a new recombination model for device simulation including tunneling

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 39, NO. 2, FEBRUARY 1992 33 1

10-@

A New Recombination Model for Device Simulation Including Tunneling

~ N A = ~ X I O ’ @ N~=10’@

! \,

G. A. M. Hurkx, D. B. M. Klaassen, and M. P. G. Knuvers

Abstract-A new recombination model for device simulation is presented. This model includes both trap-assisted tunneling (under forward and reverse bias) and band-to-band tunneling (Zener tunneling). The model is formulated in terms of analytical functions of local variables which makes it easy to imple- ment in a numerical device simulator. The trap-assisted tunneling effect is described by an expression that for weak electric fields reduces to the conventional Shockley-Read-Hall (SRH) expression for recombination via traps. Compared to the conventional SRH expression, the proposed model has one extra physical parameter, vis. the effective mass m*. For m* = 0.25m0 the model correctly describes the experimental observations as- sociated with tunneling, including the distinctly different temperature behavior of trap-assisted tunneling and band-to-band tunneling. The band-to-band tunneling contribution is found to be important at room temperature for electric fields larger than 7 X lo5 V/cm. It is shown that for dopant concentrations above 5 X 10” or, equivalently, for breakdown voltages below approximately 5 V, the reverse characteristics are dominated by band-to-band tunneling.

1. INTRODUCTION ECENT developments in both bipolar and MOS tech- R nologies, such as lateral downscaling, shallow-junc-

tion formation, and the use of self-alignment techniques, have led to an increase in electric field strength around p-n junctions in these devices. In bipolar transistors it is particularly the emitter-base junction at the emitter pe- riphery where the maximum electric field can reach values as high as lo6 V/cm, while in MOS transistors such strong fields can occur at the drain. In addition, the high intrinsic-base dopant concentration possible in Si/SiGe/Si heterojunction bipolar transistors also gives rise to strong electric fields at the intrinsic emitter-base junction.

It is a well-known fact that in a strong electric field, tunneling of electrons through the bandgap can significantly contribute to carrier transport in a p-n junction [ 11, [2]. Both transitions directly from band to band (Zener tunneling) and transitions via traps (trap-assisted tunneling) can be important. Tunneling not only adversely af- fects the leakage currents (e.g., the so-called “Zener breakdown”) but it can also lead to an anomalously high

Manuscript received February 28, 1991. Part of this work was funded by ESPRIT Project 2016. The review of this paper was arranged by As- sociate Editor D. D. Tang.

The authors are with Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands.

IEEE Log Number 9104679.

nonideal current under forward bias (forward-biased tunneling) [2]-[4]. The latter is shown in Fig. 1 , where the current at 0.3 V forward bias and at room temperature is plotted versus the zero-bias depletion layer width for lit- erature data and for our own measurements [3], [5]-[7]. Details of this figure are given in Section IV. Other char- acteristic features of this high nonideal forward current are a reduced temperature dependence and a high nonideality factor [3], [4].

For CAD purposes it is of crucial importance that these effects are properly taken into account in a numerical device simulator. Since these effects can basically be con- sidered as the generation or recombination of electron- hole pairs, they must be incorporated into the recombination term in the electron and hole continuity equations. Existing models for trap-assisted tunneling [2], [3] give a semi-empirical relation between the current density and a certain exponential function of the applied voltage. These models, however, suffer from the following drawbacks:

Since these models describe tunneling by means of a current density, they are only suitable for post-processing calculations and cannot be incorporated into the continuity equations.

0018-9383/92$03.00 O 1992 IEEE

332 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 39, NO. 2, FEBRUARY 1992

They only describe the voltage dependence of the current density, while its magnitude must be obtained from experiments.

The predicted temperature dependence is too weak

In this paper we present a recombination model which takes into account band-to-band tunneling in reverse-bias and trap-assisted tunneling in both forward and reverse bias. In situations of a weak electric field (i.e., lowly doped junctions) the model reduces to the conventional Shockley-Read-Hall expression for recombination via traps. In [4], [SI we have established the basic physics behind the model. In this work we concentrate on the for- mulation of the model for device simulation purposes and on the comparison of simulation results with experiments.

