EEW508

1. EEW508 V. Electrical Properties at Surfaces • Measurement of work function • Thermionic Emission • Photoemission • Field Emission

2. EEW508 V. Electrical Properties at Surfaces The Fermi-Dirac probability distribution function The electron in the conduction band follow Fermi-Dirac statistics with the probability of occupancy being given by The Bose-Einstein probability distribution function μ is the chemical potential, k is the Boltzmann constant, and T is temperature.

3. EEW508 V. Electrical Properties at Surfaces Density of state curve for the conduction band of a free electron metal The electron in the conduction band follow Fermi-Dirac statistics with the probability of occupancy being given by At T=0K, all of the states for which    will be filled and all states for which  >  will be empty. The value of  corresponding to  =  is called the Fermi energy. At temperature above 0 K, the probability of occupancy of states just above F becomes finite. Surface Science; An Introduction J. B. Hudson (1992)

4. EEW508 V. Electrical Properties at Surfaces Measurement of work function – a. Thermionic emission At high temperature, states with -F > will have a probability of occupancy given by Where - ( +) represents the energy in excess of that needed to reach the vacuum level. Any electron with - ( +) >0 is essentially outside of the crystal and can be collected as a free electron. Such electrons are called thermionic electrons and the process by which they are produced is called thermionic emission.

5. EEW508 V. Electrical Properties at Surfaces Measurement of work function- a. Thermionic emission Richardson-Dushman equation The expression relating thermionic current j to temperature Where A = 4  mk2e/h3 ~ 120 A /cm2 deg2 To obtain work function from a thermionic emission measurement, one plots ln (j/T2) as a function of 1/T Plot of Richardson-Dushman equation For different faces of tungsten

6. EEW508 V. Electrical Properties at Surfaces Thermionic emission at metal-semiconductor Schottky diode assumes that electrons, with an energy larger than the top of the barrier, will cross the barrier provided they move towards the barrier For thermionic emission over the barrier, the current density of Schottky contacts as a function of applied voltage is given by Where F = Area, Φn = Schottky Barrier Height, η = Ideality Factor, Rs = Series Resistance, respectively. A* [Effective Richardson Constant] = 4  m* k2e/h3 The effective mass of the conduction electrons in GaN, m* = 0.22m0, gives an effective Richardson constant A* (GaN) = 2.64 x 104 A/cm2K. The effective Richardson constant for TiO2 is A = 24 A/cm2K. Energy diagram of metal-semiconductor Schottky diode

7. EEW508 V. Electrical Properties at Surfaces Measurement of work function- b. Photoelectron emission Any photon incident on the sample surface can interact with an electron in the solid to transfer its energy to the electron. If this energy is greater than the difference between the initial energy of the electron and the vacuum energy, then the electron may escape from the solid. This process is called photoelectron emission. The ejected electrons are called photoelectrons. If the minimum photon energy, hvo, can cause electron ejection, the work function () is the same with hvo Vacuum level hvo  Fermi level

8. EEW508 V. Electrical Properties at Surfaces Photoelectron emission- Fowler law The total electron current generated in this process is given by the Fowler equation Where B is a parameter that depends on the material involved. If measurements are made of the photoemitted current as a function of photon energy, extrapolation allows the determination of vo, thus 

9. EEW508 V. Electrical Properties at Surfaces Measurement of work function- c. Field emission If a strong electric field is present at the surface with the surface electrically negative wrt the surroundings, the effective surface barrier will be modified with smaller height and width. In this case, a quantum mechanical calculation indicates that electrons can escape through the barrier by quantum mechanical tunneling and appear in the space outside the metal as field-emitted electrons

10. EEW508 V. Electrical Properties at Surfaces Measurement of work function- c. Field emission Fowler-Nordheim equation The current arising from this process is given by the Fowler-Nordheim equation as Where F is the electric field strength, and  is a slowly varying function of F, of the order of unity. A plot of this relation, ln (j/F2) vs (1/F) can be used to determine work function

11. EEW508 V. Electrical Properties at Surfaces Field Emission (Fowler-Nordheim equation) If we change this equation to the more feasible form where,  is work function in eV, I is the emission current in A, V is the applied voltage in V. In this equation, is called the field enhancement factor, and defined as E=  V Likewise, I=JA, where A is the effective emitting area of electron source in cm2.

