chap 11. Charge storage and charge-control MODEL

BIPOLAR TR 에서는 space charge 와 injected minority charge 등이 존재하는데 우선 injected minority charge 에 대해 생각해보자.

Emitter charge = [eqn: Q_E]                              E          B         C

Base charge = [eqn: Q_B]

collector charge = [eqn: Q_C]

F              R

* Forward bias 의 경우

[eqn: Q_E] 와 [eqn: Q_C] only

Reverse bias의 경우는 [eqn: Q_C]와[eqn: Q_B] only.

우선 [eqn: V_BC =0]인 경우를 보면 diode때와 같이

[eqn: Q_E = qA_E p_n e^{qV_BE / kT } X_E /2]      ,

[eqn: Q_BF = qA_E n_p e^{qV_BE / kT } X_B /2]

current equation을 [eqn: Q_B]와[eqn: Q_E] 견지에서 쓰면

[eqn: I_C = { qAD_n n_p e^{qV_BE / kT } } over X_B = Q_BF / X_B^2 / 2D_n]  ,

[eqn: I_C = { qAD_p p_n e^{qV_BE / kT } } over X_B = Q_E / X_E^2 / 2D_p]

Define [eqn: tau_B == X_B^2 / 2D_n] and   [eqn: tau_E == X_E^2 / 2D_p]

then,       [eqn: Q_E ~propto ~ Q_BF ~propto ~ e^{qV_BE / kT }]

전형적으로

[eqn: Q_E ~<< ~ Q_BF ~~~~~BECAUSE n^+ >> p]

따라서 [eqn: Q_E + Q_BF = q_F approx Q_BF] 즉  [eqn: I_C = q_F / tau_F]  where   [eqn: tau_F approx X_B^2 / 2D_n]

[eqn: I_C = q_F over tau_F]

[eqn: Q_E]       [eqn: Q_BF]

now, what about [eqn: I_B] ?

Base

[eqn: I_B = q_F over tau_BF]

[eqn: I_BF == q_F over tau_BF ~,~ tau_BF = q_F over q_E cdot tau_E]

이러한 [eqn: I_C 와~ I_BF] 를  [eqn: q_F~ 및 ~tau_F ~, ~ tau_BF] 로 표현하는 것이 바로 forward bias 경우의 charge control relationship이라 한다.

obviously, [eqn: beta_F = I_C over I_BF = tau_BF over tau_F ~~therefore ~tau_BF = beta_F tau_F]

[eqn: [~ q_F simeq Q_BF ~,~ tau_F = x_B^2 over 2D_n ~]~,~ tau_BF = beta_F tau_F]

Reverse 특성을 보면,

[eqn: V_BE ~=~ 0 ~이고 ~V_BC >0 ~의~상황에서]

[eqn: Q_C ~=~ qA_E p_n e^{qV_BC / kT} cdot L_p ~~~Q_BR ~=~ qA_E n_p e^{qV_BC /kT} cdot x_B over2]

[eqn: I_E ~=~ Q_BR over tau_B ~,~ I_BR ~=~ Q_C over tau_C ~~where ~tau_C ~=~ L_p^2 over D_p]

[eqn: I_E ~simeq~ q_R over tau_R ~,~ where ~~ tau_BR ~=~ q_C over { q_R tau_C }]

[eqn: I_B ~simeq~ q_R over tau_BR ~~FOR ~q_R ~APPROX~ q_C ~,~ tau_BR ~approx~ L_p^2 over D_p]

[eqn: therefore ~beta_R ~=~ tau_BR over tau_R]

↓ [eqn: I_C] , [eqn: V_BC]

[eqn: I_B] →

+

[eqn: V_BC]

_

[eqn: I_C]

[eqn: ↑V_BE]

[eqn: ↑ I_B]

[eqn: V_CE]

First-order [eqn: beta_F] is constant linear

linear [eqn: I_C] vs [eqn: I_B]

Space charge vs injected minority charge

[eqn: q_F] 와 [eqn: q_R] 은

[eqn: q_F =I_C tau_F]   [eqn: V_BE > 0] , [eqn: V_BC ≤0]

[eqn: q_R = I_E tau_R]   [eqn: V_BC > 0] , [eqn: V_BE ≤0]

이 값들을 space charge 값들과 비교

transient device response를 위해 [eqn: q_EV] , [eqn: q_VC] 를 정의.

