# ssp_15 - 5 Continued 5 5.1.3 Thermionic Emission of...

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5-21 5.1.3 Thermionic Emission of Electrons from Metals Heating of metal electron emission no infinite potential well (see Fig. 5.1). Fig. 5.10 (a) schematic drawing of a diode circuit observing thermionic emission of electrons from the heated cathode C (A = anode). (b)Qualitative behaviour of the current-voltage curve at two different temperatures T 1 and T 2 > T 1 . As a consequence of their thermal energy, electrons can even overcome a countervoltage (A negative with respect to C) j s = saturation current: all emitted electrons are accelerated to anode. Work function: φ = E Vac - E F barrier to overcome futher requirement: v > 0 Generalization of j = en v , v = drift velosity (5.31)
5-22 > + > = = k k v E E x x x x F k d k v T k E f k Z V e k v V e j 0 ) ( 3 ) ( ) ), ( ( ) ( ~ ) ( φ (5.32) .... dk k 3 3 ) 2 ( 2 ) ( ~ π V k Z = density of states in k - space. k d k v T k E f e j x k v E E x x F 3 0 ) ( 3 ) ( ) ), ( ( ) 2 ( 2 > + > = (5.33) E k m mv x x x == h 22 2 2 1 2 v k m x x = h = min , 3 ) ), ( ( 4 x k x x z y x T k E f k dk dk dk m e j h (5.34) E - E F > φ >> k T Fermi statistics Boltzmann statistics: 1 1 222 EE kT kT E kT k mkT k mkT k mkT F FF x y z e ee e −+ + + ≈= h h h

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5-23 4 4 4 4 4 4 43 4 4 4 4 4 4 42 1 h 4 4 4 4 1 1 h h h h h h kT E m E mkT k x k kT E mkT k x x mkT k z mkT k y x F F x x F x z y e mkT e dk e e k dk e dk mkT e dk m e j φ π + + = = 2 2 ) ( 2 2 min , 2 2 2 3 2 2 2 2 2 2 1 2 4 j me h kT e x kT = 4 3 2 () (5.35) j x = saturation current j s (Richardson-Duschman equation). Derivation: all e - with E k m E x F =≥ + h 22 2 can escape Quantum mechanical treatment : reflection at potential steps j j kT E transmission reflection F = + << 1 (5.36)
5-24 Effect of external field: Fig. 5.11 Schematic representation of the thermionic emission of free electrons (density n ) from a metal. An electron in the potential well must overcome the work function barrier φ = E vac - E F in order to reach the energy level E vac of the vacuum and escape from the crystal. An important part of the work function is assumed to be the Coulomb potential between the escaping electron andits positive image charge in the metal (image potential). If an electric field is applied, then φ is reduced by an amount ∆φ . Reductions of the work function of 1 eV as shown here can only be achieved with extremely strong external fields of 10 7 - 10 8 V/cm Image potential: Vx e x B () =− 2 0 42 πε E = ( E ,0,0) V E ( x ) = -e Ex , x = 0 : surface, ( F = e E = -gradV) V = V B + V E

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ssp_15 - 5 Continued 5 5.1.3 Thermionic Emission of...

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