Digital Integrated Circuit Design

Transcription

1 Digital Itegrated Circuit Desig Lecture 4 PN Juctio -tye -tye Adib Abrishamifar EE Deartmet IUST Diffusio (Majority Carriers)

2 Cotets PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig Schottky Diode Thermal Equilibrium Eergy Bads Trasort Methods IUST: Digital IC Desig 2/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

3 PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios IUST: Digital IC Desig 3/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

4 PN Juctio Formatio -tye -tye Diffusio (Majority Carriers) Diffusio Drift Miority Carriers Field Created Cotact Potetial V 0 Ucovered Charge Recombiatio IUST: Digital IC Desig 4/98 E c E v Deletio Regio E F qv o E c E v LECTURE 4 : PN Juctio Adib Abrishamifar 2008

5 PN Juctio Formatio -tye Vacuum -tye Vacuum Diffusio Drift Electro qχ Affiity qχ qχ Vacuum E c E F E v IUST: Digital IC Desig E c E v 5/98 E F E c E v Deletio E F qv o E c Regio E v LECTURE 4 : PN Juctio Adib Abrishamifar 2008

6 PN Juctio Formatio -tye -tye Diffusio Drift E c E v E F E c E v E F E c E v Deletio Regio E F qv o E c E v Field Created -E () E o IUST: Digital IC Desig 6/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

7 PN Juctio Issues Sace charge i -tye material created by ucomesated accetor atoms (Na- Na ) Sace charge i -tye material created by ucomesated door atoms (Nd- Nd ) Electric field created i directio Fermi level must by costat throughout the device Diffusio is limited by cotact otetial Drift removes miority carriers from deletio regio (trasitio regio). No et curret ca flow across the juctio at equilibrium, thus J(drift) J(diff) E c E v Diffusio Drift Miority Carriers Deletio Regio Field Created E F Cotact Potetial V 0 qv o E c E v IUST: Digital IC Desig 7/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

8 PN Juctio i Equilibrium E e -tye -tye N(E) f(e) E c E e N(E) f(e) E c E v E F qv o N(E) [1-f(E)] E h IUST: Digital IC Desig 8/98 E v W e Eh LECTURE 4 : PN Juctio Adib Abrishamifar 2008 N(E) [1-f(E)]

9 Carrier Relatioshis Majority Carriers e qv / kt o qv / kt e o Majority Carriers e qv / kt o Hole cocetratio i -tye material Hole cocetratio i -tye material Electro cocetratio i -tye material Electro cocetratio i -tye material Miority Carriers e qv / kt o Miority Carriers e qv / kt o IUST: Digital IC Desig 9/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

10 PN Juctio Equatios Hole cocetratio i -tye material Electro cocetratio i -tye material qv / kt e Ei EF ktl i IUST: Digital IC Desig o E c E i E v qv o 10/98 E F qv o E c Hole cocetratio i -tye material Electro cocetratio i -tye material kt l q kt l q E i EF Ei ktl i E v LECTURE 4 : PN Juctio Adib Abrishamifar 2008 V V o o

11 PN Juctio Sace Charge The sace charge i both tyes of materials must be the same (Q+ Q- ) Sace charge oly due to ucomesated atoms sice all other carriers have bee swet out of the trasitio regio Q - -qa 0 N a Charge Desity qn d Q + qa 0 N d The maimum electric field occurs at 0 E() is egative withi the trasitio regio W de q Na d ε 0 E () -qn a E o 0 de q Nd d ε IUST: Digital IC Desig 11/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

12 Juctio Relatioshis W 2εV 1 1 o 2ε kt NaN d l q Na Nd q i Na Nd W V o kt q NN a d l 2 i qv / kt e o 0 0 o o W Nd W 2εV o N d Na + Nd 1 + ( Na / Nd) q Na( Na Nd) + W Na W 2εV o N a Na + Nd 1 + ( Nd / Na) q Nd ( Na + Nd) IUST: Digital IC Desig 12/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

13 PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios IUST: Digital IC Desig 13/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

