Transport Phenomena in Micro- and Nanoscale Functional Materials and Devices

By Joao B. Sousa, Joao O. Ventura and Andre Pereira

Transport Phenomena in Micro- and Nanoscale Functional Materials and Devices offers a pragmatic view on transport phenomena for micro- and nanoscale materials and devices, both as a research tool and as a means to implant new functions in materials. Chapters emphasize transport properties (TP) as a research tool at the micro/nano level and give an experimental view on underlying techniques. The relevance of TP is highlighted through the interplay between a micro/nanocarrier’s characteristics and media characteristics: long/short-range order and disorder excitations, couplings, and in energy conversions. Later sections contain case studies on the role of transport properties in functional nanomaterials.

This includes transport in thin films and nanostructures, from nanogranular films, to graphene and 2D semiconductors and spintronics, and from read heads, MRAMs and sensors, to nano-oscillators and energy conversion, from figures of merit, micro-coolers and micro-heaters, to spincaloritronics.

• Presents a pragmatic description of electrical transport phenomena in micro- and nanoscale materials and devices from an experimental viewpoint
• Provides an in-depth overview of the experimental techniques available to measure transport phenomena in micro- and nanoscale materials
• Features case studies to illustrate how each technique works
• Highlights emerging areas of interest in micro- and nanomaterial transport phenomena, including spintronics

• Language: English
• Publisher: Elsevier Science
• Release date: Mar 23, 2021
• ISBN: 9780323461245

Joao B. Sousa

João Bessa Sousa is Emeritus Professor at Department of Physics & Astronomy of Faculty of Sciences of University of Porto. His teaching and research activities span solid sate & low temperature physics, micro & nanotechnologies, transport phenomena. Degree in Electrotechnical Engineering at Univ. Porto (1957-63) and Ph.D. at Univ. Oxford, Clarendon Laboratory (1965-68; Superconductivity). More than 260 published scientific articles. Awarded with the Order Santiago Espada, one of Portugal highest civil honours; also the Prize for Scientific Excellence by the Foundation for Science and Technology (2005). Effective Member of Portuguese Academy of Sciences. Co-founder and later President of Portuguese Physical Society. Former member of NATO Research Grants Scientific Committee, of Physical Society (London), of Condensed Matter Division of European Physical Society; of Linacre & Wolfson Colleges, Oxford. Co-founder and first President of Institute of Physics of Materials of University of Porto (IFIMUP).

Book preview

Transport Phenomena in Micro- and Nanoscale Functional Materials and Devices

João B. Sousa

Department of Physics and Astronomy, Faculty of Sciences, University of Porto, Porto, Portugal

João O. Ventura

Institute of Physics for Advanced Materials, Nanotechnology and Photonics (IFIMUP), University of Porto, Porto, Portugal

André Pereira

Department of Physics and Astronomy, Faculty of Sciences, University of Porto, Porto, Portugal

Table of Contents

Cover image

Title page

Copyright

Contents

Preface

Book Contents

Acknowledgments

Chapter 1. A brief overview on transport phenomena

Abstract

1.1 Driving forces and fluxes

1.2 Thermoelectric phenomena

1.3 Electrochemical potential

1.4 Electron work function

1.5 Contact potentials

1.6 Measurements of contact potential and of work function

1.7 Microscopic insights into the thermoelectric effects

References

Chapter 2. Calculation principles for transport coefficients

Abstract

2.1 Expressions for transport fluxes

2.2 Boltzmann transport equation

2.3 Isothermal electrical conductivity

2.4 Thermoelectric, electrical, and thermal transport phenomena

2.5 Limits of the diffusive regimes

References

Chapter 3. Illustrative microscopic calculations for electronic systems

Abstract

3.1 Basic microscopic formulae

3.2 Impurity scattering

3.3 Lattice thermal vibrations and electrical resistivity

3.4 Physical features and constrains on electron–phonon collisions

3.5 Phonon-induced ρ(T) dependence

References

Chapter 4. Heat transport by phonons and electrons

Abstract

4.1 Thermal conductivity of insulators and semiconductors

4.2 Thermal conductivity of metals

4.3 Illustrative results on K(T) in metals

4.4 Closing remarks: physical and conceptual issues

References

Chapter 5. Advanced topics

Abstract

5.1 Kubo formalism applied to transport phenomena

5.2 Molecular dynamics simulation of lattice thermal conductivity

5.3 Thermal conductivity phonon spectroscopy

References

Chapter 6. Transport phenomena in thin films and nanostructures

Abstract

6.1 Electrical resistivity in thin films

6.2 Ballistic electron transport and Landauer formalism

6.3 Phonons in nanowires and heat conduction

References

Chapter 7. Electrical measurements

Abstract

7.1 Introduction

7.2 Electrical resistivity

7.3 Ac methods for transport measurements

References

Chapter 8. Thermal and thermoelectrical measurements

Abstract

8.1 Thermal conductivity

8.2 Seebeck effect

8.3 Experimental and technical notes

References

Chapter 9. Transport effects under magnetic fields

Abstract

9.1 Magnetic field generation

9.2 Magneto-transport

9.3 Magneto-thermal conductivity

9.4 Thermomagnetic effects (Nernst–Ettingshausen and Seebeck)

References

Chapter 10. Noncontact techniques

Abstract

10.1 Electrical conductivity measurements using the eddy current method

10.2 The magnetorefractive effect as a tool to measure magnetoresistance

10.3 Measuring electrical properties using terahertz-time domain spectroscopy

10.4 Thermal conductivity: the transient thermal-reflectance method

References

Chapter 11. Microscopic information provided by transport measurements

Abstract

11.1 Transport as a versatile and sensitive research tool

11.2 Transport properties on magnetic materials: basic mechanisms and models

11.3 Universal behavior of transport properties near a phase transition

11.4 Magnetocaloric effect and relations with magnetotransport

11.5 Relaxation phenomena and spectroscopic-like transport behavior

11.6 Thermoelectrics: physical principles and materials engineering for energy conversion

References

Chapter 12. Transport at the nanoscale

Abstract

12.1 Transport phenomena in 2D graphene

12.2 Spintronics

12.3 Discontinuous metal–insulator magnetic multilayers

12.4 Resistive switching in nanostructured materials

References

Index

Copyright

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Preface

Transport fluxes result from net movements of particles (e.g., electrons, atoms, ions) or of collective excitations (e.g., phonons, magnons, electron pairs) exhibiting packet-like wave propagation under nonuniform conditions in a medium. Such flows ensure mass, charge, spin, energy, or momentum transport. They require driving agents such as temperature or concentration gradients, external electric or magnetic fields, exerting forces on each carrier with specific direction and strength which shape the flux characteristics in each case. The presence of physically different driving forces causes hybrid fluxes such as thermoelectric, thermomagnetic, or thermo-magneto-galvanic, according to their coupling strengths and influence on the carrier motions (currents).

