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1- Luryi Serge Transistors Silicon of Physics Plan — Transistor Junction

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1- Luryi Serge Transistors Silicon of Physics Plan — Transistor Junction Serge Luryi - 1 - Physics of silicon transistors Plan Ð Bipolar junction transistor · homojunction and heterojunction · Doping considerations · Transport factor and current gain · Frequency dependence of gain in a microwave transistor · Base shrinkage and ®nite output conductance · Ebers-Moll model and small-signal equivalent circuit Ð Elements of small-signal analysis · Two port; common terminal con®gurations · Power gain, Mason theorem Ð Transistor Principles: PETs and FETs Ð Field effect transistor · Two-dimensional channel; sheet conductance · Conduction Laws by dimensional analysis · Gated diode · Short channel effects · MOS structure: relevant energies · Band bending; threshold evaluation · Inversion layer structure · Pinch-off effect in MOSFET · Concept of quasi-Fermi level (imref) Ð Discussion · Why silicon ??? Serge Luryi - 2 - Bipolar Junction Transistors Junction Transistor np n 10 16 10 20 10 17 Veb Vcb Figure 1: Schematic diagram of a npn transistor in equilibrium and under applied bias. By Kirchhoff law: = + ∼ I E I C I B ∼ I C Neglecting recombination in the base and parasitic injection of holes into emitter,² the collector current ¯ows through a much larger impedance than the emitter current, whence the power gain. _______________ ² In a homojunction transistor, injection of holes into the emitter is suppressed compared to np 0 N D (emitter) the useful injection of electrons into the base only by the factor ____ = ____________ . pn 0 N A (base) This means that the base must be lightly doped compared to the emitter and hence base resistance is a concern. The fundamental trade-off: thicker base for lower base resistance, thinner base for faster diffusion across the base. Serge Luryi - 3 - Bipolar Junction Transistors Ie Ic np n Ib Φ 1 Veb Φ 2 Vcb ∆Ε V Heterojunction bipolar transistor (HBT) Figure 2: Homojunction and heterojunction npn transistor In homojunction transistor at the base-emitter junction − β Φ (e ) ∝ 1 J E e − β Φ (h ) ∝ 2 J E e (h ) N (base) _J___E = ____________A (e ) J E N D (emitter) (e ) J E Emitter ef®ciency η ≡ __________ ∼ 1 − doping ratio . (e ) + (h ) J E J E In heterojunction transistor: (h ) J E N A (base) − β ∆ ____ = ____________ × e E V (e ) J E N D (emitter) Serge Luryi - 4 - Bipolar Junction Transistors Back to (homo)junction transistor. We understand why the base doping must be much lower than emitter doping. Now why the collector doping must be much lower than emitter doping ? N D (emitter) >> N A (base) >> N D (collector) Threefold answer: Ð For W B to have little dependence on V cb (need high output impedance) Ð to lower base-collector capacitance Ccb Ð to lower the ®eld in base-collector junction (increase breakdown voltage) Ie Ic np n Ib Φ 1 Φ 2 Vce Ic (mA) Ib = 0.6 mA 60 0.5 50 0.4 40 0.3 30 0.2 20 0.1 10 Vce Figure 3: Common-emitter transistor characteristics. Base current is stepped up by increment of 0.1 mA and the emitter current increases by much larger amounts. Here current gain β ∼ 100 . Serge Luryi - 5 - Bipolar Junction Transistors Base transport factor and current gain. From the continuity equation: n n′′ − ____ = 0 2 L D ___x − ___x n (x ) = A e L D + B e L D Boundary conditions at x and at W : 2 ni β n (0) = ____ e V eb N A 2 ni β n (W ) = ____ e V cb ∼ 0 N A − ⁄ n (0) sinh (W x ) L D ☛ = − ⁄ = _______________________ n (x ) A sinh (W x ) L D ⁄ sinh (W L D) − ⁄ ∂ cosh (W x ) L D ☛ = ____n = _e________D n (0) ___________________ Jn (x ) e D ∂ ⁄ x L D sinh (W L D) The base transport factor alpha: 2 Jn (W ) 1 W α ≡ _______ = ____________ ∼ 1 − _____ ⁄ 2 Jn (0) cosh (W L D) 2 L D J(0) J(W) JB ∂ α β = ____I C β = ______ Current gain: ∂ . By Kirchhoff's law, − α I B 1 Combined with emitter ef®ciency η, we take instead of α the product α η . Serge Luryi - 6 - Bipolar Junction Transistors Frequency dependence of the current gain. β (dB) 104 40 static gain 103 30 102 20 f T 10 10 slope 10 dB/dec 1 0.1 1 10 100 1000 frequency f (GHz) Figure 5: High-frequency gain. At high frequencies the current gain rolls-off at 10 dB/per decade (i.e. as 1⁄f ). This behavior is quite universal and has nothing