14 Forward Biased PN Juctio + - V f The otetial barrier is decreased by V f The deletio regio decreases i size The device is o loger i a equilibrium coditio so the Fermi level is ot costat across the device The quasi-fermi levels are searated by qv f. The field withi the deletio regio decreases Some carriers i the high-eergy tail of the distributio have eough eergy to diffuse to the oosite side E c E v -tye φ (Diff) J (Diff) φ (Drift) J (Drift) E E F F -tye φ (Diff) J (Diff) φ (Drift) J (Drift) V 0 -V f q(v 0 V f ) E c E v IUST: Digital IC Desig 14/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

15 Forward Biased Juctio E e + - V f -tye -tye E e N(E) f(e) E c N(E) f(e) qv f E c E F E F E v N(E) [1-f(E)] E v N(E) [1-f(E)] E h IUST: Digital IC Desig 15/98 W e W f E h LECTURE 4 : PN Juctio Adib Abrishamifar 2008

16 Drift Curret + - V f The drift curret is isesitive to the height of the otetial barrier ad alied voltage -tye -tye The drift curret is ot limited by how fast the carriers are swet but how may are swet Thermally geerated EHPs created withi a diffusio legth of the trasitio regio create the drift curret Drift curret due to geerated EHPs is called geeratio curret The total curret through the juctio is the sum of the diffusio ad drift currets E c E v φ (Diff) J (Diff) φ (Drift) J (Drift) E E F F φ (Diff) J (Diff) φ (Drift) J (Drift) V 0 -V f q(v 0 V f ) E c E v IUST: Digital IC Desig 16/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

17 Juctio Relatioshis W ( ) 2ε Vo V 1 1 2ε kt NN a d l q Na Nd q i Na Nd W V o kt q NN a d l 2 i e qv ( V)/ kt o 0 0 o o ( V V) W N 2 d W ε o N d Na + Nd 1 + ( Na / Nd) q Na( Na + Nd) ( V V) W N 2 a W ε o N a Na + Nd 1 + ( Nd / Na) q Nd ( Na + Nd) IUST: Digital IC Desig 17/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

18 Ecess Miority Carriers + - V f -tye -tye E c qv f E c E F E F E v E v Ecess Miority Electro Carriers ( ) ( ) Ecess Miority Hole Carriers IUST: Digital IC Desig 18/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

19 Ijected Miority Carriers + - V f -tye Majority Carriers Carriers -tye Majority Carriers Ijected Miority Carriers IUST: Digital IC Desig 19/98 Thermally Geerated Miority Carriers Ijected Miority Carriers LECTURE 4 : PN Juctio Adib Abrishamifar 2008

20 Miority Carriers + - V f -tye -tye ( ) ( 0 ) δ Δ e IUST: Digital IC Desig Majority Carriers e / kt ( qv 1) f Δ e / L qv ( o Vf )/ kt 20/ Carriers Δ Majority Carriers ( ) e 0 Thermally Geerated Miority Carriers qv ( o Vf )/ kt ( qv ) f / kt Δ e 1 δ ( ) Δ e Δ LECTURE 4 : PN Juctio Adib Abrishamifar 2008 / L

21 Miority Currets + V - f -tye -tye D I( ) qa ( ) L δ IUST: Digital IC Desig Δ ( ) ( 1) qv / kt / L δ e e f D J ( ) q ( ) L δ Ecess Miority Electro Carriers 21/ ( ) Δ Δ ( ) ( ) ( 1) Δ D J ( ) q ( ) L δ D I ( ) qa ( ) L δ Ecess Miority Hole Carriers qv / kt / L δ e e f LECTURE 4 : PN Juctio Adib Abrishamifar 2008

22 PN Juctio Curret I + - V f -tye -tye i Total Curret I I I ( 0) + I ( 0) I I ( ) I I ( ) I ( ) I I ( ) D I ( ) qa ( ) L δ D I ( ) qa ( ) L δ IUST: Digital IC Desig 22/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

23 Curret Mechaism I + - V f -tye -tye i I Majority Drift Majority Drift & Diffusio Majority Diffusio Majority Drift & Diffusio Majority Drift Miority Diffusio Recombiatio Majority Diffusio Miority Diffusio Recombiatio IUST: Digital IC Desig 23/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