Carrier propagation depends on the type and availability of relevant quantum states (and occupancy) at each temperature, pressure, and volume. On the other hand, wave-like propagation depends on the sample internal structure (long- and short-range order), being quite sensitive to defects which cause scattering; the inherent carrier recoils, elastic or inelastic, oppose propagation (resistive processes). In turn, carrier–lattice and carrier–carrier interactions (e.g., electrons and phonons in charge and heat flow) allow direct energy conversions inside a solid, without mobile parts. One can also couple charge, field, and spin to manipulate spin transfer between systems (Spintronics). In contrast to thermodynamic equilibrium, transport deals with the dynamic responses of a system to a variety of external stimuli, transferring information or physical quantities, and transducing among them. Thus, multiactuated and multiresponse functions can be performed in a single system.

Transduction usually has low efficiency (apart Joule conversion) due to weak cross-couplings, for example, between electric (E) and magnetic field (H) effects. Whereas E exerts a parallel force on a charge, permanently accelerating it (if free) and continuously transferring energy, the magnetic force is perpendicular to the charge velocity, preventing direct work on the charges (although magnetic moment precession may arise). Thus, while charge flow provides electrical currents up to many Ampères, magneto-electro voltages (e.g., Hall effect) only fall in the μV–mV ranges. Similarly, thermoelectric conversion typically provides voltages in the 1–220 μV range, while usual magnetic cooling provides ΔT-decreases of 1–10 K around room temperature. In spite of this, the extraordinary progress in the accurate detection and manipulation of very small signals (down to near-noise levels) has brought into light the practical relevance of transport cross-couplings.

Another route exploits system downsizing, dimensionality reduction, and micro-nanostructuring. Here transport appears rather resilient, even providing extra flexibility and improved characteristics. For example, electric and thermal flows can be strictly controlled by the sample dimensions (at nanoscale), exhibiting quantum ballistic behavior, and also universal features related to fundamental physical constants. In high-quality multilayer structures, coherent phonon propagation is observed (so far in limited paths), functioning as waveguides for vibrational transport and manipulation (phononics). Progress in this area may eventually overcome the present heat bottleneck in high-density technologies.

Transport resilience to downsizing reflects three key features. On the one hand, charge, heat, and spin flows in solids involve huge carrier numbers at ordinary temperatures, due to the high number of electron, phonon, and magnon (magnetic media) excitations, up to the order of atomic concentrations. Second, one can nowadays detect extremely low transport flows, such as, the arrival of single or few electrons. Phonon and magnon individual arrivals can also be detected in dedicated experiments. Finally, downscaling can preferentially reduce the contribution of certain carrier types in transport, when relevant sample dimensions are reduced below selected ranges of carrier mean free paths (mfp). This is the emergent field of mfp spectroscopies (charge, phonon, spin), identifying particular mfp ranges worth improvement in materials optimization. Also, to support first-principles calculations (avoiding incorrect assumptions), deepening our views on transport and inherent microscopic roots.

Excellent books have been written on transport phenomena in condensed matter, both at introductory and frontier levels. Flows prevail in science and technology, being core subjects in many fields beyond physics. A broad account of this subject, integrating experimental and theoretical views, has been the trigger to write this book. Being solids discrete and highly dense (~10²⁸ atoms m−3), powerful statistical methods and microscopic knowledge are required to rigorously describe flows. From the authors’ experiences, teaching this subject at undergraduate or first-year graduate level demands a blended approach, combining theory and experimental views, of strictly formal, approximate, or even of inquiring nature. Also stimulating creative thinking, student confidence, and conceptual perception, when possible giving complementary views on a problem, with level gradation from macro to atomic level processes. The book also contains detailed descriptions of experimental setups to measure transport coefficients; practical lab-type details and materials; applications in research and technology, as well as many illustrative research results, are displayed throughout.

Book Contents

The main areas covered in this book can be grouped into three parts:

1. General Introduction to Transport Phenomena (Chapters 1–6); J. B. Sousa

2. Experimental Techniques and Setups

Measurement of electrical transport coefficients in bulk, thin film, and nanomaterials (Chapter 7); J. Ventura, J. B. Sousa

Thermal and thermoelectric measurements (Chapter 8); J. B. Sousa, A. Pereira

Transport measurements under a magnetic field: magneto-electric-thermo coefficients (Chapter 9); A. Pereira, J. B. Sousa

Noncontact techniques for electrical, thermal, and magnetoresistance measurements in bulk, micro, and nanomaterials (Chapter 10); J. Ventura

3. Advanced Research Topics

Transport phenomena in thin films, quantum nanowires and structures (Chapter 6); J. B. Sousa, J. Ventura

Microscopic information provided by transport properties (Chapter 11); J. B. Sousa

Solid-state energy conversion: thermoelectric and magnetocaloric (Chapter 11); A. Pereira, J. B. Sousa

Resistive switching, learning features, artificial intelligence (Chapter 12); J. Ventura

Spintronics: transport in spin valves and discontinuous metal–insulator magnetic multilayers (Chapter 12); J. Ventura

Graphene physics: Insights on its extraordinary electrical and thermal conductivities (Chapter 12); J. B. Sousa

Chapter 1 deals with the basic aspects of linear transport phenomena within the general scope of thermodynamics of irreversible processes. It provides a general definition of driving forces, fluxes, and transport coefficients, with associated tensor symmetries. Boundary effects and contact potentials between distinct media are also considered. The Seebeck, Peltier, and Thomson effects are analyzed, establishing their thermodynamic relations.

Chapter 2 introduces the carrier distribution function (f) for electrons under an electric E-field, f(r,t;k,E), enabling to calculate transport coefficients. It reflects the effects of the driving force (−eE) on the electron quantum states (k) at mean r-positions (local averages) and of collisions causing a mean collision time τ. The inherent balance equation (Boltzmann, BTE) is solved under steady-state conditions, giving f(r,t;τ,k,E;T) at each T. Formulae for electrical (σ) and thermal (k) conductivities, Seebeck (S) and Peltier (π) effects are derived, identifying relevant factors for materials improvement. Formal simplicity prevails throughout.

In Chapter 3, instead of using a phenomenological electron mfp, τ is microscopically calculated for electron–impurity and electron–phonon scattering, obtaining the electrical resistivity dependences on point defects and on temperature, respectively. Comparison with experiment includes metals, alloys, and compounds. In the two last cases (and for order–disorder effects) the formalism is generalized using occupation-site operators.

Chapter 4 contains the BTE approximation to calculate heat conduction, obtaining the electron and phonon k(T) contributions from T ~ 0 to >> 300 K. The relevant microscopic scatterings (e-ph, ph-e, ph-ph, e-defect, ph-defect, ph-boundaries) are worked out, and k(T) is obtained in each case. Comparisons with experimental k(T) results in metals, semiconductors, and insulators are critically examined.

