2. Crystals and Current Carriers

Transcription

1 miconductor Devices Summary Andreas Biri, D-ITET Constants & Various Constants / , 38.6, # # $ % Silicon (@300K) -> 4 valence electrons & ' # ( ) & ' # ( * # ( *, # ( -.8. /0 0 2 ( % : ;.2 < 4.05 Quantum physics h ? A B 2C. A h 2C hd 2. / /0 F3/ 0 2 (/% Electronics G H I&Jh M $KL,H(N&OP(J7.7JQ 5R (9 : S'T +: V3/ (N&%JL&J : V3/ X: OY, : O OY 2. Crystals and Current Carriers mi-conductor: Conductivity controllable over orders of magnitude by means of : Impurities ( doping ), light, temperature, EM-fields Coordination number: number of nearest neighbors K 3M 8 L K M 2 L K 3M 4 L K 2M 4 L Simple Metals: coord. number > # of valence electrons Transition Metals: bonds covalent-like, harder Covalent Bonding: hybridization of s- & p-orbitals, stiff -> tetrahedral bonding: coord. number 4, 8N states s-orbitals : 2 allowed states ; p-orbitals : 6 allowed states Partially filled/empty bands conduct currents! Band gap: between valence and conduction band Intrinsic carriers No doping, pure semiconductor, created by heat & A & ' ~ M : ; : ; : 67]7(N&.2, 300 Extrinsic carriers Donors (n-type): give electrons ( P, As, Sb ) Acceptors (p-type): give holes ( B, Al, Ga, In ) Overall, solid is neutral: one fixed charge, one free A & ' a, *` & & ' a * b Fermi Dirac Statistics F(E): probability of finding an electron with energy E 5:9 + 5cc d9/ 5cc d 9 : : h Fermi level i j : energy where 5: : h 9 M 2 Probability of finding a hole: k5:9 5:9 n & X l5:9 Y m5: S'T 9 O: S'T c o cr,a X p l5:9q Y m5: S'T 9 O: S'T n m&%7jq Nl 6JLJ: m5: S'T 9 8C 2 h 5 9 / 5 : S'T 9 / 7&J7( &KQ: : S'T At 2 5A 49 +pa u q +5Av 9 2 i) Fermi levels in all regions will lign up ii) Far away from transition, Fermi level is like without junction ( material doesn t know ) iii) At Equilibrium/Steady-State, : h must be flat (constant) so that no current will be flowing Carrier concentration * C 9 / h, *, 4 25C 9 / h & * + c o c d A *, c d c r & ' *, * + c xm & ' c d c w & ' c w c d, : ; : + :, Constant product: & A & ' Carrier Freeze-Out : T << 0 C Extrinsic Region : donors ionized Intrinsic Region : doping irrelevant

2 3. Carrier transport Diffusion current: concentration gradients T m T O& OY, Drift current: electric field Total current: from high to low concentration V m V OA OY holes with field, electrons against it T & y :zt, T & y:zt + m T O& OY, Conductivity V A y :zt V A y:zt m V OA OY { '}/,/3/ ~ : ~ & y T +A y V Einstein relation m T y T, m V y V In PN Junction: only diffusion currents (flat bands) O& OY & V35,d/ 9, OA I T OY A T35,d/ 9 I V / m T O& OY m V OA OY 5, d/ 9 / m T & V3 I T + m V A T I V 5, d/ 9 Reverse breakdown i ) Band-to-Band Tunneling (Zener) applies when both sides are heavily doped ii ) Avalanche Multiplication strong electric field creates large kinetic energy to the carriers, so that they ionize others via collision 4. PN Junction Built-in voltage Band-bending that balances drift & diffusion currents ' ln * b *` 5& ' 9 2 : ˆ4 Forward Bias: reduce band bending, less difference more minority carriers -> minority carrier injection Reverse Bias: increase band bending, less minority Band-bending presence of an electric field Conduction Band Edge: : V3/ of electrons Valence Band Edge: : V3/ of holes Diode currents: minority carriers & V 5& '9 *`, Šw * b A T3 5& '9 *` * b, Šw Under Forward-Bias: Shockley Boundary Conditions & V *` 5, Šw, d 9 A T * b 5, Šw, d 9 & V3, d A T3, d Poisson Equation O: OY H H 4 Œ ' X :5Y9 Depletion approximation : ˆ4 :5Y 09 * by V 4 *`Y T OY Y V +Y T Ž 2 + p * ' ˆVV u q b *` Neutrality: * b Y V *`Y T (same areas) One-Sided junction: only depletion on lightly-doped side Ž 2 Depletion capacitance *` ', *` * b O O 5 ',ˆVV u 9 Ž * b *` 2 ' 5* b +*`9