24 Quasi Fermi Levels The equilibrium Fermi level, E F, is slit ito quasi-fermi levels, F ad F F ad F are searated i the deletio regio by qv I a forward biased deletio regio 2 ( F F )/ kt 2 qv / kt i i e e -tye + - V f -tye The majority carrier quasi-fermi level is relatively costat The miority carrier quasi-fermi level varies sigificatly E F F F E F IUST: Digital IC Desig 24/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

25 PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios IUST: Digital IC Desig 25/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

26 Reverse-Biased PN Juctios - V r + IUST: Digital IC Desig The otetial barrier is icreased by V r The deletio regio icreases i size The device is o loger i a equilibrium coditio so the Fermi level is ot costat across the device The quasi-fermi levels are searated by qv r The electric field withi the deletio regio icreases The oly curret is geeratio curret caused by EHPs created withi oe diffusio legth of the deletio regio 26/98 -tye E c E v φ (Diff) J (Diff) φ (Drift) J (Drift) E F qv r E F -tye V 0 + V r q(v 0 + V r ) φ (Diff) J (Diff) φ (Drift) J (Drift) E c E v LECTURE 4 : PN Juctio Adib Abrishamifar 2008

27 Reverse-Biased Juctios E e - + V r -tye -tye N(E) f(e) E c E e N(E) f(e) E F E c E v E F N(E) [1-f(E)] E v N(E) [1-f(E)] E h W e E h IUST: Digital IC Desig 27/98 W r LECTURE 4 : PN Juctio Adib Abrishamifar 2008

28 Drift Curret - V r + The drift curret is isesitive to the height of the otetial barrier ad alied voltage -tye -tye IUST: Digital IC Desig The drift curret is ot limited by how fast the carriers are swet but how may are swet Thermally geerated EHPs created withi a diffusio legth of the trasitio regio create the drift curret Drift curret due to geerated EHPs is called geeratio curret 28/98 E c E v φ (Diff) J (Diff) φ (Drift) J (Drift) E F qv r E F V 0 + V r q(v 0 + V r ) φ (Diff) J (Diff) φ (Drift) J (Drift) E c E v LECTURE 4 : PN Juctio Adib Abrishamifar 2008

29 Sace Charge - V r + The sace charge i both tyes of materials must be the same (Q+ Q- ) Sace charge oly due to ucomesated atoms sice all other carriers have bee swet out of the trasitio regio The maimum electric field occurs at 0 E() is egative withi the trasitio regio The electric field ad total sace charge icreases uder reverse bias IUST: Digital IC Desig 29/98 -tye Charge Desity Q - - qn d Q + qa 0 N d qa 0 N a W e de q Na d ε E () -qn a E o -tye W r de q N d ε LECTURE 4 : PN Juctio Adib Abrishamifar 2008 d

30 Juctio Relatioshis W ( ) 2ε Vo V 1 1 2ε kt NN a d l q Na Nd q i Na Nd W V o kt q NN a d l 2 i e qv ( V)/ kt o o o ( V V) W N 2 d W ε o N d Na + Nd 1 + ( Na / Nd) q Na( Na + Nd) ( V V) W N 2 a W ε o N a Na + Nd 1 + ( Nd / Na) q Nd ( Na + Nd) IUST: Digital IC Desig 30/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

31 Ijected Miority Carrier + - V r -tye Majority Carriers ( qv r / kt 1) qv ( o Vr)/ kt ( ) / L ( ) Δ Δ e e 0 0 δ e -tye Carriers Majority Carriers qv r / kt Δ e 1 Δ Δ ( ) ( ) qv ( o Vr)/ kt 0 0 e ( ) δ Δ e / L IUST: Digital IC Desig 31/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

32 Miority Currets + - V r -tye -tye 0 0 Majority Carriers Majority Carriers D D I I ( ) qa ( ) L δ ( ) qa ( ) L δ D Carriers D J ( ) q ( ) J ( ) q ( ) L δ L δ qv / r / kt L δ ( ) ( e 1) e qvr / kt δ ( ) ( e 1) e Δ Δ / L IUST: Digital IC Desig 32/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