Chapter 5 goes beyond the collision time (τ) approach, which misses a fundamental process at nanoscale: the unavoidable correlation between nearby particles (atoms or electrons), due to their extended wavefunctions, interparticle potentials, and inertia (no instantaneous position/velocity changes). Thus, correlations in positions, velocities, and accelerations are intrinsic, fluctuating in space and time at system characteristic lengths and ω frequencies. The statistical Kubo formalism specifically connects such correlations (Γ) to each transport coefficient (and for each j-type current), through a time current–current correlation Γ(t)=<j(o)·j(t)>. As fluctuations are always present at finite T (in and out of equilibrium), Γ(t) can be calculated in thermodynamic equilibrium! As an application, k(T) is calculated for insulators, as well as its ω-spectra (k-phonon spectroscopy), so both can be compared with experiment.

Chapter 6 analyses transport in the limit of nanosize (d) samples, that is, 0D-dot, 1D-nanowire, and 2D-film. Here, the carrier ψ-wavefunction is intrinsically spread (λo length) according to its energy [εo=f(λo)], thus becoming confined when d<λo. Then, stationary states only occur with quantized higher εn-energies giving d=n·λn. To evidence the gradual approach to such a regime, the electrical resistivity ( ent ) is first analyzed in thin films, using the diffusive Fuchs–Sondheimer theory. When d<T) and k(T) dependences are calculated in such a quantum regime, evidencing a universal behavior in each case, independent of sample type and nanosize, as T 0 K.

Chapters 7 and 8 cover relevant techniques to measure electrical transport coefficients of bulk, thin, and nanosize materials. This include electrical resistivity rod-type and van der Pauw methods; thermal conductivity steady-state (rod and plate samples) and transient or ac methods (flash-type and 3ω); Seebeck effect measurements (integral and differential; steady-state and quasi-steady-state; dynamic 2ω and 3ω); polyvalent transport modular systems (thermal conductivity, Seebeck effect, electrical resistivity). Hindered experimental details are critically analyzed, particularly when measuring nanoscale samples and devices. The different techniques are supported by focused accounts of the physical principles behind each methodology. The aim is to provide the reader the basics of each setup, together with its potential, applicability, and limitations.

Chapter 9 addresses several aspects concerning magnetic fields, including their generation over wide intensity ranges (10−3–10² T), specific setups and their impact on the electronic and thermal properties of materials. Experimental techniques are described to study magneto-Seebeck, magneto-thermal conductivity, Nernst and Ettingshausen effects. Applications in science and technology are also considered.

Chapter 10 describes how to extract electrical and thermal conductivity information in a noncontact manner in bulk and thin-film samples. Based on electromagnetic induction, the eddy current method gives detailed information on the electrical conductivity. On the other hand, optical probing can retrieve electrical, magneto-electrical, and thermal properties of samples. The magneto-refractive effect takes advantage from the dependence of the refractive index on the electrical conductivity to probe the magnetoresistance of thin films. Terahertz time-domain spectroscopy is an ultrafast optical technique able to study the dynamics of charge carriers in semiconductors and metals and their frequency-dependent electrical conductivity. Finally, the transient thermal reflectance method (also a pump-probe technique) is described to measure the thermal properties of both bulk and thin-film samples by locally heating the surface with ultrashort laser pulses and retrieving the corresponding time-dependent reflectively. Particular emphasis is placed throughout the chapter in providing the physical principles of each method, its detailed experimental implementation and main limitations.

Chapter 11 deals with a simple question: transport being measured at macro- or microscales, which micro-nanoscopic knowledge can be obtained? First, its response to carrier charge, spin, energy or mass and to multiple driving, static or time-dependent (E, H, T,…). Second, an extreme sensitivity to scattering events at the nanoscale, allows useful energy conversions in a solid; also, a better understanding of the carrier and medium quantum states and corresponding transitions responsible for scattering. Such multivalence is illustrated for the cases of ent (T) near a spin-reorientation transition and of S(T), ent (T) near a magnetic transition; also, near a first-order magneto-structural transition causing a strong magnetocaloric effect. The underlying microscopic processes are examined at some extent; applications in medicine and magnetic cooling are also considered.

Chapter 12 demonstrates how transport can be an insightful tool to retrieve unique information at the nanoscale, in four of the most interesting research topics at present. (1) First, graphene is a model example of how nanomaterials can give rise to new physics and devices. Being made of a single atomic layer of carbon, graphene displays extraordinary thermal conductivity and electron mobility. (2) Spintronics takes advantage of the electron spin as a new degree of freedom to obtain new functionalities, particularly for read heads in hard disk drives and nonvolatile magnetic random access memories. (3) Discontinuous metal–insulator magnetic multilayers offer a set of nontrivial physical phenomena, including superparamagnetism, tunneling, and hopping or Coulomb blockade. (4) Finally, resistive switching nanostructures allow mimicking basic learning features of biological systems, opening new opportunities in the field of artificial intelligence.

Acknowledgments

This book could not have been written without the extensive research work (and teaching experiences) locally accumulated since the establishment of the Group on Transport Properties, in 1970 (João B. Sousa) at Centro de Física da Universidade do Porto, headed by Prof. J. Moreira Araújo, in the Faculty of Sciences of the University of Porto (FCUP); presently integrated in IFIMUP, the Institute of Physics for Advanced Materials, Nanotechnology and Photonics of the University of Porto, headed by Prof. J. Pedro Araújo at the Physics and Astronomy Department of FCUP. We are also grateful for the research support provided over the years, particularly by Fundação para a Ciência e Tecnologia and European Programs. Special thanks go to the publishers and editors at Elsevier, for their encouragement and patience in all stages of development of this book.

João B. Sousa acknowledges with deep gratitude to his D.Phil. Supervisor Prof. Kurt Mendelshon and to Dr. John Lowell as Group Leader at Clarendon Laboratory, University of Oxford (1965–68). He also acknowledges the unforgettable atmosphere and scientific drive at this institution, promoting intimate links among experimental physics and technical skills (hands-on facilities), open teaching and concern for deep physical understanding. In particular to Profs. R. Peierls, H. M. Rosenberg, R. Elliott, B. Bleaney, Drs. G. Garton and D. Hukin (Materials Preparation), T. Smith (Electronics). A special gratitude and heartfelt thanks for the continuous support, friendship, and scientific contributions of Drs. Fátima Pinheiro, Rafaela Pinto, José M. Moreira, M. Manuela Amado, M. Elina Braga, as members of the Group of Transport Properties for decades. To Mr. Magalhães, M. Pinheiro, E. Pinheiro, and Engineers Jaime Bessa and Francisco Carpinteiro, for the Group technical support, design, and research-equipment construction along the years. To all my Ph.D. students from whom I much learned and benefitted.

