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Lecture 7 Work Function; Electron Emission

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Lecture 7 Work Function; Electron Emission Lecture 7 Work Function; Electron Emission Outline: 1. Work Function 2. Electron Emission A. Thermionic Emission B. Field Emission C. Secondary Electron Emission 3. Measurements of Work Function References: 1) Zangwill, p.57-63 2) Woodruff & Delchar, pp. 410-422, 461-484 3) Luth, pp.336, 437-443, 464-471 4) A. Modinos, “Field, Thermionic and Secondary Electron Spectroscopy”, Plenum, NY 1984. Lecture 7 1 7.1 Work Function: Uniform Surfaces • The “true work function” e φ of a uniform surface of an electronic conductor is defined as the difference between the electrochemical potential µ of the electrons just inside the conductor, and the electrostatic (− Φ ) potential energy e o of an electron in the vacuum just outside •µ is work required to bring an electron isothermally from infinity to solid − Φ e o ∂G Energy φ µ = (5.1) e ∂n µ e T ,P φ = − Φ − µ µ e e o (5.2) µ − eΦ φ = −Φ − (5.3) I o e distance • Note: µ is function of internal AND surface/external (e.g., shifting charges, dipoles) conditions; • We can define quantity µ which is function of internal state of the solid µ = µ + eΦ Average electrostatic potential inside Chemical potential of electrons: I Lecture 7 2 Work Function • The Fermi energy [E F], the highest filled orbital in a conductor at T=0K, (− Φ ) µ is measured with respect to e I and is equivalent to . • We can write: φ = − Φ + Φ − µ e e o e I (5.4) µ φ = ∆Φ − (5.5) e • ∆Φ depends on surface structure and adsorbed layers. The variation in φ for a solid is contained in ∆Φ . • What do we mean by potential just outside the surface??? Lecture 7 3 Potential just outside the surface The potential experienced by an electron just outside a conductor is: k e e Nm 2 V (r) = − e = − = 99.8 ×10 9 (5.6) πε 4r 16 o C Φ For a uniform surface this corresponds to o in (5.1): V (r) → 0 as r → ∞ [in mV range for r ≥10 3 Å] In many applications, an accelerating field, F, is applied: k e V (r) = −Fr − e (5.7) 4r dV = 0 dr r=r0 e 2/1 1 9.1 ×10 −5 r = k = m (5.8) o e 4 F 2/1 F 2/1 − 2/1 δφ = () 2/1 = × 5 2/1 δφ ∝ F keeF 79.3 10 F Volts ( F in Volts/m) (5.9) = 4 = × −7 = δφ = For F 10 Volts/mLecture (100 7 V/cm), ro 9.1 10 m 1900 Å 48.3 mV Selected Values of Electron Workfunctions* Element φ (eV) Element φ (eV) Element φ (eV) Ag 4.26 Cu 4.65 Si 4.85 Ag (100) 4.64 Cu(100) 4.59 Ru 4.71 Ag (110) 4.52 Cu(110) 4.48 Ta 4.25 Ag (111) 4.74 Cu(111) 4.98 Ta (100) 4.15 Ba 2.52 Ir (110) 5.42 Ta (110) 4.80 C 5.0 Ir(111) 5.76 Ta (111) 4.00 Ce 2.9 K 2.30 Ti 4.53 Cr 4.5 LaB 6 2.66 W 4.55 Cs 2.14 Mo 4.60 Zr 4.05 Units: eV electron Volts; *Reference: CRC handbook on Chemistry and Physics version 2008, p. 12-114. Lecture 7 5 Work Function: Polycrystalline Surfaces Consider polycrystalline surface with “patches” of different Φ i, oi j, … k l workfunction, and different value of surface potential m m o p th Φ At small distance ro above i patch electrostatic potential is oi At distances large w/r/t/ patch dimension: Φ = Φ = o ∑ fi oi , fi fractional area of i th patch i So mean work function is given by: φ = φ e ∑ ief i (5.10) i - at low applied field, electron emission controlled by: eφ - at high field (applied field >> patch field) φ electron emission related to individual patches: e i On real surfaces, patch dimension < 100Å, if ∆φ~ 2 eV then patch field F ~ 2V/(10 -6 cm ) ~ 2 x 10 6 Volts/cm. work required to bring