The new equations of p-n junction carrier injection level.

I. INTRODUCTION

The p-n junction is still the basic building block of many semiconductor devices. The parameter, which indicates the operating conditions of forward biased p-n junction, is injection level of minority carriers. It is important to know the injection level of analyzed junctions since it determines the characteristics of device and validity of models used for analysis. For example, if injection level is low the voltage drop across the quasi-neutral regions of the p-n junction can be neglected and exponential (Shockley) equation-based models can be used for simulation. If injection level is high, the operating conditions of p-n junction are changed greatly and more complex models should be used for adequate simulation of such devices.

The injection level is ratio of excess minority carrier concentration to equilibrium majority carrier concentration [1]-[3]. Equilibrium majority carrier concentration of real pn junctions is very close to concentration of impurity, consequently, the injection level of holes and electrons kp and [k.sub.n] can be presented as follows:

[k.sub.p] = [p.sub.n]([x.sub.n]) - [p.sub.n0]/[N.sub.D] (1)

[k.sub.n] = [n.sub.p](-[x.sub.p]) - [n.sub.p]0/[N.sub.A], (2)

where [p.sub.n]([x.sub.n]) and [n.sub.p](-[x.sub.p]) are hole and electron (minority carrier) boundary concentrations at the edges of the depletion region x=[x.sub.n] and x=-[x.sub.p] in n and p region, respectively, [p.sub.n0] and [n.sub.p0] are minority carrier boundary concentrations at the equilibrium when voltage drop across the depletion region [U.sub.d]=0 and [N.sub.D], [N.sub.A] are impurity (donor and acceptor) concentrations.

The minority carrier boundary concentrations can be expressed by the commonly used exponential boundary conditions:

[p.sub.n]([x.sub.n]) = [p.sub.n0] exp([U.sub.d]/[V.sub.T]) (3)

[n.sub.p](-[x.sub.p]) = [n.sub.p0] exp([U.sub.d]/[V.sub.T]) (4)

where [V.sub.T] is thermal potential.

Using (1), (2) and (3), (4), [k.sub.n] owing that [p.sub.n0] = [n.sup.2.sub.i]/[N.sub.D] and [n.sub.p0] = [n.sup.2.sub.i]/[N.sub.A], (ni is intrinsic carrier concentration) the hole and electron injection levels:

[k.sub.p] = [n.sup.2.sub.i]/[N.sup.2.sub.D] [exp([U.sub.d]/[V.sub.T])-1], (5)

[k.sub.n] = [n.sup.2.sub.i]/[N.sup.2.sub.A] [exp([U.sub.d]/[V.sub.T])-1], (6)

The (5) and (6) are important since they allow us to estimate the injection level of minority carriers at given Ud, and [U.sub.d] can be related with current of the junction.

Unfortunately, the equations (3), (4) and consequently, equations (5), (6) are valid only at low-level injection when [p.sub.n]([x.sub.n]) - [p.sub.n0] << [N.sub.D] and [n.sub.p](-[x.sub.p])-[n.sub.p0] << [N.sub.A]. On the other hand, the junctions of the semiconductor devices especially those used in the integrated circuits and power semiconductor devices operate at high-level injection of minority carriers. Because of this it is of interest to relate the injection level of minority carriers with [U.sub.d] for case of highlevel injection.

II. The Equations of Injection Level Valid at Arbitrary injection

The boundary conditions that relate the minority carrier boundary concentrations with [U.sub.d] at arbitrary injection level including high-level injection, derived on basis of commonly accepted Boltzmann relations and quasi-neutrality conditions are following [4]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

where [V.sub.B] is the potential barrier of junction.

On basis of equations (1), (2) and (7), (8), taking into account that [p.sub.n0] = [n.sup.2.sub.i]/[N.sub.D] and [n.sub.p0] = [n.sup.2.sub.i]/[N.sub.A], the hole and electron injection levels:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (9)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)

In contrast to exponential equations (5) and (6), the high-level injection of minority carriers is taken into account in derived new equations (9) and (10). The validity of (9) and (10) is limited by the validity of Boltzmann relations and quasi-neutrality conditions. The analysis of obtained equations shows that they coincide with equations (5), (6) at low-level injection when [[N.sub.D] exp([U.sub.d]/[V.sub.T]))]/[N.sub.A], and [[N.sub.A] exp([U.sub.d]/[V.sub.T])]/[N.sub.D]] << exp([V.sub.B]/[V.sub.T]), i.e. when [U.sub.d] is significant lower than [V.sub.B].

