A Unified Model of Early Voltage for Bipolar
Transistors at low temperatures.

Onno W. Purbo
Inter University Center on Microelectronics
Institute of Technology Bandung
Bandung 40132, INDONESIA
Tel: 62-22-250-4256, FAX: 62-22-250-5442
onno-itb@kesemek.cs.ui.ac.id, yc1dav@itbgtw.itb.ac.id

Early voltage (VA) of ZMR-SOI Polysilicon Emitter Transistor (PET) is studied as a function of temperature. A unified model of VA(T) is developed taking into account various effects such as carrier recombination in the neutral base, the Emitter-Base (EB) space charge layer and Poole‑Frenkel effect in the EB space charge region; the effect of a finite saturation velocity; carrier multiplication in the CB space charge region.

1.Introduction.
Early voltage is one of the important parameters to assess a bipolar transistor's performance [1-2]. Few research studies [1-5] have been devoted to explain the origin of various effects in the Early voltage of high performance bipolar transistors. To the best of our knowledge, only Dillard and Jaeger [2] have attempted to model the Early voltage as a function of temperature.
In this work, we report a unified model of the Early voltage as function of temperature VA(T) of high performance bipolar transistors. These include (a) carrier recombination, namely, in the neutral base [3], in the emitter-base (EB) space charge layer [7] and Poole‑Frenkel effect [8] in the EB space charge layer [9]; (b) the effect of a finite carrier velocity vd [4-6]; (c) carrier multiplication in the collector-base (CB) space charge region [10-11] and (d) temperature effects on each of these parameters.

2.	Theory
2.1.	Physical parameters for low temperature conditions.
	Okuto and Crowell's carrier ionization rate [19] are used. It is a fitting of the theoretical results of Baraff's 
[20] as well as measurements. Taking the ionization rate, the carrier multiplication factor Me and Mh are integrated [21]. The carrier multiplication in the CB junction is important to consider in high performance PETs and may lead to the well known reverse base current (RBC) [11-12]. The temperature dependent Aurora-Hauser-Roulston mobility model of electron and hole [22] are used. An empirical SHR life-time model [18-23] is used. The temperature dependent effective intrinsic carrier concentration (nie) of Slotboom [25] and the Slotboom-deGraaff's BGN model [26] (which assumed to be independent of temperature) are used.
	The effective doping concentration has to be calculated taking into account carrier freezed-out effect [21]. The acceptor Ea and donor Ed energy level is modeled following Chrzanowska-Jaeger [27]. The Joyce-Dixon approximation to Fermi-Dirac statistics is used as in [28]. As expected [21], for a certain doping concentration the quasi-Fermi potential will move closer to the respective band edge at lower temperature.

2.2.	Early Voltage in High Performance Bipolar Transistor.
	Assuming an npn high performance bipolar transistor, the Early voltage (VA) is defined as [7]
									(1)
The output transconductance, go, can be written as [7]
								(2)
Assuming Vbe is nearly constant (or a constant), then
										(3)
To a good approximation, Eqn.(3) is determined by a zero‑current Poisson solution and is independent of the origin of the base current. Cjc can then be described as [1]
									(4)
As shown in Eqn. (4), W/Vcb depends mainly on the doping at the edge of the depletion region and Cjc. It is essentially independent on the impurity profile in neutral base and Vcb. At high current density conditions, nc is required to sustain a certain finite saturation velocity vd. The existence of finite nc at the edge of CB depletion layer will affect Cjc [29]. Moreover, in the case of high defect density material, the C-B depletion layer might be affected by the distributed traps in the band [15-17]. This will be described in more detail at the end of this section.
	The output transconductance, go, in Eqn. (2) essentially depends on Jc/W. In the next few paragraphs, we will focus on go involving several physical effects. In npn transistor, the electron minority carrier current in the base Jn is
							(5)
Solve Jn for n(x), we obtain
							(6)
Depending on the existence of nc, the possible boundary conditions of n(x) at the edges of the deletion layers in the base are as follows
								(7)
while n(W) may be zero(0) or equal to nc depending on the current density.
	Solving C1 in Eqn.(6) (n(x)) and using the boundary conditions at n(W) (Eqn.(12) and Eqn.(13)) and n(0) (Eqn.(7)), the complete expressions for Jn and n(x) can be obtained. In the case of n(W)=0 (Eqn.(13)),
									(8)
							(9)
While for n(W)=nc (Eqn.(12)) with nc=Jn/q.vd [7]
							(10)
							(11)
							(12)
with

