Use of the gummel number in calculating the diffusion current density in Schottky diodes

00361101/82/01006~04sO3.0/0 Pergamon Press Ltd.

Solid-Stole Ebctronicr Vol. 25, No. I, pp. 63-66. 1982 Printed in Crest Britain

USE OF THE GUMMEL NUMBER IN CALCULATING THE DIFFUSION CURRENT DENSITY IN SCHOTTKY DIODES M. KLEEFSTRA Electrical Engineering Department, Delft University of Technology, Delft, The Netherlands (Receioed 4 March 1981;in revised form 11 May 1981)

Abstract-It is shown that the theories for the diffusion current density in Schottky diodes and in p-n diodes of finite length are equivalent. Writing :he final expression for the diffusion current density in the Schottky diode with the Gummel number as one of the main controlling parameters facilitates their comparison with p-n diodes. As an illustration the theory is applied to a metal-thin p layer-n layer Schottky diode.

The final expressions given for the current density, however, look very different. The diffusion theory for the Schottky diode is commonly given for the simple case of homogeneously doped semiconductor [ 1,2]. Cases of a nonhomogeneously doped semiconductor or the presence of a small layer of opposite doping are left out of consideration. Changes in the bandgap E,, which may be introduced by heavy doping, are not taken into account. Recently majority-carrier diodes of the types metalthin p layer-n layer and n t - thin p layer-n layer have been proposed and partially evaluated [3-71. In principle, such diodes have some advantages: The height of the barrier can be tailored with the aid of the thickness and doping level of the thin p-type layer. There is no imageforce barrier lowering, and because the barrier is not as sharp as in classical Schottky diodes, tunneling will be less likely to occur. It should, of course, be determined to what extent diffusion hampers the thermionic emission current in such diodes. These reasons incited us to consider the diffusion theory in such a way that the presence of a thin p-type layer on the n layer and the transition to the p-n diode are easily accounted for. The assumptions made will be the same as those in the existing theories. The Gummel number of the region which determines the diffusion will prove to be extremely useful.

NOTATION Richardson’s constant electron diffusion coefficient EC conduction band edge EF Fermi level 4 bandgap E” valence band edge JD electron saturation current density according to diffusion theory J” electron current density JT electron saturation current density according to thermionic emission theory NA acceptor impurity concentration ND donor impurity concentration N, effective density of states for electrons T absolute temperature V applied voltage vb, built-in voltage w electrostatic potential W thickness of layer determining electron diffusion d thickness of p-type layer h Planck’s constant Boltzmann’s constant effective mass of electrons n electron density tlf electron densitv times hole densitv_ at eauilibrium . P hole density I 4 absolute magnitude of electronic charge 0th thermal velocity of electrons distance i electric field h mean free path for electrons CL electron mobility PENdistance between conduction band edge and Fermi level at the metal-semiconductor interface App,, increase in barrier height through the presence of a p-type layer A D

Ink”

2.DIFFUSION THEORY FOR SCHO'ITKY DIODESOF ARBITRARY DOPING

In Fig. 1 the electron energy levels are given for a metal-semiconductor structure. The n layer to the right of W is not depleted. The current-voltage characteristic of this structure will be calculated under the following conditions:

LINTRODUCTION

The thermionic emission theory is generally accepted for metal-semiconductor contacts on moderately doped semiconductors where the charge carriers have a high mobility, such as silicon and gallium arsenide. The diffusion theory is applicable foi Schottky diodes on lightly doped semiconductors with a low potential barrier. Both theories have been extensively reviewed by Rhoderick[l]. In the derivation of the diffusion theory for Schottky diodes no reference is made to the theory of the p-n diode of finite length, and vice versa, although the starting equations are the same in both cases.

The recombination-generation rate of charge carriers is infinitely large at point x = 0, the metal-semiconductor interface. The recombination-generation rate of charge carriers is negligibly small between x = 0, and x = IV, The transport of electrons is solely determined by drift and diffusion between points x = 0 and x = IV. 63

64 the “electric

field ~t‘n

Using

the latter

stein

relation

electron

by the electron\”

expre\\ion

(WC’ ,Ippendi\)

D = ICTICI

den\il

current

equal\

and the [
p 10 eliminate

/L yield\

the

:

ii t Multiplying

both member\

integrating current

hole current

is solely

determined

by diffusion

with

relation

constant

applied All

I*. Maxwell-Boitzmann

to the charge these

doping

to the image s = 0 can

When,

no

E,

be

vary

bandgap

where

either

As E,

and

Equation

can be p
(61 gibes

he useful

~ II type diode.

the limit

second

the t‘lect:on

term

numerical

./,, i\ constant

case. between

tlitfusion

current

handgap

&

interface

\,,. Jrn-

and might

in the

the

second = 0

term

then reduces

Schottky

numerator

mu\t

In the cast

in the

and the fr\t

of the already

for

t)pt‘

stated

houndar)

in the

diodes.

