ECE606: Solid State Devices Lecture 17 Schottky Diode Gerhard Klimeck [email protected] Klimeck – ECE606 Fall 2012 – notes adopted from Alam 1 Presentation Outline 1) Importance of metal-semiconductor junctions 2) Equilibrium band-diagrams 3) DC Thermionic current (simple derivation) 4) Intermediate Summary 5) DC Thermionic current (detailed derivation) 6) AC small signal and large-signal response 7) Additional information 8) Conclusions Reference : Semiconductor Device Fundamentals, Chapter 14, p. 477 Klimeck – ECE606 Fall 2012 – notes adopted from Alam 2 Metal-semiconductor Diode Symbols M N ND Metal Wire N Klimeck – ECE606 Fall 2012 – notes adopted from Alam 3 Applications of M-S Diode …. STM (scanning tunneling microscope) Detectors on semiconductor CNT (carbon nanotube) www.fz-juelich.de/ibn/index.php?index=674 Original Bipolar Transistors Originally, Gelena (PbS), Si as semiconductor and Phospor Bronze for metal (cat ’s whisker) Klimeck – ECE606 Fall 2012 – notes adopted from Alam 4 Presentation Outline 1) Importance of metal-semiconductor junctions 2) Equilibrium band-diagrams 3) DC Thermionic current (simple derivation) 4) Intermediate Summary 5) DC Thermionic current (detailed derivation) 6) AC small signal and large-signal response 7) Additional information 8) Conclusions Klimeck – ECE606 Fall 2012 – notes adopted from Alam 5 Topic Map Equilibrium DC Small Large Circuits signal Signal Diode Schottky BJT/HBT MOS Klimeck – ECE606 Fall 2012 – notes adopted from Alam 6 Drawing Band Diagram in Equilibrium… ∇•= −++ − − D qpnN( D N A ) Equilibrium ∂n 1 = ∇•J −r + g ∂t q N N N =µ E + ∇ J Nqn N qD N n DC dn/dt=0 Small signal dn/dt ~ jωtn ∂p 1 =− ∇•J −r + g Transient --- full sol. ∂t q P P P =µ E − ∇ J Pqp P qD P p Klimeck – ECE606 Fall 2012 – notes adopted from Alam 7 Band-Diagram 1. E F flat in equilibrium 2. E C/E V in Metal 3. Vacuum level in Metal N D 4. Ec/Ev in semiconductor 5. Vacuum level in semiconductor 6. Connect vacuum levels Vacuum level Φ 7. duplicate the connections M χ down to Ec/Ev EC EF Since N A is large, xp is negligible .. EV = NxAp Nx Dn Charge(metal) = charge(semiconductor) Klimeck – ECE606 Fall 2012 – notes adopted from Alam 8 Built-in Potential: bc @Infinity neglect qV bi Φ M χ Φ B ∆ ∆ + χ + = Φ =(Φ−χ ) − ∆ ≡ Φ − ∆ qV bi M qV bi M B N non- =Φ − k T ln D B B degenerate NC Klimeck – ECE606 Fall 2012 – notes adopted from Alam 9 Analytical Solution (Simple Approach ) Depletion Approximation E ρ qN D integrate Actual Xn x x Xn Q qN x E =D = − D n max ε ε QD = Area negligible Ks0 K s 0 QM =-QD NDXn integrate φ φB q Φ =qV E max bi C x Xn EV E x φ = − max n x max Xn 2 Notice how we neglect xp here, is Klimeck – ECE606 Fall 2012 – notes adopted from Alam there a proof? 