Effcet of Tunneling

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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. H ur kx,

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 analyt-

ical functions of local variables which makes it easy to imple-

ment in a numerical device simulator. The trap-assisted tun-

neling effect is described by a n expression that f or weak electric

fields reduces to the conventional Shockley-Read-Hall SRH)

expression for recombination via traps. Compared to the con-

ventional SRH expression, the proposed model has one extra

physical parameter, vis. the effective mass m . or

m =

0 25m0

the model correctly describes the experimental observations as-

sociated with tunneling, including the distinctly d ifferent tem-

perature behavior of trap-assisted tunneling and band-to-band

tunneling. The band-to-band tunneling contribution is found to

be important at room temperature fo r electric fields larger than

7 X lo5

V/c m. It is shown that for dopant concentrations above

5

X

1 ” or, equivalently, for breakdown voltages below

approximately 5 V, the reverse characteristics are dominated

by band -to-band tunneling.

1.

INTRODUCTION

ECE NT developments in both bipolar and MOS tech-

R ologies, 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 add ition, the high

intrinsic-base dopant concentration possible in Si/SiG e/Si

heterojunction bipolar transistors also gives rise to strong

electric fields at the intrinsic emitter-base junctio n.

It is a well-known fact that in a strong electric field,

tunneling of electrons through the bandgap can signifi-

cantly contribute to carrier transport in a p-n junctio n [11,

[2]. Both transitions directly from band to band (Zener

tunneling) and transitions via traps (trap-assisted tunnel-

ing) 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. ang.

The authors are with Philips Research Laboratories, 56 00 JA Eindhoven,

The Netherlands.

IEEE Log Number 9104679.

nonideal current under forward bias (forward-biased tun-

neling) [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 fo r lit-

erature data and for our own measurements [3], [5]-[7].

Details of this figure are given in Section IV . O ther char-

acteristic features of this high nonideal forward current

are a reduced temperature dependence and a high non-

ideality factor [3],

[4].

For C AD purposes it is of crucial importance that these

effects are properly taken into account in a num erical de-

vice simulator. Since these effects can basically be con-

sidered as the generation or recombination of electron-

hole pairs, they must be incorporated into the recombi-

nation 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. T hese

models, however, suffer from the following drawbacks:

Since these models describe tunneling by means of a

current density , they are only suitable fo r post-processing

calculations and cannot be incorporated into the continu-

ity 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

s the contribution of transitions via traps (in-

cluding the conventional SRH recombination mechanism)

and Rbb, s the band-to-band tunn eling contribution. In the

following sections we discuss these two terms

in

detail,

while in Section IV a comparison is made between sim-

ulation results and experiments.

11. MOD ELING RAP-ASSIST EDU N N E L I N G

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 be-

tween 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 den-

sities of electrons and holes which have the capture rates

c

and

c p ,

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 car-

riers and the emission probability per unit of time are in-

creased by tunneling.

In a weak electric field the carrier densities at a certain

location in a depletion lay er 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 in-

creased due to the finite probability of carriers tunneling

into the gap. For instance an electron at location

x l

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 tun-

neling distances are relatively short, an d, henc e, this tun-

neling effect becomes important. In order to obtain an

neut ra l

n

neut ra l p

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) tun-

neling 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

x 1 =

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, in-

cluding the tunneling contribution, from the solution of

the effective-mass Schrodinger equation for a linear po-

tential. 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 ma ss.

The first term on the right-hand sid e of the above expres-

sion is the conventional density of electrons in the con-

duction band, while the second term is the tunneling con-

tribution [SI. The physical meaning of

Ai2[y x

x l ) ] A i 2

(0) is the probability that an electron at

x 1

will

tunnel to a trap at

x .

The integration in

3)

is performed

over all locations

x 1

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 minim um 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 mini-

mum at the n side,

6

> 0 and tunneling to x can occur

only over a part of the depletion layer. This will be dis-

cussed in more detail below. For holes, a similar expres-

sion can be derived.