The

he r~aluated

c~f constant

numerator

term is e:tjily

just

of a meta-/I

c‘a+e has not been considered theories

calculation.

dE,

a minimum

Thij

diffusion

a maximum

p layer. both under the influence

hecau\e

I0J

when ;I thin highly doped 11 layer i\ prejent

by x,,,, where

reaches

\\i

0 and M’. or in the image foric

Gty for the care of ;I non-constant

existing

ap-

clec[ron

U:

due

a reasonable

in an n layer or ,!I?, reaches

close to the metal in the

or

due to heavy narrowing

into account

used.

betueen

at the metal-semiconductor

the limit x = 0 can be replaced

close to the metal

of the

carriers.

111 is taken

longer

point

be

p ~n1.1

in the

result\

. [‘I(,” + /XI‘ ‘I ),,, (/, ‘rPdkc ~~~~ 1’cl\ I /I

.I,, = yl)

are neglected.

independent

may

however,

force[l,

proximation is the

of charge

of the bandgap

effects.

hold

may

x = 0 and x = I+’ and of whether

is depleted

value

statistics

Edge effects

should

level between

The

X,

carriers.

conditions

not this layer doping

is valid for electrons

the hole densit!

I,‘> l/f

The inlegralion

D = (k’i7q)~

uith

0 to .I - M

to

the right of point x = U’. The Einstein

.r

den\it!

Fig. 1. Electron energy for a Schottky diode uith an applied forward voltage. The dashed line indicate5 the hole quasi-Fermi level. Image force effects are now shoan.

The

from

vanishes

evaluated

h)

bandgap hrcauw

with the aid

conditions,

Equation

(61

to

of the image force. From level

these conditions

for holes

metal

and can be taken

x = W. as was already The now

it follows

is determined

voltage

be derived

from

current

as constant indicated

current

level

between

in the

.K = 0 and

in Fig. I.

characteristic

of the

the equation

giving

diode

can

the electron-

constant

The

total

Dg].

(II

E, the gradients

bandgap

are equal and the electric

in the E, and /?,

field is given

by

y dx

12)

q ds

Maxwell-Boltzmann

statistics

may

be applied

field is then related to the hole density as

9

(3)

pdx

When

the bandgap by a term

is not constant,

the gradient

in E,

dE,/dx from the gradient in E,. and thw

quantity.

iaje[X/

where

way

in the known

N-/I-/I

Because

the Gummel

he relatively

small.

and

the

narrow

number

thermionic

of ;I depleted current

emission

will

/I laqer v ill

density

will he

hr

current

the

The same hold> for transistor\

base[9.

In considering

it ix

number.

the diffusion

mechanism.

a very

tranjijtor

a9 the Gummrl

b’ith

IO].

Schottky

diodes

way.

because

it i\ customary instead

to \hritc

of multiplying

eqn (5) with the hole density /J it i\ multiplied with :I factor e UL.1rl:k r ( where \‘(I) equals the electrostatic hole

[ I, I]. But in the ca\e of ;I Lonctant

density

essentially

p

potential

is proportional

the same operation

of a non-constant differs

I he wnie

\arne

generally

potential

E=kTl!L,

cullrent den\i~).

controlling

in much the

eqn (7) in a different Because

/J d.r I\ the main

appear!

limiting dx

\\

called the ha\e charge or h;rje number.

high

E,_dv_ I~J%_ 1d6,

the electric

I

hole number

factor of the \aturation

density:

.I,, = q[npE+

With

that the quasi-Fermi

by the Fermi

cannot

bandgap

e ““““’

ic performed.

an unambiguous

be defined

can then he conveniently

to

(see .Appendix)

used.

handgap

the

so that In the case electrostatic and eqn Ih)

Gummel’s number and Schottky diodes Finally, some Gummel numbers for metal-n silicon Schottky diodes with homogeneous doping and constant bandgap have been calculated with the abrupt depletion approximation and are given in Fig. 2. Because Gummel numbers increase with increasing forward voltage, diffusion will hamper the thermionic emission current density more and more at high forward voltages. Because the diffusion theory should be applied mostly in cases of a low potential barrier combined with a low doping level case to the metal contact the effect of the image force is small and may often be neglected. So the simple formula (7) can very often be applied. 3.TRANSITIONFROM DIFFUSIONTO THERMIONIC EMISSION

The familiar expression for the thermionic emission saturation current density J7 is: JT =

AT2 e-‘WBN~~~,

(8)

In order to compare the thermionic emission and diffusion current densities, it is convenient to write the expression for the thermionic emission current density in a less conventional way, introducing the substitutions A = 4rqm”k’ h’

(9) (10)

q%N = qv,, - q(E, - EC)

65

and taking to the right of W (Fig. 1) that n = ND yields JT = q

--$&

NDe-qVbilkT

(12)

Taking the approximation of a constant electric field E,,, (in the layer between x = 0 and x = W) which equals the maximum electric field [l, 21 and using our expression (11) makes it possible to give a simple formula for the diffusion saturation current density JD: (13)

The only differences in expressions (12) and (13) are the effective velocities, respectively (u,J~(6~) and pE,,,, at which electrons are crossing the barrier. It is clear then that thermionic emission will limit the current for the case (14)

It can be shown (1) that formula (14) is equivalent to Bethe’s criterion stating that the potential changes at least (kT/q) within the mean free path A. For the thermionic emission theory to be applied, when a more exact criterion than (14) is wanted, a better expression for the diffusion saturation current density, such as the one given by (7), should be used, or

(II)

7

When the Gummel numbers of Fig. 2 are used, it that the ratios (&,,/u&‘(~P) and appears [DnF/Jowp dx]: [NDvl,,/~(6~) emqVbi’kT are equal to within a few percent. This is not surprising because the region where the electric field is close to its maximum gives the main contribution to the Gummel number, as is clear from Fig.