3 4. Generation and Recombination Recombination brings the system back to equilibrium Non-equilibrium concentration: & & + &,A A + A, & A Recombination rate (even at Non-equilibrium): Thermal generation rate G 5& A9 ^/0 G /0 5 & T A T 9 External excitation (e.g. Light) gives additional term: ^ ^ +^/0 OA T OJ ^ +^/0 G Direct recombination Direct recombination across the bandgap results in the emission of a photon with energy : ; h l Net generation rate U 5 & A & ' 9 ^ G ^/0 Under low-level injection: A T & T, A & T A V, V & T : Minority carrier lifetime ( how fast decay) Example: Lesson 5, p.7 - Light ON ^ A T A T - Light OFF: V A T A T3 + V^ ^ 0 OA T OJ ^/0 G A T A T A T 5J9 A T + V ^ / M V Indirect recombination (Neamen: p.223) G-R Centers in the Gap ( defect states near midgap) These traps facilitate the return of an electron G/R centers: most effective if : / near intrinsic : ' A A. /0 ~ * / +š 2& ' coshš : / : ' & T V * / Density of Recombination Centers ~ Recombination Center cross section T. /0 ~ T & ' 5c c w 9/ Electron emission prob. V. /0 ~ V & ' 5c wc 9/ Hole emission probability Gˆ &* / 5 l9. /0 ~ T Electron capture rate G T * / l Electron emission rate G A* / 5l9. /0 ~ V Hole capture rate G { V * / 5 l9 Hole emission rate Surface recombination: dangling bonds at surface Continuity equation O& OJ O T OY +5^T G T 9, OA OJ O V OY +5^V G V 9 O& V OJ & O:zt Vy T OY +y T:zt O& V OY +m O & V T OY +^T & V & V T OA T OJ A O:zt Ty V OY y V:zt OA T OY +m O A T V OY +^V A T A T V Steady State: Quantities are Time Independent Zero Field: fields in neutral regions are approx. zero Generation: deficiency of minority carriers Recombination: excess of minority carriers Exp: Steady State surface Generation Long diode: semi-infinite, exponential decay A T 509 (N&%J, A T 5Y 9 A T A T 5Y9 A T + A T 509 A T 4 M V OA T V 5Y T 9 m OY m VA T 5,d 9 4 Œ I V T p Y V q m T O& V OY 4 m T& V I T 5,d 9 Short diode: finite, linear decay A T 509 (N&%J, A T 5Y 9 A T sinh Y I V A T 5Y9 A T + A T 509 A T ª sinh I V Minority Carrier Diffusion Length: Quasi-Fermi Levels I V Fm V V Under bias, the equilibrium Fermi level splits into 2 distinct quasi- Fermi levels that describe carrier statistics in each diode region &5Y9 * + 5 c o 549c dœ9/, A5Y9 *, šc d c r 549 / &5Y9A5Y9 * + *, c x 5c dœc d 9/ : ht : hv h

4 Carrier Profile through Depletion Region Generation currents Reverse bias Carrier Deficit -> T X ^ OY ±e + T Forward bias Carrier Excess -> m T Generation current & ', ^ & ' * b I T + m V *`I V & ' + & ' Recombination current ˆ4 ~ * / & ' 5,d 9 A T +& T +2& ' 2. /0~ * / & ', d/ Real PN Junction Diode a) Recombination in Deplention Region b) Ideal Injection 5 ; 60 M O( 9 c) High-Level Injection d) ries resistance effects (ohmic loss) e) Generation in Deplention region Capacitance in depletion region Depletion capacity per unit square / ( b $ Non idealities, :OA]J7N& 7OJh &5Y9* + c o549 c dœ A5Y9*, c d c r 549 X OY & ' 2 he m ² * b I T + m V *`I V & ' ³, d/ +, d & ' 2, d/ Ideality Factor : characterizes Diode Forward Current Ideality Materials with longer recombination lifetime have better ideality h he, d + ~ exp Reverse Breakdown of Diode: i ) Band-to-Band Tunneling (Zener) applies when both sides are heavily doped ii ) Avalanche Multiplication strong electric field creates large kinetic energy to the carriers, so that they ionize others via collision Ohmic losses Ohmic losses reduce the internal voltage that actually appears across the depletion; at low current levels negligible º º,»/ ¼±/ &5Y9 A5Y9 * + *, c x c dœ c d & ', d

