# About Modeling the Reverse Early Effect in HICUM Level 0

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1 About Modeling the Reverse Early Effect in HICUM Level 0 6 th European HICUM Workshop, June 12-13, 2006, Heilbronn Didier CELI, STMicroelectronics 1/21 D. Céli

2 Purpose According to the bipolar models, the reverse Early effect is modeled in various ways From device physics in models such as HICUM, MEXTRAN or VBIC. With more or less important approximations for models like SPICE Gummel-Poon (SGP) or HICUM Level 0. The question is to know if these approximations are justified and which are their impacts on the model accuracy. Application to HICUM/L0 2/21 D. Céli

3 Outline Introduction Why Reverse Early effect is so important? How is it implemented in BJT models? Experimental results and HICUM/L0 limitation Summary 3/21 D. Céli

4 Introduction Despite its name, the reverse Early effect affects strongly the collector current in forward mode. Difficult to observe on forward Gummel plots, its effect being concealed by the exponential dependence of I C with V BE V BC = 0 V V AR = 1.5 V V AR = 2.5 V V AR = 5 V V AR = 100 V I C [A] V AR The slope decreases with an increase of the reverse Early effect (lower reverse Early voltage) V BE [V] 4/21 D. Céli

5 But clearly visible on the forward current gain... Normalized Current Gain b/b F V BC = 0 V V AR Normalized Collector Current I C /I KF V AR = 1.5 V V AR = 2.5 V V AR = 5 V V AR = 100 V The forward current gain decreases with an increase of the reverse Early effect (lower reverse Early voltage). This effect is not always well known from designers. We can observe that the forward current gain is divided by 2 at I C = I KF. It is a way to directly extract the forward knee current I KF of the SGP model, from the forward current gain corrected of the reverse Early effect. 5/21 D. Céli

6 ...or better on the normalized collector current I C I S e V BE V BC = 0 V Normalized Collector Current I C /I S e V BE / V AR V BE [V] V AR = 1.5 V V AR = 2.5 V V AR = 5 V V AR = 100 V We can notice, that in fact this normalized collector current is, at low and medium V BE (linear part of the characteristic), simply the inverse of the normalized hole charge Q p /Q p0. The reverse Early effect can be directly quantified from the slope of this normalized collector current. It is the only way to directly extract (linear regression), without any ambiguity, the reverse Early voltage V AR of the SGP model. 6/21 D. Céli

7 Why the Reverse Early Effect so Important? Whatever the technology, for vertical (conventional) BJTs, the reverse Early effect is not negligible (low equivalent reverse Early voltage on the order of 1 to 5V). Why? Initially, the inverse Early voltage V AR was introduced, in the SGP model, in order to take into account the base width modulation with V BE. In fact, in advance bipolar technologies, the base doping is very high, and consequently, the base width is less affected by a V BE variation. Therefore, low reverse Early voltages, for this kind of technologies, can not be explained only by a modulation of the base resulting from a variation of the BE space-charge layer width. 2 possible explanations The reverse Early voltage does not depend only on the base concentration, but also on the shape of the base doping (Si BJTs). Reverse (effective) Early voltage can be also effected by the Ge profile in the base of HBTs. 7/21 D. Céli

8 Effect of the Shape of the Base Doping on I C (V BE ) Si BJTs Net Doping [cm -3 ] GHz 0.5 mm BiCMOS E B C N A (V BE2 ) N A (V BE1 ) V BE + The collector current depends on the injected electron in the base, at the EB depletion layer boundary I C V BE v T I n b0 ( V BE ) e or C I C* = n b0 ( V BE ) If V BE increases e V BE v T N A ( V BE2 ) > N A ( V BE1 ) 2 n Since i n b , we have n b0 ( V BE2 ) < n b0 ( V BE1 ) N A SCL and therefore I C* ( V BE2 ) < I C* ( V BE1 ) VBE2 >V BE1 V BE Depth [mm]] Similar effect than the reverse Early effect (base width modulation with V BE ), where I C* decreases with V BE. 8/21 D. Céli

9 Effect of the Ge Base Profile on the I C (V BE ) HBTs Emitter V BE + Base In HBTs, the collector current depends on the reduction Δ EG of the base bandgap due to the introduction of Ge into the base Δ EG (V BE2 ) Δ EG (V BE1 ) graded Ge profile I C* I C e V BE = e v T Δ EG v T In graded Ge base profile (triangle or trapezoidal), if V BE increases Δ EG ( V BE2 ) < Δ EG ( V BE1 ) SCL and therefore I C* ( V BE2 ) < I C* ( V BE1 ) x E (V BE2 ) x E (V BE1 ) x This effect is responsible of the very low effective reverse Early voltage (decrease of I C* with V BE ) of S i G e HBTs with graded Ge base profile. 9/21 D. Céli

10 How the Reverse Early Effect is Implemented in BJT Models? Depending on the model, several approaches more or less physics based Physics Based and Complexity Accuracy HICUM/L2 SGP HICUM/L0 Model HICUM/L2 SGP HICUM/L0 Model 10/21 D. Céli

11 HICUM Level 2 Low injection, V BC = 0 V I S e I C Q jei 1 + h jei Q p0 = I S e h jei q jei h jei weighting factor in HBTs q jei normalized BE depletion charge 11/21 D. Céli

