Lecture contents. Metal-semiconductor contact

Transcription

1 1 Lecture contents Metal-semiconuctor contact Electrostatics: Full epletion approimation Electrostatics: Eact electrostatic solution Current Methos for barrier measurement

2 Junctions: general approaches, conventions 2 metal-semiconuctor contact p-n homojunctions heterojunctions => - Formation of potential barriers - Different from bulk material Formalism inclues the following phenomena: Electrostatics (Gauss law) 0 0 replace F cm Poisson equation SI CGS Continuity equations: n Gn Rn 1 t q J n p Gp Rp 1 t q J p Current equations Drift an iffusion currents Einstein relation (in non-egenerate semiconuctor) Thermionic current Tunneling current J J n p nqn qd nq qd p n p kbt D J nq k T n q n n B n p

3 Metal an semiconuctor: Schottky approach 3 q M 4. 75eV Two systems are isolate from Muller, Kamins 2003 E 0 M S : The Vacuum level. It represents the energy of a free electron : work function of the metal (constant of the material) : work function of the semiconuctor (epens on oping) : electron affinity (constant for semiconuctor) Consier the case where M S, an the two materials come in contact. On average, the electrons in the metal will tipically have lower energy than in the semiconuctor (lower E F ). Thus there will be a transfer of electrons from the semiconuctor into the metal (holes are ignore).

4 Formation of metal/semiconuctor interface 4 Ban iagram before contact establishe Charge reistribution at contact Ban iagram of M-S contact (Schottky) Barrier height Built-in potential From Van Zeghbroeck, 1996

5 Formation of metal/semiconuctor interface: Fermi-level pinning (Cowley-Sze) 5 Ban structure before contact: electrons trappe in the interface states create epletion zone an ban bening Ban structure after contact: If trap ensity is very high the alignment of the Fermi levels will be accomplishe by the transfer of electrons from the traps into the metal, instea of from the semiconuctor into the metal: q E q B G 0 (1) The trap ensity is finite, therefore, height of the potential barrier is somewhere between the (1) an Schottky moel: q B q( ) M From Colinge & Colinge 2005 M-S contact properties are etermine by potential variation in the semiconuctor (not metal) Usually ifference in work-functions oes not etermine the contact barrier ue to eistence of interface states (Fermi level pinning) this behaviour is technology epenent! However, the basic formalism for electrostatics currents is still vali

6 Metal-semiconuctor contact 6 M-S barrier height of n-gaas vs. work function of metal In GaAs there is harly a epenence of M-S contact barrier properties on metal Most of the barrier are within 0.2 ev though metal work functions are within 0.8 ev From Murakami, 1993

7 Schottky-barrier heights 7 From Sze, 1981

8 Electrostatics: Full epletion approimation 8 Charge ensity in MS contact We consier semiconuctor fully eplete up to (onors are ionize, no electrons) Integrate charge ensity => fiel qn () s 2 2 qn s Integrate fiel => potential (built-in + applie) qn ( ) i Va 2 s 2 Integrating from 0 to : From Van Zeghbroeck, 1996

9 Applying potential 9 Depletion with Capacitance (per unit area) Compare to the Debye length: L D qn kt q 1 2 Ban iagram Potential (built-in + applie) From Van Zeghbroeck, 1996

10 Charge ensity in general case Electrostatics: eact solution 10 In n-type semiconuctors without acceptors an holes, consiering zero potential eep into nonegenerate an fully ionize semiconuctor : Poisson equation yiels: Solution (fiel vs. potential): Compare to full epletion approimation: Capacitance:

11 Electrostatics: eact solution 11 Capacitance Full epletion approimation: Depletion with (effective): 2 V V i Va qn i a t 2 qn Numerical solution neee for () calculation Fiel an potential may be also taken from full epletion approimation V a qn qn 2 2 From Van Zeghbroeck, 1996

12 Electrostatics: eact solution 12 Results of numerical solution From Van Zeghbroeck, 1996

13 Applicability: Diffusion an rift are vali if concentration is not changing at a mean-free path: This requirement is stronger than Current: iffusion theory: 1 v th Drift-iffusion ominates in low-ope low-mobility semiconuctors m m n n e i 1 k T B i kbt e n ; n 13 Jrift J iff in equilibrium Electron flues currents are opposite Forwar bias Reverse bias Applying rift-iffusion equation in the semiconuctor The current: from Muller, Kamins 2003

14 Current: iffusion theory: 2 14 The rift/iffusion current: ep Multiplying both sies by, an integrating from 0 to : Bounary conitions for electron ensity an potential: kt q q q n kt kt kt n n J e qn e qd e q q q kt kt kt n ( ) n J e qd ne qd ne Using also: kt D e qn ( ) V 2 2 i a s n N C e ( E E C f )/ KT ( ) 0 n(0) N C e q / KT B (0) 0 from Muller, Kamins 2003 n( ) N N C e q / KT n ( ) i V a B n V a