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

The emission of electrons and holes from a trap is en-

hanced by the phonon-assisted tunneling effect (see Fig.

2(b)) [ lo], [4]. Instead of thermal emission ov er 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 tun-

neling through the remaining potential barrier (transition

P ’ P ”).

Following the approach of Vincent

et al .

[ l o ] ,

the expression for the enhancement of the emission p rob-

ability is given by an integral over the trap depth of the

product of a Boltzmann fa ctor, 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. F or electrons the emission probabil-

ity reads

e Ai

(0) d E

i 2 ( 2 m * y - 2 h - 2 E )

(4)

where

e,,Os

the emission probability in the absence of an

electric field. Again, the value of A E,,

depends on the rel-

ative position of the trap level and the conduction-band

minimum at the neutral n side. F or the situation sketched

in

Fig. 2(b), tunneling at all levels between

ET

and

E ,

is

possible,

so A E , = 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)

n ? =

r,,+ 1

en0 n

p - p t = r p + i

e p o P

where we have introduced the field-effect functions rn nd

rp .

ollowing the sam e derivation used to obtain the con-

ventional SRH expression from

( 2 )

[9], we arrive at

where

The quantity F is the local electric field. Analytical ap-

proximations 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 posi-

tion 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 po-

sition-dependent. For the determination of these integra-

tion 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 conduc-

tion-band minimum

E,(x)

and the conduction-band mini-

mum 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 in-

tegration 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 po-

tential ( i. e. , the isri ns ic Ferm i level) , the local value of

the bandgap and E , which is the relative position of the

trap level with respect to the intrinsic Ferm i 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, re-

spectively, 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

- q (x), respectively. The quantities +,(x) and +p x) are

the local quasi-Fermi levels of electrons and ho les. In re-

verse bias these levels are not constant but their relative

position with respect to the trap level is such that they do

provide the co rrect 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 inter-

action between the free carrier and the trap, the effective

trap depth is lowered (s ee 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 [l o] . 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- o r acceptor-like only o ne of the two

field-enhancement factors in

(6)

increases due to the

Poole-Frenkel effect [9]. The maximum influence of t he

Poole-Frenkel effect on

Rtrap

ccurs 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. Th is 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 va-

lidity of using a linear potential to calculate the tunneling

probability. This can be investigated by replacing the tun-

neling probability evaluated for a constant field

Ai2

y(x

x l ) ] 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 t he carrier in

the gap, which is determined by the actual potential dis-

tribution between xI, nd

x .

Fig. 3 shows the numerically

calculated value of r or a trap in the middle of a for-

ward-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 elec-

tric 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 t 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

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 A E , , = 0.4 e V

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 tun-

neling prob ability, while the other lines a re 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. T o 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 F ig. 7), which is quite a plau-

sible one.

111. MOD ELIN G AND -TO-B ANDUNNELING

For the band-to-band tunneling contribution Rbbrwe

base ourselves on the theoretical work of Keldysh and

Kane [121-[ 141. Since silicon is an indirect semiconduc-

tor whose direct bandgap is much larger than its indirect

gap, indirect transitions including electron-phonon inter-

action are predominant. Keldysh calculated the transition

rate on the basis of a solution of the time-dependent

Schrodinger equation , including electron-phonon inter-

action. His results were later adopted by Kane to obtain

an expression for the tunneling current density per unit of

energy d J b b r / d E [14]. Both directly from the work of

Keldysh, and from Kane's work by using the relation

the following expression for Rbbtcan be obtained:

In [12]-[14] it can be found that U = 2 for direct transi-

tions and

U

= 5 / 2 for indirect transitions, including elec-

tron-phonon interaction. Since silicon is an indirect semi-

conductor, 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 o Rbbrhe tunneling of an electron

at a certain energy, say E l (see Fig. 4), from x 1 to

x

is

represented by the generation of an electron-hole pair in



HURKX er

a l . :

A NE W M ODE L

FOR

DE VI CE S I M UL AT I ON I NCL UDI NG T UNNE L I NG

335

neut ra l p

. - - - - - - - - -

neut ra l

n

4

t

A

_ _ _ _ _ _ - - -

x xp x

Fig. 4 . Schematic energy-band diagram

of

a reverse-biased

p-n

junc t ion.