	In ideal cases, go, can be calculated from Jn/W (see Eqn. (2) and (4)) [3]. In the case of n(W)=0, Jn/W is
						(13)
and for n(W)=nc, Jn/W is
						(14)
	The existence of carrier recombination and carrier multiplication will perturb the charge equilibrium (in the poisson equation) as well as the carrier current. This in turn will change the width of the depletion regions as well as Vbe to sustain an equilibrium state. In other words, the term 1/Vt Vbe/W in Jn/W will exist in the presence of carrier recombination and carrier multiplication. There are two main factors which affects the neutral base width W variation:
  1.	carrier recombination in EB space charge and neutral base; and carrier multiplication in the CB space charge, through changes in 1/Vt Vbe/W.
  2.	the existence of finite nc.
	A closer look into the calculation of 1/Vt Vbe/W, it is performed through the evaluation of the majority carrier base current Jb taking into account recombination in the EB depletion layer, recombination in neutral base and carrier multiplication in the CB junction, such that:
						(15)
Jbo is the base saturation current. For PETs, the tunnelling mechanisms at poly-mono silicon interface at the Emitter is dominating Jbo [7]. To calculate VA, it is required to know Vbe. Vbe can be calculated from Eqn. (15) knowing the externally applied Jb.
	In most cases, we externally fix Jb during VA measurements and, thus, Jb/W0. From the derivative of Jb, we found that:
						(16)
Thus, the contribution of 1/Vt Vbe/W in VA can be evaluated from Eqn. (16).
	nc is a finite injected electron into the CB depletion region required to sustain a certain Jn under a finite vd. The existence of nc will disturb the charge equilibrium in the CB depletion layer and, thus, alters the depletion layer width as well as the electric field within and, thus, the carrier multiplication in the CB junction. Thus, depending on nc and the applied Vcb, Wbc will be varied [5]. Note that nc is affected by Jn and vd [5-7]. Moreover, Cjc will also be affected due to the existence of a finite nc [29].
	Takes a closer look on the 1/Vt Vbe/W term in Jb/W for several cases, namely:
  1.	Finite non-negligible carrier recombination in the neutral base [3].
  2.	Finite nc [4-5].
  3.	Finite carrier recombination in the EB depletion layer [7].
  4.	Finite carrier recombination with Poole-Frenkel effect [8] in the EB depletion layer [9].
  5.	Carrier multiplication in the CB junction [21].
	In the case, both recombination in the neutral base region (case 1) and nc (case 2) occurs simultaneously, 1/Vt Vbe/W term can be derived by taking into account n(x)/W (from Eqn.(9) and (12)) and Jb/W (Eqn. (16)), such that
						(17)
In the case, low current density (nc=0),
						(18)
with

	In the third case, finite carrier recombination in the EB depletion layer, with [21],
						(19)
Assume a graded EB junction in forward biases, we have dnedpe. Since change in the neutral base width due to Vbe variation is affected only by dpe (W-dpe), we can approximate 1/Vt Vbe/W from
						(20)
Thus, with the existence of EB recombination, 1/Vt Vbe/W in Eqn. (13)-(14) should take into account Eqn. (20).
	The fourth factor is the Poole-Frenkel effect at low temperature [8]. In a high electric-field EB depletion region, moving electrons may be trapped and, thus, affect the charge equilibrium as well as the charge emission rate [8] and capture rate [9]. Assuming donor like traps in the mid-band, the electron carrier life-time will be shortened by the amount of the applied electric-field such that,
								(21)
The shorter carrier lifetime especially at low temperature conditions will increase Jebr (Eqn.(27)) and Jebr/W (Eqn.(20)) and, thus, modify the output transconductance in Eqn. (13)-(14).
	The last (fifth) factor we wish to consider namely, the existence of carrier multiplication in the CB depletion layer will affect both Jn [7,21] and Jb [11,12]. An increase in Jn will increase go. Assuming negligible change at the edges of depletion region due to carrier multiplication Me, the term 1/Vt Vbe/W will be affected through,
						(22)
Taking into account Eqn.(22), go in the presence of carrier multiplication but without recombination would be,
						(23)
where Jn/W' is the case with no carrier multiplication (Eqn.(13) or (14)). Jn(II) is the electron majority carrier at the collector with multiplication.

4.	Summary.
	A unified model of the Early voltage as a function of temperature was presented taking into account several effects such as carrier recombination effects, namely, in the neutral base, in the EB depletion layer and due to an enhanced Poole-Frenkel effect; effect of finite velocity saturation and carrier multiplications in CB depletion region. These effects would be significant in high performance bipolar transistors.

5.	Acknowledgements.
	The author would like to thank Prof. C.R.Selvakumar (Univ. of Waterloo) for his encouragement. The author wishes to thank the Government of Indonesia and the Canadian International Development Agency (CIDA) for granting the overseas scholarship.

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