If junction operates at high-level injection when [U.sub.d] is so close to [V.sub.B] that assumptions exp([U.sub.d]/[V.sub.T]) [approximately equal to] exp([V.sub.B]/[V.sub.T]) and sh[([V.sub.B]-[U.sub.d])/[V.sub.T]] [approximately equal to] ([V.sub.B]-[U.sub.d])/[V.sub.T] can be made, the equations (9), (10) simplifies us follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (11)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (12)

It is seen (11), (12) that [k.sub.p] and [k.sub.n] dependence on [U.sub.d] at high-level injection, when [U.sub.d] is close to Vr, become not exponential and is determined by function [V.sub.T]/([V.sub.B]-[U.sub.d]). By this is meant that voltage drop across the depletion region can not exceed [V.sub.B], i.e. the inequality [U.sub.d] < [V.sub.B] is valid. On the other hand, according to the exponential equations (5) and (6), the wrong conclusion can be drawn that there is no limitation on [U.sub.d].

On basis of (5) and (6) can be estimated that ratio of hole and electron injection levels at low-level injection is independent of [U.sub.d] and is determined by square of impurity concentrations ratio

[k.sub.p]/[k.sub.n] = [([N.sub.A]/[N.sub.D]).sup.2]. (13)

The equation of ratio [k.sub.p]/[k.sub.n] obtained on basis of (9) and (10)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (14)

shows that in general case the ratio [k.sub.p] / [k.sub.n] is [U.sub.d] dependent (14) and varies in range from value presented by equation (13) at low-level injection when exp[-([V.sub.B]-[U.sub.d])/[V.sub.T]] << [N.sub.D] /[N.sub.A], [N.sub.A]/[N.sub.D] to value [k.sub.p]/[k.sub.n] [approximately equal to] [N.sub.A]/[N.sub.D] at high-level injection when [U.sub.d] is so close to [V.sub.B] that exp[([V.sub.B]-[U.sub.d])/[V.sub.T]] is close to unity. Equation (14) is derived taking into account fact that the fractional term in the angle brackets of equations (9) and (10) in case of forward-biased junction is much more higher than unit.

III. ANALYSIS OF CONCRETE P-N JUNCTIONS

To examine the derived equations of injection level, three silicon p-n junctions with different impurity concentrations (Table I) were analyzed. The analysis was performed at room temperature assuming that junctions are abrupt and homogeneously doped.

Using data presented in Table I and widely [k.sub.n] own relation [V.sub.B] = [V.sub.T]ln[[N.sub.A][N.sub.D]/[n.sup.2.sub.i]], the dependences of [k.sub.p] and [k.sub.n] on [U.sub.d] on the basis of equations (9) and (10) were computed (Fig. 1). Additionally, the [k.sub.p] and [k.sub.n] dependences using exponential equations (5), (6) were calculated for junction No.1 (Fig. 1). It is seen that at low [U.sub.d] the dependences calculated using (9), (10) and (5), (6) coincide. When [U.sub.d] becomes high and increases, equations (9), (10) give a higher rise of injection level as compared with equations (5), (6). At high values of Ud the dependences of [k.sub.p] and [k.sub.n] computed using (9), (10) become non-linear on log scale, i.e. they become nonexponential.

It is of interest to evaluate the current density of the junction at given injection level. The calculation of approximate values of current densities for analyzed junctions at given [k.sub.p] and [k.sub.n] was provided according to the following scheme. Firstly, the [U.sub.d] for given [k.sub.p] and [k.sub.n] using graphs presented in Fig. 1 were estimated. Secondly, using classical equations [J.sub.sp] = [en.sup.2.sub.i][D.sub.p]/([W.sub.n] [N.sub.D]), [en.sup.2.sub.i][D.sub.p]/([L.sub.p] [N.sub.D]) and [J.sub.sn] = [en.sup.2.sub.i][D.sub.n]/([W.sub.p] [N.sub.A]), [en.sup.2.sub.i][D.sub.n]/([L.sub.n] [N.sub.A]) the hole and electron saturation current densities were calculated for case when junctions are with short ([W.sub.p] [W.sub.n] = 3x [10.sup.-4] cm) and long ([W.sub.p] > [L.sub.n], [W.sub.n] > [L.sub.p]) quasi-neutral regions, where [W.sub.p] and [W.sub.n] are lengths of p and n quasi-neutral regions, [L.sub.p] and [L.sub.n] are hole and electron diffusion lengths. The data for calculations were taken from [5].