각각 BE 와 BC space charge에 zero bias 에서 positive bias 로 옮길 때 더해지는 transient charge 라고 함

기본적으로 [eqn: INT dQ_SC = INT C(V)dV]

[eqn: q_VE = INT _{ 0}^{V_BE }[C_JE / ( 1- V over psi _i ^BE )^m,BE ]dV]     [eqn: q_VC = INT _{ 0}^{V_BC }[ C_JC / ( 1- V over psi _i ^BC )^m,BC ] dV]

[  ] 내는 bias dependent junction capacitance 임

[eqn: psi_i]는 built-in voltage , m은 0.3∼0.5 ( 0.5는 abrupt junction의 경우 )

+q

[eqn: q_F]                    [eqn: k_OR] small [eqn: V_BE]> 0, [eqn: q_V_E] > [eqn: q_F]

large [eqn: V_BE] > 0,  [eqn: q_V_E]  < [eqn: q_F]

[eqn: q_V_E]

[eqn: V_BE]

-q

complete DC charge-control model and Equations

[eqn: I_C = q_F over tau_F - q_R (1 over tau_R + 1 over tau_BR )]

[eqn: I_E = q_F over tau_R - q_F (1 over tau_F + 1 over tau_BF )]

[eqn: I_B = q_F over tau_BF + q_R over tau_BR]

For Time-dependent case

[eqn: i_C = I_C (dc) - dq_R over dt - dq_VC over dt]

[eqn: i_E = I_E (dc) - dq_F over dt - dq_VE over dt]

[eqn: i_E = I_E (dc) - dq_F over dt + dq_R over dt + dq_VE over dt + dq_VC over dt]

Saturation

Biploar device에서 saturation 이란 [eqn: V_BE >0 ~, `V_BC >0] 인 상태로 통상의 E-B (Forward bias), B-C(Reverse bias)와 달리 두 juction이 모두 Forward bias가 걸리는  상태이다.   즉 [eqn: q_F >0 ~,~ q_R >0] 이 된다. circuit의 관점에서 보면 BJT가 saturation 되면 [eqn: I_C]는 [eqn: I_C ^max] 로 더 이상 바뀌지 않으며 I_B 만 늘어나기 때문에 [eqn: I_B > I_C ~/~ beta_F ~가 ~된다.]

[eqn: therefore]Edge of saturation (EOS) [eqn: == I_BEOS = I_C ~/~ beta_F]

[eqn: total ~q ~=~ q_F ~+~q_R]

[eqn: I_C 가 ~fixed ~therefore] EOS 이상에서는 Slope가 변화하지 않는다.

자세한 내용은 M&K 책의 Bipolar section을 강조할 것.

[eqn: q_F (EOS)]

[eqn: i_BS ~==~ i_B (total) ~-~ i_C / beta_F ~~,~~ q_S ~==~ saturated ~charge~=~q_F (tot) ~+~ q_R (tot) ~-~ i_C tau_F]

First order approximation 에 의해

[eqn: i_BS ~=~ q_S over tau_S ~+~ dq_S over dt]

where  [eqn: tau_s ~=~ {1 over tau_F + 1 over tau_R + 1 over tau_BR } over { 1 over {tau_F tau_BR} + 1 over { tau_R tau_BF} +1 over { tau_BR tau _BF}} ~~,~~ beta_F ~=~ tau_BF over tau_F ~and~ beta_R ~=~ tau_BR over tau_R]

Then [eqn: tau_s ~=~ {beta_F (beta_R +1)tau_F + beta_F beta_R tau_R } over { 1+ beta_F +beta_R }]

전형적으로 [eqn: beta_F ~=~ 50 ~,~ beta_R ~=~ 3 ~,~ tau_F ~=~ 0.1ns ~,~tau_R ~=~ 1ns]

[eqn: tau_s ~=~ { (50)(40)(0.1ns) ~+~ (50)(3)(1ns) }over{ 50 +3 +1} ~simeq ~3ns]

소자의 관점에서 보면,

EOS를 넘어서 과잉으로 저장된 charge (excess saturation charge)는 steady state 하에서  [eqn: q_s ~=~ i_BS tau_S] 로 나타낼 수 있다.

Final Comments on Curve-Tracer Figures

[eqn: I_C]                                         [eqn: I_C]

1                                             1

2                      [eqn: I_B](fixed)            2                   [eqn: V_BE]( fixed)

3                                           3

[eqn: V_BC]                                              [eqn: V_CE]

SC                              SＧ          SC                         3  SC

2

６                ５                                   1

[eqn: V_BC] 가 forward bias로 되면 [eqn: I_BR] 이               [eqn: V_BE] 가 고정된 경우,

주도되어 [eqn: V_BE]의 bias가 감소된다.                   saturation에 의해 [eqn: I_B]가 큰 값으로

증가하게 되고 따라서 [eqn: I_BR]이 증가                                                      하며 [eqn: I_C]는 감소한다.

