The width of the non-steady state transition region in deep level impurity measurements

Solid-StateElectronicsVol. 26, No. 10, pp. 987-990, 1983 Printed in Great Britain.

0038-I101/83 $3.00+ .00 © 1983 Pergamon Press Ltd.

THE WIDTH OF THE NON-STEADY STATE TRANSITION REGION IN DEEP LEVEL IMPURITY MEASUREMENTS S. D. BROTHERTON Philips Research Laboratories, Redhill, Surrey, England (Received 4 November 1982; in revised form 10 January 1983)

Abstract--The important influence of the transition region on the interpretation of deep level impurity measurements is well established and these effects are usually evaluated numerically. A simple analytic expression is developed for the width of the non-steady state transition region which is shown to be in good agreement with the numerical results. It is also demonstrated that the expression is consistent with second order effects noted in experimental trap concentration measurements. l. INTRODUCTION

It is widely recognised that when making deep level measurements on semiconductor diodes it is necessary to take account of the fact that the trap occupancy may be rapidly changing near the edge of the space charge region. This is due to the Debye tail of free carriers extending into the space charge region from the neutral material. This region is usually referred to as the transition region, and in steady state its width, ho, is given by the expression (2es6o(ETho = \qno EF)] 1/2 for a step junction into uniformly doped material, with a free carrier concentration no on the lightly doped side of the junction. The situation is illustrated in Fig. 1. In measurements where the width of the transition region is changing with time the above steady state expression is inappropriate and, with the exception of the work of Ports[l], it has been customary to deal with the problem numerically. Examples of this are in the measurement of capture cross section by the diode shortcircuiting technique[2,3] and in other C-V measurements of a deep level centre [4, 5]. In this paper an approximate analytical expression is introduced for the width of the non-steady state transition region, which is shown to be in good agreement with numerical calculations. In addition, its influence in experimental measurements is illustrated. 2. NON-STEADY STATE TRANSITION REGION WIDTH

of the depletion approximation and the Debye tail of free carriers. In analogy with the depletion approximation, and with the underlying assumption of the normal expression for ho, it will be assumed that the continuous distribution of charge on the deep level centre can be adequately represented by a step-function distribution terminating at the point at which the trap occupancy is 0.5. This is then defined to be the width of the non-steady state transition region. A further assumption is that the deep level concentration, NT, is sufficiently smaller than the shallow level impurity concentration, N ~ so that carrier trapping within the transition region does not significantly alter the parabolic potential distribution within this region and hence the Debye tail of free carriers is not time dependent. If the period of growth of the transition region is restricted to times shorter than the trap thermal emission time constant, ~',, then its time dependence is amenable to an analytical representation. (A time scale short compared with the thermal emission time is experimentally encountered in capture cross section measurements and low temperature C-V profiling.) With these assumptions, i.e. (i) t a r , ( L - x) 2 (ii) n = no e x p - ~ x< L x>L

n=no

(1) (no = No - NT)

then considering only carrier capture in the space charge region, at any point x,

2.1 Analysis Consider a diode under reverse bias as illustrated in Fig. 1. Let the bias be reduced abruptly so that the space charge region width rapidly decreases by a distance greater than )to and thereafter the space charge width remains constant. In this situation the traps in the space charge region will be uniformly empty at t = 0 and will progressively fill with majority carriers until a new steady state transition region is formed. This process is illustrated schematically in Fig. 2. The free carrier distribution shown in Fig. 2(a) is based upon the assumption

dNTdt - ncn(NT- NT-)

from which NT-

NT = (1 -

e-nC

n')

(2)

If eqn (1) is substituted into eqn (2), and this equation is equated to 0.5 an expression is obtained for the point x 987

988

S. D. BROTHERTON then,

)t = ( 2 k T ~ o , noCnt'\ 112 ', qno m--~ff-) . Ec

E~ Er

The usual expression for the steady state transition region width is )to

Ev

I

(3)

(2Eo~(ET-EF)~ ''2 -\ qno /

=

(4)

Using the equation of detailed balance

L

e,, = c,Nc e x p -

Fig. 1. Illustration of the band diagram on the lowly doped side of a reverse biassed abrupt p+n junction.

