A novel approach for justification of box-triangular germanium profile in SiGe HBTs

  • Centre for Applied Research in Electronics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi 110016, India

Key words: SiGe HBTsbase transit timeoptimum base dopingshifted Ge profilebox-triangular germanium profile

Abstract: This work presents a unique and robust approach for validation of using the box-triangular germanium profile in the base of SiGe heterojunction bipolar transistors, where the methodology considers the simultaneous optimization of the p-type base doping profile in conjunction with the germanium profile in the base. The study analyses the electron motion across the SiGe base in SiGe HBTs, owing to different accelerating electric fields. The analysis first presents a figure of merit, to achieve the minimum electron transit time across the base in conjunction with the increased current gain in n—p—n-SiGe HBTs, which shows a general trend vis-v-vis the advantage of a trapezoid germanium profile, but with additional accuracy as we considered simultaneously optimized p-type base doping. The effect of minority carrier velocity saturation is then included to make the study more detailed. The analysis then investigates the shifted germanium profile in the base to further minimize the base transit time. Finally, it is shown that a shifted germanium profile eventually evolves into a box-triangular Ge-profile in the SiGe base, which could simultaneously minimize the base transit time and reduce emitter delay by virtue of the high current gain. The analysis verifies that for an average Ge-dose of 7.5% Ge across the base, a box-triangular germanium profile in conjunction with an optimum base doping profile has an approximately identical base transit time and a 30% higher current gain, in comparison with an optimum base doping and triangular Ge-profile across the whole base.

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1.   Introduction
  • The SiGe heterojunction technology based solid state devices are popular for their high cut-off frequency ($f_{\rm t})$ and maximum frequency of oscillation ($f_{\rm max})$[1, 2]. The minority carrier's (electron) transit time ($t_{\rm B})$ across the base is the most important parameter related to the speed of n-p-n bipolar transistors. In homojunction bipolar transistors, an accelerating filed for electrons can be achieved by tailoring the distribution of p-type dopants in the base region[3, 4]. This accelerating field in the base aids the electrons motion towards the collector and provides a minimum $t_{\rm B}$. Apart from the dopant distribution in the base, the germanium profile in the p-type base of n-p-n-SiGe HBTs can also be used to create the desired accelerating electric field to achieve the minimum base transit time ($t_{\rm Bmin})$ for electrons.

    The design of Ge profile to get $t_{\rm Bmin}$ in SiGe HBTs (without analyzing the p-type base doping profile effect) has been studied comprehensively in References [5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. Salmon et al.[8] and Kim et al.[9] considered a non-uniform base doping profile (resembling Gaussian) while analyzing the Ge profile effect on $f_{\rm t}$. Kwok et al.[18] gave $t_{\rm B}$ dependence on the Ge profile only for uniform and Gaussian doping profiles, whereas Chang et al.[11] optimized the Ge profile in the base for an imprecise optimized base doping profile[3]. Julian et al.[12] used different combinations of base doping and Ge profile to get $t_{\rm Bmin}$. Richey et al.[13] and Patri et al.[19] considered specific (exponential, epitaxial and uniform) base doping for different Ge profiles to optimize current gain ($\beta$) and $t_{\rm B}$. Joshi et al.[5] proposed a geometric programming approach to simultaneously achieve the optimized base doping and Ge profile in SiGe HBTs. However, no rationalization of the box-triangular germanium profile is performed in Reference [5] and the procedure involves a complex geometric programming optimization method, and requires a dedicated GP solver tools like MOSEK GP solver[14]. Recently published papers for justification of the trapezoid or box-triangular germanium profile in SiGe HBTs conventionally consider some fixed linear, uniform, Gaussian or exponential base dopant profile[20, 21]. Then they argue the advantages of a box-triangular germanium profile over some other graded or triangular or uniform Ge-concentration in the base, which is a somewhat limited approach for a comprehensive insight, as it ignores the simultaneously significant effect of the base dopant profile in $t_{\rm B}$ minimization. Consequently, no literature exist where an optimized base doping profile is inherently considered to achieve $t_{\rm Bmin}$ in SiGe HBT, and then a rationalization of the box-triangular germanium profile is done while examining the effect of varying the Ge profile for achieving higher $\beta$ and $t_{\rm Bmin}$.