10 Complete Analytical Solution + qN x E ()0 = D n ε Charg ks 0 e − qN M xp + x E ()0 = ? n ε - ks 0 Position x = p ⇒ ND xn NM xp E-field − + Position E(0) x E ( 0 ) x qV =n + p bi 2 2 Potentia 2 l qN x 2 qN x =D n + M p ε ε 2ks0 2 k s 0 Position Klimeck – ECE606 Fall 2012 – notes adopted from Alam 11 Depletion Regions ND xp xn ε ε = 2ks 0 NM 2 k s 0 1 NDxn N M xp x =V → V n ()+ bi bi qNNNDMD qN D ∞ 2 2 NM qN x qN M xp qV =D n + 2k ε N bi 2k ε 2 k ε x =s0 D V → 0 s 0s 0 p ()+ bi q NMN M N D This is why we can neglect xp Klimeck – ECE606 Fall 2012 – notes adopted from Alam 12 Presentation Outline 1) Importance of metal-semiconductor junctions 2) Equilibrium band-diagrams 3) DC Thermionic current (simple derivation) 4) Intermediate Summary 5) DC Thermionic current (detailed derivation) 6) AC small signal and large-signal response 7) Additional information 8) Conclusions Klimeck – ECE606 Fall 2012 – notes adopted from Alam 13 Topic Map Equilibrium DC Small Large Circuits signal Signal Diode Schottky BJT/HBT MOS Klimeck – ECE606 Fall 2012 – notes adopted from Alam 14 Band Diagram with Applied Bias… ∇•= −++ − − D qpnN( D N A ) Band diagram … ∂n 1 = ∇•J −r + g This works for doping-modulated ∂t q N N N Semiconductors. J =qnµ E + qD ∇ n N N N Does not work for heterostructures ∂p 1 (when the conduction band is not =− ∇•J −r + g ∂t q P P P continuous) =µ E − ∇ J Pqp P qD P p Metal-Semiconductor is a HS Need theory of thermionic Klimeck – ECE606 Fall 2012 – notes adopted from Alam emission 15 Depletion Regions with Bias ND xp xn 2kε N 2 k ε 1 x=s0 M V → s 0 () VV − n ()+ bi bi A qNNNDMD qN D Forward bias: x decrease 2kε N n x=s0 D V → 0 Reverse bias: x increase p ()+ bi n q NM N M N D Klimeck – ECE606 Fall 2012 – notes adopted from Alam 16 Band-diagram with Bias Forward Bias qV bi -V EC-EF VA qV bi EC-EF Reverse Bias VA EC-EF Klimeck – ECE606 Fall 2012 – notes adopted from Alam 17 I-V Characteristics N-type Forward Bias VA Metal Ec I Semi. EF EF M ESF 1,2,3 EV 4 – generation current I 5 – impact ionization Reverse Bias 6,7 EF M 4 1 – diffusion control 5 Ec 2 – ambipolar EF 3 – resistance drop ESF 6 – defects – trap assist 7 – Esaki EV Klimeck – ECE606 Fall 2012 – notes adopted from Alam 18 Current Flow Concept Barrier Barrier smaller Partition problem in The same! M<=S increased M=>S current qV -V 2 currents bi A E -E Φ C F unchanged! B M=>S S<=M VA Forward Bias qV Φ bi B Barrier EC-EF The same! Barrier bigger M=>S current M<=S decreased Φ unchanged! B q(V bi +V A) VA In Equilibrium: M=>S current EC-EF and Independent Reverse Bias are equal of bias! Klimeck – ECE606 Fall 2012 – notes adopted from Alam 19 Left Boundary Condition We use thermionic emission q(V bi -VA) because: EC 1. EC is not continuous here V 2. There’s no minority carriers! A J( V ===0) 0 J (0) − J (0) T A m→s s→ m (detailed balance) = ⇒ Jm→ s (0) J s→ m (0) J (V ) =J( V ) − J( V ) T A m→ s A s→ m A M=>S current = − independent of bias Jm→s (0) J s→ m(V A ) = − JVTA() J sm→ (0) JV smA → () Klimeck – ECE606 Fall 2012 – notes adopted from Alam 20 Semiconductor to Metal Flux V n −q bi = −s kT υ = J s→ m (0) q e th Jm→ s (0) J → (0) = m s 2 V− V Φ −q bi A q B n n − s kT m kT J→ ( V ) = − q e υ J → (V ) = − q e υ s m A th m s A 2 th M 2 n υ − qV bi qV A Only the ones = −s th kT × kT Only half of above the q