33 Curret Mechaisms I + - V r -tye -tye i I Majority Drift Majority Drift & Diffusio Miority Drift Majority Drift & Diffusio Majority Drift Miority Diffusio Recombiatio IUST: Digital IC Desig 33/98 Miority Drift Miority Diffusio Recombiatio LECTURE 4 : PN Juctio Adib Abrishamifar 2008

34 Quasi Fermi Levels The equilibrium Fermi level, E F, is slit ito quasi-fermi levels, F ad F. F ad F are searated i the deletio regio by qv I a reverse biased deletio regio ( F F)/ kt e 0 2 i No carriers i deletio regio The majority carrier quasi- Fermi level is relatively costat The miority carrier quasi- Fermi level varies sigificatly E F -tye - + V r F F -tye E F IUST: Digital IC Desig 34/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

35 PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios IUST: Digital IC Desig 35/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

36 Short Quasi Neutral Regios A PN juctio ca be modeled as a short diode whe the legth of the quasi-eutral regio, is much smaller tha the miority carrier diffusio legth Width of the quasi-eutral regio i the -tye material. -tye W -tye W Width of the quasi-eutral regio i the -tye material. IUST: Digital IC Desig 36/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

37 Ecess Miority Hole Curret Desity ( ) 0 ( qv / kt ) f δ ( ) e 1 J( ) Δ δ ( + W ) 0 J ( ) 0 δ ( ) ( ) + W 0 Δ W 0 +W 0 0 +W qd f J ( ) e 1 W ( qv / kt ) IUST: Digital IC Desig 37/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

38 Miority Currets + - V f L << W -tye W << L -tye ( ) ( 1) qv / kt / L δ e e f D J ( ) q ( ) L δ I ( ) AJ ( ) ( ) ( ) ( 0 + W) δ ( ) Δ W qd qv f / kt Δ J ( ) e 1 W Δ ( ) I ( ) AJ ( ) IUST: Digital IC Desig 38/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

39 Miority Currets + - V f L << W -tye W << L -tye i I I ( 0 ) + I ( 0 ) I I ( ) I I ( ) qad f I ( ) e 1 W ( qv / kt ) D I ( ) qa ( ) L δ I ( ) I I ( ) IUST: Digital IC Desig 39/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

40 PN Juctio Overview PN Juctios i Equilibrium Forward-biased PN Juctios Reverse-biased PN Juctios Short Quasi-eutral Regios P + N Juctios IUST: Digital IC Desig 40/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

41 P + N Diodes + - V f + -tye -tye Heavily Doed Majority Carriers Ijected electro miority carriers i the + -tye material are isigificat comared to the ijected hole miority carriers i the tye material Carriers Lightly Doed Majority Carriers Ijected hole miority carriers IUST: Digital IC Desig 41/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

42 P + N Miority Currets + - V f + -tye -tye I ( ) Igore Ecess Miority Electro Carriers IUST: Digital IC Desig 42/98 ( ) Δ ( ) ( ) ( 1) Δ D J ( ) q ( ) L δ D I ( ) qa ( ) L δ Ecess Miority Hole Carriers qv / kt / L δ e e f LECTURE 4 : PN Juctio Adib Abrishamifar 2008

43 P + N Juctio i Equilibrium E e + -tye -tye N(E) f(e) E c E e N(E) f(e) E c E v E F qv o N(E) [1-f(E)] E h IUST: Digital IC Desig 43/98 E v W e Eh LECTURE 4 : PN Juctio Adib Abrishamifar 2008 N(E) [1-f(E)]

44 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 44/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

45 Poisso s Equatio The electric field at (,y,z) is related to the charge desity ρq(n d -N a --) by the Poisso s Equatio: ε is the ermitivity ε o 8.85E ε E r ρ ε, where, 14, ε ε Fd/cm, 11.7 for silico o E ε r, ad with E IUST: Digital IC Desig 45/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

46 Poisso s Equatio For -tye material, N (N d -N a ) > 0, o N, ad (N d -N a +-) -δ +δ + i2 /N For -tye material, N (N d -N a ) < 0, o -N, ad (N d -N a +-) δ-δ- i2 /N So eglectig i2 /N q E ( δ δ), for - tye ad ε - tye material with ecess carriers. IUST: Digital IC Desig 46/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