12. EEW508 V. Electrical Properties at Surfaces Field Emission (Fowler-Nordheim equation) • The range of electrostatic field for field emission is 3 - 7 x 10 7(V/cm). The field E at the surface of a free sphere of radius r and potential V is E = V / r • At the surface of an actual tip, the field is reduced from this value by the presence of the cylindrical or conical shank, is given to a very good approximation by • E = V / kr, where k ~ 5 • In other words, for the sharp field emission source, • is 1/kr and is not significantly dependent on the distance between the tip and anode. • In case of the planar field emission source as like diamond, DLC(Diamond like carbon), and boron nitride,  is inversely proportional to the tip-anode distance.  increases from 3.27 x 105 to 7.98 x 105 cm -1 as the tip approaches an anode from distance of 30 m–500 nm. Park et al. Rev. Sci. Instru. (1999).

13. EEW508 V. Electrical Properties at Surfaces Measurement of work function- c. Field emission Field emission microscopy The contrast observed in FEM arises from the variation in work function with crystallographic orientation over the tip. Field emission microscopy image of a clean Rh(111) This technique along with F-N plot, allows the measurement of the variation of work function with orientation. FEM(field emission microscopy) has been used to measure adsorption and surface diffusion processes.

14. VI. Mechanical Properties at Surfaces

15. EEW508 VI. Mechanical Properties at Surfaces Mechanical properties of a surface • Static properties: hardness, young’s modulus, yield strength and adhesion – how the surface deforms when it is indented or stretched by external load • Dynamic property: friction – how the surface resists the relative motion against another surface • Questions on • Why materials are hard or soft? • Why some surfaces are slippery while others are not? • How surfaces deform?

16. EEW508 VI. Mechanical Properties at Surfaces tensile stress and strain In a tensile test, a specimen with the initial length l0 and cross sectional area A0 is subjected to a tensile load, L. The length of specimen, l, increases as the load increases. The tensile stress, ,is defined as The strain ,  , is defined as

17. EEW508 VI. Mechanical Properties at Surfaces The Poisson effect When a sample of cube of a material is stretched in one direction, it tends to contract in the other two directions perpendicular to the direction of stretch. This phenomenon is called the Poisson effect Poisson’s ratio  characterize Poisson effect Where x is the transverse strain (normal to the applied load) and y is the axial strain (in the direction of the applied load)

18. EEW508 VI. Mechanical Properties at Surfaces The Poisson’s ratio of materials For most materials, the stretch in one direction causes the contraction in transverse directions  positive Poisson’s ratio Some materials like cork shows very little transverse expansion when stretched or compressed  Poisson’s ratio very close to zero Some materials such as polymer foams have a negative Poisson’s ratio Because the cross sectional area, A, changes under the tensile load (Poisson effect), the true tensile stress is defined as

19. EEW508 VI. Mechanical Properties at Surfaces A typical stress-strain curve for ductile materials • Elastic deformation: atoms are slightly displaced away from their equilibrium position. Young’s modulus, E, (or elastic modulus) describes this region. E = d  / d  • Once the stress is increased beyond the yield point, the specimen will undergo the plastic deformation. The stress at the yield point is called yield strength y • (3) Stain hardening: the tensile stress needs to increase appreciably for additional strain to occur. The maximum stress reached in the strain-hardening region defines tensile strength (or ultimate strength) of materials, u • (4) Necking: necking occurs with the cross-sectional area in a localized region of the specimen decreasing rapidly. • (5) Fracture : the specimen will rapture at the fracture stress

20. EEW508 VI. Mechanical Properties at Surfaces A typical stress-strain curve for Brittle materials brittle materials fracture at much lower strains, but often have relatively large elastic moduli and ultimate strength in comparison to ductile materials Schematic appearance of round metal bars after tensile testing. (a) Brittle fracture (b) Ductile fracture (c) Completely ductile fracture