In our model the total net recombination rate is given

[41.

by

where Rtrap is the contribution of transitions via traps (including the conventional SRH recombination mechanism) and Rbb, is the band-to-band tunneling contribution. In the following sections we discuss these two terms in detail, while in Section IV a comparison is made between simulation results and experiments.

11. MODELING TRAP-ASSISTED TUNNELING The net recombination rate via traps is determined by

the density of carriers captured per unit of time and the probability per unit of time of emitting a free carrier from a trap. To obtain an expression for Rtrap we start with the following general phenomenological expression for the net recombination rate resulting from a dynamic balance between the net rate of captured electrons and that of holes. This expression reads (see, e.g., [9])

where NT is the trap density, while n, and p t are the densities of electrons and holes which have the capture rates c, and cp, respectively. The quantities e,l and ep are the respective probabilities per unit of time for the emission of an electron or a hole. Both the density of captured carriers and the emission probability per unit of time are increased by tunneling.

In a weak electric field the carrier densities at a certain location in a depletion layer are given by the conventional density of free carriers in the conduction and valence bands. However, in a strong electric field the density of carriers at a certain location within the depletion layer increased due to the finite probability of carriers tunneling into the gap. For instance an electron at location xl in Fig. 2(a) has a certain probability of tunneling to a trap at x, where it has a chance of being captured. In highly doped junctions, which have a narrow depletion layer, the tunneling distances are relatively short, and, hence, this tunneling effect becomes important. In order to obtain an

neutral n 1 1 neutral p

0

1 (b)

Fig. 2. Energy-band diagram of a depletion layer around a forward-biased junction (a) and around a trap in a reverse-biased (b) junction. In (a) tunneling of an electron from x, to a trap at location x is indicated. In (b) tunneling-enhanced emission of an electron from a trap is indicated. The solid line in (b) denotes the potential well of the trap without Coulomb interaction and the dashed line with Coulomb interaction.

expression for the tunneling current, in [2], [3] only the probability of tunneling directly through the depletion layer, i.e., the transition from x1 = 0 to x = W , is con- sidered. For this reason the temperature dependence of these models is too weak [4]. In [SI we have derived an expression for the carrier density in a depletion layer, including the tunneling contribution, from the solution of the effective-mass Schrodinger equation for a linear potential. For electrons this expression reads

(3)

where Ai is the Airy function. In y = (2qFm*h-2)”3, F is the average electric field and m* is the effective mass. The first term on the right-hand side of the above expression is the conventional density of electrons in the conduction band, while the second term is the tunneling contribution [SI. The physical meaning of Ai2[y(x - xl)] /Ai2 (0) is the probability that an electron at x1 will tunnel to a trap at x. The integration in (3) is performed over all locations x1 from which electrons can tunnel to location x. The value of 6 depends on the relative position of the trap level and the conduction-band minimum at the neutral n side. In Fig. 2(a) the trap level at location x is below the conduction-band minimum and, hence, 6 = 0. When the trap level is above the conduction-band minimum at the n side, 6 > 0 and tunneling to x can occur only over a part of the depletion layer. This will be discussed in more detail below. For holes, a similar expression can be derived.

HURKX er al . : A NEW MODEL FOR DEVICE SIMULATION INCLUDING TUNNELING 333

The emission of electrons and holes from a trap is enhanced by the phonon-assisted tunneling effect (see Fig. 2(b)) [lo], [4]. Instead of thermal emission over the entire trap depth E, - ET, which is the only escape mechanism possible in the absence of a field, carriers can also be emitted by thermal excitation over only a part of the trap depth (transition P + P‘ in Fig. 2(b)), followed by tunneling through the remaining potential barrier (transition P ’ -+ P ”). Following the approach of Vincent et al. [lo], the expression for the enhancement of the emission probability is given by an integral over the trap depth of the product of a Boltzmann factor, which gives the excitation probability of a carrier at the trap level to an excited level E, and the tunneling probability at that energy level from the trap to the band. For electrons the emission probability reads

e Ai (0) d E 1 Ai2(2m*y-2h-2E)

(4) where e,,O is the emission probability in the absence of an electric field. Again, the value of A E,, depends on the relative position of the trap level and the conduction-band minimum at the neutral n side. For the situation sketched in Fig. 2(b), tunneling at all levels between ET and E, is possible, so AE, = E, - ET.