47 Quasi Fermi Eergy I equilibrium, the equilibrium carrier cocetratio ad the Fermi Eergy level are give by o o Ef Efi Ef Efi kt l, ad e i i kt If the cocetratio is ot i equilibrium,,the the o +δ quasi Fermi level must be used IUST: Digital IC Desig 47/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

48 Quasi Fermi Eergy The Quasi Fermi level is defied Ad the carrier desity is o + δ Ef Efi kt l i o + δ Ef Efi e kt i IUST: Digital IC Desig 48/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

49 Quasi Fermi Eergy For holes, the Quasi Fermi level is defied as o + δ Efi Ef kt l i Ad the carrier desity is o + δ i Efi Ef e kt IUST: Digital IC Desig 49/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

50 Deletio Charge Distributio +qn d - - c c Q -qn a [Coul/cm 2 ] ρ -qn a +Q qn d [Coul/cm 2 ] Charge eutrality > Q + Q 0, > N a N d IUST: Digital IC Desig 50/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

51 Iduced E-field i the Deletio Regio The sheet diole of charge, due to Q ad Q iduces a electric field Charge eutrality ad Gauss Law* require that E 0 for - c < < - ad E 0 for - < < c ε S E ds V ρdv εa > E < d ( ' ' ) + Q A Q 0 IUST: Digital IC Desig 51/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

52 Iduced E-field i the Deletio Regio -cotact E N-cotact -tye charge eutral regio O- O- O- O+ O+ O+ -tye charge eutral regio Eosed Accetor Ios Deletio regio (DR) W Eosed Door ios - c - 0 c IUST: Digital IC Desig 52/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

53 1-Dim Solutio of Gauss Law - c - E -E ma c IUST: Digital IC Desig 53/98 W N E E E eff E 0, 2ε N ( V Va) a q - N ε q Na ε 0, a a bi qn N N + < eff d N d ( + ) ( ) >, N,,, a 0 W LECTURE 4 : PN Juctio Adib Abrishamifar 2008 < < N < d < + 0,,,,

54 Arbitrary Doig Profile If the et door cocetratio, N N(), the at, the etra charge ut ito the deletio regio is δq -qn( )δ The icrease i field, δe -(qn/ε)δ, by Gauss Law (at, but also costat). So δv a -( + )δe (W/ε) δq Further, sice N( )δ N( )δ gives, the dc/d as... IUST: Digital IC Desig 54/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

55 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 55/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

56 Cotiuity Equatio Aly the Cotiuity Equatio i CNR 0 t d dt 1 q J, - c < < ad 0 t d dt + 1 q J, < < c IUST: Digital IC Desig 56/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

57 Cotiuity Equatio Sice d 2 ( ) d d δ 2 2 ( δ ) d E 2 ad D D 0 δ J τ δ τ i the 0, qd 0, for CNR, for d d - J c givig < < < qd c <, d d ad IUST: Digital IC Desig 57/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

58 Ideal Juctio Theory (cot.) δ δ with ad Defie ( ) Ae ( ) B.C. δ Ce L δ D De Be ad ( ) ( ) δ, - < < < < ( V V ) e a t ( ) δ ( ) 0, (cotacts) c 2 L L o + τ + L L o c L 2, c D τ. So c. 1, IUST: Digital IC Desig 58/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

59 Ecess Miority Carrier Distributio Fuctio For δ δ ( ) ( ) < < ad for o - o sih c c sih, W [( ) L ] c sih W < < [ L ] [( + ) L ] Va c [ ] V e t 1 L sih W c, a e W, V V t c 1, IUST: Digital IC Desig 59/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

60 Carrier Ijectio ( ) δ o e 1 ~V a /V t V a V t l(carrier coc) l N a l N d l i δ ( ) o e 1 ~V a /V t V a V t l i2 /N d l i2 /N a - c - 0 c IUST: Digital IC Desig 60/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

