248 Facta Universitatis ser.: Elect. and Energ. vol. 9,No.2 (1996) doping of donor (N D ) and acceptor (N A ), respectively. With the degenerate appro

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1 FACTA UNIVERSITATIS (NIS) Series: Electronics and Energetics vol. 9, No. 2 (1996), 247{254 NEW INVESTIGATION ON SILICON BIPOLAR TRANSISTOR AT LOW TEMPERATURES Xiao Zhixiong and Wei Tongli Abstract. The intrinsic carrier concentration is calculated with the nonparabolic energy bands at low temperatures. Besides, the current gain of a silicon bipolar transistor is quantitatively modeled at 77K and 300K, and the temperature dependence of the mean nonideal coecient of the base current is also analyzed. The obtained results are in agreement with experimental data. 1. Introduction The intrinsic carrier concentrtation, the current gain and the non-ideal coecient are fundamental parameters in silicon bipolar transistors. Their detailed investigation is useful for the undestanding of the low temperature device physics. 2. Calculation of the intrinsic carrier concentration On the basis of Chakravarti et al.'s work [1], the intrinsic carrier concentration n in with the non-parabolic energy bands could be deduced as 2F n ind(a) = n i 1=2 ( p nc(v ) exp(nc(v ) ) + 5KT D(A)F 3=2 ( nc(v ) ) p exp(nc(v ) ) 1=2 (1) where n i is the intrinsic carrier concentration with the non-degenerate approximation. D =1=E gd A =1=E ga. F 1=2() and F 3=2() are Fermi integrals with orders of 1/2 and 3/2 respectively. T is the absolute temperature and K is the Boltzmann constant. E gd and E ga are silicon bandgaps with Manuscript received February 21, The authors are with Microelectronics Center, Southeast University, Nanjing , P.R. China. 247

2 248 Facta Universitatis ser.: Elect. and Energ. vol. 9,No.2 (1996) doping of donor (N D ) and acceptor (N A ), respectively. With the degenerate approximation, the intrinsic carrier concentration n id could be expressed as n idd(a) = n i F1=2 ( C(V ) ) 1=2 exp( C(V ) ) (2) Figures 1 and 2 show curves of n idd(a) =n ind(a) as a function of N D (N A ). It is clear that n id deviates from n in at high concentrations. Figure 1. n idd =nind versus N D at dierent T. Figure 2. n ida =nina versus N A at dierent T.

3 X. Zhixiong and W. Tongli: New investigation on silicon Modeling of the current gain For a common n + pn silicon bipolar transistor, minority{carrier concentrations in the base and in the emitter could be approximately written, respectively, as n B (x) =n B (0)(1 ; x ) W B p E (x) =p E (0)(1 ; x (3) ) W E where W E and W B are widths of the emitter and the base, respectively. n B (0) and p E (0) are minority{carrier concentratiions at edges of the emitter{ base junction. The collector current I C may be written as I C = W B R 0 qa E n 2 i exp( qv BE KT ) C B (x)+n B (x) n2 i dx D nb (x) n 2 ib (x) (4) where A E is the emitter area. V BE is a voltage on the emitter{base junction. C B (x), D n B(x) andn ib (x) are the eective majority{carrier concentration, the electron diusion coecient and the eective intrinsic carrier concentration in the base, respectively. q is the electronic charge. The ideal base current I BI could be written as. I BI = W E R 0 qa E n 2 i exp( qv BE KT ) C E (x)+p E (x) n2 i dx D pe (x) n 2 ie (x) (5) where C E (x), D pe (x) and n ie (x) are the eective majority{carrier concentration, the hole diusion coecient and the eective intrinsic carrier concentration in the emitter, respectively. The non{ideal base current could be written as [2] I R = qa E W n i exp( qv BE 2KT )exp(qp E 2KT ) 2 p p n (6) where W is the width of the emitter-base space charge region. p and n are minority-carrier lifetimes in the emmiter and in the base, respectively. E is the electric eld intensity in the space charge region. " is the silicon dielectric constant. = q="

4 250 Facta Universitatis ser.: Elect. and Energ. vol. 9,No.2 (1996) The base recombination current I RB could be written as I RB = qa B n 2 i W 2 B exp( qv BE KT ) 2 W RB 0 [C B + n B (x)] n dx (7) where A B is the base area. At a medium collector electric intensity, the base widening WB 0 may be written as [3] W 0 B = W C (1 ; J C0 J C ) (8) where W C is the collector width. J C is the collector current density and J C0 is the collector current density corresponding to the beginning of the eective base widening eect [4]. The contribution of the Erly eect may be expressed by C early [5]. C early =1+ d c 2(W B + W 0 B ) (9) where d c is the width of the base-collector space charge region. The current gain H FE could be expressed as H FE = I E I BI + I R + I RB C early W B W B + W 0 B (10) With the current crowding eect, n B (0) and p E (0) may be written as n Bcr (0) = n B (0) S e S eff1 S e (11) p Ecr (0) = p E (0) S eff2 where S e is the half emitter width. S eff1 and S eff2 are eective emitter half widths for the forward and the inverse injections [6], respectively. By n B (0) and p E (0), the emitter currentcrowding eect makes the conductivity modulation eect easily happen. Figure 3 shows curves of n B (0) and p E (0) as a function of V BE at 77K. When the emitter current crowding eect is not considered, they exponentially increase with V BE, but when it is considered, n B (0) and p E (0) increase much faster than the exponential function at high injection levels. Figure 4 showes H FE as a function of I C with V CE =1V at 77K.