In order to make (3) and (4) suitable for implementation into a numerical device simulator, we must express these tunneling effects in terms of analytical functions which depend on local variables only. For a linear potential it can be shown that both the carrier concentration and the emission probability are enhanced by the same factor, i.e.

( 5 4

(5b)

- en - - ? = - r,, + 1 en0 n

- - ep - p t = - r p + i epo P

where we have introduced the field-effect functions rn and rp. Following the same derivation used to obtain the conventional SRH expression from (2) [9], we arrive at

where

The quantity F is the local electric field. Analytical approximations for the integral in (7) are given in the Ap- pendix.

Because the conduction-band minimum E&) and the valence-band maximum E&) are a function of the position x in the depletion layer, the absolute value of the trap level ET(x) is also position-dependent. This implies that also the integration intervals A E,(x) and A Ep(x) are position-dependent. For the determination of these integration intervals we must distinguish between two situations: For the situation of a trap at location x in Fig. 2(a), which is important in forward-biased junctions, tunneling can occur only at an energy level between the local conduction-band minimum E,(x) and the conduction-band minimum at the neutral n side E,,, because below E,, there are no states available from (and into) which an electron can tunnel. In the case where the trap level Ej-(x) lies above E,, (most important in reverse bias, see Fig. 2(b)) the integration interval is the whole trap depth, i.e. , A E,(x) = E,(x) - ET(x). For holes, a similar criterion holds. The expression for the integration intervals can be written as

AEn(x) = - Ern, ET@) 5 Ern

= E&) - ET@), ET(x) > Ern (9a)

and AEp(x) = Evp - Ev(x17 ET(x) > Evp

= - ET(x) 5 Evp. (gb)

For device simulation the quantities E&) , E&), and ET(x) can easily be determined from the electrostatic potential (i.e., the isrinsic Fermi level), the local value of the bandgap and E , which is the relative position of the trap level with respect to the intrinsic Fermi level. At high dopant concentrations the Fermi level in a neutral region nearly coincides with the corresponding band edge. For this reason and because under low and medium fonvard-

where E = ET - Ei , i.e., the difference between the trap level and the intrinsic level. The quantities r, and rp are the recombination lifetimes of electrons and holes, respectively, while nie is the intrinsic carrier concentration. For weak electric fields, rn,p << 1 and (6) reduces to the conventional SRH recombination formula.

Using the asymptotic behavior of the Airy function Ai(y) - exp (-(2/3)y3I2), the expressions for r, and r, can be written as

bias conditions, where tunneling is important, the quasi- Fermi levels are approximately constant in the depletion region, E,, and Eup can be replaced by -q+,(x) and - qq$(x), respectively. The quantities +,(x) and +p(x) are the local quasi-Fermi levels of electrons and holes. In reverse bias these levels are not constant but their relative position with respect to the trap level is such that they do provide the correct criterion for the integration intervals.

Using the analytical approximations for the integral in (7), together with (8) and (9), (6) is readily suitable for incorporation into a numerical device simulator. How-

334 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 39, NO. 2. FEBRUARY 1992

ever, considering the validity of (6), together with (7), two questions may arise.

The first one concerns the Poole-Frenkel effect, which is the mechanism whereby in the case of Coulomb interaction between the free carrier and the trap, the effective trap depth is lowered (see dashed line in Fig. 2(b)). This is not taken into account in the above expressions. At a strong electric field this effect is much weaker than the tunneling effect, but at a weak field it can greatly enhance the emission probability [lo]. This Poole-Frenkel effect occurs in processes where the trap is neutral when it is occupied by a carrier (either a hole or an electron) and charged when the carrier is being emitted. However, since a trap is either donor- or acceptor-like only one of the two field-enhancement factors in (6) increases due to the Poole-Frenkel effect [9]. The maximum influence of the Poole-Frenkel effect on Rtrap occurs when one of the two field-enhancement factors in (6) becomes so large that Rtrap is determined solely by the slowest process, i.e., the other term in the denominator. That term only experiences the tunneling effect. This implies that the overall influence of the Poole-Frenkel effect on the net recombination rate is fairly limited (maximum a factor of 2 for midgap states and equal lifetimes for electrons and holes).