Last, but not least, all my love and thanks to my wife Maria José for so many days and weekends with my endless writings in her company. Her patient indulgence and uncompromised support, strength, and unspoken encouragement allowed me to overcome difficult periods when book tasks appeared unsurmountable. To my sons Rui and João, for all the happiness and soothing given to me.

João O. Ventura would first and foremost like to thank Prof. João B. Sousa for his constant dedication and motivation in all things science-related. His accumulated knowledge and endless desire to always know more are a true inspiration. Without him, this book would never be possible. To the long friendship of colleagues at IFIMUP and FCUP (particularly João Pedro Araujo, André Pereira, Francisco Carpinteiro). A special word to José Miguel who will always be among us. To present and past collaborators and students, all contributing to extend our knowledge every day. Within the context of this book, a special thanks to José Diogo Costa, Catarina Dias, Leandro Martins, and Cátia Rodrigues. Finally, I would like to thank my family for all the support during these years. The final word is for my daughter Rita.

André Pereira would like to thank Prof. João B. Sousa for being a person who motivated these studies of transport phenomena and also for motivating me for writing this book. I also thank my IFIMUP friends from long date, who discussed a lot about physics, especially João Pedro Araújo, João O. Ventura, and José Miguel (who unfortunately left us). Obviously, everyone who works or worked day by day with me helps to achieve the high scientific quality that we have always pursued and for the pleasant days of work with them. My family (parents) and sister because they have always been very supportive. My little daughter Íris, for the love she shows and for many moments of happiness that she provides me every day. To all the other friends who always supported me throughout life.

Chapter 1

A brief overview on transport phenomena

Abstract

This chapter covers linear transport, thermodynamic functions, and electrochemical potentials, focusing on electric, thermal, and thermoelectric phenomena, within the scope of thermodynamics of irreversible processes. Besides bulk response(s), boundary and contact effects are highlighted, with the underlying microscopic features. The principles of relative and absolute work function (ϕ) measurements are analyzed, as they are basic in modern Kelvin microscopies and nanospectroscopies. The measurement of thermoelectric coefficients is also analyzed. Physical insights are given throughout the chapter.

Keywords

Thermodynamics of irreversible processes; electrical and thermoelectrical effects; measurement principles; surface and interface phenomena

Transport phenomena are dynamic processes caused by internal or externally driven imbalances in matter. The equilibrium properties of systems with many particles and with linear sizes much larger than interparticle distances, when averaged over periods much longer than the particle (atoms, ions, electrons, …) collision times, follow the laws of Thermodynamics. On the other hand, external perturbations cause transport phenomena, for example, of particles (jN), charge (j), heat (jQ). These perturbations, such as thermal (ΔT) or electrical (ΔV) finite differences in bulk media, when scaled-down to the interparticle mean distance, usually give minute microscopic disturbances, δT ent T and δV ent Vi where Vi is the interparticle potential energy. This justifies a local equilibrium approach, where the system responds (j fluxes) linearly to the driving forces, and gradients in such examples. This chapter covers linear transport, thermodynamic functions, and electrochemical potentials, focusing on electric, thermal, and thermoelectric phenomena, within the scope of thermodynamics of irreversible processes. Besides bulk responses, boundary and contact effects are also highlighted, including the work function ϕ, contact potentials and the underlying electronic structure features. Such effects play a key role in surface physics and in technological applications, from nanodevices to nanomaterials. The principles of relative and absolute work function (ϕ) measurements are analyzed, as they are the basis of modern Kelvin nanospectroscopies. Physical insights are given on the microscopic processes underlying thermoelectric effects—Seebeck, Peltier, Thomson—which are a key factor in the development of new materials for energy conversion and cooling with higher efficiency. Experimental and measurement principles are also critically analyzed. Formal treatments, microscopic views, and experimental details naturally appear throughout the chapter.

1.1 Driving forces and fluxes

Transport phenomena deal with the flow of physical quantities in a medium (e.g., electric charges, heat, particles) due to specific perturbations (driving forces), such as electric fields, temperature, or particle concentration (na) gradients. In bulk media (na ~ 10²³ atoms cm−3) such effects are commonly small. To demonstrate this, let us consider large thermal gradients (∇T~10² K mm−1) or electrical fields (E~10² V mm−1) along a copper rod. In spite of such magnitudes, at the microscopic level the resulting perturbations are extremely small: and between neighboring atoms (a~3 Å). At T~300 K, one has relative perturbations δT/T~10−7 and δV/(Ei·a)~10−4, where Ei are the microscopic (mainly ionic) fields acting on the electrons, say Ei~10⁵ V m−1 (neglecting electron shielding). A better assessment uses the electron mean free path (le~400 Å in Cu) instead of interatomic distances a, giving δT/T~10−5 and δV/Eile~10−4. These weak perturbations allow to consider small volumes ( ; still containing many atoms, N~10⁶) in local (r) equilibrium and with proper thermodynamic entities, for example, temperature, energy, or entropy, T(r), u(r), s(r) [1,2].¹,²

In these conditions, a linear approximation describes well the link between the flux components (Ji) and driving forces (Xj):

(1.1)

where Ji represents a flow per unit time and surface (along i-direction), Xj are driving forces, that is, j-components of gradients of electrical potential ( ), temperature ( ), concentration ( ), and Lij are the cross transport coefficients that, in nonhomogeneous media, are written as Lij(r). As the Xj forces vanish in thermodynamic equilibrium, the neglect of higher-order terms (LijkXjXk,…) means systems near equilibrium [1]. However, they are in a dissipative state (heat release, δQ/dt), with continuous entropy production dS/dt≅(1/T)·δQ/dt. Onsager has shown that dS/dt can be expressed as a sum of bilinear terms of the form [2,3]:

(1.2)

Thus if one calculates dS/dt and decouples each bilinear term in a properly defined flux Ji [2], the remaining part is the conjugate driving force Xi. An illustration of this procedure is given in Section 1.2. The transport coefficients Lij are symmetrical due to the time-reversal symmetry of physical laws, in the absence of magnetic fields (H) or bulk rotations (Ω):

(1.3)

Otherwise, one has Lij(H)=Lji(−H) and Lij(Ω)=Lji(−Ω) [1,3]. Transport phenomena in the presence of magnetic fields or in magnetic materials will be discussed in Chapter 9, Transport Effects Under Magnetic Fields.

1.2 Thermoelectric phenomena

Applying an electric field ( ) and a temperature gradient ( ) along a homogeneous conducting rod causes electrical (j) and heat (jQ) fluxes. In a steady state, using and as driving forces results in first-order [4]:

(1.4a)

(1.4b)

where Lij(0) are the transport coefficients when one chooses E and as driving forces (see below). Measuring the fluxes and driving forces under distinct experimental conditions one can obtain the Lij(0) coefficients, as illustrated with two simple configurations of physical interest:

1.  : Isothermal electrical conductivity (σ) and Peltier coefficient (Π)

Eq. (1.4a) gives j=L11(0) E≡σ·E, where σ=L11(0) is the electrical conductivity. From Eq. (1.4b) one has jQ=L21(0)E=[L21(0)/L11(0)]·j≡Π·j, where Π is the Peltier coefficient. Thus a flux of charge at finite temperature T also carries heat (particle’s thermal energy).