15 Current: iffusion theory: 3 15 Integrating: Estimating enominator: Assuming J q q( V ) qb qb qv kt kt kt kt kt qdn NC e e e qdn NCe e q q kt kt n B n a a ( ) ( 1) e e 0 0 From electrostatic solution: 2 ( ) V qn qn qn i a 2 i a 2 2 V i a V kt Leaving the linear ominant term s s s q 2( i a) q V q 2( i Va) kt kt kt kt kt e e ( e 1) 2 q( V ) 2 q( V ) 0 0 i a i a V 2 2 s i a qn Finally for the current ensity: with saturation current J qva J SDep 1 kt J SD 2 2qN n c i Va q B q D N kt s ep kt

16 Current: iffusion theory: 4 The saturation current can be written as a rift current at the metalsemiconuctor interface : qb qva qv kt kt kt ma ma J qn e ( e 1) q n ( e 1) C a 2 2 s i a ma Using also: V qn qn s Schottky ioe rift-iffusion current can be written slightly ifferently, given that J SD epens on V a Log-Linear Plot for Al/Si ioe 2 qb qva qva q Dn NC 2 qn ( i Va ) kt kt ' nkt S kt s J e ( e 1) J ( e 1) where n ieality factor is a fitting parameter (on t mi it up with concentration!)

17 Current: thermionic theory: 1 17 Thermionic emission ominates in semiconuctors with high mobility an high oping. Current is ue to electrons with energy higher than the barrier. Although the potential barrier is larger than kt/q at room temperature there eists a non-zero probability that some electrons gather enough energy to overcome the barrier. Current from semiconuctor to metal: v E Electron ensity After substituting non-egenerate 3D-DOS an averaging, current from semiconuctor to metal: q B J s m A* T 2 ep kt with Richarson constant 2 4 qm* k m* A A* 120 h m cm K qv ep kt a

18 For non-egenerate semiconuctor ensity of states times istribution function is: Current: thermionic theory: 2 18 minimal velocity of an electron in the quasi-neutral n-type region to cross the barrier From Van Zeghbroeck, 1996

19 Current: thermionic theory: 3 19 Current from metal to semiconuctor oes not epen on applie voltage an at zero bias equals to Total current: J 2 q B Jms J sm Va 0 A* T ep kt qva J ST ep 1 kt with J ST A* T 2 ep qb kt Richarson constant 2 4 qm* k m* A A* 120 h m cm K

20 Aitional factors affecting the current Quantum mechanics, i.e. An electron with E > b may be QM reflecte or an electron with E < b may tunnel through the barrier when biase Injection of minority carriers (holes from n-type semicomnuctor) at high reverse bias Often phenomenological equation is use: J J ' S ( e qva nkt 1) Log-Linear Plot for Al/Si ioe Linear-Linear Plot J Reverse Bias Forwar Bias V from Muller, Kamins 2003

21 Current: Tunneling 21 Dominates in highly-ope semiconuctors an at low temperatures Quantum mechanic tunneling is escribe by wave function Transmission coefficient (triangular barrier): E J thermionic Electron ensity Current is calculate by integrating over ensity of states*istribution function, similarly to the thermionic current but with tunneling probability J tunnel n Jt q v E E

22 Tunneling ominates in highlyope semiconuctors an at low temperatures Current: tunneling vs. thermionic 22 From Sze, 1981

23 Schottky effect = Image force barrier lowering 23 Barrier height slightly epens on current Charge near a metal surface is attracte to the surface with force q F 4 Maimum of the barrier occurs at 2 2 m q 16 Reuction of potential barrier: q 2 m 4 From Sze, 1981

24 Measuring the barrier height: internal photoemission 24 Internal photoemisison: current through Schottky ioe is measure as a function of incient photon energy Illumination through a thin metal is usually use Photorsponce is given by Fowler theory (thermionic), where quaratic epenence is asymptotic at h b 3kT Can be use to stuy image-force lowering of barrier From Sze, 1981

25 Measuring the barrier height: C-V 25 Slope gives carrier ensity in semiconuctor Deep levels can be probe (or even ientifie by DLTS) From Sze, 1981

26 Measuring the barrier height: I-V activation energy 26 Arrenius plots of Al - n-si ioe current of current at a fie forwar voltage Slope of log(i s /T 2 ) vs. 1/T (for thermionic contact) gives activation energy corresponing to the barrier height Does not nee knowlege of contact geometry ev for Al-n-Si is measure From Sze, 1981

27 Ohmic contacts 27 Ohmic contacts shoul have negligible resistance relative to bulk or spreaing resistance of semiconuctor evice Since M-S contact usually has a potential barrier ue to interface states there are two ways to obtain low resistance junctions: Reuce barrier height by choice of metal an processing techniques Reuce the barrier with by oping From Sze, 1981

28 Ohmic contacts Figure of merit for Ohmic contacts = Specific contact resistivity tunneling thermionic 28 R C 1 J 2 cm V V 0 When thermionic current ominates: R C k q B ep * qa T kt When tunneling current ominates (high oping level) R C 4 m* ep B N D From Sze, 2002