SUMMARY

1. Base Transport model

[eqn: I_E][eqn: rightarrow]  E                     C  [eqn: leftarrow][eqn: I_C]

-          -

[eqn: I_s over beta_R (e^{ V_BC / V_T }- 1 )]  [eqn: V_BE] [eqn: V_BC]     [eqn: I_s over beta_R (e^{ V_BC / V_T }- 1 )]            [eqn: V_T == kT over q]

+   +

B

[eqn: I_E]

[eqn: I_C = I_s ( e^ { V_BE / V_T }- e^{V_BC / V_T } )- I_s over beta_R ( e^{V_BC / V_T } -1 )]

[eqn: I_E =- I_s over beta_F ( e^{V_BE / V_T } -1 )- I_s ( e^ { V_BE / V_T }- e^{V_BC / V_T } )]

[eqn: I_B = I_s over beta_F ( e^{V_BE / V_T } -1 ) + I_s over beta_R ( e^ { V_BC / V_T }- 1 )]

[eqn: I_s == qAn_i^2 over {int rho over D_n dx}]

2. space charge

[eqn: p_p_o]

[eqn: n_n_o]        for [eqn: p^+ n] diode

[eqn: p_n_o]             [eqn: X_p << X_n ~~ THEREFORE X_d approx X_n]

[eqn: n_n_o]

[eqn: X_d]

applied voltage [eqn: V_a] 가 [eqn: X_d]에 영향을 주며 따라서 [eqn: q_V_d]에 영향을 줌

[eqn: CDOT]Junction capacitance

[eqn: dQ over dV = C_j (V) = C_jphi ( 1- V over psi_i ) ^ -m]

for step junction

[eqn: C_{j phi} = left [ {q epsilon_si N_D N_A } over 2psi_i ( N_D + N_A ) right ] ^ +m]     m=0.5  , [eqn: psi _i = V_T ln {N_D N_A } over n_i^2]

[eqn: dQ over dV = c(V)]  , [eqn: dQ = c(V) dV]     ,[eqn: Q= int from V=V_1 to V=V_2 C_ {j phi } (1 - V over psi_i ) ^ -m dV]

[eqn: Q ~=~ C_j Phi(- PSI_i )( 1 over {1-m} ) {left[ left(1- V over PSI_i Right)^(1-m) Right]}`_{V_1 } ^{ V_2 } { }`_{ } ^{ }]

[eqn: V_D] [V]	[eqn: C(V)] [fF]	[eqn: qV_D left( Int_{0}^{V_D}C(V) dV right)] [fC]
-0.7	71	-56
-0.3	82	-26
0	95	0
+0.2	106	+20
+0.4	120	+43
+0.6	158	+71

*sample calculation of stored space charge & Juction capacitance[eqn: 10 times 10 ~mu^2 ~,~N_D =10^17 ~,~N_A = 10^19]

3. Injected minority carrier charge

[eqn: I_n ~=~ qA D_n over L_n n_p_o (e^V/V_T ~-~ 1)]

[eqn: I_p ~=~ qA D_p over L_p p_n_o (e^V/V_T ~-~1)]

Since [eqn: p^+ n ~(e.g ~ N_A ~=~ 10^19 ~,~N_D ~=~ 10^17 ) ~~p_n_o >>n_p_o ~~ therefore ~I simeq I_P]

Stored minority carrier charge

[eqn: Q_n ~=~ qAn_p_o (e^V/V_T -1) L_n ~,~ Q_p ~=~ qAp_n_o (e^V/V_T - 1) L_p ~,~~ Q simeq Q_p ~~since ~p^+ n]

characteristic time [eqn: tau] such that [eqn: I ~=~ Q over tau]

[eqn: tau~==~ Q over I ~=~ {qAL_p p_n_o (e^V/V_T -1)} over { qA D_p over L_p p_n_o (e^V/V_T -1)} ~=~ L_p^2 over D_p ~=~ {tau_p D_p }over D_p ~=~ tau_p ~( life ~time ~for ~holes)]

Diffusion capacitance

[eqn: Q ~=~ qAL_p p_n_o (e^V/V_T -1) ~therefore dQ over dV ~=~ {qAL_p p_n_o (e^V/V_T -1) }over V_T ~=~ Q over V_T]