The band-to-band tunneling mechanism is indicated. Band-to-band tunnel-

ing 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 influ-

ence 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,

in Fig. 4,

because in this region there are no final states into which

an electron can tunnel. Fo rx

>

xp n Fig. 4, this function

also equals zero because there are no initial states from

which electrons can tunnel. For

x, < <

xp

or, equiva-

lently, when the tunneling energy El lies between

Ef,

nd

ESP,

his function equals unity.

As

in the case of trap-as-

sisted tunneling, the quantities

Efi,

nd

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. Th e reason fo r this is that the

finite saturated drift velocity of the carriers causes a n in-

crease in the ca me r densities. When the generated elec-

tron 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)

ives 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 den-

sity has increased to a certain fraction (e.g . , l o p3 ) 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

c m - ' l 2

*

V p 5 / *

s - ' ,

respectively [4], [15]. The prefactor

B

is taken to be

temperature-independent. Th e quantity F,, which is pro-

portional to

where

Eg

is the bandgap [12]-[14], de-

pends on the temperature due to the temperature depen-

dence of this bandgap. In order to have an idea of the

electric field strength above which band-to-band tunnel-

ing becomes important at room temperature, we compare

Rbbt

with the ratio

n r e / r ,

which is a measure of the gen-

eration rate via traps. When we take for this ratio a real-

istic value (at room temperature) of

IO"

c m P 3

*

s- ' , we

find that band-to-band tunneling becomes important at a

field strength above 7

X lo5

V / c m .

IV. SIMULATIONESULTS

N D A

COMPARISONITH

EXPERIMENTS

To give an impression of the model behavior, Fig.

5

shows

1D

simulations of diodes in reverse and forward

bias. T he diodes are step junctions with

No = lo2'

c m p 3 ,

while

N A

is varied. In these simulations conventional

models for the mobilities, bandgap n arrowing, recomb i-

nation 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 temper-

ature-independent lifetimes. From the reverse character-

istics, 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

(Zene r 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 con-

centrations above a few times 10' ' cm p3 , which is due to

trap-assisted tunneling. This is in agreement with exper-

imental observations in [3], [5]-[7] and with our own ex-

periments, 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. T he junc tion is formed by the diffusion

of boron into a heavily doped, homogeneous n-type sub-

strate. The doping profiles are determined from

C-

I/mea-

surements and from the resistivity of the substrate. Diodes

A , B , and

C

have a zero-bias depletjon layer width of ap-

proximately 200, 270, and 400

A

respectively. The

magnitude of the calculated curves depends on the life-

times, while both the slope (i.e., the nonideality factor)

and the temperature depen dence are given by the effective

mass

m .

ince the values of the lifetimes are unknown,

we have taken a constant value for T,

=

r,, =

T

for each

diode. Fo r 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 life-

times 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 - ~ ,espectively.



336

IEEE TRANSACTIONS

ON

ELECTRON DEVICES. VOL. 39, NO. 2 . FEBRUARY 1992

0.0

5

10

15

reverse vo l tage (VI-

(a)

0.0 0.2 0.4 0 6 0.8

f o rw a rd vo l t a ge ( V I-

(b)

Fig.

5 .