to do with either recombination or emitter ef®ciency. The characteristic cut-off frequency f T is de®ned by the condition of unity current gain (β → 1 ) and is mainly due to the propagation delay τ of minority carriers² through the base. 1 f = ____ T 2π τ In general, τ = W ⁄v , where v is the average velocity of minority carrier propagation. For diffusive transport, W 2 τ = ____ << τ . D 2 D C _______________ ² Do not confuse this τ with the minority carrier lifetime (which is typically much longer). τ Let us denote the minority-carrier lifetime by C ("capture" time). Serge Luryi - 7 - Bipolar Junction Transistors Let us derive an expression for α (f ) . Begin with the continuity equation: ∂n ∂2 n n ___ = D ____ − ___ ∂ 2 τ t ∂ x C Take = + δ i ωt n (t ) n 0 n e = δ = δ where n 0 n 0 (x ) is the static solution and n n (x ) is the harmonic variation amplitude at frequency ω ≡ 2π f . δ Both the static equation for n 0 and the dynamic equation for n are of similar form: ∂2 n _____0 − − 2 = L D n 0 0 ∂ x 2 2 ∂ δn − _____ − L 2 δn = 0 ∂ x 2 where 2 ≡ τ L D D C − − i ω L 2 = L 2 (1 + i ω τ ) ∼ ___ D C ωτ C >> 1 D The solutions to both equations are similar too (in form): − = _______W x n 0 (x ) A sinh L D W − x δn (x , ω) = B sinh _______ L α = ____________1 whence we ®nd (in analogy to 0 ⁄ cosh (W L D) ∂ J (W , ω) 1 α (ω) ≡ __________ = ___________ ∂ J (0 , ω) cosh (W ⁄ L ) Serge Luryi - 8 - Bipolar Junction Transistors ω ω τ At suf®ciently high 's (nothing "spectacular", just C >> 1 ) we have W W 2 ___ = √2iω τ where τ = ____ L D D 2D + = √ω τ √ = i π ⁄ 4 = _1____i 2i D remember i e √2 + ωτ − 1 ωτ ( 1 i D ) for D << 1 α ω = ___________1 = ( ) − ωτ − i ωτ ωτ cosh (W ⁄ L ) 2 e D e D for D >∼ 1 2 − 1 Im modern transistors, typically, W <∼ 1,000 A and D ∼ 50 cm s . Therefore, at frequencies easily accessible to the measurement (up to, say 20 τ − 12 ωτ GHz) one has, typically, D <∼ 10 s and D << 1 . − ω τ α ω ∼ _________1 ∼ i D ( ) + ωτ e 1 i D − ω τ − ωτ ⁄ α e i D − i e i D 2 β (ω) = ______ = __________ = ____________ − α − i ω τ ωτ ⁄ 1 1 − e D 2 sin ( D 2) β ω = _______________1 ∼ _____1 ( ) ωτ ⁄ ω τ 2 sin ( D 2) D Uisng heterostructures, it is possible to design an HBT such that α (ω) does not spiral in signi®cantly, even for large ω 's ± remaing close to the circle exp (i ωτ) for phase angles φ = ω τ as large as φ = 2π . Such coherent transistors,² are capable of "life after death", showing current gain above f T and power gain above f max. _______________ ² A. A. Grinberg and S. Luryi, "Coherent transistor", IEEE Trans. Electron Devices ED-40, pp. 1512-1522 (1993). S. Luryi, A. A. Grinberg, and V. B. Gor®nkel, "Heterostructure bipolar transistor with enhanced forward diffusion of minority carriers", Appl. Phys. Lett. 63, pp. 1537-1539 (1993). Serge Luryi - 9 - Bipolar Junction Transistors Base shrinkage and ®nite output conductance. Emitter Base Vce Collector Ib = 0.6 mA Ic (mA) 60 0.4 Early voltage 40 0.2 20 Vce VA Figure: Base shrinkage and Early effect. Neglecting recombination, the injected minority current must be constant in → → the base, ∇ . J = 0 . If the current is by diffusion only, then β V eb dn n (0) − n (W ) np 0 e ___ = const = _____________ ∼ _________ dx W W W shrinks with the collector bias. Hence, at a ®xed base current, one has an increasing collector current (Early effect) V ∝ + ____cb I C 1 V A Finite output conductance is detrimental; we live better without it. The Early voltage depends on the base width W and the Gummel number = n G N A W . For W >> than the BC junction depletion width, one has e N W 2 e n W ∼ _________A ≡ _______G V A ε ε Serge Luryi - 10 - Bipolar transistor, continued Ebers-Moll model α α I I R N I F i i E C E C REx I F I R RCx CCx i B RBx B E B C Figure 1: Dashed line indicates the "intrinsic" portion of the device, excluding "parasitic" extrinsic elements. The model for the diode blocks can be further speci®ed to include the internal junction capacitance. Serge Luryi - 11 - Bipolar transistor, continued Small-signal equivalent circuit of an abrupt junction HBT α i α CE B E C i i E C C E C C R REx E RCx RB CCx i B RBx B ∆Ε V Figure 2: This model is good for "ballistic" propagation of carriers across the base. For diffusive propagation, the intrinsic portion must be adjusted,² _______________ ² A.
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