    Therefore, in the present work, we have given a more detailed and more robust analysis for validation of box-triangular germanium profile in SiGe HBT base, which points out the advantage of the box-triangular germanium profile without any doubt. The various studies on Ge profiles[15, 16] corroborates that the triangular Ge profiles are best suited to get $t_{\rm Bmin}$. Therefore, in the first step, to get $t_{\rm Bmin}$ in SiGe HBTs, we simultaneously optimized the base dopant profile while considering a triangular germanium profile in the base. In this process, we arrived at a figure of merit (FOM) defined as the ratio of $\beta$ and $t_{\rm B}$ for different linear Ge profiles, for a fixed Ge-dose. This analysis provides the preliminary justification for the use of the trapezoid germanium profile in SiGe HBTs. We then included the important effect of minority carrier velocity saturation in SiGe HBTs in our study, to get more accurate results. Then in the second step, to further minimize $t_{\rm Bmin}$, we progress from the triangular Ge-profile to the shifted germanium profile. In the third and final step, we further modified the shifted Ge-profile according to our current gain and frequency requirements. This eventually leads us to the box-triangular germanium profile in SiGe HBTs, which provides the optimum base transit time and current gain, and thus reduced emitter delay and/or reduced base sheet resistance, while providing the highest cut-off frequency. For all three steps, we inherently included the simultaneous optimization of the base dopant distribution. Therefore, this methodology for justification of the box-triangular germanium profile (in SiGe HBTs) is indeed novel and more vigorous as far as a literature survey done by authors is concerned.

2.   Theory
  • The bandgap of Ge (0.66 eV) is significantly smaller than that of Si (1.12 eV). By incorporating Ge into the base region of a Si bipolar transistor, the bandgap of the base region, and hence the accompanied device characteristics, can be modified[22]. The drift field for electrons in the base can be maximized by having the largest Ge concentration near the base-collector junction and the smallest near the emitter-base junction[23]. The SiGe HBTs realized using such a triangular Ge profile are customarily knows as graded-base-bandgap transistors. For constant base doping, the ratio of $t_{\rm B}$ for SiGe HBT ($t_{\rm BSiGe})$ and Si BJT ($t_{\rm BSi}))$ devices can be given as[15]:
    \begin{split} \frac{t_{\rm BSiGe} }{t_{\rm BSi} }\cong {}& \frac{2kT}{\chi \Delta E_{\rm G(BC-BE)} } \left\{ 1-\frac{kT}{\Delta E_{\rm G(BC-BE)} }\right. \\[3mm]& \times \left. \left[1-\exp \left(-\frac{\Delta E_{\rm G(BC-BE)} }{kT}\right) \right] \right\}, \end{split}(1)
    where
    $t_{\rm BSi} =\frac{1}{2}\frac{W_{\rm B}^2 }{D_{\rm nB} }\left( {1+\frac{2D_{\rm nB} }{v_{\rm d} W_{\rm B} }} \right), $
    where $\nu_{\rm d}$ is the drift velocity of minority carriers ($\nu_{\rm d}$ $=$ $\nu_{\rm dsat}$ for minority carriers attaining saturation velocity inside the base region), $W_{\rm B}$ is base width, $D_{\rm nB}$ is electron diffusion coefficient in the base, $K$ is Boltzmann's constant, $T$ is absolute temperature in Kelvin, and $\chi$ is the strain induced increase in electron mobility with increasing Ge %. The total bandgap reduction due to the grading of the Ge across the uniform neutral base is given as $\Delta E_{\rm G(BC-BE)}$ $=$ $\Delta E_{\rm G(BC)}$ - $\Delta E_{\rm G(BE)}$, where $\Delta E_{\rm G(BE)}$ and $\Delta E_{\rm G(BC)}$ are the Ge induced reduction in the bandgap of the SiGe base at the emitter-base and collector-base edge of the quasi-neutral base. The total bandgap narrowing $\Delta E_{\rm G(BC-BE)}$ ($=$ $\Delta E_{\rm G, SiGe})$ is a function of the total Ge-slope (Ge') across the SiGe base[16]:

    Equation (1) in conjunction with Equation (2) states that $t_{\rm Bmin}$ will be obtained for the maximum Ge-slope inside the quasi-neutral base corresponding to the Ge-dose inside base[15].

    The effective intrinsic-carrier concentration ($n_{\rm ieB, SiGe})$ for SiGe HBTs with a triangular Ge-profile is given as[15]:
    $n_{\rm ieB, SiGe}^2 =\gamma \left( {n_{\rm i}^2 \exp \frac{\Delta E_{\rm gB} }{kT}} \right) \exp \left( {\frac{\Delta E_{\rm G, SiGe} }{kT}\frac{x}{W_{\rm B} }} \right), $(3)
    where $\gamma$ is a constant accounting for the reduction in the effective density of states with increasing Ge %, $n_{\rm i}$ is intrinsic carrier density ($=$ 1.5 $\times$ 10$^{10}$ cm$^{-3})$, $\Delta E_{\rm gB}$ is the bandgap narrowing parameter in the base, `$x$' defines the location inside the base, and in case of a triangular Ge profile $\Delta E_{\rm G(BC-BE)}$ $=$ $\Delta E_{\rm G(BC)}$ $=$ $\Delta E_{\rm G, SiGe}$.

    Similarly, the carrier diffusion coefficient ($D_{\rm nB, SiGe})$ in SiGe HBT for a triangular Ge-profile in the base is expressed as:
    $D_{\rm nB, SiGe} =D_{\rm nB} \left( {1+\frac{K_{\rm SiGe} {\rm Ge'}x}{W_{\rm B} }} \right), $(4)
    where $K_{\rm SiGe}$ $=$ 3[19].

    Accordingly, the modified electric field profile inside the base is given as:

    At low injection currents, $p_{\rm p}$ $\approx$ $N_{\rm B}$ (base doping). Equation (5) shows that the electric field due to the heavy doping effect (second term) always tends to counteract the electric field due to the dopant distribution (first term). As the first term reduces the base transit time, the second term increases it. The third term in the R.H.S. of Equation (5) is due to the Ge-concentration profile and it would assist in reducing the base transit time.

    An iterative methodology has already been described by authors to obtain the optimum base doping profile in homojunction bipolar devices[24]. The methodology has been modified in the present study to optimize the base-doping profiles in SiGe HBTs, by including the effect of Ge induced bandgap narrowing for intrinsic carrier concentration and carriers diffusivity coefficients, as mentioned in Equations (3) and (4), respectively. The fixed point iteration algorithm solves for the minimum value of $t_{\rm B}$ (as mentioned in Reference [24]), however, now it also takes into account the particular triangular-Ge profile and corresponding $n_{\rm ieB, SiGe}$, $D_{\rm nB, SiGe}$ values in the algorithm. Once the minimum value of $t_{\rm B}$ is found using a high level computer language program, then we extract the corresponding base doping profile values from the program. The extracted values of the base doping profile at every point of the base width (discretized into 1000 intervals) would then give us the corresponding optimized base doping profile for that particular Ge-concentration profile.