61 Miority Carrier Currets J J ( ) q ( ) 2 i N L d IUST: Digital IC Desig 2 qi D N L a D qd cosh qd cosh 61/98 ( δ ) [( ) L ] sih W d d ( δ ) d d [ L ], c e [( + ) L ] Va c [ ] V e t 1 L sih W, for for - c < V < 1 LECTURE 4 : PN Juctio Adib Abrishamifar 2008 a V t < < c

62 Evaluatig the Diode Curret Assumi g o ge/rec i DR, the J J where s / s J ( ) + J ( ) J s q 2 i J N s D + a / d J / L s / coth W Va Vt J s e 1 with defiitios [ L ] / / IUST: Digital IC Desig 62/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

63 Secial Cases For the Diode Curret Log diode : J J s s 2 i 2 i a a Short diode : q q D N L D N W W W >>, ad, ad L << J, s L J, s or or q W 2 i q W 2 i >> D d N L D d L << N W L IUST: Digital IC Desig 63/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

64 Ideal Diode Equatio J s J s, + J s, hole curret + electro curret J s, q i 2 D coth(w /L )/(N d L ) q i 2 D /(N d W ) W << L, short q i 2 D /(N d L ) W >> L, log J s, q i 2 D coth(w /L )/(N a L ) q i 2 D /(N a W ) W << L, short q i 2 D /(N a L ) W >> L, log J s, << J s, whe N a >> N d IUST: Digital IC Desig 64/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

65 Diode Coductace I JA J Aed V V ( ) ( ) D s a t s a t where ed() [e() -1] ( V) di Ie ( ) s V D Q t gd V Q. If V a >V t, dva Vt V Q t I ed V V I ( ) DQ the g ( ) d V Q, where I DQ ID V Q. V ( ) 1 V The diode resistace, rd V Q g I d t DQ IUST: Digital IC Desig 65/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

66 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 66/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

67 Charge Distributio i a (1-Sided) Short Diode Assume N d << N a δ W c - The sih ecess miority carrier distributio becomes liear for W << L δ ( ) 0 ed(v a /V t ) Total charge Q qδ ( )W /2 δ ( ) Q c IUST: Digital IC Desig 67/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

68 Charge Distributio i a (1-Sided) Short Diode Assume Quasi-static charge distributios Q qδ ( )W /2 dδ ( ) (W/2)*{δ (,V a +δv)-δ (,V a )} δ δ (,V a +δv) δ (,V a ) δq Q c IUST: Digital IC Desig 68/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

69 Caacitace of a (1-Sided) Short Diode Q d dv a So, Whe Q' A, qaδ r d V a ( V ) C ( V ) Q > ( 2 V, t d A )W C Q diode area. d ( V ) Q τ d dv a I trasit DQ V qa 2 t W 2D Defie c 0 2 W δ q J I ed DQ V C t d τ d dq dv ( V V ) a trasit 2 W 2D a t. IUST: Digital IC Desig 69/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

70 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 70/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

71 Geeral Time Costat For all diodes, log g d ( V ) g Q C D a dq dv VQ a, ad ( J ) There is always a characteristic time so that τ di dv d J + A dv total diode caacitace C or τ g short, a C C the coductace VQ + dq dv C a g + g, ad the IUST: Digital IC Desig 71/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

72 Geeral Time Costat For the short diode side, 2 W 2D, ad τ hysical charge trasit times. For the log diode side, τ,tras τ W 2D ad, the the resective mi. carr. life - times. τ 0 2 τ τ τ,tras τ 0, IUST: Digital IC Desig 72/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

73 Geeral Time Costat Practical diodes are usually oe - sided The effective trasitio time average give by C d g d τ F 1 τ F τ 1 mi + τ is 1 trasit the ad IUST: Digital IC Desig 73/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

74 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 74/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

75 Diode Equivalet Circuit (Small Sigal) I D di dv D D VQ g d 1 r d I Q ηv t I Q η is the ractical ideality factor r d C IUST: Digital IC Desig C d V Q τ, diffusio ( τ 75/98 tr τi Q ηv t V D for short,, r diff τ ηv I Q mi t for log) LECTURE 4 : PN Juctio Adib Abrishamifar 2008

