Second International Conference on Electrical and Computer Engineering ICECE 2002, 26-28 December 2002, Dhaka, Bangladesh Analytical Expressions of Collector Current Density and Base Transit Time in an Exponentially Doped Base Bipolar Transistor for All Levels of Injection Md. Ziaur Rahman Khan and M.M. Shahidul Hassan Department of Electrical & Electronic Engineering, BUET, Dhaka –1000. Abstract - Analytical expressions of collector current density and base transit time are developed for an exponentially doped base modern bipolar junction transistor (BJT). The present model incorporates dopant dependence of carrier mobility, bandgap narrowing and finite velocity saturation effects in the calculation of collector current density and base transit time. This model is applicable for all levels of injection before the onset of Kirk effect. 1. Introduction Modeling of collector current density and base transit time is essential for design of high-speed bipolar transistor. The two integral relations for the current flow through the base region of a bipolar transistor, and for the transit time, were generalized to the case of a hetero structure bipolar transistor with a nonuniform energy gap in the base region by H. Kroemer  in 1985. J. H. Van den Biesen  in 1986 used a regional analysis to study the transit time of the BJT as a function of base emitter bias. He subdivided the total transit time from emitter to collector contact into five components. But no closed form solution in  was obtained. J. S. Yuan  in 1994 studied the effect of the base profile on the base transit time of the bipolar transistor for all levels of injection. He proposed equations for the minority carrier distribution within the base for different types of base doping. Using boundary conditions and the proposed equation he numerically evaluated the base transit time. So the equation forms for Jn and ô B in his work are not concise to express Jn and ôB as the function defined by some existing models. K. Suzuki  in 1993 proposed electron current density Jn and base transit time (ô B) models of uniformly doped bipolar transistor for high level of injection. He incorporated the electron velocity saturation effect in the collectorbase depletion region. Later, P. Ma  had simplified the equations, but they are not simple enough to give a physical insight into device operation. P. Ma, L. Zhang in  improved the work of  considering the velocity saturation of the electron in the depletion region of base-collector and the electrical field dependence on the minority carrier mobility. However, the method is based on iteration techniques. M. M. Shahidul Hassan and A. H. Khandoker  ISBN 984-32-0328-3 developed a mathematical expression for Jn and ô B for uniform base doping density for all levels of injection. But the model ignored the velocity saturation of electron at base-collector depletion region. D. Rosenfeld  derived an analytical formula of base transit time through a dopant-graded base considering the dependence of mobility on the doping level. But the equation is derived ignoring the bandgap narrowing and velocity saturation effect. M. M. L. Jahan and A. F. M. Anwar  developed an analytical expression of base transit time for exponentially doped base. But the expression is applicable for low injection level only. In our present model, analytical expressions for the collector current density and base transit time for an exponential doped base are obtained considering dopant dependence of carrier mobility, bandgap narrowing effects and velocity saturation at the base-collector junction. The expressions are applicable for all levels of injection. 