3.   Analyses of Ge-slope effect on electron transit time
  • In the present study, a practically reasonable profile in the base is conceived by forcing a condition that the peak base doping never exceeds the doping at the emitter edge of the base and it is never less than the doping at the collector edge. A peak base doping $N_{\rm Bmax}$ of 1 $\times $ 10$^{19}$ cm$^{-3}$ at the emitter edge of the base and a minimum base doping $N_{\rm Bmin}$ of 2 $\times $ 10$^{17}$ cm$^{-3}$ at the collector edge of the base has been chosen to include the band gap narrowing effect in the entire base region for a neutral base width of 0.1 $\mu $m[24]. Figure 1 shows the various Ge profiles considered in the analysis, where a fixed Ge-dose equivalent to uniform 12.5 % Ge inside the base of SiGe HBT is considered to maintain the stable strained SiGe base for 0.1~$\mu $m base width[25].

    The fixed Ge dose is then divided inside the base in a trapezoidal profile manner. A linear Ge ramp having 0 % Ge at the emitter-base junction with 25 % Ge at the base-collector junction (Ge-slope of 25 %), and a uniform box type 12.5 % Ge profile (Ge-slope of 0 %) inside the base are taken as two extreme cases of a trapezoidal Ge profile in the present study. In addition, the analysis is also done for Ge-slopes of 21 %, 15 %, 9 % and 5 % across the base. The associated bandgap narrowing effect due to the grading of the Ge across the neutral base are: $\Delta E_{\rm G(BC-BE)}$ $=$ 172 meV (for 25 % Ge slope), 144.48 meV (for 21 % Ge slope), 103.2 meV (for 15 % Ge slope), 61.92 meV (for 9 % Ge slope), and 34.4 meV (for 5 % Ge slope). The iteration methodology is applied for the different Ge profiles shown in Figure 1, and the results for the corresponding optimized base doping profile are shown in Figure 2.

    The results predict that an increasing Ge-slope (for a fixed Ge dose in the base) leads towards an optimized base doping profile with lower base Gummel numbers until a Ge-slope of $\geqslant $ 21 % is reached. Nevertheless, $t_{\rm B}$ shows a continuous reduction with an increasing Ge-slope. In addition, to analyze the effect of increasing Ge dose (and the corresponding Ge slope) on $t_{\rm B}$, three different linear Ge profiles with 25 % Ge, 40 % Ge and 60 % Ge at the base-collector junction (not shown in the figure), corresponding to 12.5 %, 20 % and 30 % average Ge concentration in the base, respectively, are studied (while relaxing the critical thickness limit constraint). The bandgap reductions due to the grading of the Ge % across the neutral base are 172, 275.2 and 412.8 meV for 25 %, 40 % and 60 % Ge slopes across the base, respectively. The optimized base dopant profiles for these Ge-slopes have been found to be identical to that for the 21 % Ge-slope profile in Figure 2. Nevertheless, $t_{\rm B}$ of 0.58, 0.35 and 0.21 ps has been calculated for 25 %, 40 % and 60 % Ge-slopes across the base, respectively. Therefore, $t_{\rm B}$ shows a continuous decrease with increasing Ge-dose (and Ge-slope) in the base.

4.   Figure of merit
  • In trapezoid germanium profile SiGe HBTs, high current gain could be traded-off to allow an increase in base doping, thereby achieving the high maximum oscillation frequency[19]. An increased base doping in trapezoid Ge profile SiGe HBTs also has an additional advantage of suppression of the reverse-Early effect, which remains an issue for triangular Ge profile SiGe HBTs. The ratio of $\beta$ between a SiGe HBT and a Si BJT is given as[15]:
    $\frac{\beta _{\rm SiGe} }{\beta _{\rm Si} }\cong \gamma \chi \frac{\dfrac{\Delta E_{\rm G(BC-BE)} }{kT}}{1-\exp \left[{\dfrac{-\Delta E_{\rm G(BC-BE)} }{kT}} \right]} \exp \frac{\Delta E_{\rm G(BE)} }{kT}, $(6)
    where
    $\beta _{\rm Si} \approx \frac{W_{\rm E} D_{\rm nB} N_{\rm E} }{W_{\rm B} D_{\rm pE} N_{\rm B}}, $
    where the parameter $\chi$ is the strain induced increase in minority carrier mobility in the base of SiGe HBT, $D_{\rm pE}$ is hole diffusion coefficient in the emitter, and $W_{\rm E}$ and $N_{\rm E}$ are the emitter region thickness and doping, respectively. The first term on the R.H.S of Equation (6) is due to the Ge ramp inside the base and the second term is due to the Ge concentration present in the base at the emitter-base junction edge.

