Chapter 2. Small-Signal Model Parameter Extraction Method

Transcription

1 Chapter Sall-Signal Model Paraeter Extraction Method In this chapter, we introduce a new paraeter extraction technique for sall-signal HBT odeling. Figure - shows the sall-signal equivalent circuit of a SiGe HBT under forward active ode operation. The odel is based on the well-known hybrid- equivalent circuit with a three-eleent substrate network adding a parallel RC block connected in series with collector-substrate capacitance []. The equivalent circuit is divided into two parts, the inner part (in the dashed box) containing the as-dependent intrinsic eleents, and the outer part with the ostly as-independent extrinsic eleents.. Extraction of Extrinsic Eleents To extract the equivalent circuit paraeters, pad parasitics are carefully reoved fro easured S-paraeters through a de-ebedding procedure using an open test pattern. Soe reaining parasitics not reoved in the de-ebedding procedure such as parasitic capacitances, pad inductances and series resistances are relatively sall, but lead eventually to the errors in extracting intrinsic eleents. Thus, their values should be deterined carefully. After reoving the residual parasitics, the extraction of substrate network paraeters and subsequently the intrinsic paraeters can then be perfored... Extraction of pad inductances and series resistances As reported in [3], the extraction of series resistances and pad inductances is ade by 5

2 asing the SiGe HBTs in over-driven I B as condition where the base-collector and base-eitter junctions are in the forward as condition. At high base current density, the base-eitter and base-collector dynaic resistances tend to be nearly zero and the junction capacitances becoe quite large. In such as condition (collector voltage, V CE =0 and driving large forward base current, I B ), the SiGe HBTs can be represented as a siple T-equivalent circuit consisting only of external base, eitter and collector series resistances and pad inductances as shown in Fig. -. The Z-paraeters of the equivalent circuit in Fig. - are defined by the following equation: ( ) Z Z R + R + jω L + L R + jωl bx e b e e e Z Z = Re + jωle Rc + Re + jω( Lc Le) + (-) The series resistances are deterined at low frequency fro the real parts of the Z-paraeters and are show as Rbx Re( Z Z) R c = Re( Z Z). (-) Re Re( Z) At high base current densities, base-eitter and base-collector dynaic resistances becae very sall and Re(Z Z ), Re(Z Z ), and Re(Z ) decrease linearly as a function of /I B, as shown in Fig. -3(a). The Y-axis interception for infinite large base currents yields the values of R bx, R c and R e. The pad inductances can be described by the following equations: ωlb I( Z Z) ωl I( c = Z Z). (-3) ω Le I( Z) 6

3 Using (-3), one can easily get the values of L b, L c and L e fro the slope when plotted I(Z Z ), I(Z Z ), and I(Z ) versus ω at high base current density, as shown in Fig. -3(b)... Extraction of Parasitic Capacitances The parasitic capacitances can be estiated fro the SiGe HBTs under cutoff ode operation [4]. Under such as conditions (V CE =0 and reverse and/or low forward base voltage, V BE ), the effects of series resistances, dynaic resistance, and transconductance are negligible copared with those of the capacitance eleents at low frequency range. The SiGe HBTs equivalent circuit of Fig. - is then reduced to capacitance eleents only, and can be represented by the equivalent circuit shown in Fig. -4. Fro the Y-paraeters analysis of this circuit, we get I( Y + Y ) = ω(c + C ) (-4) bep I( Y ) = ω(c +C +C ). (-5) bcp bcx In (-4) and (-5), the base-eitter junction capacitance C, and the total base-collector junction capacitance, C bcx +C, are considered to be as dependent, while the parasitic capacitances C bep and C bcp are as independent eleents. Both the base-eitter and total base-collector junction capacitances can be described by the well-known expression ( / ) j C = C 0 V V where Cj is the junction capacitance, C j0 is the junction capacitance at j j j pj zero as, V j is the applied reverse voltage, V pj is built-in voltage of the junction, and j is the exponent. Fig. -5 shows the extraction of the parasitic capacitances, C bep and C bcp, are carried out 7