12 SPICE Gummel-Poon Low injection, V BC = 0 V 1 q jei = Q p0 C jei ( V) dv 0 In SGP model, C jei is assumed to be constant and equal to its average value C jei I S e I S e I S e Q p0 I C h jei q jei 1 h = jei C jei V with V AR = BEi h jei C jei Q V p0 AR Moreover, In SGP model it is assumed (questionnable for advanced HBTs) V AR >> V , and finally I AR V C AR V IS e AR 12/21 D. Céli

13 HICUM Level 0 Low injection, V BC = 0 V Same approximation than in SGP model, it is assumed that V AR >>. Therefore 1 V BEi V AR can be considered as the first two terms of the Taylor series expansion of e V AR I C V IS e V AR V V T T V AR I S e e = I S e AR V We can consider /V AR << 1, T can be rewritten as V AR V AR V, AR which gives the final expression of I C V V VT BEi AR M CF I C I S e = I S e with M CF = V AR 13/21 D. Céli

14 Few comments concerning the relation between M CF and V AR As M CF = and V AR >> V AR therefore M CF is very close to 1. In order to reproduce accurately the reverse Early effect, M CF must be given at least with 5 digits. V AR (V) M CF M CF (4 digits) V AR (V) M CF (5 digits) V AR (V) As M CF = , it could be temperature dependant (assuming V AR temperature independent). V AR To be investigated in more detail (to a next HICUM workshop!...) Temperature ( C) V AR (V) M CF /21 D. Céli

15 Experimental Results HICUM/L2, SGP, HICUM/L0 1 High current parameters not fully extracted I S = A C 10 = A.C Q P0 = fc H JEI = I S = A V AR = V 1 I S = A M CF = I C /[I S.exp(V BE / )] I C /[I S.exp(V BE / )] I C /[I S.exp(V BE / )] V BC = 0 V V BC = 0 V V BC = 0 V V BE [V] V BE [V] V BE [V] HICUM/L2 SGP HICUM/L0 Despite the different approaches used to model the reverse Early effect, all models give similar accuracy (negative slope well described) on the normalized collector current (or current gain), at V BC = 0 V. But it is no more valid for positive V BC, in the saturated region. 1. ST solver 15/21 D. Céli

16 Gummel PLots with Positive V BC 10-6 S C I C [A] I C < 0 V BC = 0.5 V I C > 0 B PNP B E NPN NPN I BC I T I SC C V BC = 0 V I C I BC E I TS I BE S V BE [V] DC low current simplified equivalent circuit I C = I T I BC I SC I SC 0 because Vsc = 0 V BE V BC at V BC = V BE (0.5 V) but not in HICUM/L0 I T e e = 0 I C I BC V BE V BC M CF V M T CR V T I T e e 0 16/21 D. Céli

17 Limitation of HICUM/L0 and solution to minimize this modeling issue V BC = 0.5 V V BC = 0.5 V I C, I B [A] SGP, HICUM/L2 HICUM/L0 I C, I B [A] SGP, HICUM/L2 HICUM/L0 M CR = M CF HICUM/L0 M CF, M CR = M CF V BE [V] V BE [V] In saturated region, and especially for V BE = V BC, the shift between HICUM/L0 and other models increases with M CF (high reverse Early effect, low reverse Early voltage). To minimize this effect, a solution consists of set M CR equal to M CF. Therefore, in this case, the transfer current I T is equal to zero at V BE = V BC V BE V BC M CF V M T CR V T I T e e = 0 if M CF = M CR This correction is to the detriment of the accuracy on the reverse Gummel plot (slope to low). 17/21 D. Céli

18 The same behavior can be observed on the forward output characteristics. Here again, M CR needs to be set to M CF in order to fit accurately the characteristics at low V CE I C [ma] 3 2 I C [ma] M CR = M CF 1 0 M CF, M CR = V CE [V] V CE [V] 18/21 D. Céli

19 Summary The modeling of the reverse Early effect in HICUM/L0 has been investigated. Its formulation is a simplification of the SPG model which leads to the introduction of a non-ideality factor M CF on the forward transfer current. We have demonstrated that in some bias conditions (saturated region), this model can be not enough accurate. To overcome this model issue, 2 possibilities: To use the same non-ideality factor in forward and reverse (M CR = M CF ). Inconsistent with a correct description of the reverse Gummel characteristics. To come back to the SGP model formulation by using a reverse Early voltage V AR instead of a nonideality factor M CF. One pending question: as M CF is linked to V AR via, can we assume M CF temperature independent? 19/21 D. Céli

20 Addenda (1/2) An other important limitation of HICUM/L0, in the saturated region, is the lack of parasitic substrate PNP. With the present model it is not possible To describe the bias dependence of the forward Gummel plot (especially I B) with positive V BC. I C = I T I BC I SC 10-5 I B = I TS + I BE + I BC 10-6 NPN I BC I T C I C [A] V BC = 0 V B I SC V BC = 0.5 V I TS 10-9 I BE E S V BC = 0 V V BE [V] To accurately model the output characteristics at low V CE. 20/21 D. Céli

21 Addenda (2/2) Therefore, the implementation of the parasitic substrate PNP, in HICUM/L0 is always requested (using a flag if needed). NPN S C C I BC PNP I T B NPN B I TS I SC E E I BE S 21/21 D. Céli

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