The second question which may arise concerns the validity of using a linear potential to calculate the tunneling probability. This can be investigated by replacing the tunneling probability evaluated for a constant field Ai2 [y(x - xl)] / A i 2 (0) (or its asymptotic approximation in (7)) by the WKB expression [ 111

Tt = exp /-2 i:, IK(X')I d x ' j (10)

which is valid for an arbitrary potential. In ( lo) , I K ( x ' ) ( is the absolute value of the wave vector of the carrier in the gap, which is determined by the actual potential dis- tribution between xI, and x . Fig. 3 shows the numerically calculated value of rn for a trap in the middle of a forward-biased, linearly graded junction versus the depletion layer width (i.e., the doping gradient) for three cases:

1) the tunneling probability evaluated for a constant field is used and for F the local electric field at the trap is taken (dotted line);

2) the tunneling probability evaluated for a constant field is used and for F the average electric field in the depletion layer is taken (dashed line);

3) the tunneling probability as given by (10) is used (solid line).

From Fig. 3 we observe that the choice of the local electric field gives results which agree better with the WKB calculations than the results obtained with the average field. At this point is should be noted that both (7) and (lo), as well as the expression of Vincent et al. [lo] ac- counting for the Poole-Frenkel effect, are obtained in a one-dimensional (1D) approach. A three-dimensional (3D) numerical treatment of these problems show that a

t i c 0

c 3

0

*

c c

1014

1012

10'0

108

106

1 o4

102

0 200 400 600 800 1000

depletion width (A) - Fig. 3 . The field-effect function I',, in the case of AE,, = 0.4 eV versus the depletion-layer width of a forward-biased, linearly graded junction for two temperatures. The solid lines are obtained by using (10) for the tunneling probability, while the other lines are obtained by using the tunneling probability for a constant electric field (dashed lines: average field; dotted lines: local field).

small variation of the effective mass in the 1D expressions can account for the 3D effects. To account for the above- mentioned effects, the value of the effective mass to be used in (8) is obtained from a comparison of simulations with experiments. Using the local value of the electric field in (8) , the experimentally obtained value of m* is 0.25mo (see Section IV and Fig. 7), which is quite a plausible one.

111. MODELING BAND-TO-BAND TUNNELING For the band-to-band tunneling contribution Rbbr we

base ourselves on the theoretical work of Keldysh and Kane [ 121-[ 141. Since silicon is an indirect semiconductor whose direct bandgap is much larger than its indirect gap, indirect transitions including electron-phonon interaction are predominant. Keldysh calculated the transition rate on the basis of a solution of the time-dependent Schrodinger equation, including electron-phonon interaction. His results were later adopted by Kane to obtain an expression for the tunneling current density per unit of energy dJbbr/dE [14]. Both directly from the work of Keldysh, and from Kane's work by using the relation

the following expression for Rbbt can be obtained:

In [12]-[14] it can be found that U = 2 for direct transitions and U = 5 /2 for indirect transitions, including electron-phonon interaction. Since silicon is an indirect semiconductor, we use U = 5/2. In (1 1) rl, is the electrostatic potential, while in (12) E' and E' are the Fermi levels at the neutral n and p side, respectively. In the above trans- formation from d Jbbt to Rbbr the tunneling of an electron at a certain energy, say El (see Fig. 4), from x1 to x2 is represented by the generation of an electron-hole pair in

HURKX er a l . : A NEW MODEL FOR DEVICE SIMULATION INCLUDING TUNNELING 335

neutral p

.---------

neutral n 4' I Et"

' A 1 _ _ _ _ _ _ _ - - -

x, xp x - Fig. 4 . Schematic energy-band diagram of a reverse-biased p-n junction. The band-to-band tunneling mechanism is indicated. Band-to-band tunneling is only possible in the region x,, 5 x < x,,.

the middle of the gap (xI + x2)/2. The function D(F, E, Eb,, ESP) accounts for the relative position of the Fermi levels Efi, and Efp in the neutral regions and for the influence of the motion of the electron perpendicular to the electric field on the tunneling probability [ 141. In forward bias, it accounts for the well-known peak in the tunneling current observed in Esaki diodes. An expression for D(F, E , Eb,, Eh), which is valid in zero and reverse bias and which is suitable for implementation in a device simulator can be obtained from [14]. This gives

(13) 1 -

exp [(-E, - q$)/kTl + 1 .