2.  0, j=0: Thermal conductivity (K) and Seebeck coefficient (S)

From Eq. (1.4a) one has E=−(L12/L11)· . Thus a temperature gradient induces an electric field in the sample and, in a steady state, it cancels charge flow (j=0). This field relates to the Seebeck coefficient, S=−(L12/L11), and one has an electric field induced by a temperature gradient:

(1.4c)

Thus for a rod-shaped sample, electrically insulated at ends 1 and 2 (j=0), a voltage appears between them:

(1.4d)

If T1>T2 ( and under free electron transport, more electrons diffuse from 1 to 2 than in reverse. The hot end becomes positive (ΔV>0) and S<0, as observed in simple metals like Na and K [5], which closely follow the free-electron model. Inserting Eq. (1.4c) in Eq. (1.4b) gives jQ=−(L12L21−L11L22)/L11·∇T≡−K·∇T, where K is the sample thermal conductivity.

Experiment and theory show that the Seebeck (S) and the Peltier (Π) coefficients are connected by a simple (Kelvin) relation [6] (see also Section 2.4):

(1.5)

One should notice the difference between ad hoc choices of Xj (as E and ∇T components above) and those imposed by thermodynamics which ensure the Lij symmetry, as physically required. As seen in Eqs. (1.4a) and (1.4b) above, the experimental Lij(0) coefficients can be obtained from σ, S, Π, and K measurements, giving in particular L12(0)=−σ·S and L21(0)=σ·Π. The Kelvin relation [Eq. (1.5)] gives L21(0)=−T·L12(0), thus the Lij symmetry is not satisfied. Using thermodynamics, one can derive as driving forces E/T (instead of E) and (instead of ∇T), and then Lij=Lji, as can be readily checked [3,4].

Many treatments of transport phenomena are available, ranging from fundamental, applied, introductory or formal levels, in the scope of statistics and thermodynamics of irreversible processes; for example, Refs. [1–4,7–13]. The techniques to measure transport coefficients will be detailed in Chapter 4, Heat Transport by Phonons and Electrons.

1.3 Electrochemical potential

The simple description given earlier does not include surface effects nor contact phenomena between different materials. For example, mismatches in the electron energy levels (or of other charge carriers) lead to interfacial adjustments and electric potential differences and to heat effects when an electric current flows [4,6,7]. To describe such effects, one introduces the so-called carrier(s) electrochemical potentials μ, so that variations of μ in a sample or across the contacts introduce driving forces . In particular, the μ(T) dependence induces thermoelectric effects, since one has .

A bulk sample contains many particles (e.g., atoms, ions, electrons) and, in equilibrium, they occupy energy levels according to temperature, particle numbers, interactions, etc. Thus, changes in particle numbers modify the thermodynamic functions, so that minimum energies are needed to remove or add different types of particles from or into a system, underlying the concepts of electrochemical potential (μ) and work function (ϕ).

1.3.1 Thermodynamic definition of μ

For a body in equilibrium at constant temperature (T) and pressure (P) and with N particles of a given type, one defines [14]:

(1.6a)

(1.6b)

where G is the system Gibbs free energy, U, S, and V the internal energy, entropy, and volume, respectively, and j labels particles of different species, whose Nj numbers remain constant; indeed , a sum over species. The smallest N-variation is 1, so that Eq. (1.6a) is equivalent to:

(1.7)

When two initially noninteracting bodies (1, 2), with μ1≠μ2 and Gibbs functions G1, G2, are brought in close proximity, allowing particle transfer [14,15], the dominant rate is from 1→2 when μ1>μ2 and it lasts until μ1=μ2; in the reverse direction when μ1<μ2. Then, G=G1+G2 is minimized and corresponds to thermodynamic equilibrium at T, P [14]. In the transient period before equilibrium, one can write:

(1.8)

Since the total number N1+N2 is constant, one has dN2=−dN1. Therefore:

(1.9)

As one must have dG<0, this implies dN1<0 when μ1>μ2, that is, particles flow from high to low electrochemical potential, and equilibrium occurs when μ1=μ2 (dG=0). From Eq. (1.6b), one sees that changes of G (in any system at constant P, T) result from changes in U, S, or T:

(1.10)

where ΔU=U(T,P;N+1)–U(T,P;N) and equivalently for ΔS and ΔV. In common solids at low pressures (e.g., atmospheric), the high lattice rigidity turns PΔV negligible, and one writes:

(1.11)

where F(T,V)=U–TS is the so-called Helmholtz free energy and ΔF=F(T,V;N+1)–F(T,V;N). When the N-changes occur adiabatically (ΔQ=TΔS=0), results μ=ΔU. The same happens with electrons in metals at low/moderate temperatures, kBT<<EF (Fermi energy), even under nonadiabatic conditions, since the electronic entropy is very small [11,16]. In fact, the Pauli exclusion principle prevents electron disorder in most of the occupied electron states below μ. Only a small fraction of the electrons, ΔN/N~kBT/μ is thermally excited since μ ~ few eV in metals and kBT~25 meV at 300 K.

1.3.2 Estimating μ in a metal (free electron jellium model)

Thermodynamics establishes general relations between macroscopic quantities such as V, T, P (defining the equation of state), PΔV (expansion work), TΔS (heat), ΔU (kinetic and potential; changing through heat or work exchanges), in the presence or absence of external fields (E, H,…). Quantum and Statistical Physics quantifies the microscopic processes underlying μ, as well as the work function ϕ [15]. As an example, let us estimate μ for electrons in simple metals (see also Section 1.4).

The free electron jellium model is a simple quantum model [17–20] that considers N conduction band free electrons in a metal with volume Ω=L³. Each electron senses an average positive electrical potential <V0>, dominated by the ionic-charges potential (over electron–n repulsions and exchange effects, which are included in the electron quasiparticle concept). The ψk(r) electron states have quantized kinetic energies ∈(k)=(ħ²/2m)k², velocities v(k)=(ħ/m)k and λ(k)=2π/k wavelengths, with k=(2π/L)(n1, n2, n3), ni being positive/negative integers or zero [19]. Introducing the electron spin (σ) results in ψ(k,σ), with no more than two electrons (↑,↓) per k-state (Pauli principle). The lowest energy level (∑0) occurs for k=0, with ∑0=−e<V0>~−10¹ eV, that is, only the potential energy due to the ionic attractions is considered in ∑0 (Fig. 1.1).³

Figure 1.1 Simplified free-electron energy diagram, with the Fermi energy EF and corresponding level at ∑F=∑0+EF; ∑0 is the lowest electron energy level, dominated by the ionic attractions in each metal (zero kinetic energy); electron–electron charge and spin interactions are not explicit in the free-electron model, but hindered in the quasiparticle concept.