[eqn: Q~=~ I tau_p ~~so ~~ dQ over dV ~=~ {tau_p I}over V_T]

Small Signal Conductance

[eqn: I= qAD_p over L_n p_no( e^{V/V_T}-1),~ dI over dV = g_D = I over V_T ~or~r_D = V_T over I]

Small Signal Model

[eqn: R_s]                   [eqn: r_D ∝ 1 over I_DC]

[eqn: V_DC]        [eqn: r_D]   [eqn: C_D]  [eqn: C_j]        [eqn: C_D ∝ I_DC]

∼ [eqn: V_s]                               [eqn: C_j =f(V_DC )]

Sample calculation of stored minority carrier charge & Diffusion Capacitance

[eqn: V_D]           [eqn: C_D]         [eqn: q_D]         [eqn: C_j]                            [eqn: C_D]

0              0           0

0.2           0.275fF      0.0071fC      106fF                         [eqn: C_j]

0.4            621fF         16fC       120fF

0.6            1403pf        36pC       158fF

4. Charge Control Model

목적→ Derive a set of equations for terminal currents as a function of Q and [eqn: dQ over dt]

방법→ ① Define [eqn: Q_BF], [eqn: Q_E ,~ Q_BR ,~ Q_C] as a total excess charge in respective regions in                  forward mode( [eqn: Q_BR ,~Q_C] )

② Approximate these charges with two base charges [eqn: q_F] ( Forward mode ) & [eqn: q_r] (             Reverse mode )

③ Express terminal currents as function of these two charges and space region                  charge

active mode excess minority charge and currents

[eqn: □I_nC]

[eqn: Q_BF]

[eqn: Q_E]                            [eqn: V_BC =0]

[eqn: X_E]          [eqn: X_B]

[eqn: I_C = { qA_E D_n n_p_o ( e^{ V_BE / V_T } - 1 ) } over X_B]   ,    [eqn: I_BF = { qA_E D_p p_n_o ( e^{ V_BE / V_T } - 1 ) } over X_E]

[eqn: Q_BF = qA_E X_B over 2 n_p_o ( e^ { V_BE / V_T} -1)]    ,    [eqn: Q_E = qA_E X_E over 2 p_n_o ( e^ { V_BE / V_T} -1)]

[eqn: I_C = Q_BF over tau_B]  where   [eqn: tau_B == X_B^2 over 2D_n]

[eqn: I_BF = Q_E over tau_E]  where   [eqn: tau_E == X_B^2 over 2D_p]

Now define  [eqn: q_F == Q_BF + Q_E]    ( [eqn: n^+ p], so [eqn: q_F approx Q_BF] )

[eqn: I_C = q_F over tau_F]  where  [eqn: tau_F = tau_B]

[eqn: I_BF = Q_E over Q_E 2D_p over X_E^2 = q_f over tau_BF]   (note : [eqn: beta_F = I_C over I_BF = tau_BF over tau_F = Q_F over Q_E tau_E 1 over tau_F] )

Reverse mode excess minority charge and currents

[eqn: I_nE]

[eqn: rightarrow]

[eqn: Q_C]

[eqn: Q_BR]

[eqn: I_E ={ qA_C D_n n_p_o ( e^{ V_BC / V_T } - 1 ) } over X_B]        [eqn: Q_C ={ qA_C L_p p_n_o ( e^{ V_BC / V_T } - 1 ) }]

[eqn: I_BR ={ qA_C D_p} over L_p p_n_o ( e^{ V_BC / V_T } - 1 ) }]       [eqn: Q_BR = qA_C X_B over 2 n_p_o ( e^ { V_BC / V_T} -1)]

[eqn: I_E = Q_BR over {X_B^2 /2D_n } = Q_BR over tau_B]    ,     [eqn: I_BR = Q_c over { L_p^2 / D_p } = Q_c over tau_c] , [eqn: tau_c = L_p^2 over D_p]

now define [eqn: q_R = Q_BR + Q_c ~~(Q_C > Q_BR ~~ BECAUSE L_p > X_B ~, ~ p_n_o > n_p_o ~ THEREFORE ~ q_R approx Q_C )]

[eqn: I_BR = q_R over tau_BR]  where [eqn: tau_BR == L_P^2 over 2D_p]  ,   [eqn: I_E = I_BR beta_R = q_R over tau_BR beta_R = q_R over tau_R]

where [eqn: tau_R = tau_BR over beta_R]