    For the different Ge profiles considered in Figure 2, we have defined a figure-of-merit (FOM) as the ratio of current gain and base transit time ($\beta$/$t_{\rm B})$. A maximum value for this FOM is desired to get the maximum advantage of reduced $t_{\rm B}$ and for trading off the high $\beta$ to decrease the base sheet resistance. Another important aspect of this FOM in the present day SiGe HBT devices comes from the fact that emitter delay will start playing a significant role, because a very low $t_{\rm B}$ can be obtained in these devices by using a very thin base and providing a ramp in the Ge concentration in the base region. To keep the emitter delay at a low value, $\beta$ of the device must be kept high[19]. Further, a high $\beta$ gives a higher collector current for the same bias conditions, and increases the cut-off frequency[26]. So in present day high frequency SiGe HBT devices, $\beta$/$t_{\rm B}$ can be treated as a basic FOM. Table 1 gives the values of the base transit time, current gain, and $\beta$/$t_{\rm B}$ for different Ge % slopes inside the base for a fixed equivalent dose of uniform 12.5 % Ge in the base. Table 1 also shows that a maximum in the FOM ($\beta$/$t_{\rm B})$ is achieved between a Ge-slope of about 11 % to 13 % Ge slope inside the base for the currently chosen base structure and constraints. Ge profiles having a slope below 11 % Ge show a high $\beta$ but an increased $t_{\rm B}$, while profiles having a Ge-slope above 13 % Ge show reduced $t_{\rm B}$ but the corresponding $\beta$ increase is lower.

    A plot of the variation of FOM ($\beta$/$t_{\rm B})$ as a function of Ge % difference between the emitter-base junction and the base-collector junction is shown in Figure 3. It is thus proposed that in the present case, a Ge %-slope of 11 % to 13 % should be kept, and the rest of the Ge-dose must be used to raise the Ge-concentration at the emitter-base junction for a higher $\beta$. This would ultimately lead to a trapezoid germanium profile in the base of SiGe HBT (with the simultaneous optimum base dopant profile), and hence gives the rationalization for the hypothesis that the trapezoid germanium profile in the base is best suited for high frequency SiGe HBT devices. However, the analysis must include the effect of minority carrier velocity saturation in high frequency SiGe HBTs to completely validate the use of the box-triangular germanium profile in its base.

5.   Velocity saturation effect in SiGe HBT
  • Inclusion of velocity saturation effects (VSE) in the base transit time calculation of high frequency SiGe HBTs is necessary to achieve accurate conclusions[5]. It has been shown that the inclusion of minority carrier VSE for $t_{\rm B}$ calculations in homojunction and heterojunction bipolar transistors increases the $t_{\rm B}$ value[27]. Nonetheless, the results in Figure 4 show a tendency of decreasing influence of VSE on an increase in $t_{\rm B}$ value, as the Ge-slope across the base is increased.

    A percentage increase of 30.6, 28.6, 26.9, 26.0 and 25.7 is observed in the $t_{\rm B}$ value for increasing slopes of 0, 5, 10, 15 and 25 percent Ge-slope across the base, respectively. Therefore, it can be stated that a higher Ge-slope inside the base, in conjunction with an optimum base doping profile, partially compensates for an increase in the $t_{\rm B}$ value due to velocity saturation of minority carriers.