[k.sub.n] owing the [J.sub.sp] and [J.sub.sn], the minority carrier hole and electron current densities of the junction at the edges of the depletion region [J.sub.p] and [J.sub.n] at given [U.sub.d] were computed assuming that their dependences on [U.sub.d] in respect of [J.sub.sp] and [J.sub.sn] are determined by law of boundary conditions (7) and (8), respectively.

The total current density of junction at given [U.sub.d] and, consequently, at given [k.sub.p] and [k.sub.n] was estimated as sum J = [J.sub.p] + [J.sub.n]. The J values at injection levels [k.sub.p], [k.sub.n] = 0.1; 1 for analyzed junctions with short and long quasi-neutral regions are presented in Table II and Table III. The value [k.sub.p], [k.sub.n] = 0.1 can be considered as limiting value of low-level injection of minority carriers holes and electrons, i.e. as limiting value for validity of Shockley equation-based models. Value [k.sub.p], [k.sub.n] = 1 corresponds with high-level injection [3]. It is seen (Table II and Table III) that at given values of [k.sub.p] and [k.sub.n] the current density of junctions with short quasi-neutral regions is higher than that of junctions with long quasi-neutral regions. The reason for this is the fact that [J.sub.sp] and [J.sub.sn] are higher for junctions with short quasi-neutral regions.

The p-n junctions of silicon diodes and BJT's often operate at current densities that may be as much as tens of A/[mm.sup.2] [6]. For emitter junction of n-p-n integral transistors, as an example, the current density can reach 100 A/[mm.sup.2] and more [7]. It is apparent that values of current density mentioned above exceed considerably the low-level injection limiting values presented in Table II and Table III.

The experimental voltage-current characteristic of base-emitter junction of p-n-p lateral transistor used in integrated circuit of voltage comparator [8] is presented in Fig. 2. The UrE in Fig. 2 is voltage applied to the base--emitter junction and IE is emitter current. The area of the junction is 1.3x[10.sup.4] [mm.sup.2,] the maximum operating current [I.sub.Emax] = 5mA. This junction corresponds with junction No.2 with short quasi-neutral regions analyzed above. using results presented in Table II, the approximate values of current that correspond with injection levels [k.sub.p], [k.sub.n] =0.1; 1 are marked in Fig. 2.

It is seen that the low-level injection condition for holes ([k.sub.p] < 0.1) and electrons ([k.sub.n] < 0.1) is violated at about 0.01 and 0.35 mA, respectively. This fact shows that for the adequate simulation of circuit based on such transistors the model, which takes into account the high-level injection, should be used.

IV. CONCLUSIONS

1. In contrast to exponential equations (5) and (6), the derived equations (9) and (10) allow us to relate the injection level of minority carriers with [U.sub.d] for case of high-level injection.

2. In general case the ratio of hole and electron injection levels is [U.sub.d] dependent and varies in range from [([N.sub.A]/[N.sub.D]).sup.2] at low-level to approximately [N.sub.A]/[N.sub.D] at high-level injection.

3. The analysis of concrete p-n junctions shows that junctions of silicon semiconductor devices often operate at injection levels that are much greater than low-level injection limiting values.

http://dx.doi.org/ 10.5755/j01.eee.19.2.3467

REFERENCES

[1] S. M. Sze, K. Ng. Kwok, Physics of semiconductor devices. John Wiley, New Jersey, 2006, p. 832. [Online]. Available: http://dx.doi.org/10.1002/0470068329

[2] B. L. Anderson, R. L. Anderson, Fundamentals of Semiconductor devices. Mc Graw Hill, N. Y., 2005, p. 271.

[3] G. Massobrio, P. Antognetti, Semiconductor device modeling with SPICE. McGraw-Hill, N.Y., 1993, p. 479.

[4] A. Baskys, "The model of the p-n junction depletion region v-i characteristic considering the dependence of concentration of majority carriers on voltage", in Proc. of the Int. Baltic Electronics conf. BEC 2006, Tallinn, 2006, pp. 45-46.

[5] A. Dargys, J. Kundrotas, Handbook on physical properties of Ge, Si, GaAs, and InP, Science and encyclopedia publishers, Vilnius, 1994, p. 264.