DC charge controll summary

[eqn: I_C ~=~ q_F over tau_F ~~ I_BF ~=~ q_F over tau_BF]

[eqn: tau_F ~=~ x_B^2 over 2D_n ~,~ tau_BF ~=~ beta_F tau_F]

[eqn: I_E ~=~ q_R over tau_R ~~ I_BR ~=~ q_R over tau_BR]

[eqn: tau_BR ~=~ L_p^2 over D_p ~~tau_R ~=~ tau_BR over beta_R]

Combined Forward & Reverse

Charege control DC model

[eqn: I_C ~=~ q_F over tau_F ~-~ q_R ( 1 over tau_R ~+~ 1 over tau_BR )]

[eqn: I_E ~=~ q_R over tau _R ~-~ q_F ( 1 over tau_F ~+~ 1 over tau_BF )]

[eqn: I_B ~=~ q_F over tau_BF ~+~ q_R over tau_BR]

Time dependent charge control model

[eqn: i_B ~=~ q_F over tau_BF ~+~ q_R over tau_BR ~+~ dq_F over dt ~+~ dq_R over dt ~+~ dq_VE over dt ~+~ dq_VC over dt][eqn: i_C ~=~ q_F over tau_F ~-~ q_R ( 1 over tau_BR ~+~ 1 over tau_R ) ~-~ dq_R overdt ~-~ dq_VC over dt]

[eqn: i_E ~=~ q_F over tau_F ~-~ q_R ( 1 over tau_BR ~+~ 1 over tau_R ) ~-~ dq_F over dt ~-~ dq_VE over dt]

FORWARD ACTVE MODE EXAMPLE

[eqn: beta_F =50]

[eqn: tau_F =0.1ns]

[eqn: tau_BF = 50 TIMES 0.1 =5ns]

[eqn: t_1]            t

appropriate diffusion equation : [eqn: I_B = q_F over tau_BF + dq_F over dt]

[eqn: q_R =0]  [eqn: DELTA V_EB ~small~, ~DELTA q_VE ~ ~DELTA q_VC □small~,~ DELTAq_R =0]

[eqn: DELTA V_BC ~small]

Solve for [eqn: q_F (t)] then ,

[eqn: q_F (t)=Ae^{-t/ tau_BF }+I_B2 tau_BF]       so [eqn: A=I_B1 tau_BF -I_B2 tau_BF]

[eqn: q_F (0^- )=I_B1 tau_BF =25pC]             [eqn: q_F (t)=-25e^{-t/ tau_BF } +50pC]

[eqn: q_F (□)=I_B2 tau_BF =50pC]

[eqn: □I_C (t)= {q_F (t)} over tau_F = -0.25e^{-t/ tau_BF } +0.5mA]

5. Saturation Model (DC)

[eqn: beta_F =50~,~ tau_F =0.1ns~~so~ tau_BF =5ns]

[eqn: beta_R =3~,~ tau_R = 1ns]        [eqn: tau_BR =3ns]

□

[eqn: I_B =] 10□A                [eqn: I_C =] 0.5nA

50□A                      2.5nA

100□A                       5nA

200□A                      10nA                               i.e.     [eqn: q_s]

500□A                      10nA

Define   [eqn: q_s == q_R + q_F - q_FEOS]

[eqn: i_BS == i_Btotal - i_BEOS]

then

[eqn: i_BS = q_s over tau_s]   at steady state   and  [eqn: i_BS = q_s over tau_s + dq_s over dt]

Example

[eqn: beta_F = 50 ~~ beta_R = 3]

[eqn: tau_F = 0.1 ns ~~~ tau_R = 1ns]

then

[eqn: tau_s = { beta_F ( beta_R + 1 ) tau_F + beta_F beta_R tau_R } over { 1 + beta_F + beta _R } = 3ns]

[eqn: i_BS = q_s over tau_s + dq_s over dt]

[eqn: i_BS1 = 500 mu A - 200 mu A = 300 mu A]

[eqn: THEREFORE q_S1 = 300 mu A CDOT3ns = 900 pC]

[eqn: i_BS2 = -50 mu A - 200 mu A = -250 mu A]

[eqn: THEREFORE q_S2 = -250 mu A CDOT3ns = -750 pC]

so,  [eqn: q_s (t) = (900 + 750 ) e^ { -t/tau_s } -750 pC]

[eqn: q_s (t) = 0] when [eqn: e^ { -t / tau_s } = 750 over {900 + 750 }]

[eqn: t = tau_ s ln 1650 over 750 = 3ns CDOT 0.788 = 2.37 ns]