    A further insight into this phenomenon is obtained by investigating the variation of electron apparent velocity `$v (x)$' inside the base, for 0 %, 5 %, 10 % and 15 % Ge-slopes, as shown in Figure 5. The electron saturation velocity inside the base is taken as 1 $\times $ 10$^{7}$ cm/s, as shown in Figure 5. The results show that in the case of an optimized base doping profile having a 0 % Ge-slope (homojunction or uniform-Ge in the base) the apparent velocity `$v (x)$' saturation occurs at $x$ $=$ 0.846$W_{\rm B}$. Whereas, the apparent velocity saturates at lower values of fractional base width of 0.800$W_{\rm B}$, 0.727$W_{\rm B}$ and 0.576$W_{\rm B}$ with a 5 %, 10 % and 15 % Ge-slope across the base region, respectively.

    Two distinct phenomenons observed in Figure 5 are:

    (1) For increasing the Ge-slope in the base, a higher drift velocity for electrons is achieved for smaller values of distance traveled in the base (before the velocity saturation). This will reduce $t_{\rm B}$ with increasing the Ge-slope, as shown in Figure 4.

    (2) For increasing the Ge-slope in the base, the earlier attainment of electron's saturation velocity for lower fractional base widths, will increase the base transit time when VSE is included in $t_{\rm B}$ calculations. This effect is also clearly visible in Figure 4.

6.   From shifted Ge-profile to box-triangular Ge-profiles
  • The shifted-Ge linear profile further minimizes $t_{\rm B}$ of SiGe HBs[28]. This advantage can be traded-off to increase $\beta$ or reducing the base resistance of the SiGe HBT device, by configuring a trapezoidal (or box-triangular) Ge-profile or by increasing the base doping, respectively. In cases where $t_{\rm B}$ happens to be lower than the emitter delay term, the optimum shifted-linear Ge-profile must be changed to the box-triangular type inside the base, while keeping $t_{\rm B}$ comparable to the emitter delay term. The selection of such a profile will increase $\beta$ and reduce the emitter delay term[19]. An increased $\beta$ for SiGe HBT as given in Equation (6) can also be traded off to reduce the base sheet resistance, which is necessary to achieve a high $f_{\rm max}$ and to lower reverse Early effect.

    To illustrate our point, we have taken the case of Ge-dose equivalent to uniform 7.5 % Ge across the base. Figure 6 shows that when a linear Ge-profile of 15 % Ge slope across the total base width is used, the $t_{\rm B}$ (with VSE) for the optimized base doping profile is 1.18 ps. Figure 6 also shows that when for the same dose, the Ge-profile is shifted inside the neutral base (by about 17.3 % of $W_{\rm B}$ from the emitter-base edge, and the respective Ge-slope is raised to about 18.137 % in the remaining 82.7 % of $W_{\rm B})$, $t_{\rm B}$ is reduced to 1.06 ps. However, in both cases shown in Figure 6, the Ge-concentration is 0 % at the emitter-base edge. So, the second term on the R.H.S in Equation (6) is unity and $\beta$ for both the structures is likely to be the same.

    In case the Ge-profile is shifted inside the neutral base by about 17.3 % of $W_{\rm B}$ from the emitter-base edge, and the respective Ge-slope is kept to about 15.7 % in the remaining 82.7 % of the base width, $t_{\rm B}$ increases to 1.18 ps on account of the lower slope of Ge concentration in the base. This is identical to a $t_{\rm B}$ of 1.18 ps for a linear Ge-profile of a 15 % Ge-slope across the total base width. However, for this 15.7 % Ge-slope profile, a box-triangular profile having approx. 1.0 % Ge-concentration at the emitter-base junction can be kept for the same Ge-dose as for a 15 % Ge-slope across the total base width. The finite Ge-concentration at the emitter-base junction effectively increases the $\beta$ for this Ge-profile by $\approx $ 30 %, while keeping approximate the same $t_{\rm B}$ in comparison with the linear Ge-profile of a 15 % Ge-slope across the total base width. This clearly shows that even when base dopant optimization is simultaneously done, a box-triangular Ge-profile in the base of SiGe HBT would be more beneficial than a triangular Ge-profile, for the best possible FOM. This conclusion is more accurate and more detailed than previously considered work, where some fixed base dopant profile was used[20, 21] while comparing the box-triangular Ge-profile with other Ge-profiles.