76 Small-Sigal Equivalet Circuit V a C diff r diff C del C diff ad C del are both charged by 1/2 Va C del Cj Cjo 1, Va V bi V Q V a V Q IUST: Digital IC Desig 76/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

77 Switchig Diode Trasiets Poisso s Equatio Cotiuity Equatio Charge Distributio i a (1-Sided) Short Diode Geeral Time Costat Diode Equivalet Circuit Diode Switchig IUST: Digital IC Desig 77/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

78 Diode Switchig Cosider the chargig ad dischargig of a PN diode (N a > N d ) W d << L For t < 0, aly the Thevei air V F ad R F, so that i steady state I F (V F -V a )/R F, V F >> V a, so curret source For t > 0, aly V R ad R R I R (V R + V a )/R R, V R >> V a, so curret source IUST: Digital IC Desig 78/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

79 Diode Switchig I R F ( ) F t < 0 IF RF F: t < 0 Sw R: t > 0 V V F,V R >> V a + V F R R D for t < 0, + V R I( t ) I s I Q V F R F V a V R F F IUST: Digital IC Desig 79/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

80 Diode Charge for t < 0 ( ) ( V /V,V e t ) F d d o I F qad F o,, sice J 2 i N D qd d d Q' ( ),diff q,vf o c W τ TR I F τ TR 2, WN 2D IUST: Digital IC Desig 80/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

81 Diode Charge for t >>> 0 (Log Times) ( ) ( ) V F / V t, t < 0 d d IS qad, o e sice J qd d d J s o (, t ) c IUST: Digital IC Desig 81/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

82 Equatio Summary d d R,t 1 qd Is AqD J,, I s J A s d d F,t> 0 I F qad, I F V R F F For t flows small, but > to discharge 0, Q a curret, I R V R /R R IUST: Digital IC Desig 82/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

83 Sashot for t Barely > 0 o ( ( ) ) V F / V t, t < 0 oe (, t 0) > Total charge removed, Q dis I R t, c t s d d ( ) d d I F qad I R qad IUST: Digital IC Desig 83/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

84 I(t) for Diode Switchig I D I F -0.1 I R t s t s +t rr t For -I R 0 < t < t s, d d I R qad I s q 2 i L is a costat, D ( ) tah W / L IUST: Digital IC Desig 84/98 Q discharge I R LECTURE 4 : PN Juctio Adib Abrishamifar 2008 t s

85 Diode Switchig i(t) If t2 is refereces VF V1 t < t 2 i(t) IF R V V t t 2 i(t) IR R R 1 i(t) I F V f V R R V(t) Charge Cotrol Eressio t < t i 2 F t t i Q τ P 2 R + τ P P P Q dq dt P Q(t) i τ +Ce(-t τ ) I R o t 2 P R P P t s t 3 t t t IUST: Digital IC Desig 85/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

86 Diode Switchig t t Q (t) i τ i τ +C C τ (i i ) 2 P F P R P P F R [ τ ] Q (t) τ i + (i i )e(-t ) P P R F R P F t t3 QP 0 t3 τ PL -i R F t3 t2 ts tstorage τ PL -i R i i i i R R IUST: Digital IC Desig 86/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

87 Diode Switchig Icludig the effects of deletio layer ca Note To chage the charge takes time, this time is the trasitio time t t trr t s +tt Short Base Diode Log Base Diode W τ P 2D L τ P D 2 P 2 P P P P Q P P N 0 W N Q P P 0 W IUST: Digital IC Desig 87/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

88 Schottky Diode Thermal Equilibrium Eergy Bads Trasort Methods IUST: Digital IC Desig 88/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

89 Metal Semicoductor Cotacts Theories for metal-semicoductor (MS) cotacts came of oit cotact rectifiers used i radar receivers Curret carried by majority carriers oly ad it is very high seed Alicatios: Microwave Diodes (f 100 GHz) Gates i MESFETs Otical, uclear article detectors Digital Itegrated Circuit Desig IUST: Digital IC Desig 89/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