5 X. Zhixiong and W. Tongli: New investigation on silicon Figure 3. n B (0) p E (0) versus V BE at 77K. Figure 4. H FE versus I C at 77K with V CE =1V. The modeled result is in agreement with experimental data. When V CE = 1V, J C0 = 1540A=cm ;2, the related I C = 3:08times10 ;2 A. In comparison with this value with that of I C corresponding to the decrease of H FE in Figure 3, it is clear that the eective base widening eect is not the main reason for the decrease of H FE. At 77K, the conductivity modulation eect and the emitter crowding eect are main factors for the decrease of H FE.

6 252 Facta Universitatis ser.: Elect. and Energ. vol. 9,No.2 (1996) At 300K, the conductivity modulation eect and the emitter current crowding eect have little inuence on n B (0) and p E (0). Figure 5 showes W B 0 as a function of I C with V CE = 1V. W B 0 rapidly increases with I C. Figure 6 showes the curve ofh FE with V CE =1V at 300K. Figure 5. W 0 B versus I C at 300K with V CE =1V. Figure 6. H FE versus I C at 300K with V CE =1V. When V CE = 1V J C0 = 1044A=cm ;2, the related I C = 2:09 10 ;2 A. This value is in agreement with its experimental value. At 300K, the eective base widening eect is the main reason for the decrease of H FE.

7 X. Zhixiong and W. Tongli: New investigation on silicon Temperature dependance of the mean non{ideal coecient Figures 7 and 8 show calculate curves of the mean non-ideal coecient n mean of the total base current I B at low injection level as a function of temperature at dierent N B and N E. It is clear that its temperature dependance relies on N B and N E. When N B is high and N E is low, it has a negative temperature coecient, otherwise it has a positive temperature coecient. Figure 7. n mean versus T at dierent N B. Figure 8. n mean versus T at dierent N E.

8 254 Facta Universitatis ser.: Elect. and Energ. vol. 9,No.2 (1996) Stock et al.'s data [7] show that n mean (83K) = 1:29 n mean (296K) = 1:33 with N B = cm ;3 n mean (296K) = 1:40 n mean (83K) > 2:74 with N B = cm ;3. Woo et al.'s data [8] show that n mean (79K) = 1:10 n mean (300K) = 1:39 with W B = 0:7m G B = 1: cm ;2 and N E = cm ;3 for a metal contacted transistor. 5. Conclusion It is appropriate to adopt the non{parabolic energy bands at high concetrations. The current gain is determined by the conductivity modulation eect and the emitter current crowding eect at 77K, but at 300K, it is mainly determined by the eective base widening eect. The temperature dependance of the mean non-ideal coecient of the base current relies on the emitter and the base concetrations at low injection levels. REFERENCES 1. A. N. Chakravarti and B. R. Nag: Generalized Einstein relation for degeneratesemiconductors having non{parabolic energy bands. Int. J. Electronics, vol.37, No.2, 1974, pp. 281{ J. C. S. Woo, J. D. Plummer and J. M. C. Stork: Non{ideal base curent in bipolar transistors at low temperatures. IEEE Trans. Electron Devices, ED-34, No.1, 1987), pp. 130{ A. Blicher: Field{Eect and Bipolar Power Transistor Physics. Academic press, ch.7, G. Rey, F. Dupuy and J. P. Bailbe: A unied approach to the base widening mechanisms in bipolar transistors. Solid{State Electronics, vol.18, 1975, pp J. M. Early: Eect of space{charge layer widening in junction transistors. Proc. IRE, vol.40, 1952, pp S. K. Ghandhi: Semiconductor Power Devices Physics of Operation and Fabrication Technology. John Wiley & Sons.Inc J. M. C. Stork, D. L. Harame, B. S. Meyerson and T. N. Nguyen: Base pro- le design for high{performance operation of bipolar transistors at liquid{nitrogen temperature. IEEE Trans. on Elect. Devices, ED{36, No.8, 1989, pp. 1503{ J. C. S. Woo and J. D. Plummer: Optimization of silicon bipolar transistors for high current gain at low temperatures. IEEE Trans. on Elect. Devices, ED{35, 1988), pp. 1311{1321.