . . . . . . . . . . . . . . . . 316
11.7 Conductance fluctuations . . . . . . . . . . . . . . . . . . 319
11.8 Atomic chains: Parity oscillations in the conductance . . . 322
11.9 Concluding remarks . . . . . . . . . . . . . . . . . . . . . 331
11.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
12. Spin-dependent transport in ferromagnetic atomic
contacts 335
12.1 Conductance of ferromagnetic atomic contacts . . . . . . 336
12.2 Magnetoresistance of ferromagnetic atomic contacts . . . 343
12.3 Anisotropic magnetoresistance in atomic contacts . . . . . 347
12.4 Concluding remarks and open problems . . . . . . . . . . 353
Transport through molecular junctions 355
13. Coherent transport through molecular junctions I: Basic
concepts 357
13.1 Identifying the transport mechanism in single-molecule
junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
13.2 Some lessons from the resonant tunneling model . . . . . 364
13.2.1 Shape of the I-V curves . . . . . . . . . . . . . . . 366
13.2.2 Molecular contacts as tunnel junctions . . . . . . 368
13.2.3 Temperature dependence of the current . . . . . . 369
13.2.4 Symmetry of the I-V curves . . . . . . . . . . . . 371
13.2.5 The resonant tunneling model at work . . . . . . 373
13.3 A two-level model . . . . . . . . . . . . . . . . . . . . . . 374
13.4 Length dependence of the conductance . . . . . . . . . . . 377
13.5 Role of conjugation in -electron systems . . . . . . . . . 381
13.6 Fano resonances . . . . . . . . . . . . . . . . . . . . . . . 382
13.7 Negative differential resistance . . . . . . . . . . . . . . . 385
13.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 388
13.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 389
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Contents xvii
14. Coherent transport through molecular junctions II:
Test-bed molecules 391
14.1 Coherent transport through some test-bed molecules . . . 392
14.1.1 Benzenedithiol: how everything started . . . . . . 392
14.1.2 Conductance of alkanedithiol molecular junctions:
A reference system . . . . . . . . . . . . . . . . . 395
14.1.3 The smallest molecular junction: Hydrogen
bridges . . . . . . . . . . . . . . . . . . . . . . . . 401
14.1.4 Highly conductive benzene junctions . . . . . . . . 405
14.2 Metal-molecule contact: The role of anchoring groups . . 408
14.3 Tuning chemically the conductance: The role of
side-groups . . . . . . . . . . . . . . . . . . . . . . . . . . 412
14.4 Controlled STM-based single-molecule experiments . . . . 416
14.5 Conclusions and open problems . . . . . . . . . . . . . . . 420
15. Single-molecule transistors: Coulomb blockade and
Kondo physics 423
15.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 423
15.2 Charging effects in transport through nanoscale devices . 425
15.3 Single-molecule three-terminal devices . . . . . . . . . . . 429
15.4 Coulomb blockade theory: Constant interaction model . . 432
15.4.1 Formulation of the problem . . . . . . . . . . . . . 432
15.4.2 Periodicity of the Coulomb blockade oscillations . 435
15.4.3 Qualitative discussion of the transport
characteristics . . . . . . . . . . . . . . . . . . . . 436
15.4.4 Amplitudes and line shapes: Rate equations . . . 439
15.5 Towards a theory of Coulomb blockade in molecular transistors
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445
15.5.1 Many-body master equations . . . . . . . . . . . . 447
15.5.2 A simple example: The Anderson model . . . . . 449
15.6 Intermediate coupling: Cotunneling and Kondo effect . . 451
15.6.1 Elastic and inelastic cotunneling . . . . . . . . . . 451
15.6.2 Kondo effect . . . . . . . . . . . . . . . . . . . . . 453
15.7 Single-molecule transistors: Experimental results . . . . . 456
15.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
16. Vibrationally-induced inelastic current I: Experiment 473
16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 473
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xviii Molecular Electronics: An Introduction to Theory and Experiment
16.2 Inelastic electron tunneling spectroscopy (IETS) . . . . . 475
16.3 Highly conductive junctions: Point-contact spectroscopy
(PCS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
16.4 Crossover between PCS and IETS . . . . . . . . . . . . . 490
16.5 Resonant inelastic electron tunneling spectroscopy
(RIETS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
16.6 Summary of vibrational signatures . . . . . . . . . . . . . 499
17. Vibrationally-induced inelastic current II: Theory 501
17.1 Weak electron-phonon coupling regime . . . . . . . . . . . 501
17.1.1 Single-phonon model . . . . . . . . . . . . . . . . 502
17.1.2 Ab initio description of inelastic currents . . . . . 512
17.2 Intermediate electron-phonon coupling regime . . . . . . . 520
17.3 Strong electron-phonon coupling regime . . . . . . . . . . 524
17.3.1 Coulomb blockade regime . . . . . . . . . . . . . . 524
17.3.2 Interplay of Kondo physics and vibronic effects . . 532
17.4 Concluding remarks and open problems . . . . . . . . . . 534
17.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 535
18. The hopping regime and transport through DNA
molecules 537
18.1 Signatures of the hopping regime . . . . . . . . . . . . . . 538
18.2 Hopping transport in molecular junctions: Experimental
examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 541
18.3 DNA-based molecular junctions . . . . . . . . . . . . . . . 546
18.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 552
19. Beyond electrical conductance: Shot noise and thermal
transport 553
19.1 Shot noise in atomic and molecular junctions . . . . . . . 554
19.2 Heating and heat conduction . . . . . . . . . . . . . . . . 560
19.2.1 General considerations . . . . . . . . . . . . . . . 561
19.2.2 Thermal conductance . . . . . . . . . . . . . . . . 562
19.2.3 Heating and junction temperature . . . . . . . . . 565
19.3 Thermoelectricity in molecular junctions . . . . . . . . . . 569
20. Optical properties of current-carrying molecular
junctions 579
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20.1 Surface-enhanced Raman spectroscopy of molecular
junctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 580
20.2 Transport mechanisms in irradiated molecular junctions . 583
20.3 Theory of photon-assisted tunneling . . . . . . . . . . . . 585
20.3.1 Basic theory . . . . . . . . . . . . . . . . . . . . . 586
20.3.2 Theory of PAT in atomic contacts . . . . . . . . . 590
20.3.3 Theory of PAT in molecular junctions . . . . . . . 592
20.4 Experiments on radiation-induced transport in atomic and
molecular junctions . . . . . . . . . . . . . . . . . . . . . . 594
20.5 Resonant current amplification and other transport phenomena
in ac driven molecular junctions . . . . . . . . . . 601
20.6 Fluorescence from current-carrying molecular junctions . 604
20.7 Molecular optoelectronic devices . . . . . . . . . . . . . . 608
20.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . 613
20.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 614
21. What is missing in this book? 617
Appendixes 621
Appendix A Second Quantization 623
A.1 Harmonic oscillator and phonons . . . . . . . . . . . . . . 624
A.1.1 Review of simple harmonic oscillator quantization 624
A.1.2 1D harmonic chain . . . . . . . . . . . . . . . . . 626
A.2 Second quantization for fermions . . . . . . . . . . . . . . 628
A.2.1 Many-body wave function in second quantization 628
A.2.2 Creation and annihilation operators . . . . . . . . 630
A.2.3 Operators in second quantization . . . . . . . . . 632
A.2.4 Some special Hamiltonians . . . . . . . . . . . . . 634
A.3 Second quantization for bosons . . . . . . . . . . . . . . . 637
A.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . 638
Bibliography 639
Index 699 |
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