2. Derivation of the model equations The base width of a modern high speed bipolar transistor is very thin. So the carrier recombination within the base can safely be neglected . In the absence of recombination, current density within the base becomes constant. The base transit time for all levels of injection is derived solving the current transport equation for electron profile n(x) which is given by  dn ( x) − J n ( x) = q Dn ( x) dx + q µ n ( x )n ( x) E ( x ) (1) where Dn (x) is the electron diffusion coefficient , E(x) is the electric filed, Jn (x) is the current density and µn (x) is the electron mobility. The electric field in the base is given by  2 kT 1 dp 1 d n ie ( x) E (x ) = − 2 (2) q p dx n ie ( x) dx where p is the hole concentration in the base and n ie is the effective intrinsic carrier concentration and it is given by  2V g Vt 2 2 N A( x) n ie ( x) = nio Nr 120 (3) where n io is the intrinsic carrier concentration in the base, NA (x) is the base doping profile, Vg and Nr are constants. The exponential base doping profile is  ηx N A ( x) = N A (0 ) exp( − ) Wb (4) where Wb is the base width and η is the slope of base doping and NA (0) is the peak base doping. The electron mobility in the base is given by  µmax − µmin (5) µn ( x) = µmin + γ N A( x) N ref If this mobility equation is used directly in (1) the differential equation becomes intractable. So with reasonable accuracy (5) is simplified to the following form a (6) µn ( x) = 18 N A − − C N γA ( x) N =1 where a, A, B and C are constants. ) If the electric filed is substituted in (1) the current density equation becomes  2 n ie J n = − q D n ( x) n( x ) + The total charge storage of the injected carrier n(x) in the base per unit area can be written as Wb Qlb = q ∫ nl ( x) dx (8) 0 Substituting the values of different parameters in (7) and using the conditions for low injection i.e. NA (x)>>n(x), the equation of current density for low injection region becomes (9) where vs is the saturation velocity of electron, n lo is the normalized carrier concentration at base-emitter junction for low level of injection, η ) 4 nie2 (0) n ( 0 ) = −0 .5 N A ( 0 ) + 0 .5 N A ( 0) 1+ exp V BE 2 N A (0) V t and at x=Wb , n l (W b ) = J ln q Similarly evaluationg the value of carrier concentration for low injection and using (8), the stored charge in the base for low injection region becomes Qlb = W b L nlo (10) η where ( ) A 1 L = q N A ( 0 ) 1 − L1 eηp − 1 p p ( ) ( ) A 1 1 + q N A ( 0 ) L1 η + L1 f 1( N ) e− ηNγ − 1 + eηp − 1 p N γ p ∑( 18 f 1( N ) = B N =1 1 N − C ) (γN + p ) (N A Nγ (0) ) and 2q v s E L1 = Jno Similarly substituting the values of different parameters in (7) and using the conditions for high injection i.e. NA (x)<<n(x), the equation of current density high injection region becomes J hn = q v s G n ho (11) where, n ho is the normalized carrier concentration at base-emitter junction for high level of injection N A ( 0 ) J no and G = s f Jno + Fq ( 0 ) vs N A A F = f s − 1 s ∑( 18 +B 2 akTη N A (0 ) Jno = , Wb p=1-2s, s=Vg /Vt ) ( d n(x )[n(x ) + N A ( x )] (7) 2 ( x ) dx NA n ie ( x ) N ( 0 ) f p Jno A E = , f =e , Jno − Dq v s N A ( 0) p f 18 1 Np Nγ −γNp p N A (0 ) f − 2 B ∑ ( −C ) − f 0 (γN + p ) The boundary condition used to solve the equation are at x=0, ( x) J ln = q v s E n lo ( 2A 1− p vs 1+ ∑( D= −C N =1 ( ) 1 N γ N A ( 0) ) (γN − s ) f − (γN − s ) − 1 and the equation of stored charge density for high level of injection is Qhb = W b M nho η and 121 (12) 0 .5 where A 1 A M = q N A (0) 1 + M 1 (1 − e−ηs) − M 1 η s s s 1 + q N A (0) M 1 f 2 ( N ) Nγ q vs G Jno f 2( N ) = B ∑ (− C N γ A ) We have solved (7) for two cases, low and high level of injection. But in the intermediate region of injection, the equation is not analytically tractable. However a general expression for Jn and Qbn is possible to obtain by exploring their behaviors in low and high injection region. According to (9) and (11) current density for low injection and high injection region are J ln = q v s E n lo J hn = q v s G n ho is normalized carrier concentration for all n( 0 ) N A ( 0) levels of injection and is given by Comparing (13) with (9) and (11), we can write for W =G of injection becomes W ( L + 2M no) τ B = ηq b ( E + 2 G ) no vs E + 2G no 1 + 2 no o o o (14) η -3 8.6 8.4 8.2 8.0 7.8 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Base Emitter voltage (V) 1.3 Fig. 1(a) Variation of base transit time on base-emitter voltage b lb 18 N A ( 0 ) = 2 x 1 0 c m , η= 3 and W b =150nm 7.6 According to (10) and (12), stored charge density for