This function virtually equals zero for x < x, in Fig. 4, because in this region there are no final states into which an electron can tunnel. Forx > xp in Fig. 4, this function also equals zero because there are no initial states from which electrons can tunnel. For x, < x < xp or, equivalently, when the tunneling energy El lies between Ef, and ESP, this function equals unity. As in the case of trap-assisted tunneling, the quantities Efi, and Eh can be replaced by -q&(x) and -q4p(x), respectively. However, from numerical simulations it is found that when the current density in reverse bias is very high, the above replace- ment gives incorrect results. The reason for this is that the finite saturated drift velocity of the carriers causes an increase in the camer densities. When the generated electron and hole densities are in the order of the intrinsic carrier density, the quasi-Fermi levels lie very close to the intrinsic level E, (E, = -q$). In that case, the replace- ment of Ef, and Eh by -q&(x) and -q$p (x) gives a value of D significantly less than its actual value 1. This can be remedied simply by putting D = 1 at those mesh points where the magnitude of the electron or hole current density has increased to a certain fraction (e.g., lop3) of qnrcUs, where U , is the saturated drift velocity.

The quantities F, and B at room temperature are found to be 1.9 X lo7 V/cm and 4 X l O I 4 cm-'l2 * Vp5/*

s- ' , respectively [4], [15]. The prefactor B is taken to be temperature-independent. The quantity F,, which is pro- portional to where Eg is the bandgap [12]-[14], depends on the temperature due to the temperature dependence of this bandgap. In order to have an idea of the electric field strength above which band-to-band tunneling becomes important at room temperature, we compare Rbbt with the ratio nre / r , which is a measure of the generation rate via traps. When we take for this ratio a real- istic value (at room temperature) of IO" cmP3 * s-', we find that band-to-band tunneling becomes important at a field strength above 7 X lo5 V/cm.

IV. SIMULATION RESULTS AND A COMPARISON WITH EXPERIMENTS

To give an impression of the model behavior, Fig. 5 shows 1D simulations of diodes in reverse and forward bias. The diodes are step junctions with No = lo2' cmp3, while N A is varied. In these simulations conventional models for the mobilities, bandgap narrowing, recombination lifetimes, and impact-ionization rates are used, as can be found, for instance, in [16]. Furthermore, we have used E&) = E, (x) (i.e., "midgap" states) and temperature-independent lifetimes. From the reverse characteristics, shown in Fig. 5(a), we can observe that for dopant concentrations above 5 X lOI7 cmP3 or, equivalently, for breakdown voltages below 4Eg/q-6Eg/q, the reverse characteristics are dominated by band-to-band tunneling (Zener tunneling). This is in agreement with the criteria mentioned in standard textbooks (e.g., [17]). From the forward characteristics given in Fig. 5(b) we see that the nonideal current increases significantly for dopant concentrations above a few times 10'' cmp3, which is due to trap-assisted tunneling. This is in agreement with experimental observations in [3], [5]-[7] and with our own experiments, as will be shown below.

Fig. 6 shows a comparison of simulation results with measurements on different diodes having linearly graded junctions. These diodes have a large junction area (204 x 204 pm2), and sidewall effects are eliminated by the use of guard rings. The junction is formed by the diffusion of boron into a heavily doped, homogeneous n-type substrate. The doping profiles are determined from C- I/ measurements and from the resistivity of the substrate. Diodes A , B , and C have a zero-bias depletjon layer width of approximately 200, 270, and 400 A , respectively. The magnitude of the calculated curves depends on the lifetimes, while both the slope (i.e., the nonideality factor) and the temperature dependence are given by the effective mass m*. Since the values of the lifetimes are unknown, we have taken a constant value for T, = r,, = T for each diode. For each diode the value of T is chosen such that at T = 294 K the magnitude of the simulated curve, using the new model, fits the measurements. The resulting lifetimes are 0.6, 2.5, and 20 ps for diodes A , B , and C , which have a substrate doping concentration of around 2 X 1019, 7 X lo1', and 1.9 x 10" ~ m - ~ , respectively.