[6] T. T. Mnatskanov, D. Schroder, A. Schlogl, "Effect of high injection level phenomena on the feasibility of diffusive approximation in semiconductor device modeling", Solid-St Electron., vol. 42, no. 1, pp. 153-163, 1998. [Online]. Available: http://dx.doi.org/10.1016/S0038-1101(97)00265-7

[7] M. Linder, F. Ingvarson, K. O. Jeppson, J. V. Grahn, S. L. Zhang, M. Ostling, "On DC modeling of the base resistance in bipolar transistors", Solid-St Electron., vol. 44, pp. 1411-1418, 2000. [Online]. Available: http://dx.doi.org/10.1016/S0038-1101(00) 00075-7

[8] A. Baskys, R. Navickas, C. Simkevicius, "The fast differential amplifier-based integrated circuit yield analysis technique", Acta Physica Polonica A., vol. 119, no. 2, pp. 259-261, 2011.

A. Baskys (1), M. Sapurov (2), R. Zubavicius (2)

(1) Department of Computer Engineering, Vilnius Gediminas Technical University, Naugarduko St. 41, LT-03227, Vilnius, Lithuania; phone: +370 5 2744767

(2) Electronic Systems Laboratory, Center for Physical Sciences and Technology, A. Gostauto St. 11, LT- 01108 Vilnius, Lithuania, phone +370 5 2613989 algirdas.baskys@vgtu.lt

Manuscript received March 18, 2012; accepted April 26, 2012.

This work was supported by the Agency for Science, Innovation and Technology (MITA) under High technology development programme project No. 31V-37.
```

TABLE I. IMPURITY CONCENTRATIONS OF JUNCTIONS.

[N.sub.2]                     1             2             3

[N.sub.A], [cm.sup.-3]   [10.sup.16]   [10.sup.17]   [10.sup.18]
[N.sub.D] [cm.sup.-3]    [10.sup.16]   [10.sup.16]   [10.sup.15]

TABLE II. CURRENT DENSITY OF SILICON JUNCTIONS WITH SHORT
QUASI-NEUTRAL REGIONS  ([W.sub.p], [W.sub.n] = 3 x [10.sup.-4] cm)
at given  injection levels [K.sub.P] and [K.sub.N].

Junction current density,[A/[mm.sup.2]]

[N.sub.A], [N.sub.D]               [k.sub.p]           [k.sub.n]
No   [1/[cm.sup.3]]
0.1               1       0.1    1.0

1    [N.sub.A] = [10.sup.16]   0.23              2.3     0.23   2.3
[N.sub.D] = [10.sup.16]
2    [N.sub.A] = [10.sup.17]   0.072             0.82    2.7    18.6
[N.sub.D] = [10.sup.16]
3    [N.sub.A] = [10.sup.18]   6.7x[10.sup.-3]   0,067   25     130.6
[N.sub.D] = [10.sup.15]

TABLE III. CURRENT DENSITY OF SILICON JUNCTIONS WITH LONG
QUASI-NEUTRAL REGIONS ([W.sub.P] > [L.sub.N],
[W.sub.N] > [L.sub.P]) at given injection levels [K.sub.p] and
[K.sub.N].

Junction current density, [A/[mm.sup.2]]

[k.sub.p]
[N.sub.A], [N.sub.D]
No   [1/[cm.sup.3]]                  0.1                  1

1    [N.sub.A] = [10.sup.16]         0.0016             0.016
[N.sub.D] = [10.sup.16]

2    [N.sub.A] = [10.sup.17]   6.8 x [10.sup.-4]        0.008
[N.sub.D] = [10.sup.16]

3    [N.sub.A] = [10.sup.18]   5.7 x [10.sup.-5]   6 x [10.sup.-4]
[N.sub.D] = [10.sup.15]

Junction current
density,
[A/[mm.sup.2]]

[k.sub.n]

[N.sub.A], [N.sub.D]       0.1    1.0

No   [1/[cm.sup.3]]

1    [N.sub.A] = [10.sup.16]   0.002   0.02
[N.sub.D] = [10.sup.16]

2    [N.sub.A] = [10.sup.17]   0.028   0.2
[N.sub.D] = [10.sup.16]

3    [N.sub.A] = [10.sup.18]   0.37    2.7
[N.sub.D] = [10.sup.15]
```
COPYRIGHT 2013 Kaunas University of Technology, Faculty of Telecommunications and Electronics
No portion of this article can be reproduced without the express written permission from the copyright holder.