e e them goes to 2 barrier goes to qVA the right the right = kT J s→ m (0) e qVA q(Vbi -VA) = kT Jm→ s (0) e VA EC-EF Φ n υ − qB qV A = − qm th M ekT e kT 2 −q Φ Φ qV qn υ mB A =− =m th M kT kT − JT J s→m(0)J s → m ( V A ) e e 1 Klimeck – ECE606 Fall 2012 – notes adopted from Alam 2 21 Diffusion vs. Thermionic Emission ∆n Fn Fp Check that both gives the same result for a diode… Thermionic Emission theory is a more general approach, can be used for device so small that no scattering F n occur (can’t use diffusion!). Fp Klimeck – ECE606 Fall 2012 – notes adopted from Alam 22 Intermediate Summary Schottky barrier diode is a majority carrier device of great historical importance. There are similarities and differences with p-n junction diode: for electrostatics, it behaves like a one-sided diode, but current, the drift-diffusion approach requires modification. The trap-assisted current, avalanche breakdown, Zener tunneling all could be calculated in a manner very similar to junction diode. Klimeck – ECE606 Fall 2012 – notes adopted from Alam 23 Presentation Outline 1) Importance of metal-semiconductor junctions 2) Equilibrium band-diagrams 3) DC Thermionic current (simple derivation) 4) Intermediate Summary 5) DC Thermionic current (detailed derivation) 6) AC small signal and large-signal response 7) Additional information 8) Conclusions Klimeck – ECE606 Fall 2012 – notes adopted from Alam 24 Energy Resolved Thermionic Flux 1 Energy < m*υ 2 Occupation 2 min ~ DOS (non-denegerate) will be bounced back −υ ∞ ∞ min Ω −() − β J= − qdk dkedk E E F υ s→m ∫ ∫ ∫ π 3 xy z x −∞ −∞ −∞ 4 Only movement in the υ x direction will contribute min to the current flux q(Vbi -VA) VA EC-EF x Klimeck – ECE606 Fall 2012 – notes adopted from Alam 25 Thermionic Flux from Semi to Metal .. E ∞ ∞ −υ min E Ω −()− β C E E F E J= − q dk dk dk e υ F sm→ ∫ ∫ ∫ π 3 xyz x −∞ −∞ −∞ 4 1 1 1 E− E =()EE− + ()EE−= mmm*2υ + *2 υ+ *2 υ + () EE − F CCF 2 xyz 2 2 C F ∞ ∞ −υ ** * (mυ 2+ mυ 2+ m υ 2 ) min υ υ υ − x y z β −()− β Ω dm( x) dm( y) dm( z ) = qe Ec E F e 2 υ ∫ ∫ ∫ π 3 x −∞ −∞ −∞ 4 ℏ ℏ ℏ 3m *υ 2 υ 2 υ 2 * ∞−y β ∞ − m* z β −υ − m* x β qΩ() m min −()E − E β 2 2 2 J= ec F ededυ υ de υ υ s→ m π 3ℏ 3 ∫y ∫ z ∫ x x 4 −∞ −∞ −∞ Klimeck – ECE606 Fall 2012 – notes adopted from Alam 26 Thermionic Current … 2q υ =()V − V min m* bi A 3m *υ 2 υ2 υ2 * ∞−y β ∞ − m* z β −υ − m* x β qΩ() m min −()E− E β 2 2 2 J= ec F ededυ υ de υυ s→ m π 3 3 ∫y ∫ z ∫ x x 4 ℏ −∞− ∞ −∞ 1 − − β π π e q(Vbi V A ) 2 * 2 π ()− − β β β 4 qm k 2 EF E C qV bi qVAq V A J→ = Te eAe = s m h3 0 β =−=qVA − JJT sm→ J ms → Ae0 ( 1) Klimeck – ECE606 Fall 2012 – notes adopted from Alam 27 Some insight of the Thermionic Current … Compare to the current of p-n diodes… β =−=qVA − JJT sm→ J ms → Ae0 ( 1) Schottky diode 2 2 D n Dp n β =−n i +i qV A − JT q () e 1 p-n diode Wp N A Wn N D Both of them depends exponentially on V A, however current of p-n diodes depends more on temperature (since ni depends strongly on Eg), where the Schottky diode doesn’t have that dependence.