    To complete the $\beta$ enhancement study in SiGe HBTs, it is important to analyze the effect of a linear 15 % Ge-ramp across the total base width (say profile-2) and a 15.7 % Ge slope across 0.827$W_{\rm B}$ after a Ge-shift of 0.173$W_{\rm B}$ from the emitter-base edge (profile-1), on the first term on the R.H.S of Equation (6). This Ge-slope induced \textit{$\beta $ }enhancement term can also lead to an increase in the total current gain. Assuming similar base doping profiles, a simplified expression for the ratio of $\beta$ enhancement due to the Ge-slope in any two different linear Ge-slope profiles across a similar base width can be given by using Equations (1) and (6), as:
    $\frac{\beta _1 }{\beta _2 }\cong \frac{1-\dfrac{\chi \Delta E_{\rm G2, SiGe} t_{\rm BR2} }{2kT}}{1-\dfrac{\chi \Delta E_{\rm G1, SiGe} t_{\rm BR1} }{2kT}}, $(7)
    where $\Delta E_{\rm G1, SiGe}$ and $\Delta E_{\rm G2, SiGe}$ are the bandgap reduction due to the grading of Ge across the neutral base in SiGe HBTs with two different Ge-slopes (Ge-slope 1 and Ge-slope 2, respectively) in the base. The $t_{\rm BR1}$ and $t_{\rm BR2}$ are the ratio of base transit time of SiGe HBTs w.r.t. a particular Si BJT, as given in Equation (1), for Ge-slope 1 and Ge-slope 2 profiles, respectively.

    Profile-1 and profile-2 have almost identical $t_{\rm B}$ of 1.186 and 1.188 ps, respectively, therefore, we can safely assume that $t_{\rm BR1}$ and $t_{\rm BR2}$ are almost identical in the present case. However, as the two Ge-profiles cover different lengths of the neutral base region, the relative effect of the Ge-ramp on the $\beta$ enhancement is not exactly clear for the two profiles. However, an insight can be gained by comparing the apparent velocities of minority carriers inside the base for both these profiles. This is due to the fact that $\beta$ enhancement owing to the Ge-slope inside the base region is directly dependent on the efficient minority carrier movement in the base and hence on their drift velocity. Therefore, as the $t_{\rm B}$ for the two profiles are almost identical, the current gain term dependent on minority carrier drift velocity can be assumed to be identical for these two profiles.

    The effects of electron apparent velocity on $t_{\rm B}$ for two profiles under consideration are shown in Figure 7. Figure 7 predicts that due to the absence of any Ge-slope near the emitter-base junction edge in the base for profile-1, the minority carrier accelerates to a high velocity ($>$ 9 $\times $ 10$^{6}$ cm/s) inside the base somewhat later when compared with profile-2. However, the minority carriers in profile-1 get accelerated to their saturation velocity earlier due to a higher Ge-slope after a distance of 0.173$W_{\rm B}$ from the emitter-base junction edge. The result of these two facts has led to an effectively identical $t_{\rm B}$ for profile-1 and profile-2 in the present case and can be assumed to achieve identical $\beta$ enhancement due to their respective Ge-ramp and optimized base doping profiles in the base.