5 5. Bipolar Junction Transistor (BJT) BJT is a Minority Carrier Device and acts as an ideal current source ( º +3 /3 does not vary with +½ ) Ideal currents - Injection from Emitter into Base - No Generation/Recombination in Base Layer - neglect Diode Leakage Current For NPN: m T½ m Vc I Àc ½ *`c * b½ º ½ º V½ º Vc m À I Vc A c,¾ º + º T½ m ²½ ½ & ½,¾, ¾o Common Emitter current gain Emitter/Base Junction (in active mode) Forward-Biased: Minority Carrier Injection Base/Collector Junction (in active mode) Reversed-Biased: Minority Carrier Extraction º + º V+ º Vc m V½ I Tc º ½ º Tc º Tc m Tc * bc *`½, +½ 0 º + Á º c º ½, Á M Á Emitter doping must be higher than base doping: º V+ º Tc * bc *`½ Doping Ration most powerful factor to reach gain Gummel-Characteristics: 60 M O( L7& 7& º/ Transconductance Collector current: º + º Ãr ¾ ÄÅ Constant carrier densities in the depleted regions Assumed no recollection or generation Transconductance: ¼ o / Non-ideal currents & c & c,y ; & c 5 Y c 9 & c, ¾ & c 5Y9 & c +5 & c 5 Y c 9 & c º Tc º ½ m Tc O& c OY m Tc I Tc & c, ¾ A ½ 509 A ½, ¾, A ½ 5 9 A ½, ¾o A ½ 5Y9A ½ 5 9+p A ½ 509 A ½ 5 9q š Y NPN PNP º V½ m V½ OA T½ OY m V½ ½ A ½, ¾, ¾o

6 Early effect: In practice, the º + depends on ½+. The depletion region becomes wider with increasing BC reverse bias, decreasing the undeplented base width, which increases º +. To avoid this, the base doping must be higher than collector: *`½ * b+ Power Gain from Amplifier ^ º 3Æ/ G 3ˆ{ º 'T G 'T G G º + G 'T / For power gain: G 3Æ/ b ^b G G 'T Ç b+ +c /º + G b + +c /º + G + È Used power: É +c º + Cost of power gain: É` +c º + É 3Æ/ Maximum Power gain (with impedance matching) l e ^V l, 8CG ½ ½+ l ÊbË Ž l e 8CG ½ ½+ Cutoff frequency Unity Current Gain Cutoff frequency: 5 l e 9 Measured with Short-Circuit load ( G 0 ) h } 5B9 º + K Ì 5B9 º ½ +ÍB K Ì Ì h } 5B 09 K Ì Transistor behaves as a Low-Pass 5B9h } 5B9 l Î 2C Ì K Ì, +Í p l / l Î q Cutoff Frequency: l e3 Ï Ì + Ð K Ì Common-Base (BC) Current Gain Á5B9 5B9 5B9+ º + Á º c +Í 5 l / l Ñ 9 l Ñ 5 +9 l Î, Á M + Additional Delay terms ½ : Base Transit Time (diffusion across the base) + : Collector Signal Delay Time ( through depletion) ½ ½ ½, m T 2. -ˆ/ +, l l Ñ l Ñ l 2C Cutoff frequency including delay terms l e Ò l Î Á l Ñ 2C e Where Ó Ô is the transit time / sum of all delays e Ì + Ì Common-emitter delay term: ÌM Kirk-Effect ( Base spreading ) At high currents, the electron density & + in the collector becomes comparable to the donor density (npn BJT). Therefore, it cannot be neglected for the calculation of the E-Field in the collector: :5Y9 Ö 5*`+ & + 9Y+: { V /'3T Base spread (increases ½, reduces ) S + Ø Ž +½/ +{ 5& + /*`+ 9 Ù Kirk effect threshold current Ú. -ˆ/ *`+ Ç + 2 +½ *`+ + È