90 Basic Termiology E 0 E 0 Φ M χ S Φ S E C E F E F metal E 0 Vacuum level Φ M Metal workfuctio IUST: Digital IC Desig 90/98 semicoductor E V E 0 Vacuum level Φ S Semicoductor workfuctio χ S Electro affiity Φ S χ χ S S + + k ( E E ) B C LECTURE 4 : PN Juctio Adib Abrishamifar 2008 F N T l N C D

91 Schottky Diodes Average eergy of the electros i a SC is larger tha i the metal. Whe merged together, the eergy differece leads to trasfer of electros from the SC ito the metal E 0 must be cotiuous across the iterface (Φ M ad χ S are fudametal roerties of the material) Uder zero-bias coditios E F must be costat Φ M Φ B E 0 qvbi Φ S Φ B Schottky barrier height: Φ Φ χ B M S V bi Built-i voltage: qvbi Φ M Φ S IUST: Digital IC Desig 91/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

92 Schottky Diode Electrostatics Solve 1D Poisso equatio 2 d ϕ qn D 2 d ksε0 Boudary coditios ϕ( 0) 0, ϕ( ) V, F(W) W bi 2k qn ks ε s qn W bi Fial eressios ϕ( ) V F( ) D qn 2k D ε V 0 0 bi s D ε 0 ( W ) ( W ) 2 0 ρ() ϕ() F () F ma + W qn D qv bi V bi E F IUST: Digital IC Desig 92/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

93 Schottky Diode Thermal Equilibrium Eergy Bads Trasort Methods IUST: Digital IC Desig 93/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

94 Trasort Methods A: Trasort of electros over the otetial barrier B: Quatum-mechaical tuelig through the barrier C: Recombiatio i the SCR D: Hole ijectio from the metal ito the SC A B C D IUST: Digital IC Desig 94/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

95 (A) Trasort Over the Potetial Barrier Thermioic emissio theory due to Bethe (valid whe the iterface barrier reresets a imortat imedimet for the curret flow) Diffusio theory due to Schottky ( low mobility ad relatively low-doed semicoductors). Curret limited by diffusio ad drift i the SCR rather tha the M-SC barrier Thermioic-diffusio theory due to Crowell ad Sze Criterio for the validity of the thermioic emissio theory λ v T τ m > d T k qf T B 2 N s 0 B Dμ > ma 6Vbi k ε k T m * / q IUST: Digital IC Desig 95/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

96 IUST: Digital IC Desig LECTURE 4 : PN Juctio LECTURE 4 : PN Juctio Adib Abrishamifar /98 T k qv T k T k qv T k E B M S B A B B B A B B B F e e T A e e T h qk m d v q J / / 2 * / / * 4 Φ Φ +Φ π T k M V S S M B B A e T A J J / 2 * 0 Φ ( ) 1 / / 2 * Φ T k qv T k S M M S tot B A B B e e T A J J J * ) / 120( * /K A/cm m m A Thermoioic Emissio Theory Curret flow from the semicoductor ito the metal Curret flow from the metal ito the semicoductor Total curret desity

97 Descritio of the diffusio theory Curret is limited by drift ad diffusio rocesses withi the SCR J where: qn F c ma μ ma Φ Fial eressio for the curret F qn DW k ε s 0 qn e 2qN B D / k ( qv / k T e 1) A B 2 Φ / k T qv / k T J qn μ D bi A c e B B e A B 1 ksε0 Bottleeck i ot the emissio over the barrier, but the trasort through the SCR. Valid for low-mobility SCs Sice most semicoductors have high mobility, thermioic emissio theory will be sufficiet to elai the IV-characteristics B T ( V V ) k s bi ε 0 A ( V V ) ( ) IUST: Digital IC Desig 97/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008

98 (B) Quatum Mechaical Tuelig I Schottky barriers o highly-doed SCs, the deletio regio is so arrow that the electros ca tuel through the barrier ear the to, where the barrier is thi eough. This rocess is called thermioic-field emissio I degeerate SCs, electros ca tuel through the barrier ear the fermi level, ad the tuelig curret will be domiat. Such a mechaism is called field emissio E tuel qv A qv A thermioic-field emissio field emissio IUST: Digital IC Desig 98/98 LECTURE 4 : PN Juctio Adib Abrishamifar 2008