21. EEW508 VI. Mechanical Properties at Surfaces Dislocation and plastic deformation Dislocation : one-dimensional defects in materials Plastic deformation is carried out by the movement of dislocation in materials. There is compressive stress in the upper region of the dislocation while the tensile region in the lower region. As the external stress increases beyond the yield strength of a material, more dislocation will be generated. At the certain concentration of dislocation, dislocations will hinder the further movement of each other. To cause further plastic deformation, increase of the external stress is needed. Edge dislocation

22. EEW508 VI. Mechanical Properties at Surfaces Hall-Petch relation (d> 1m) In polycrystalline materials, small size grains with crystalline structures are surrounded by grain boundaries. The grain boundary serves as an obstacle to the movement of dislocations so that the number of dislocations piled up in the grain. For a small grain, a larger external stress is needed to generate the same number of dislocations as that in a larger grain. Hall-Petch relation: the grain size dependence of the yield strength Where 0 , k are material constants and d is the grain size. This relation is effective polycrystalline materials with the grain size of mm~micrometer Hall-Petch relation for 99.999% Al

23. EEW508 VI. Mechanical Properties at Surfaces Inverse Hall-Petch effect (d< 1m) Combined Hall-Petch plot for Cu from different sources. When the grain size is less than 25 nm, the inverse Hall-Petch relation was reported by Chokshi et al. For nanostructured materials, plastic deformation is mainly carried out by the sliding of the grain boundries.

24. EEW508 VI. Mechanical Properties at Surfaces Indentation • When the surface is indented by a hard indenter • Small load – elastic deformation – no bond breaking, thus reversible process • High load- plastic deformation – chemical bonds in the close vicinity of the indenter tip begin to break - permanent deformation Indenter profile left after the indentation by a square diamond indenter (Vickers test)

25. EEW508 VI. Mechanical Properties at Surfaces Hardness The indentation hardness, H, is given by H=P/A where P is the applied pressure, and A is the surface area of indent. The indenter is either the Vickers diamond pyramid (Vickers test) or the Knoop elongated diamond pyramid (Knoop test) Vickers indenter penetrates about twice as deep as Knoop indenter Pa = N/m2

26. EEW508 VI. Mechanical Properties at Surfaces More information http://www.memsnet.org/material/ Macroscopic hardness measured by the tests with large indenters is usually much lower than that intrinsic hardness for single crystal. One reason for this discrepancy is that most materials have lots of defects inside bulk. The plastic deformation in the macroscopic indentation tests usually initiates at those defect sites. The macroscopic hardness depends on the defect density of a material.

27. EEW508 VI. Mechanical Properties at Surfaces Study of friction – tribology (‘tribo’ means ‘to rub’) Surfaces in contact and moving parallel to each other exhibit phenomena that are encountered in our everyday life. These phenomena include friction, slide, wear, and lubrication (the process to modify or reduce friction). The consequences of friction and wear have enormous economic impact, and are of great concern from both a national-security and quality-of-life point of view. improved attention to friction and wear would save developed countries up to 1.6% of their gross national product, or over $100 billion annually in the U.S. alone. ‘‘Tribology: Origin and Future,’’ H.P. Jost, Wear 136 , 1–17 (1990).

28. EEW508 VI. Mechanical Properties at Surfaces Static friction (Stiction) of nanoscale objects Friction, adhesion, and wear are important issues at nanometer scale because of large surface area-to-volume ratio Vital issues in reliability of MEMS or NEMS devices. MEMS: microelectromechanical systems NEMS: nanoelectromechanical systems

29. EEW508 VI. Mechanical Properties at Surfaces History of Tribology Egyptians using lubricant to aid movement of Colossus, El-Bersheh, circa 1800 BC The use of a sledge to transport a heavy statue By Egyptians, and there is one man pouring the liquid 60 tons – 172 people.