At T=0 K, the N-lowest electron energy states are filled up to a maximum EF(N), the so-called Fermi energy, . The corresponding level is at ∑F=∑0+EF(N), taking zero-energy for an electron at rest in the external vacuum at . Each k-state corresponds to a phase volume (2π/L)³, so that N electrons fill a volume (4π/3)kF³/2, leading to EF at T=0 K [19]:

(1.12)

where N/Ω≡ne is the electron concentration, a is the interatomic distance and one considers 1 free electron/atom. Using ne=2.65 × 10²² cm−3 for the simple metal Na, one finds EF(0)=3.23 eV, and for Cu (n=8.45×10²² cm−3) results EF(0)=8.45 eV [19]. For free electrons, EF(0) only depends on the metal electron concentration. In nonsimple metals (e.g., transition metals), the electron band features may lead to effective electron masses m* quite different from the electron intrinsic mass (m) [11,19], as well as exchange and e–e effects which modify the free-electron picture.

Recalling that μ=ΔU−TΔS+PΔV, when the number of free electrons changes from N to N+1 at T=0 K the internal energy rises by ΔU=EF(0)~eV,⁴ while the entropic TΔS term is zero since S→0 as T→0 (third law of thermodynamics). Also, the change in volume is very small (ΔV≈a³~10−29 cm³) and at atmospheric pressure PΔV is negligible (<10−10 eV). Thus μ(0)=∑0+EF, which means that the added electron state sits at the Fermi level ∑μ=∑F (Fig. 1.1).

For T≠0 K, it can be shown [11] that, when EF~eV (in typical metals), EF decreases very little with temperature, ΔEF/EF~−[kBT/EF(0)]². Thus at room temperature, EF(T) only differs from EF(0) by a relative factor of ~10−5.⁵

1.4 Electron work function

1.4.1 A simplified approach

The work function ϕ is the minimum energy required to remove an electron from the bulk to outside the material surface, and letting it at rest at a distance D~μm,⁶ at vacuum zero-energy (see below). A good account on the physics underlying ϕ is given in Refs. [21,22]. Restricting here to metals, in orders of magnitude one has ϕ≈+e<V0>−EF (Fig. 1.1), but this simplified picture needs further elaboration [23].

First, only bulk electron energies are considered above, missing the extra energy (Es) for the electron to cross the surface. In fact, the sample does not terminate abruptly at the ionic surface plane, since the electron cloud (associated to |ψk|²) spills out with an exponential ψ-decay in the outside vacuum, over a subnanometer distance δ. This produces a small excess of negative charge outside and an equal positive excess left behind (incomplete ion-shielding). An electric dipole then forms at the surface, with its electric field Ed perpendicular to the surface, giving a potential difference ΔV=Edδ between the inner (+) and outer (−) dipole layers (Fig. 1.2A).

Figure 1.2 (A) Ions give a mean constant positive-charge density ρi inside the metal and zero outside (last ionic plane at x=0). Electrons give a comparable but negative charge density (ρe) well inside metal, apart from a small ρe(x) oscillatory behavior on the boundary approach, caused by electron wave function ψ normalization (Friedel oscillations). Also, near the metal surface, ρe markedly decreases inside and slightly extends outside due to the δ-spill of ψ. A surface charge-dipole forms across the surface [24]. (B) Estimation of ϕ using a surface-dipolar term (energy Es~eE·δ) plus Ei outside due to the image electron attraction.

An energy Es=eΔV is thus required for an electron to cross the surface, and then the metal gets positively charged (q=+e), attracting such electron backward (Fig. 1.2B). Thus to move it to infinite, an extra energy Ei(∞) is required. A classical charge–image calculation [22] shows that at a distance D~μm the Ei(D) image-energy only negligibly differs from Ei(∞), and one takes Ei=Ei(D). Summing up results an improved expression for ϕ [20]:

(1.13)

where φ is an electric potential eventually existing outside the surface (beyond D) due to external fields. In practice, differences in ϕ from metal to metal arise from their electronic structures (∑0 level and EF energy in Fig. 1.2B) giving ϕ0=−∑0−EF, where ∑0=−e<V0> reflects the intensity of the ionic potentials and EF depends on the concentration (ne) of free electrons, [Eq. (1.12)]. One therefore has:

(1.14)

The dipolar term Es depends on the surface structure and on the atomic spacing (open structures tend to lower Es), whereas Ei~constant for an image charge approximation [22]. Both Es and Ei are of the order of ~eV.⁷ One notes that the electrochemical potential is positioned in the energy diagram at ∑μ=−ϕ, and common ϕ values are of a few eV. Generally, the work function ϕ depends on temperature, external E-fields (e.g., in electronic emission), surface structure, and coatings (used to lower ϕ) [25]. These effects and the underlying many-body problem (not considered here) still prevent precise ϕ-calculations, say below ~eV depending on the metal.

1.4.2 Quantum surface effects

In the previous section, we assumed that to extract an electron from a metal (at the Fermi level) it first requires an energy Es to cross the surface dipole barrier (eΔV=eEd·δ), leaving the metal positively charged (+e), thus exerting an attractive force f on the electron. Classically, for such particle at a distance x from the surface, the image–charge method (second charge +e at –x) gives [22,24]:

(1.15)

where n is the unit vector along X. Thus to take the electron from x to along n direction requires an energy

(1.16)

Eq. (1.16) shows that the image method is inapplicable at small x, since Ei diverges. It happens that the electron has an associated wave function ψ(x) with λe~ħ/(mv)~1/kF, and when ψ overlaps with the metal ions (and other electrons) at small x, quantum interference occurs and must be explicitly considered when x<x0≅λ. A quantum treatment is also necessary to describe electron transfer across the surface dipole layer. Quantum surface physics is thus mandatory to reliably obtain ϕ of a metal (or other material). Here, one retains a few basic conclusions from such treatments [26]:

• Electrons to escape from a metal must have high kinetic energies ( ), EF few eV, to cross its surface dipole barrier (Es) and reach a distance d0 where they are at rest (see below) and sufficiently outside to neglect quantum interactions with the metal. Then, for x≥d0 the classical image method [Eq. (1.16)] can be applicable to obtain ϕ:

(1.17)

• The electron displacement from inside the metal (v=vF) to d0 outside (where it is at rest) implies the total conversion of its initial kinetic energy into potential energy. As the electron reduces its momentum by Δp=ħkF, the uncertainty principle ΔpΔx≈h enables one to estimate d0 (putting Δx=d0):

(1.18)

• Using kF=(3π²)¹/³ne¹/³ for free electrons [19], where ne is the electron concentration, results:

(1.19)

Substituting this expression in Eq. (1.17), one obtains:⁸

(1.20)

with c=(e²/16πε0)=5.78 × 10−29 J m.