    Therefore, profile-1 may deliver an approximately 30 % increase in total $\beta$ in comparison with profile-2 due to the finite (1 %) Ge-concentration at the emitter-base junction edge. This increase in $\beta$ would reduce the emitter-delay term by approx. 30 % in profile-1 in comparison with profile-2[19]. However, an important point to note here is the requirement of only 1 % Ge in the box region to achieve the required $\beta$ enhancement by about 30 %. The result contrasts somewhat to the studies on trapezoid and/or box-triangular profiles[20, 21], where a high Ge % is inherently considered in the box-region, while comparing the box-triangular germanium profile with uniform or linear germanium profiles, to show the significant advantage in current gain and cut-off frequency. The less strain on the Ge % requirement in the box-region in the present study can be attributed to the fact that our analysis primarily concentrates on keeping the minimum $t_{\rm B}$ identical while comparing the robustness provided by the simultaneously optimized base dopant profile, which ensures that our analysis is always close to the global minimum in base transit time.

    The optimum base doping profiles for profile-1 and profile-2 are shown in Figure 8. The shifted Ge-profile (profile-1) shows an optimum base doping with higher base Gummel number in comparison with profile-2, which is an additional advantage for achieving a lower base sheet-resistance for the corresponding optimized base doping. Therefore, a shifted-Ge profile can be used for achieving higher $f_{\rm max}$ by reducing $t_{\rm B}$, and/or increasing the $\beta$ (thus reducing the emitter delay by using the box-triangular SiGe base) and/or increasing the base Gummel number (thus reducing the base sheet-resistance).

7.   Conclusions
  • This paper presents a systematic approach for justification and validation of the use of the box-triangular germanium profile in the base of n-p-n SiGe HBTs. To achieve the minimum possible base transit time, we have simultaneously considered the optimum base doping profile while looking for the optimum germanium profile. The study first considered the trapezoid and linear Ge-slope profiles in the base, presumably the best option to reduce $t_{\rm B}$ in SiGe HBTs, in conjunction with optimum base doping. The analysis shows a reduction in $t_{\rm B}$ with increasing the Ge-slope inside the base, while the corresponding optimized base doping profile shows a trend towards a lower base Gummel number. For an optimized base doping profile, (with a Ge-dose equivalent to a uniform 12.5 % Ge-concentration in the base), the dependence of FOM ($\beta$/$t_{\rm B})$ on the Ge distribution in the base predicts a best compromise for the Ge slope varying between 11 %-13 % across the base region, while rest of the Ge-dose may determine the pedestal height of the Ge-concentration at the emitter-base junction. This analysis provides the preliminary justification for the use of the trapezoid or box-triangular germanium profiles in SiGe HBTs, as has also been reported in conventional approaches (but without the optimized base dopant distribution)[20, 21]. Therefore, the present work is more accurate and provides robust validation for the use of trapezoid or box-triangular germanium profiles in SiGe HBTs as we have also included the effect of the optimized base dopant on $\beta$ and $t_{\rm B}$.

    The inclusion of the minority carrier velocity saturation effect on the base transit time calculations is known to give a higher value of $t_{\rm B}$. However, the present analysis also shows a tendency of a lower $t_{\rm B}$, as the Ge-slope across the base is increased, even after the inclusion of VSE in $t_{\rm B}$ calculations. The reduction in $t_{\rm B}$ has also been investigated by shifting the onset of Ge % (by a distance $X_{\rm G}$ away from the emitter-base junction) for increasing the Ge-slope in the remaining base region. For an average Ge-dose of 7.5 % Ge across the base, the shifted-Ge profile is observed to reduce $t_{\rm B}$ by about 10 % in comparison with a linear Ge slope across the base width $W_{\rm B}$. Finally, it is shown that an evolution of the shifted germanium profile for the simultaneous optimization of $\beta$ and $t_{\rm B}$ in SiGe HBTs would eventually lead to a box-triangular Ge-profile along with an optimized base doping profile. For an average Ge-dose of 7.5 % Ge across the base, an optimized shifted-Ge profile in conjunction with an optimum base doping profile shows an approximately identical $t_{\rm B}$ and a 30 % higher $\beta$ in comparison with a linear Ge-profile without shifted Ge-slope across the base.

Figure (8)  Table (8) Reference (28) Relative (20)

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