an electron from infinity to solid Lecture 7 6 Workfunction Factors that influence work function differences on clean surfaces: • Adsorbed layers • Surface dipoles (cf. Zangwill, p 57) • Smooth surface: electron density “spillover” • Electron density outside rough surface • For tungsten eφ, eV W plane 5.70 (110) 4.93 (211) 4.39 (111) 4.30 (116) Lecture 7 7 Work function change upon adsorption • Charge transfer at interface: electro positive (K, Na, …) or electro negative (Cl, O, F, …) • Model dipole layer as parallel plate capacitor: nµ − ∆φ = µ - dipole moment [C m]; n - surface charge density [m 2- ]; ε = 85.8 ×10 12 [C /Vm ] ε o o Suppose ∆φ = 1.5V for 1 ×10 15 /cm 2 O atoms on W (100). What is µ? For molecules with a permanent dipole moment: Lecture 7 8 7.2.1Electron Sources: Thermionic Emission Thermionic emission occurs when sufficient heat is supplied to the emitter so that e’s can overcome the work function, the energy barrier of the filament, Ew, and escape from it • Richardson’s Equation : (derivation – aside) φ 2 e Current density, j: j = A 1( − r)T exp( − ) o kT 4πmek 2 Amp A = =120 4. r = reflection coefficient; o h3 cm 2 deg 2 • Richardson plot : ln(j/T 2) vs 1/T ⇒ ⇒ straight line Lecture 7 9 7.2.1 Electron Emission: Thermionic Emission • Richardson plot : ln(j/T 2) vs 1/T ⇒ ⇒ straight line • Schottky Plot φ → Φ − 2/1 e e o bF (cf. eq.5.9) ln j vs F 2/1 ⇒ straight line Lecture 7 10 7.2.2 Field Electron Emission • Electron tunneling through low, thin barrier – Field emission, when F>3 ×10 7 V/cm ~ 0.3 V/Å • General relation for electron emission in high field: ∞ = j e∫ P(EZ , F)v(EZ )dE Z 0 • P is given by WKB approximation 2 3/2 m 2/1 l P = const ×exp − (V − E ) 2/1 dz h ∫ Z 0 1 φ 1 φ 2/3 • If approximate barrier by triangle: ~ φ 2/1 ~ ∫ 2 F 2 F 2 3/2 m 2/1 φ 2/3 P = const ×exp − h F • Fowler – Nordheim eqn, including potential barrier: 2 2/3 2/3 2/1 − F φ f (y) e F j = 54.1 ×10 6 t 2 (y)exp − 83.6 ×10 7 ; where y = φ F φ Lecture 7 11 How do we get high fields: Field Emission Microscope! Get high field by placing sharp tip at center of spherical tube. Mag: R/r ~ 5cm/10 -5cm ~ 500,000 F = cV; c ~ 5/r F ~ 5 x 10 7 V/cm For V = 2,500 Volts. W single crystal wire as tip. Typical pattern on phosphor screen Lecture 7 12 Field Emission Properties Lecture 7 13 7.2.3 Secondary Electron Emission Electrons emitted from surfaces after electron bombardment In general complicated phenomenon involving several interrelated processes Generally classify secondaries into three categories: - (I) Elastic ; - (II) Inelastic; - (III) “true” secondaries (KE < 50 eV) Total coefficient for secondary emission, σ= j2/ji = r + η + δ j2 For metals, max values: r ~ 0.2 (Ep ~ eV); ~ 0.02 (large Ep) η~ 0.3 to 0.4; δ~ 0.5 to 1.8 (Ep ~ few hundred eV) σ j For insulators, can be MUCH higherLecture (~ 20!!!!) 7 1 14 Secondary Electron Emission • Establishment of stable potential for insulators and dielectric materials In practice, steady state potential reached by dielectic is due mainly to incomplete extraction of secondary electrons • For metals and semiconductors: Correlation between δ and density, ρ Lecture 7 15 7.3 Measurements of Workfunction Absolute value of φφφ A. Photoemission φ = hν −W W = full width of energy distribution B. Thermionic Emission (Richardson’s Equation) C. Field Emission Retarding Potential (FERP) Workfunction difference A. Shelton Method B. Vibrating Capacitor – Kelvin Probe http://www.kelvinprobe.info/ Lecture 7 16.
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