4 be by fitting I(Y +Y )/ω and I( Y )/ω to the expression ( V / V ) and bc ( V / V ), respectively. By varying the paraeters V and be until I(Y +Y )/ω bc ci be shows a linear behavior as a function of be ( V / V ), the extrapolated Y-axis interception be yields the value of parasitic capacitance Cbep. The obtained value of C bep is about 3.00 ff. In a siilar way, the extracted C bcp is about 0 ff.. Extraction of Substrate Network Paraeters Under the as conditions (V BE =0 and forward and/or low reverse collector voltage), the substrate network paraeters can be estiated fro the Y-paraeters analysis of the equivalent circuit shown in Fig. -. After reoving the extrinsic inductances, parasitic capacitances, extrinsic base resistance and collector resistance, the equivalent circuit of Fig. - is reduced to that of Fig. -6(a). The extrinsic inductances, L b, L c and L e are de-ebedded firstly by [ Z ] [ Z] i ( + ) jω Lb Le jωle = jωle jω( Lc + L e ). (-6) where [Z] denotes ipedance paraeters of the device under test (DUT) after de-ebedding the open test pattern. After transforing [Z i ] into its adittance atrix [Y i ], the parasitic capacitances, C bep and C bcp are reoved by ( ) jω C + C jωc Yj = [ Yi] jωcbcp jωc bep bcp bcp bcp. (-7) Converting [Y j ] to its ipedance atrix [Z j ], then R bx and R C can be subtracted by 8

5 [ Z ] k Rbx 0 = Z j 0 R. (-8) c Finally, convert [Z k ] to [Y k ] to obtain the inforation containing the frequency response of intrinsic HBT and substrate network. In this study, the substrate network Y sub, is constituted of substrate-collector depletion capacitance C sub, substrate resistance R bk and bulk capacitance C bk accounting for Si dielectric behavior. For siplicity, the influence of eitter resistance R e has been neglected where the approxiation is valid for ( ) ωrc =. We transfor the intrinsic part of the device e equivalent circuit (R, C, C ) using the well-known T Π transforations shown in detail in the right side of Fig. -6(b). Observing [Y k ] shown in Fig. -6(b), we can derive Y + Y = Y +Y (-9), k, k sub 3 where Y Y sub ( ) ( ) + ω R C C + C = + + ω + + ω + ω RbkCsub bk bk bk sub jωc sub Rbk ( Cbk Csub ) Rbk Cbk Csub ( + ) ( ) ω C C C C R 3 ω CCR 3 = + j + ω R ( C + C) + ω R C + C (-0). (-) Fro (-9), it is clear that (Y,k +Y,k ) deviates fro Y sub by an additional ter, Y 3, which is constituted of the intrinsic circuit eleents. If the extraction of substrate network paraeters is perfored on Y,k +Y,k, the conductance of substrate network will be underestiated and the susceptance of substrate network will be overestiated. To extract substrate network paraeters, Y 3 should be deterined first. Observing Y k shown in Fig. -6(a), R and C can be obtained as 9

6 R Re( Y, k + Y, k) = Re Y, k + Y, k Re( Y, k) (-) ωc = I Y, k + Y, k. (-3) Fig. -7 and -8 show the R extracted fro (-) is nearly constant and the ωc extracted fro (-3) is linear. We find R =.37 Ω and C =75.69 ff at V CE =3.0 V, and V BE =0 V for the µ SiGe HBT. In the low frequency range, Re(Y,k ) can be approxiately rewritten as (-4) ( + ) ( ) ω R C C Re( Y ) ω R C C, k = : ( ) + at low frequency + ω R C + C (-4) under the assuption, ( ) ω R C + C =, it can be expressed a typical result of (-4) at VCE=3.0V, and V BE =0V for a µ capacitance C can be extracted as 6.93 ff fro SiGe HBT.The intrinsic base-collector C β R C = where β is the slope of Re(Y,k ) versus ω plot which shows at Fig. -9. Once R, C and C are obtained, Y 3 and Y sub can be derived fro (-) and (-0), respectively. Fro (-9), it indicates that Y 3 (or intrinsic circuit eleents) indeed affects the easured (Y,k +Y,k ). Figs. 0 and show the coparison between easured (Y,k +Y,k ) and Y sub for a SiGe HBT. Due to the internal feedback signal through Y 3, Re(Y,k +Y,k ) and I(Y,k +Y,k )/ω show a deviation fro Re(Y sub ) and I(Y sub )/ω as operation frequency beyond 5 GHz and 0 GHz, respectively. To extract the substrate network paraeters, two linear equations are derived fro Y sub and are shown as 0