336 IEEE TRANSACTIONS ON ELECTRON DEVICES. VOL. 39, NO. 2 . FEBRUARY 1992

0.0 5 10 15

reverse voltage (VI - (a)

0.0 0.2 0.4 0.6 0.8

forward voltage (V I - (b)

Fig. 5 . One-dimensional simulation results for step junctions (N,] = IO”’ cm-3 while N A is indicated in the figure) in reverse bias (a) and in forward bias (b). Tunneling is included. The regions where band-to-band tunneling predominates are indicated.

From this figure we observe that the nonideal current of the highest doped diode ( A ) has a much weaker temperature dependence than predicted by the conventional SRH model, while for the lowest doped diode this difference is much less pronounced. For diode A the nonideality factor (i.e., the slope of the low-bias I-V curve) is also, espe- cially at low temperatures, much larger than 2, whereas the conventional SRH model predicts a value of slightly less than 2. In [4] this is discussed in greater detail. It is important to notice that calculations with a different value of the effective mass agree less well with both the mea- sured nonideality factor and temperature dependence of these diodes. This is illustrated in Fig. 7 where the calculated forward curve of diode A is given for three values of the effective mass.

Fig. 8 shows the reverse characteristics of these diodes at room temperature. The values of the lifetimes that are used are the same as for the forward-bias calculations. For these diodes band-to-band tunneling predominates in reverse bias. In order to show the extreme sensitivity of the band-to-band tunneling current on the electric field, in Fig. 9 the calculated reverse characteristics of diode B are shown for three slightly different doping profiles. The doping profiles differ in such a way that the corresponding zero-bias depletion capacitance is 10% higher or 10% lower than the value used to obtain Fig. 8.

Fig. 10 shows the reverse characteristics of a (lower doped) diode with a linearly graded junction at three temperatures. The zero-bias depletion layer width of this

c ? 10’0

10-12

0

, , ,,

0.0 0.2 0.4 0.6

forward voltage (V) - (a)

,I , 1 O ~ ’ 2 0.0 0.2 0.4 0.6

forward voltage (V) - (b)

0.0 0.2 0.4 0.6

forward voltage (V) - (C)

Fig. 6 . Measurements and simulation results for three diodes in forward bias at two temperatures. The solid dots are measurements, while the lines are simulation results with (solid lines) and without (dashed lines) the in- clusion of tunneling effects.

0.0 0.2 0.4 0.6

forward voltage (V) - Fig. 7. Forward J-Vcurves of diode A calculated for three different values

of the effective mass and at two temperatures.

diode is 460 A , so the electric field is less than that in diodes A-C. This can also be observed from the fact that band-to-band tunneling dominates trap-assisted tunneling

HURKX CI al . : A NEW MODEL FOR DEVICE SIMULATION INCLUDING TUNNELING

~

337

102 1 100

NI 10-2 i 3 10~4

? 10-6

108

t

U

1 8 8 1 1 1

0 1 2 3 4 5

reverse voltage (V) - Fig. 8. Measurements (dots) and simulation results with (solid lines) and without (dashed lines) tunneling for the three diodes of Fig. 6 in reverse bias and at room temperature.

t 10 lo I I

0 1 2 3 4 5

reverse voltage (V) - Fig. 9. Measurements and simulation results for diode B in reverse bias for three slightly different doping profiles. The solid dots are measurements. The solid line denotes the same simulation results as given in Fig. 8. The doping profiles corresponding to the dashed lines differ in such a way that the corresponding zero-bias depletion capacitance is 10% higher or 10% lower than the value used to obtain the curve in Fig. 8.

E

t m

E

2 YI C

U C a+

3

l o o t 1:T=294K 2: T = 338K

10 2

1 0 . ~

10-6

10-8

rn-10

r 3 :T=383K I

0 1 2 3 4 5 6

reverse voltage (V) - Fig. 10. Reverse characteristics of a diode with a linearly graded junction at three temperatures. The dots are measurements, while the lines are simulation results with (solid lines) and without (dashed lines) tunneling. For this diode impact ionization is not included.

only above 3 V reverse bias. Notice the different temperature dependence of the two regimes.