• For a monovalent metal like Na (ne=2.65 × 10²⁸ m−3) [19], one has d0=1.1 × 10−10 m and ϕ≅3.3 eV, whereas for Cu (ne=8.45 × 10²⁸ m−3) results d0=0.9 × 10−10 m and ϕ≅4.03  eV. Experiment gives ϕ=2.75 and 4.51 eV, respectively [25], which seem acceptable for an estimation based on the uncertainty principle. The electrons which are at rest at x=d0 are the ones which require the minimum work to be removed to ∞, against the image–charge attraction. Such electrons initially move perpendicularly to the surface, reaching a maximum distance d0 outside (Fig. 1.3).

• One notes that the character of the electronic orbitals in the surface atoms or ions was ignored, as well as crystallographic features, exchange and correlation effects, and the quantum nature of the dipole layer. Details on more elaborated ϕ-calculations can be found in Refs. [20,25,27].

Figure 1.3 The uncertainty principle Δp.Δx≈h allows an electron, with its kinetic Fermi energy inside metal, to move out of the surface to a maximum distance d0~Å, where it reaches rest (v=0); see text.

1.4.3 Electronic and structural effects on the work function

To illustrate electronic and structural effects on ϕ, Table 1.1 contains experimental values of ϕ, atomic volume Ω, and electron concentration ne for the noble metals Cu, Ag, and Au [25], having the same crystal lattice and valency (charge +e per ion; see below). Ag and Au have almost equal interatomic distances and electron concentrations ne, leading to similar Fermi energies EF=5.5 eV (within uncertainty). From the experimental work-function values, one can estimate the bottom of the conduction band relative to zero-energy outside the metal, E0=0−(ϕ+EF), giving E0=−9.76 eV (Ag) and −10.60 eV (Au). These close values reveal similar ion-electron mean attractions for k=0 electrons. The slightly deeper E0 value in Au may reflect a small increase in the effective ionic charge, due to poorer shielding by the Au inner-shell electrons, distributed over a larger volume than in Ag.

Table 1.1

In contrast, Cu has a much smaller interatomic distance (thus higher na), giving a large Fermi energy EF(Cu)=7.00 eV. Also, the lower E0 value (−11.65 eV) reflects stronger electron–ion interactions due to the smaller interparticle mean separation (dominating over shielding effects).

1.5 Contact potentials

For two metals (A,B) initially uncharged and with work functions ϕA and ϕB (say with ϕA < ϕB), the electrochemical potential levels are ∑μ(A)=−ϕA and ∑μ(B)=−ϕB in relation to vacuum so that ∑μ(A)>∑μ(B). However, as A and B are uncharged, both are at zero electrical potential (V0=0). When the metals are put in close proximity or in contact, more electrons flow from A→B than in reverse. In the first case, the electron transfer is by quantum tunneling (for a very small gap); and in the second case, effective atomic continuity (ψ overlap) allows electron transfer. The same happens when a conducting external wire connects A and B: electrons readily propagate along, due to their spatially extended ψk Bloch functions. Thus A gets positive and B negative, setting up a difference of electrical Volta-potential ΔV=VA−VB [19]. If ΔN electrons go from A to B (causing charges ΔQA=+eΔN and ΔQB=−eΔN), when the interface capacitance C dominates one has [20]:

(1.21)

The A→B electron transfer rises μ(A) and lowers μ(B), due to the emerging potentials VA (VB), which decrease (increase) all the electron energy levels, by −eVA and +eVB respectively. Thus ∑μ changes to new values :

(1.22)

Equilibrium occurs when , leading to:

(1.23)

For (ϕB−ϕA)~eV one obtains ΔVeq~Volt. Using Eq. (1.21) we can estimate the required ΔN-transfer to obtain equilibrium, ΔNeq=(C/e)ΔVeq. For a flat A–B interface of area S (say, 1 cm²) and a separation δc ~Å (atomic contact), if one assumes capacity C given by such apparent geometry,

(1.24)

one obtains a transfer of ΔNeq~10¹³ electrons. For A and B macroscopic (say 1 cm³; N~10²⁹ in each metal), such electron transfer is very small in relative terms, ΔNeq/N~1/10¹⁶⁹ (see also Section 5.1). However, due to the atomically discrete interfaces, the effective contact surface is much smaller than assumed, giving even smaller capacitances (~aF), and much smaller surface dipole charges. Also, quantum approaches are mandatory, and the classical surface dipole approach fails [25,26]. Further considerations on contact potentials, Fermi level equilibration, and surface charging can be found in Ref. [26]. Surface/interface physics is a rich research field that has a wide range of applications. We just mention photoemission, electrocatalysis, nanoscale devices (0D→3D dimensions), bimetallic nanosystems, microscopic optical and thermal control, nanogranular and multicomponent materials or strain-induced effects [24,26,28,29].

1.6 Measurements of contact potential and of work function

1.6.1 Absolute measurements of the work function

An absolute measurement of ϕ can be performed using the photoelectric effect (Fig. 1.4A) [30]. A light source S emits photons of known frequencies hν>ϕ, striking a metal A under study; plate B collects the photoemitted electrons, and the current i is measured by an electrometer E. By applying a potential difference V between A and B, we can retard the motion of the photoelectrons. At a particular voltage V0, the current drops suddenly to zero, which means that no electrons, not even the fastest ones (in vanishing smaller numbers), are reaching B [29]. Then:

(1.25)

Figure 1.4 (A) Absolute work function ϕ measurement using the photoelectric effect with monochromatic photons (γ; E – sensitive electrometer). (B) Same measurements using several different photon frequencies. Source: Adapted from M. Alonso, A. Finn, Fundamental University Physics, vol. 3, Addison-Wesley, 1968, pp. 11–14.

By changing the photon frequency ν, we obtain different values for the stopping potential V0, falling in a V0(ν) straight line from which one obtains ϕ (Fig. 1.4B). An alternative method to obtain ϕ measures the electric current density j emitted by A when heated to high temperatures (close to melting; thermoionic effect), according to the Richardson law [29].

1.6.2 Relative dc measurements

Consider two separate metals (1,2) with the free-electron energy diagrams shown in Fig. 1.5A, relative to a common vacuum-level ∑, with EF(1)=EF(2) for simplicity¹⁰ and ϕ1>ϕ2. This means conduction band bottom levels at ∑0(1)<∑0(2), for example, due to stronger electron–ion attractions in metal 1.

Figure 1.5 (A) Free-electron energy diagram of metals 1 and 2. (B) An interface-electric field E forms upon atomic contact, over δ=δ1+δ2, giving an electric potential difference ΔV=E·δ ≅ (ϕ1−ϕ2)/e at equilibrium.