In the case of a diode containing a p layer the Bethe criterion must be changed in its formula as indicated by Shannon[6] so that eqn (14) can no longer be used although eqn (IS) remains valid. To illustrate this case a Schottky diode with a p layer of variable thickness and doping level of NA = 10z3m-’ is considered. The distance pen between conduction band edge and Fermi level at the metal-semiconductor interface is taken as 0.7 V and this value is considered as independent of doping and type. The ratio JdJT and the increase in barrier height (as seen by electrons from the metal side) are plotted in Fig. 3 as a function of d (using the abrupt depletion approximation). Because the Gummel number increases from 2.5 x 10” me2 a+ d = 0 to 1 x 10” mm*at 10ZO 102' lC2' lo= d = 3 x lo-‘m the ratio (JdJT) decreases from 19 to about 5 before the barrier height increases significantly. N, lm-3) The decrease in the ratio (JdJT) corresponds to a decrease in the saturation current density of about 20%, Fig. 2. Gummel numbers for Schottky diodes in homogeneously which is not negligible. At thickness above 8 x lo-’ m the doped semiconductors as a function of donor impurity concentration for different barrierheights with zero applied voltage. p type layer is no longer completely depleted. But only

M. KLEEFSTRA

applied forward voltage, much care is needed to properly describe the I-V characteristics of such diodes.

O,L

4. CONCLUSIONS is shown that the diffusion theory for Schottky diodes and p-n diodes of finite length are equivalent. The use of the Gummel number gives a more compact formula for the diffusion current density. As an illustration it is shown that relatively small amounts of an opposite doping close to the metal contact increase the Gummel number so that the diffusion can hamper the thermionic emission current density. It

0,3

0.2

REFERENCES

0,)

ALPsN 1”)

I 0

-

10 dllO-‘ml

Fig. 3. Ratio Jd/r as a function of the thickness of a p layer on an II layer Schottky diode. NA = 1p3 me3 and No = 10” m-j. The change in barrierheight pBN is also indicated (both for zero

appliedvoltage). above a thickness of 20 x lo-’ m is diffusion the current limiting mechanism. One then has a p type Schottky diode in series with a p-n type junction, where the latter determines the electron current density. Taking a p type layer of 2 x 10-‘m thickness, with NA = 1 x 1022m~3,reduces the ratio at zero applied voltage (JJ&) to 1.6, meaning that the diffusion current density is only slightly higher than the thermionic emission current density. Because both the Gummel number and the barrier height are increasing with increasing

I. E. H. Rhoderick, Mefcll-semiconducror C’mtur’t\. C‘larendorr Press, Oxford (1978). 2. S. M. Sze, Physics of Semiconductor 13uricr.c. W~lr!. Ncu York (1969). 3. J. M. Shannon, Solid-St. Electron. 19. 537 (1976~ 4. A. van der Ziel, Solid-St. Electron. 20, 269 (1977). 5. S. S. Li, Solid-St. Electron. 21, 43.5 (1978). 6. J. M. Shannon. Appl. Phys. Lett. 35.63 (1979). 7. S. S. Li, J. S. Kim and K. L. Wang, IEEE Trcmt. E/rcm~r~ Dee. ED-27, 1310 (1980). X. H. K. Gummel, BeN Spst. Techn. J. 49, I I! ( 19701. 9. P. Rohr. F. A. Lindholm and K. R. .Allen, Solid St. E/rr~rn~r~ 17. 729 (1974). IO. G. Bacarani. C. Jacobini and ,A. M. Mazzonc. .G;o/id-S/ Eiecfron. 20. 5 (1977). II. M. Kleefstra and G. C. Herman. 1. .4ppl. Phy.5. 51. 4421 (1980). 12. N. K. Adam, The Physics und Chemi.ttry of .Surfuw.t. pp 300-307. Oxford University Press. London (I941 I.

APPENDIX When a change in bandgap E, takes place there i\. strictI> speaking, a change in phase, and electrostatic potential differences between two phases are indefinite quantitie\. The only important and measurable quantity in 5uch ca\e\ I, the electrochemical potential or Fermi level. This point is discussed most carefully by Adam1 121 rn III\ chapter on electrical phenomena at interfaces. Equation (5) states then that the electron current densit! I, proportionalto the gradientin the Fermi level of electron\