Finally, we return to Fig. 1, which shows a comparison between measurements (from [3], [5]-[7] and own measurements) and simulations. In this plot the forward current density at 0.3 V and at room temperature is plotted versus the zero-bias depletion width. For the data in [3] and for our own data the values of the zero-bias depletion layer width are obtained from the zero-bias depletion capacitance. For the other data we have estimated the de-

pletion layer width from simulations on junctions with a similar doping profile. Furthermore, although the junction areas are rather large, it is not explicitly mentioned in [5]-[7] that sidewall effects do not play a significant role. This means that, since the current density is obtained by dividing the current by the junction area, for these data the values of the current density are somewhat uncertain. Neverthelessb we can clearly observe that below approximately 300 A zero-bias depletion layer width or, equivalently, above a dopant concentration of a few times lo'* cmP3 for a steep junction, the nonideal current increases significantly due to tunneling. When tunneling is included in the recombination model, this increase is also given by the simulation results. In Fig. 1 this is shown by solid line 1 , which represents results for a step function (similar to the results in Fig. 5(b)) and by solid line 2 which are results for an emitter-base profile of a high-frequency process. The dashed line is obtained for step junctions by using the conventional recombination model without tunneling.

V. SUMMARY AND CONCLUSIONS In this paper we have presented a new recombination

model for device simulation which includes both trap-assisted tunneling and band-to-band tunneling (Zener tunneling). The model is formulated in terms of analytical functions of local variables, which makes it easy to im- plement in a numerical device simulator. The trap-assisted tunneling effect is described by an expression that for weak electric fields reduces to the conventional SRH expression for recombination via traps. Compared with the conventional SRH expression, the proposed model has one extra physical parameter, viz. the effective mass m*. For m* = 0.25mo, which is a quite plausible value, the model correctly describes the following experimental observations:

1) The weak temperature dependence of the nonideal forward current in heavily doped junctions.

2) The nonideality factor of such a junction which, es- pecially at low temperatures, has a value significantly larger than two.

3) The significant increase in the nonideal current for a diode wit! a zero-bias depletion layer width less than about 300 A , or, equivalently, above a dopant concentration of a few times 10'' ~ 1 1 1 ~ ~ .

The band-to-band tunneling contribution is found to be important at room temperature for electric fields larger than 7 x lo5 V/cm. We have seen that for dopant concentrations above 5 X 10'' cmP3 or, equivalently, for breakdown voltages below 4Eg /q-6Eg / q , the reverse characteristics are dominated by band-to-band tunneling. This is in agreement with the criteria given in standard textbooks.

APPENDIX In order to obtain an analytical approximation for the

we must distinguish between field-effect functions two situations:

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 39, NO. 2, FEBRUARY 1992 338

1) For

i.e., for not too large values of the electric field (e.g., at room temperature and for AEn,p = 0.5 eV this criterion corresponds to F < 9 X lo5 V/cm) the maximum contribution to the integral in (7) comes from U = U,, where 0 < U, < 1. In this case the integral can be approximated by a second-order series expansion of the function of U in the exponent of (7) around its maximum at U,. After setting the integration boundaries to - CO and a, integration yields

which, by the substitution of (8) into (Al), reduces to

with

(‘43) JiiGji$

Fr = 9h

So, in the situation where the maximum contribution to the tunneling effect comes from energy levels above the minimum level at which electrons can tunnel, the integration interval is irrelevant. Obviously, the same reasoning holds for the tunneling of holes. If this situation holds for both electrons and holes, the field-effect functions are equal, i .e., r, = rp = I’.

2) For

i.e., at strong electric fields, the maximum contribution to the integral in (7) comes from U = 1, i.e., the lowest energy level at which tunneling is possible. In that case we expand the term in the exponent of (7) to second order around U = 1. After setting the lower integration limit from 0 to --03 and subsequent integration, we arrive at the following expression:

where a = 0.375 Kn,p, b = 0.5 AE,,,/kT - 0.75Kn,,, and c = Kn,p - AE,,,/kT. Using the approximate expression for the complementary error function, as can be found in [18], the following expression is obtained:

r 1-1

+ P 7 I

and p = 0.61685, a l = 0.3480242, a2 = -0.0948798, and u3 = 0.7478556. The values of a l , u2, and u3 are from [18], while the value of p is found from the correct behavior of for Kn,p + 0.