30. EEW508 VI. Mechanical Properties at Surfaces Amanton’s Law static friction force :The force needed to beovercome in order to initiate the sliding of an object Kinetic static force: the force that must be overcome in order to maintain the sliding of an object over a surface (Static friction force) > (kinetic friction force) Leonardo Da Vinci’s scheme to test friction Where  is the friction coefficient Over a wide range of loads, the friction force is independent of the contact area between the object and the surface. Proportionality : first Amanton’s Law of Friction The independency: second Amanton’s Law of Friction These laws were discovered by Leonardo da Vinci and then rediscovered by Guillanume Amantons

31. EEW508 VI. Mechanical Properties at Surfaces Friction force is proportional to the contact area (by Bowden and Tabor) The real contacts formed at the interfaces of two surfaces The real contact area is much smaller than the apparent contact area Because the adhesion between two surfaces increases with the contact area, the friction increases with the contact area Bowden and Tabor points out that the friction force is proportional to the ‘real’ contact area Combining this with Amanton’s Law

32. EEW508 VI. Mechanical Properties at Surfaces Why is the wet surface more slippery than the dry surface? Tire loses 20-30 % of its tire-road friction on the wet road. The main reason causing the friction reduction is sealing water in the small valley, effectively reducing the real contact area between the road and the tire. Likewise the main reason that the static friction coefficient (s) is larger than kinetic friction coefficient (k) is reduction of the real contact area of dynamic parts.

33. EEW508 VI. Mechanical Properties at Surfaces Friction is energy dissipation of mechanical energy In order to keep an object at a constant velocity, v, over a surface, work must be performed continuously to the object. The work done in a unit of time is given by In most cases, the input work dissipates as heat at the interface. The friction heating can be applied to lighten a match, to weld two lower-melting-point materials, etc. Another mechanism of energy dissipation in friction is excitation of electron-hole pairs.

34. EEW508 VI. Mechanical Properties at Surfaces Stick-slip motion the static friction coefficient (s) > kinetic friction coefficient (k) This difference can lead to the stick-slip motion when the velocity v of the spring end B is small. The force exerted by the spring on the slider, Fp, can be obtained from the length of spring. In region 1, the pulling force Fp is smaller than the static friction force, s L and the slider sticks on the initial position In region 2, Fp > s L, the slider begins to slip. If Fp is smaller than k L, the slider begins to slow down. At point b, the slider stops. In region 3, the slider sticks on the surface (a) An object is pulled by a spring. Ff is the friction force. (b) The pulling force Fp vs the displacement of B. stick-slip motion of the object is indicated by the sudden change of the pulling force

35. EEW508 VI. Mechanical Properties at Surfaces Comparison between macrotribology and nano/microtribology Macrotribology Large mass Heavy load Wear (inevitable) Bulk, materials Nano/microtribology Small mass Light load No wear (few atomic layers) surface

36. EEW508 atomic/friction force microscopy VI. Mechanical Properties at Surfaces Tools for tribological study AFM and pin-on-disk tribometer on the sample specimen

37. EEW508 VI. Mechanical Properties at Surfaces Principle of friction force microscopy Stick <-- slip  AFM invented by Binnig, Quate, and Gerber in 1986 AFM has a sharp tip with a radius between 10-100 nm, and the resolutions for the displacement and force sensing can be up to 0.01 nm and 0.1 pA.

38. EEW508 VI. Mechanical Properties at Surfaces Topographical and friction images of SAM molecules on silicon n type silicon C16 silane AFM topography friction SAM molecules are common lubricating materials to reduce friction on silicon devices

39. EEW508 Friction at the single asperity Friction at the Macroscopic scale VI. Mechanical Properties at Surfaces Frictions at the different scale (nano versus macroscale) Single asperity Real contact AFM DMT: Derjaguin-Müller-Toporon JKR:Johnson, Kendall and Roberts

40. EEW508 VI. Mechanical Properties at Surfaces v F L Contact area(A) An object moving over a surface with a sliding velocity v. The friction force F is in the direction opposite to the sliding velocity v of the object