low injection and high injection region are W L Q = n (17) The variations of base transit time with different parameters are shown in Fig. 1(a) and 1(b). The Fig.1 (a) shows that the base transit time increases with the increase of base-emitter voltage. With the increase in base-emitter voltage, the minority carrier injection in the base increases. This reduces the aiding electric filed in the base for exponentially doping profile. So base transit time increases with the increase in baseemitter voltage. n >> 1 o s (16) Using (14) and (16) the base transit time for all levels 8.8 So, the empirical expression for current density then becomes, n L + 2M no no η 1 + 2 no 9.0 E + 2G n J =qv 1+2 n n n o >> 1 Qbn = W b n << 1 A best fit is found to be, W= for The empirical expression for stored charge density in the base then becomes, o for n o << 1 4. Results and Discussions For all levels of injection the expression for current density can be written as (13) J n = q v s W no W =E for A best fit is found to be, L + 2M n o H = 1 + 2 no N 1 (0 ) (γN − s ) 3. General Formulation no H =L H =M N =1 where, Comparing (15) with (10) and (13) we see that and 18 (15) η Base transit time (ps) M1 = −ηNγ 1 (e − 1) + (1 − e −ηs ) s Qbn = W b H n o lo Q hb = W b M n ho η For all levels of injection the expression for stored charge density can be written as From the Fig 1 (b) we see that the base transit time increases with peak base doping concentration. As the doping concentration in the base increases, the impurity scattering increases. This reduces the carrier mobility and increases base transit time. 122  Base transit time (ps) 9 8 7  6 5 VBE=1.0V 4  VBE=0.7V 3 10 17 10 18 -3 Peak base doping concentration (cm ) Fig. 1(b) Dependence of base transit time on peak base doping concentration  From (17) we see that the base transit time is proportional to the base width and inversely proportional to the slope of base doping (η). So base transit time increases with base width and decreases with slope of base doping (η).  5. Conclusion An empirical expression for base transit time of exponentially doped base BJT is developed which is applicable for all levels of injection. This work incorporates dopant dependent mobility, bandgap narrowing and finite velocity saturation effects. The results obtained by proposed expressions are compared with the result available in the literature and found in good agreement. References     H. Kroemer, “Two integral relationship pertaining to the electron transport through a bipolar transistor with nonuniform energy gap in the base region,” Solid State electronics, vol. 28 No. 11, pp. 1101-1103, 1985. J. H. Van Den Biesen, “A Simple Analysis of transit times in bipolar transistors,” Solid-State Electronics, vol. 29, No. 5, pp 529-534, 1986. J. S. Yuan, “Effects of base profile on the base transit time of the bipolar transistor for all levels of injection,” IEEE Trans. Electron Devices, vol. 41. No. 2, pp 212-216, Feb 1994. K. Suzuki, “Analytical base transit time model of uniformly-doped base bipolar transistors for high-injection regions,” Solid-State Electronics, vol. 36, No.1, pp 109-110, 1993. 123 P. Ma, L. Zhang, Y. Wang, “Analytical relation pertaining to collector current density and base transit time in bipolar junction transistor,” Solid-State Electronics, vol. 39, No. 1, pp 173-175, 1996. P. Ma, L. Zhang, Y. Wang, “Analytical model of collector current density and base transit time based on iteration method,” Solid-State Electronics, vol. 39, No. 11, pp 1686-1686, 1996. M. M. Shahidul Hassan, A. H. Khandoker, “New expression for base transit time in a bipolar junction transistor for all levels of injection,” Microelectronics Reliability, vol-41, pp 137-140, 2001. D. Rosenfeld and Samuel A. Alterovitz, “Carrier transit time through a base with dopant dependent mobility,” IEEE Trans. Electron devices, Vol. 41. No. 5, pp 848-849, May 1994. M. M. L. Jahan, A. F. M. Anwar, “An analytical expression for base transit time in an exponentially doped base bipolar junction transistor,” Solid-State Electronics, Vol. 39, No. 1, pp 133-136, 1996.