One-dimensional simulation results for step junctions (N,] = IO”’

cm-3 while N A s indicated in the figure) in reverse bias (a ) and in forward

bias (b). Tunneling is includ ed. 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 tempera-

ture 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

[ ]

his 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 cal-

culated forward curve of diode A is given fo r 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 re-

verse bias. In order to show the ext reme 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) diod e with a linearly graded junction at three tem-

peratures. The zero-bias depletion layer width

of

this

c

? 10’0

10-12

, ,

0.0 0.2 0.4 0.6

f orward vo l tage (V)

-

(a)

I

1O ~ ’ 2

0.0

0 2

0.4

0 6

f orward vo l tage

(V)

-

(b)

0.0

0.2

0.4 0.6

f orward vo l tage (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 vo l tage

(V)

-

Fig. 7. Forward J-Vcu rves 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

NE W M ODE L FOR DE VI CE S I M UL AT I ON I NCL UDI NG T UNNE L I NG

~

337

102

100

NI

10-2

i

3 1 0 ~ 4

?

10-6

1 0 8

t

U

1 8 8 1 1 1

0 1 2 3 4 5

reverse vol tage

(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.

10

lo

I

I

0 1 2 3 4 5

reverse vol tage (V)-

Fig.

9.

Measurements and simulation results for diode

B

in reverse bias

for three slightly different doping profiles. The solid dots are measure-

ments. 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% ower than the value used to obtain the curve in Fig.

8 .

E

t

m

E

2

YI

U

a

3

l o o t 1 : T = 2 9 4 K

2: T

=

338K

10

2

1 0 . ~

10-6

10-8

rn 10

3 : T = 3 8 3 K I

0 1 2 3 4 5 6

reverse vol tage (V)

-

Fig. 10. Reverse characteristics of

a

diode with a linearly graded junction

at three temperatures. The dots are measurements, while the lines

are

sim-

ulation results with (solid lines) and without (d ashed lines) tunneling. Fo r

this diode impact ionization is not included.

only above 3 V reverse bias. Notice the different temper-

ature dependence of the two regimes.

Finally, we return to Fig. 1,which shows a comparison

between measurements (from [3], [5]-[7] and own mea-

surements) and simulations. In this plot the forward cur-

rent 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 ca-

pacitance. For the other data we have estimated the de-

pletion lay er width from simulations on junctions with a

similar doping profile. Furthermore, although the junc-

tion 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.

Neverthe lessb we can clearly observe that below approx-

imately

300

A

zero-bias depletion layer width or, equiv-

alently, above a dopant concentration of a few times lo'*

cm P3 for a steep junction, the nonideal current increases

significantly due to tunneling. When tunne ling 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 re-

sults for an emitter-base profile of a high-frequency pro-

cess. The dashed line is obtained for step junctions by

using the conventional recombination model without tun-

neling.

V . S U M M A R YND CONCLUSIONS

In this paper we have presented a new recombination

model for device simulation which includes both trap-as-

sisted tunneling and band-to-band tunneling (Zener tun-

neling). 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-as-

sisted 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 ob-

servations:

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 tw o.

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 concen-

tration 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 lo5V/ cm . We have seen that for dopant con-

centrations above 5 X 10'' cm P3 or, equivalently, for

breakdown voltages below

4Eg

q - 6 E g

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

ield-effect functions

two situations:



IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 39, NO. 2, FEBRUARY 199238

1) For

i.e., for not too large values of the electric field (e.g., at

room temperature and for A E n , p= 0.5 eV this criterion

corresponds to F < 9 X lo5V/cm) the maximum con-

tribution to the integral in (7) comes from U = U where

0

< U, < 1. In this case the integral can be approxim ated

by a second-order series expansion of the function of

U

in

the exponent of (7) around its maximum at

U .

After set-

ting the integration boundaries to - O and a ntegration

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 integra-

tion 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 expres-

sion for the complementary erro r function, as can be found

in [18], the following expression is obtained:

r 1 1

P

and p = 0.61685,

a l

= 0.3480242,

a2

= -0.0948798,

and u3 = 0.74785 56. Th e values of

a l ,

u 2 ,and u3are from

[18], while the value of p is found from the correct be-

havior of for K n , p

0.

REFERENCES

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[17]

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*

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