REFERENCES

[ l ] E. 0. Kane, ‘‘Zener tunneling in semiconductors,” J . Phys. Chem. Solids, vol. 12, pp. 181-188, 1959.

[2] A. G. Chynoweth, W. L. Feldman, and R. A. Logan, “Excess tunnel current in silicon Esaki functions,” Phys. Rev., vol. 121, no. 3, pp.

[3] J . A. Del Alamo and R. M. Swanson, “Forward-bias tunneling: A limitation to bipolar device scaling,” IEEE Electron Device Lett.. vol. EDL-7, no. 11, pp. 629-631, 1986; also in Proc. 18th Con$ on Solid Stare Devices and Materials (Tokyo, Japan), 1986, pp. 283- 286.

[4] G. A. M. Hurkx, D. B. M. Klaassen, M. P. G. Knuvers, and F. G. O’Hara, “A new recombination model describing heavy-doping effects and low-temperature behaviour,” in Proc. Int. Electron Device Meeting (Washington, DC), 1989, pp. 307-310.

[ 5 ] H. Schaber, J. Bieger, B. Benna, and T. Meister, “Vertical scaling considerations for polysilicon-emitter bipolar transistors,” in Proc. European Solid-State Device Res. Conf. (Bologna, Italy), 1987, pp.

[6] P.-F. Lu, T.-C. Chen, and M. J. Saccamango, “Modeling of currents in a vertical p-n-p transistor with extremely shallow emitter,” IEEE Electron Device Lett., vol. 10, no. 5 , pp. 232-235, 1989.

[7] J. C. Sturm, E. J . Prinz, P. V. Schwartz, P. M. Garone, and Z. Matutinovic, “Growth and transistor applications of Si, - ,GE, struc- tures by rapid thermal chemical vapor deposition,” in Proc. 37th Nar. American Vacuum Soc. Symp (Toronto, Ont., Canada), 1990, pp. 5- 10.

[8] G. A. M. Hurkx, F. G. O’Hara, and M. P. G. Knuvers, “Modeling forward-biased tunnelling,” in Proc. European Solid-State Device Res. Conf. (Berlin, Germany), 1989, pp. 793-396.

[9] A. G. Milnes, Deep Impurities in Semiconductors. New York: Wiley, 1973.

[lo] G. Vincent, A. Chantre, and D. Bois, “Electric field effect on the thermal emission of traps in semiconductor junktions,” J. Appl. Phys., vol. 50, no. 8, pp. 5484-5487, 1979.

[ l l ] L. I . Schiff, Quantum Mechanics. Auckland, New Zealand: McGraw-Hill, 1968, pp. 268-279.

[12] L. V. Keldysh, “Behavior of non-metallic crystals in strong electric fields,” Sov. Phys.-JETP, vol. 33, no. 4, pp. 763-770, 1958.

[13] -, “Influence of the lattice vibrations of a crystal on the production of electron-hole pairs in strong electric fields,” Sov. Phys. -JETP, vol. 34, no. 4, pp. 665-668, 1958.

[ 141 E. 0. Kane, Theory of tunneling, J . Appl. Phys., vol. 32, no. 1, pp.

[ 151 G. A. M. Hurkx, “On the modelling of tunnelling currents in reverse- biased p-n junctions,” Solid-State Elecrron., vol. 32, no. 8, pp. 665- 668, 1989.

[16] H. C. de Graaff and F. M. Klaassen, Compact Transistor Modelling for Circuit Design.

[17] S . M. Sze, Physics of Semiconductor Devices. New York: Wiley, 1981, p. 98.

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684-694, 1961.

365-368.

83-89, 1961.

Vienna, Austria: Springer, 1990.

New York: Dover, 1970, p. 299. 7 *

G . A. M. Hurkx, photograph and biography not available at the time of publication.

- - k~ 4 3 ~ n , p *

D. B. M. Klaassen, photograph and biography not available at the time of (A5) publication. *

M. P. G. Knuvers, photograph and biography not available at the time of publication.

a new recombination model for device simulation including tunneling

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