The New Equations of p-n Junction Carrier Injection Level

The new equations of minority carrier hole and electron injection levels k p and k n valid at high-level injection have been derived. They relate the k p and k n and the voltage drop across the p-n junction depletion region U d . At low U d , i.e. at low-level injection the obtained equations coincide with well known exponential equations of injection level. However, at high-level injection when U d becomes high and is close to the potential barrier of junction, the derived equations give increased steepness of k p and k n dependence on U d as compared with the exponential law. The dependences of k p and k n of concrete silicon p-n junctions with different impurity concentrations have been analyzed using derived equations of injection level. DOI: http://dx.doi.org/10.5755/j01.eee.19.2.3467


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 k p and k n can be presented as follows: where V T is thermal potential.Using (1), ( 2) and ( 3), (4), knowing that p n0 = n i 2 / N D and n p0 = n i 2 / N A , (n i is intrinsic carrier concentration) the hole and electron injection levels: The ( 5) and ( 6) are important since they allow us to estimate the injection level of minority carriers at given U d , and U 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 n (x n ) -p n0 << N D and n p (-x p ) -n p0 << N 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 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 d at arbitrary injection level including high-level injection, derived on basis of commonly accepted Boltzmann relations and quasi-neutrality conditions are following [4]: where V B is the potential barrier of junction.On basis of equations ( 1), ( 2) and ( 7), ( 8), taking into account that p n0 = n i 2 / N D and n p0 = n i 2 / N A , the hole and electron injection levels: In contrast to exponential equations ( 5) and ( 6), the highlevel 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 D exp(U d / V T) )] / N A , and [N
If junction operates at high-level injection when U d is so can be made, the equations ( 9), (10) simplifies us follows: It is seen (11), (12) that k p and k n dependence on U d at high-level injection, when U d is close to V B , become not exponential and is determined by function V T / (V B -U d ).By this is meant that voltage drop across the depletion region can not exceed V B , i.e. the inequality U d < V 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 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 d and is determined by square of impurity concentrations ratio The equation of ratio k p / k n obtained on basis of ( 9) and (10) shows that in general case the ratio k p / k n is U d dependent (14) and varies in range from value presented by equation ( 13) at low-level injection when exp[- 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 known relation , the dependences of k p and k n on U d on the basis of equations ( 9) and (10) were computed (Fig. 1).Additionally, the k p and k n dependences using exponential equations ( 5), ( 6) were calculated for junction No.1 (Fig. 1).It is seen that at low U d the dependences calculated using (9), (10) and ( 5), (6) coincide.When U 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 U d the dependences of k p and k 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 p and k n was provided according to the following scheme.Firstly, the U d for given k p and k n using graphs presented in Fig. 1 were estimated.Secondly, using classical equations J sp = en i 2 D p /(W n N D ), en i 2 D p /(L p N D ) and J sn = en i 2 D n /(W p N A ), en i 2 D n /(L n N A ) the hole and electron saturation current densities were calculated for case when junctions are with short (W p, W n =3x10 -4 cm) and long (W p >L n , W n >L p ) quasi-neutral regions, where W p and W n are lengths of p and n quasi-neutral regions, L p and L n are hole and electron diffusion lengths.The data for calculations were taken from [5].9) and (10) (solid lines).The dependence for junction No1 presented by the dashed line has been calculated using equations ( 5) and (6).
Knowing the J sp and J sn , the minority carrier hole and electron current densities of the junction at the edges of the depletion region J p and J n at given U d were computed assuming that their dependences on U d in respect of J sp and J sn are determined by law of boundary conditions ( 7) and ( 8), respectively.
The total current density of junction at given U d and, consequently, at given k p and k n was estimated as sum J = J p + J n .The J values at injection levels k p , k 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 p , k 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 p , k n = 1 corresponds with high-level injection [3].It is seen (Table II and Table III) that at given values of k p and k 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 sp and J 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 2 [6].For emitter junction of n-p-n integral transistors, as an example, the current density can reach 100 A/mm 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 baseemitter junction of p-n-p lateral transistor used in integrated circuit of voltage comparator [8] is presented in Fig. 2. The U BE in Fig. 2 is voltage applied to the base -emitter junction and I E is emitter current.The area of the junction is 1.3x10 4 mm 2 , the maximum operating current I 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 p , k n =0.1; 1 are marked in Fig. 2. It is seen that the low-level injection condition for holes (k p < 0.1) and electrons (k 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 d for case of highlevel injection.
2. In general case the ratio of hole and electron injection levels is U d dependent and varies in range from (N A / N D ) 2 at low-level to approximately N A / N 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.

Fig. 1 .
Dependences of minority carrier hole and electron injection levels on voltage drop across the depletion region for silicon junctions No1-No.3calculated using equations (

Fig. 2 .
Fig. 2. Experimental voltage-current characteristic of the base-emitter junction of the p-n-p integral lateral transistor with junction area 1.3x10 -4 mm 2 .

TABLE I .
IMPURITY CONCENTRATIONS OF JUNCTIONS.

TABLE II .
CURRENT DENSITY OF SILICON JUNCTIONS WITH SHORT QUASI-NEUTRAL REGIONS (WP, WN = 3X10 -4 CM) AT GIVEN INJECTION LEVELS KP AND KN.