MTLE-6120: Advanced Electronic Properties of Materials Metal-vacuum - PowerPoint PPT Presentation

1 MTLE-6120: Advanced Electronic Properties of Materials Metal-vacuum junctions: thermal and field emission Reading: ◮ Kasap 4.9 ◮ Review Kasap 3.1.2 (Photoelectric effect)

2 Photoelectric effect ◮ Light ejects electrons from cathode ⇒ I at V = 0 ◮ V ↑⇒ I ↑ till saturation (all ejected electrons collected) ◮ V ↓⇒ I ↓ till I = 0 : all electrons stopped at V = − V 0 Light ◮ Increase intensity I : higher saturation I but same stopping V I ◮ Increase frequency ω : V higher stopping V ◮ Stopping action: eV 0 = KE max I ◮ Experiment finds eV 0 ∝ ( ω − ω 0 ) ◮ In fact eV 0 = � ( ω − ω 0 ) ◮ Different cathodes ⇒ different ω 0 V but same slope � identical to that from Planck’s law! ◮ Light waves with angular frequency ω behave like particles (photons) with energy � ω (Einstein, 1905)

3 Workfunction: energy level alignment with vacuum ◮ Minimum energy Φ required to free electron from material ◮ Photoelectric effect threshold is � ω 0 = Φ ◮ Electrons emitted with kinetic energy KE = � ω − � ω 0 ◮ Determined by alignment of energy levels across metal-vacuum interface Empty Empty Empty Empty Filled No states No states Filled Metal Vacuum Metal Vacuum

4 What determines workfunction? ◮ Electron binding in bulk material (stongly bound ⇒ higher Φ ) ◮ Equally important: surface of the metal i.e. metal-vacuum interface ◮ Energy-level alignment sensitive to details of the surface ◮ Example: work functions (in eV) of single crystalline metal surfaces Metal (110) (100) (111) Polycrystalline Al 4.06 4.20 4.26 4.1 − 4.3 Au 5.12 5.00 5.30 5.1 − 5.4 Ag 4.52 4.64 4.74 4.3 − 4.7 Cu 4.48 4.59 4.94 4.5 − 5.1 ◮ Values for polycrystalline metals averaged over facets (whose relative prominence depends on sample preparation)

5 Thermionic emission ◮ Overcome energy difference (barrier) using thermal energy ◮ Number of electrons above barrier: � ∞ � ∞ d Eg ( E ) exp − ( E − E F ) d Eg ( E ) f ( E ) ≈ k B T E F +Φ E F +Φ (assuming Φ ≫ k B T , which holds for metals even at T melt ) ◮ Can all these electrons cross? ◮ Need KE towards surface m ( v cos θ ) 2 > E F + Φ 2 ◮ Current density per state: � ev cos θ � = ev (1 − E F +Φ ) E 4 Metal Vacuum

6 Richardson-Dushman equation ◮ Current density of emitted electrons: � ∞ · ev (1 − E F +Φ d Eg ( E ) exp − ( E − E F ) ) E j = k B T 4 E F +Φ √ � √ � 3 ◮ Assuming Φ ≫ k B T and free-electron g ( E ) = 4 π 2 m E : 2 π � j = 4 πemk 2 T 2 exp − Φ B (2 π � ) 3 k B T � �� � B 0 with Richardson-Dushman constant B 0 ≈ 1 . 20 × 10 6 A/(mK) 2 ◮ Additional consideration: electrons with sufficient KE can still be reflected ◮ Include energy-dependent reflection coefficient in above consideration ◮ Modified B e � B 0 / 2 for most metals, ≪ B 0 for some d -metals (why?)

7 Electron near a metal surface ◮ Metal surface at constant potential; electric field normal ◮ Electric field outside as if due to charge and its reflection q ( � r − z ˆ z ) q ( � r + z ˆ z ) E ( � r ) = z | 3 − 4 πǫ 0 | � r − z ˆ 4 πǫ 0 | � r + z ˆ z | 3 ◮ Force on charge: − q 2 ˆ z � F = 4 πǫ 0 (2 z ) 2 ◮ Potential energy: � z − q 2 � U = − F · ˆ z = 16 πǫ 0 z ∞

8 Schottky effect Metal Vacuum Metal Vacuum ◮ Image charge effect changes energy level diagram (horizontal axis is now distance from interface) ◮ What is the energy barrier for electrons at E F ? ◮ Now consider an applied electric field E ◮ Net minimum energy level of electron is now: � e 2 e 3 E E min ( z > 0) = E F + Φ − 16 πǫ 0 z − e E z ≤ E F + Φ − 16 πǫ 0 √ ◮ Barrier reduced to Φ − β s E with Schottky coefficient � � e 3 / (16 πǫ 0 ) ≈ 3 . 79 × 10 − 5 eV/ β s = V / m

9 Field emission ◮ Electric field reduces effective barrier for electron emission ◮ Still use thermal energy, but with a lower barrier ⇒ use lower T ◮ Technically field-assisted thermionic emission ◮ Use sharpened metal tips / nanowires / nanotubes to enahance local E ◮ So far, considered electrons thermally excited across barrier ◮ Will there be a current at T = 0 ? Metal Vacuum

10 Fowler-Nordheim tunneling ◮ Consider very strong electric field E ; neglect Schottky effect ◮ Minimum energy of electron in vacuum E min ( z ) ≈ E F + Φ − e E z ◮ Electrons in metal with energy E < E F have less than minimum energy for 0 < z < E F +Φ − E e E ◮ Tunneling probability, accounting for z -KE: √ � � 3 / 2 E F + Φ − p 2 � � − 4 2 m z T ( p z ) ≈ exp − 2 d z 2 mE min ( z ) − p 2 2 m z ≈ exp � 3 e � E (based on the semi-classical WKB approximation for wavefunctions) ◮ Tunneling current: � � e 3 d � p ep z 16 π 2 � Φ E 2 exp 4 2 mφ 3 j = m T ( p z ) ≈ (2 π � ) 3 3 e � E p<p F ◮ Identical dependence with E , as thermionic emission had with T (even though one strictly classical, other quantum mechanical)