7 ( + ) I( Y ) C C C = + R = k + k (-5) sub bk bk sub bk ω Re( Ysub) ω RbkCsub Csub ω C + C Re( Y ) ω R C C ω bk sub = + R bk = + sub bk sub sub. (-6) As indicated in Fig. -, k and can be obtained fro the linear regression of (-5) and (-6). In Fig. -, C sub and R bk are calculated as 5.65 ff and 64.6 Ω fro k / and /(k ), respectively. Having deterined C sub and R bk, C bk is calculated as 5.fF fro C = ( C /( R Re( Y )) (ω R ) ) C 0.5 bk sub bk sub bk sub [5]. Fig. -3 shows the collector-voltage dependence of the extracted substrate network paraeters. The slightly increased Cbk and slightly decreased R bk probably indicate the reduction of Si bulk region due to the increase of the collector-substrate depletion width []. After extraction of substrate network paraeters, Y sub is de-ebedded through the following equation: 0 0 = k 0 Y. (-7) [ Y ] [ Y ] sub Finally, convert [Y ] to [Z ], de-ebed R e through the following equation, and re-convert [Z n ] to [Y n ] to obtain the adittance paraeters of intrinsic HBT [ Z ] [ Z ] n R R e e = Re R e (-8)..3 Extraction of Intrinsic Circuit Eleents Usually, the adittance paraeters of the intrinsic HBT in coon-eitter configuration, [Y n ] are used to extract intrinsic circuit eleents [6][7]. A uch sipler set of equations is

8 obtained, if the equivalent circuit of the intrinsic HBT is transfored to its coon-collector configuration as shown in Fig. -4 [8][9]. After two-port atrix operations, we arrive at the following ABCD-paraeters [A c ] of the intrinsic HBT A c, RY bc = + (-9) A = + RY + RY ( c, bc g + Y ) (-0) A = Y + Y + R Y Y (-) c, bc ex ex bc A = Y ( + R Y + R Y ) + Y + Y c, ex bc bc g + Y (-) where g g ( j τ ) = exp ω 0, Y = / R + jωc, Ybc = jωc, and Yex = jωcbcx. The advantage of transforing the intrinsic circuit into its coon-collector configuration is that soe circuit paraeters such as g and Y only appear in A c, and A c, and this facilitates the extraction of soe intrinsic circuit paraeters..3. Extraction of R, C, C, and C bcx Fro (-0) and (-), the well-known ABCD-paraeter forulation for extraction of intrinsic base resistance, R, given in [8] is shown as ω ( ) ( + ) + ω ( + + ) A c, R C + C Re = A ω R C C C C C C 4 c, ex ex. (-3) Since the ω 4 ter in denoinator is uch sall than the ω ter at iddle to high frequency range, (-3) is siplified to ( ) ( + + ) A c, C + C Re : R at iddle to high frequency (-4) Ac, C C Cex

9 which set up the lower liit of R. Another equation for extraction of R can be obtained fro (-9) to (-) A C = + + A C ω R C Re c, R c : R C + C be at iddle to high frequency (-5) where A c denotes ( Ac, Ac, Ac, Ac, ). It is clear that (-5) set up the upper liit of R. Typical results of (-4) and (-5) are shown in Fig. -5. We find the lower liit and upper liit of R are 7.3 Ω and 4.85 Ω, respectively, at I B =9.36 µa, and I C =.56 A and V CE =3.0 V for the µ SiGe HBT. The R estiated fro (-4) is then applied to obtain intrinsic base-collector capacitance, C, through following equation I ( A, ) = ωrc. (-6) c Fig. -6 shows the value of R C is calculated fro the slope of (-6) when plotted versus the angular frequency ω. It should be noted that the adopted R fro (-4) only serve as an initial value and it will be corrected in the accuracy iproveent procedure. By exaing (-9) to (-), we can extract intrinsic base-eitter capacitance, C, through following equation A ωr C R ωr C I A c, be be = c + ( ωr bec ) : : R ωc R ωc be C C at iddle to high frequency at iddle to high frequency. (-7) The value of C is calculated as 40.3fF fro the slope of (-7) when plotted versus /ω, as 3

