ECE- 66 SOLUTIOS: ECE 66 Homework Week 1 Mark Lundstrom (Revised 3/9/13) 1) In a forward- biased P junction under low- injection conditions, the QFL s are aroximately flat from the majority carrier region and then across the deletion region As shown in the figure below for a long diode, the minority carrier QFL then decays linearly to the majority carrier QFL Exlain why the minority carrier QFL decays to the majority carrier QFL and why the decay is linear First, recall the definitions of the QFLs: ( n = C e F n E C ) k T B ( and = V e E V F ) k B T Far away from the junction, the excess carriers should have decayed to zero The hole and electron concentrations should be the equilibrium values and: n = n i ( From the above equations, n = n i e F n F ) k B T Accordingly, we conclude that far away from the junction: F n = F Since we are in low level injection, the majority carrier QFL does not change, so the minority carrier QFL aroaches the majority carrier QFL ECE- 66 1 Sring 13
ECE- 66 HW Week 1 Continued To understand why the decay is linear, consider the quasi- neutral P- side where: Δn( x) = Δn( x )e ( x x ) L n ( n Δn = C e F n E C ) k T B F n x F n ( x) = F n x ( k B T x x ) L n = k B T ln Δn x C e ( x x )/ L n ote: As F n ( x) F, we need to be more careful and use n = n + Δn ) The diffusion caacitance is imortant for the ac resonse of a diode Consider an P + diode under forward bias A common way to estimate the low- frequency diffusion caacitance is to make the quasi- static aroximation: V A C D = dq dv A where Q given by Q is minority carrier charge obtained from the dc solution and is V A ( V A ) = Δ( x) x Comute the quasi- static diffusion caacitance for long P + diode and comare it to the low- frequency diffusion caacitance as given by Pierret in SDF, eqn (73b) For a long + P diode: Δ( x) = Δ( )e x/ L (x = is at the beginning of the quasi- neutral P region) Δ = n i I D = qa D n i L ( e qv A/k BT 1) ( e qv A/k BT 1) ECE- 66 Sring 13
ECE- 66 HW Week 1 continued The total charge stored in the quasi- neutral P- region is: Q = qa n i e qv A/k BT 1 e x/ L = qa n i D ( V A ) = Δ x V A C D = dq dv A = qa n L i ( e qv A/k BT ) q k B T ( e qv A/k BT 1) L For modest forward bias: I D = qa D n i e qv A/k BT L G = di D = qa D n i e qv A/k BT dv A L q k B T C D G = L = τ D or finally: C D = G τ One might exect the quasi- static diffusion caacitance to be the low- frequency limit of the frequency- deendent diffusion caacitance But in comaring the quasi- static answer above to the exact, low frequency diffusion caacitance, eqn (73b) of Pierret, SDF, we see that the quasi- static diffusion caacitance is times the low frequency diffusion caacitance In ractice, the diffusion caacitance deends exonentially on the forward bias, so a fact or two is not a big deal, but one should be aware of this discreancy 3) For a P+ diode, the forward biased current due to minority holes injected into a short - tye region is: J = q n D i e qv A/k BT 1 A ECE- 66 3 Sring 13
ECE- 66 HW Week 1 continued It is often the case, however, that the doing in the - tye region is not constant Work out the current equation for a P+ diode with a non- uniformly doed region by answering the following questions 3a) Obtain an exression for the electric field, E x ( x) n x, assuming quasi- neutrality, In equilibrium: J n = nqµ n E + qd n dn = so E = k B T q 1 dn n E k B T q 1 d 3b) Evaluate the hole current assuming the electric field from 3a) and show that the result is: J P ( x) = q D x d ( Δ) J = qµ E qd d J = qd d qd d J = q D d 3c) Exlain why J ( x) is constant for a short - tye region According to the hole continuity equation: t = d J q R So under steady- state conditions and for no recombination, J ( x) must be constant ECE- 66 4 Sring 13
ECE- 66 HW Week 1 continued 3d) Using the results of 3b) and 3c), solve for J in terms of Δ( ) and Δ( ) J = q D d J D ( )( ) qd D = d ( )( ) J qd D ( x) = J qd D = d ( ) ( ) ( ) ( ) ( ) = n i W ( P ) Δ( ) = J qd D ( x) n i e qv A /k B T J qd n i ( x) e qv A /k B T 3e) Comare the result to that of a uniformly doed region n Uniform doing: J qd i e qv A/k BT onuniform doing: J qd n i ( x) e qv A /k B T Difference: W ( x) which is known as the Gummel number ECE- 66 5 Sring 13
ECE- 66 HW Week 1 continued 4) Consider an P BJT with the following arameters: I En = 1 ma I E = 1 ma I Cn = 98 ma I C = 1 ma 4a) Determine the base transort factor, α T α T = I Cn I En = 98 1 = 98 α T = 98 4b) Determine the emitter injection efficiency,γ I γ = En 1 = = 99 γ = 99 I En + I E 1+ 1 4c) Determine the currents, I E, I C, I B I E = I En + I E = 1 +1= 11 I E = 11 I C = I Cn + I C = 98 +1= 981 I C = 981 I B = I E I C = 11 981= 99 I B = 99 4d) Determineα DC α DC = γ α T = 99 98 = 97 = I C I E α DC = 97 4e) Determine β DC β DC = α DC = 97 1 α DC 3 = 33 β = 33 DC ECE- 66 6 Sring 13
ECE- 66 HW Week 1 continued 5) Make sketches of the minority electron concentration in the base of an P BJT under the following conditions: 5a) Forward active region 5b) Inverted active region 5c) Saturation ECE- 66 7 Sring 13
ECE- 66 5d) Cut- off HW Week 1 continued OTE: in the above sketches we assume e qv BE /k B T 1 e qv BE /k B T for forward bias, because it takes only a small forward bias to make the - 1 negligible 6) Consider the imact of changing some key BJT device arameters on some key figures of merit For each of the 4 changes below, exlain whether the change increases, decreases, or has no effect on: i) the base transort factor, α T, ii) the emitter injection efficiency,γ, and iii) the current gain, β DC 6a) Increase the base width i) the base transort factor, α DECREASES T ii) the emitter injection efficiency,γ DECREASES iii) the current gain, β DC DECREASES 6b) Increase the base lifetime i) the base transort factor, α T ICREASES ii) iii) the emitter injection efficiency,γ DECREASES the current gain, β DC ICREASES 6c) Increase the base doing i) the base transort factor, α T O EFFECT ii) the emitter injection efficiency,γ DECREASES iii) the current gain, β DC DECREASES 6d) Increase the emitter doing ECE- 66 8 Sring 13
ECE- 66 i) the base transort factor, α T O EFFECT ii) the emitter injection efficiency,γ ICREASES iii) the current gain, β DC ICREASES ECE- 66 9 Sring 13