When the metals are put in contact (Fig. 1.5B), electrons flow from metal 2 (lower ϕ) to 1. This causes a transient current i(t) through the interface, gradually lowering ∑F(2) and rising ∑F(1) until the Fermi levels equalize and i vanishes. The attractions between excess of electrons in 1 and the ions (fixed) left in 2 prevent electrons from separating too much from the junction. An interface charged region thus forms, extending over the so-called Debye-lengths (δD) at each side [31,32]:

(1.26)

where ne is the corresponding electron concentration, ne=z·na (z=valency; na=atomic concentration=1/a³) and r0=Bohr radius≅0.53 Å. For typical metals, one has δD~Å, whereas in semiconductors (much smaller ne) results δD~μm.¹¹ As seen before (Section 1.5), the equilibrium contact potential is , but it cannot be measured with an ordinary (resistive) voltmeter connected to 1 and 2, for example, by a Cu wire. In fact, extra potentials form at the wire contacts with 1 and 2, giving zero total voltage in such closed circuit:

(1.27)

independent of the wire-metal potential.¹² Equilibrium occurs in a very short time (τ) after closing the circuit, preventing voltmeter readings. Instead, one can use a sensitive ballistic galvanometer (G) to measure the charge transfer during τ:

(1.28)

Knowing the junction capacitance CJ, one would then obtain the contact potential,

(1.29)

However, as contact occurs at atomic scale, the macro-concept of CJ capacitance fails. In practice one uses sizeable parallel plate metals (1, 2), with known areas S and d-separation, giving C=ε0S/d [33]. Consider a wire connecting C in series with G, of metal 0 (also the G-wiring), for example, copper, with work function ϕ0, and ϕ2<ϕ1 for concreteness (Fig. 1.6A). Considering ϕ0<ϕ1,ϕ2, and approximately equal initial numbers of electrons in 1 and 2, with ϕ2<ϕ1 and K-open, more electrons flow from 2→1 than from 1→0, establishing independent electrostatic equilibria. The wire open ends a, b (Fig. 1.6A) are then at different potentials, negative in this example. Closing K, electrons spontaneously go from plate 2 to plate 1 through G, causing a transient current i(t) (Fig. 1.6B) which lasts until all the three Fermi levels equilibrate, ∑F(1)=∑F(2)=∑F(0). Then, one has:

(1.30)

Figure 1.6 (A) Parallel plates of different metals (1, 2) with surface S and separation d, with ϕ1>ϕ2 and external circuit (O) with ϕ0<ϕ1, ϕ2. They form a condenser C in series with a current-detector G and with K-switch open (see text). (B) K is closed and electrons flow from 2 to 1 through G, forming a transient current i(t) which gradually charges 2 [+Q(t)] and 1 [−Q(t)]; a growing difference of potential appears across C: V21(t)=Q(t)/C. Equilibrium occurs (i=0) when , thus . (C) Circuit with an adjustable dc-source (Vs) inserted, with pole (+) connected to plate 2. When Vs is finely adjusted to the value , when K is opened and closed, no i(t) current flows in the sensitive current-detector Gi (see text).

The equilibrium difference of potential in C is and its charge Q is given by:

(1.31)

Since ϕ1, ϕ2 are intrinsic of 1 and 2,¹³ results that Q∝S/d. Measuring Q in G (charge meter) and knowing C, one obtains Vc (contact potential) and ϕ1−ϕ2, the relative work-function. Then, a sudden change in the distance d between plates (Δd), with K closed, leads to a rapid variation of Q [29], given by the integration of i(t) performed by the charge-meter G during the transient time τ, . Then, Eq. (1.31) enables to obtain (ϕ1−ϕ2):

(1.32)

In electrostatic equilibrium, all points inside each conductor (wire and plates) are at the corresponding constant potential; any mobile charges in excess, electrons in negatively charged metals (plate 1 and wire, in text) go, by mutual repulsion, to the surface giving a surface charge density ρs<0 and ρ=0 inside. In the positive case (plate 2), smaller electron numbers are still able to neutralize the ionic charges inside (imposing ρ=0), but they leave an unshielded ionic layer at the surface (ρs>0). In the 1–2 interface, the electron–ion mutual attraction (across junction) causes volume ρ densities (+/−), extending over the corresponding Debye lengths.

A better approach uses an adjustable dc source (Vs), opposing (Fig. 1.6B) and avoiding any Q-measurements [29,32]; instead of a charge-meter, one requires a sensitive current detector (Gi). Adjusting Vs one can nullify i when which turns irrelevant the actual C-value; that is, Vs just compensates the contact potential in the circuit [34]. This is the dc Kelvin method to measure contact potentials and relative differences in work functions. However, it requires sensitive dc-null detection by Gi, which is difficult due to spurious dc-potentials, rectified pickup signals, or thermal electromotive forces (emfs).¹⁴ This problem is solved using an ac-version of the same principle, as analyzed in the next section.

1.6.3 Relative measurements of the work function with the ac Kelvin method

The work function is nowadays routinely obtained from measurements of the contact potential between A and a reference material B (absolute ϕB accurately known), using ac Kelvin methods [35–38]. In particular, Kelvin probe force microscopy achieves very high spatial resolution (down to nm), providing local ϕ-values and key information on structural and electronic behavior in sample surfaces, nanostructures and even at the molecular level [39–45]. One restricts here to the basic ac Kelvin method. Connecting two metals A, B, for example, by a wire (metal C), electron transfer occurs from one metal (with lower ϕ) to the other, until equilibrium is reached when [Eq. (1.23)]. As ϕA (ϕB) are intrinsic of A (B), will also be. For metal plates with surfaces S, placed at a small distance d0 much smaller than the material’s transverse dimensions one has a capacitance C=ε0(S/d0). The constancy of (intrinsic of A–B) implies plate-charges adjusted according to the actual d0 distance between plates:

(1.33)

Fig. 1.7 contains the relevant steps to understand the ac Kelvin method [36]. As shown in Fig. 1.7A, when the initial Fermi level in A is lower than in B, that is, ∑F(A)<∑F(B), implies ϕA>ϕB. When a metallic wire connects A to B (Fig. 1.7B) electrons flow from B to A, turning it more negative and B positive, thus decreasing the ∑F(B)>∑F(A) transfer driving. The charge imbalance opposes and ultimately stops electron transfer (i=0) (eCPD in Fig. 1.7B). Then, the capacity C between A and B is charged with , where C is previously known.

Figure 1.7 The three basic steps to understand the ac Kelvin method; see text [36].

As seen in Fig. 1.7C, spurious dc-effects (Section 1.6.2) are avoided using null ac-detection at ω>>2π/τ, provided one can generate ac-currents in a circuit polarized by a dc-source. This is achieved by putting plate B under harmonic vibration (ω) with amplitude a<<d0, in relation to plate A (fixed), with B attached to an