10 shown in Fig. -7. To extract the extrinsic base-collector capacitance, C bcx, we obtain following equation by inspecting (-9) and (-) 3 3 ω ω( ) ( ) + ω ( + ) A RC C C + C I = A ω RC C C C c, bcx bcx 4 c, bcx bcx. (-8) In the low to iddle frequency range, the ω 4 ter in denoinator is uch saller than the ω ter. Also, the ω 3 ter in nuerator is uch saller than the ω ter. Thus, (-8) is siplified to A c, I : at low to iddle frequency. (-9) Ac, ω( C + Cbcx) The value of C +C bcx is calculated fro the slope of (-9) when plotted versus /ω, as shown in Fig. -8. Since the value of C has been previously deterined, C bcx is obtained by subtracting C +C bcx fro C..3. Accuracy Iproveent of R, C and C bcx In (-6) and (-9), we found that R plays a significant role on the accuracy of extracted C and subsquentlly the accuracy of extracted C bcx. However, the estiated value of R fro (-4) is lower than the real one. Thus, an accuracy iproveent procedure is necesarry and listed as follows: ) The value of R estiated fro (-4) is applied in the extraction of C through (-6). The extraction results of C is used in the extraction of C bcx through (-9). 4

11 ) The new lower liit and new upper liit of R is obtained by taking extracted C and C bcx in the calculation of (-30) and (-3), respectively. R L ( C + C + Cbcx ) ( C C ) A c, = Re A c, + at iddle to high frequency. (-30) R U C A Re c, = + C Ac at iddle to high frequency. (-3) The calculated R L is sent back to step ) to repeat the calculation to step ). Once the the difference between R L and R U is iniu, R L is treated as the final R. A typical results of proposed accuracy iproveent procedure for the µ SiGe at I B =9.36 µa, and I C =.56 A and V CE =3.0 V, is shown in Table I. The difference between R L and R U decreases very quickly in the first four iterations. The extracted R, C, and C bcx are found to be 6.50 Ω, ff, and 4.66 ff, respectively..3.3 Extraction of R, g 0 and τ The reaining unknowns are intrinsic transconductance, g 0, excess phase delay, τ, and base-eitter resistance, R which can be calculated as follows. A A R A A R i c, c = Re c, c b (-3) The value of R is estiated as.349 kω at I B =9.36 µa, and I C =.56 A and V CE =3.0V for the µ SiGe HBT, as shown in Fig. -9. The transconductance, g 0, and the excess phase delay, τ, is calculated as follows. 5

12 g Ac = Y (-33) A c ( g ) ( ) g 0 = Re + I g (-34) ( g ) ( g ) I τ = tan. (-35) Re ω Figures -0 show the extracted results for the µ SiGe HBT ased at I B =9.36 µa, and I C =.56 A and V CE =3.0 V. The value of R is estiated as.349 kω. The extracted g 0, and τ are S and.03 psec, respectively. Both of g 0, and τ are found to be nearly constant in the interested frequency range..3.4 Discrepancy between easureent and siulation Figure - and - show the coparisons between the easured and calculated S-paraeters for V CE =3V, V BE =0.83V, I C =.56A, and I B =9.36µA. Excellent agreeents over the whole frequency range were obtained. The total error in the S-paraeter calculated by following function [0][0]. E ij ij tot N eas freq i, j freq Sij eas si S S = 00 4 (-36) The residual discrepancies are about % for the whole as points. Therefore, we believe that the proposed ethod is an accurate extraction technique applicable to evaluate the process technology and optiize the transistor design. 6

13 # of iteration R L (Ω) R H (Ω) C (ff) C bcx (ff) R (Ω) Table - Typical results of proposed accuracy iproveent procedure for µ SiGe at V BE =0.8 V, V CE =3 V, I B =9.36 µa, and I C =.56 A. The initial value of R L, R H, C, and C bcx are 4.85 Ω, 7.3 Ω, ff, and ff, respectively. The extracted value of R C, C and C +C bcx are 8.9fsec, 40.3fF and 0.5fF, respectively. 7