7 Base drift field Carrier transport across the base can be aided by introducing an electric field, such as by non-flat base doping profiles / grading the doping. P-type base with width ½ with an electric field: & ½ 5Y9 T ½ m T 4 Û š ¾ accelerating field factor / grading, ½ Y ½ ½ +Û, ½5 09 ½ m T NPN base charge: ½ Ü ¾ & ½ OY 2m T½ /( Heterojunction Bipolar Transistor (HBT) Different materials and bandgaps for emitter & base Darlington Pair ½Ýe & '½ & ½Ýe c x 'c 6. MOSFET In contrast to BJTs majority devices (majority carrier) N-Channel: electrons, P-Channel: holes Depletion Mode: channel present at equilibrium Enhancement Mode: no channel at equilibrium Structure V(x): channel voltage; 509 0, 5I9` e : threshold voltage for strong inversion Basic characteristics Inversion layer has thickness X, charge density T T & ã âë p ; e 5Y9q ã âë p ; e 5Y9q & º +å º` y T âë 2 Pinch-Off, ä: 7OJh ä I 25 ; e 9 ` ` Pinch-off when ` ; e at drain side Linear region: ` < ; e º +å º` y T âë 2 Saturation region: ` ; e º`èé y T âë 2 ä I 25 ; e 9 ` ` ä I 5 ; e 9 Drain-Source voltage Þ ßà : low for uniform channel Gate-Source voltage Þ áà : large enough for channel Channel is built by minority carriers between S & D Sheet resistance G H I&Jh $KL H h7(&%% R M G 5Y9 y T âë p ; e 5Y9q

8 Band diagramm Band bending General potential: ë5y9 : ' : ' 5Y9 Bulk potential : ë ½ 5Y9 : ' : h ë ½ lnš *b M & ' Transconductance in Saturation region Oº`èé y T âë ä 5 O ; I ; e 9 Channel length modulation (L2P2) $%%P I I : channel length idependent of ` When we cannot assume that I I, we have a short-channel MOSFET whose drain-current increases with increasing `! Like Early for BJTs :](JKN& Lll7&7JQ < : : + NK lp&(j7n& ê : : h L(PP ].] : At Equilibrium, the Fermi Level must be constant! As the metal workfunction differs from the semiconductor workfunction, there will be bandbending Flatband voltage: Gate voltage that makes them flat A V & ' c wc d & ' 5 ì ¾ ì9 & V & ' c dc w & ' 5ì ì ¾9 Reducing the channel length increases: - JKL&%(N&OP(JL&( - operation speed - device density But e decreases ( threshold voltage shift) Channel Modulation Midgap: ë ë ½, & V A V & ' Depletion region width ë5y9 ë š Y, ë * b Depletion width: Ò í î ì è ²» Max. at inversion: ˆ4 Ò í î ì ¾ ²» Electric field: : - 5Y9 ²» í 5 Y 9 2 -, ë 2ë ½

9 Ideal gate voltage relationship MOS Capacitance Subthreshold swing ; 34 +ë O : 34 +ë ; : Potential drop across oxide & semiconductor , OA]J7N& (LAL(7JL&( 34 Ò * b O, O Accumulation: only majority carriers can respond to fast AC signal added delta-charge Deep depletion: DC bias changes so fast that minority carriers cannot respond. Therefore, the depletion layer keeps increasing Subthreshold regime: ; < e Subthr. Swing: how effective can it be turned on / off 6 O 5ln5º`99 O ; 7. Various / General Direct and Indirect Bandgaps 34 F2 * b ë, âë 34 âë O ; F2 * b ë âë + ë Threshold voltage ( ï à ð ï ñ ) e F2 * b 2 ë ½ âë + 2 ë ½ After that, W is maximal and stays more or less Non-ideal gate voltage relationship (voltage shift) Bands are not flat due to. Workfunction difference ë - 5ê ê 9 / 2. Fixed charges inside the oxide Cutoff frequency A transition in an indirect bandgap material must necessarily include an interaction with the crystal so that crystal momentum is conserved Material properties ; ; D + h½, ; D :7OL] LJ.N]JL h½ ë - X Y H 34 5Y9 OY 34'{ If the charge in the oxide is fixed: h½ ë - â 34, â (, 3 O º {. -, º. - /ÍB /, / - + { l e Ê 2Cp - + { q 3y T ;- e 4C I

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