E c - ET leT

and no = N~ exp

(Ec - Ev)

kT

eqn (4) may be rearranged to give Xo =

L

~X

(a)

N_;

k (b) Fig. 2. Schematic illustration of: (a) the free carrier distribution, (b) trap occupancy, within the space charge region of a reverse biassed diode.

at which (NT-INT) = 0.5, i.e.

N~

=

(

(L-

\

1 - exp - c,,tno exp - ~ 1

x)~\

and

2

noCnt'\ 112

... A = {2Lo In -ff~--/ \

v.i

(6)

NTnc. [1-- e ~. . . . . . )t].

(7)

6s~O Jo

where

In 2 = c,not

exp - ~

x(No - NT (t)) dx

V ( t ) = q--q- fL

NT-(t) =

z

(5)

2.2 Comparison with numerical analysis The situation discussed in Section 2.1, in which the change in occupancy of the deep level centre was represented by the movement of a step function, may be more accurately described by dealing numerically with the continuous distribution of charge in the space charge region. Using the depletion approximation, together with the Debye tail of the free carriers, the time dependent voltage across the space charge region is given by

= 0.5 @ L - x = ),

...O.5=exp(-c,tnoexP-2---~-~D)

hz

qno

A direct equivalence between the terms in eqns (3) and (5) is seen, with time explicitly occurring in the former and the thermal emission time constant in the latter. One notable difference indicated by these equations is that the width of the steady state transition region increases with decreasing temperature, whereas, for a given value of t, the width of the non-steady state transition region decreases.

Nr

Nr-

(2kTEseoIn noc.r,) "=

]

(In 2-~ 0.7).

Using = ( E s ~ o k T ~ 112

n is given by eqn (1) and the thermal emission rate constant, e,, is included in order to allow the system to ultimately achieve steady state. As with the analysis in the previous section, NT is assumed to be sufficiently smaller than No that the tail of carriers, as given by eqn (1), extending into the space charge region does not change with time.

The width of the non-steady state transition region Equation (6) was numerically evaluated for a range of times yielding the time dependence of V during the growth of the transition region at constant space charge width. Clearly, it is possible to postulate a step function charge distribution (as given by eqn 8) which, at any time t, will produce the same voltage, V(t), as eqn (6). Thus by defining the charge on N r to fall abruptly to zero at L - A we can write the equality:

V(t)= ~ [foLNax dx- f]c NTx dX]

(8)

(9) for spatially uniform No and Nr. Hence, using the numerically calculated values of V(t) it is possible to solve eqn (9) for A(t) and to compare these values with those calculated using eqn (3). The results are illustrated in Fig. 3 in which there is a comparison of the analytically calculated A with the value extracted from the numerical calculation. The agreement between the two is seen to be very good for times less than one emission time constant. At the other end of the time range, when t approaches one capture time constant, % the inaccuracies in eqn (3) start to increase again. At t = ~',, they rise to - 6 % compared with <1% over the rest of the time range. For times less than 0.7 rc, eqn (3) is invalid. The physical significance of this is that for these short times fifty per cent trap occupancy will not have been achieved any-

x

2.0

Numerical analysis Analytical expression for ~. X X

1.6 i

1.2]

O

_

a

s

X Clam)

~e--l= O e s

0.8

989

where in the space charge region. For times greater than one emission time constant the normal steady state expression is adequate for calculating A. 3. EXPERIMENTAL MEASUREMENTS OF N T

In this section some measurements of deep level trap concentration are presented which are intended to briefly illustrate the use of eqn (3) and the effects underlying it. Two different experimental techniques have been used to measure the gold concentration in a gold doped p+n diode. The techniques are illustrated in Fig. 4 and have been used to produce different values of A. In both cases a value of A V has been obtained at constant capacitance from the diode with the gold acceptor levels full and empty. The values of A V have been used to calculate the gold concentration using N~

2e~eoAV

q(L-~"

The measurements were: (i) low temperature short circuit: This measurement is made at a low temperature such that the thermal emission time constant is very much longer than the total measurement time, and derives from the technique first published by Sah et a/.[6]. Typically, a reverse bias is applied at room temperature to thermally empty the traps and the sample is then cooled to 77 K. At the termination of the cooling phase the transition region will be in an ill-defined state, so the voltage on the diode is abruptly reduced and the diode subsequently short circuited after an interval tx. During this period, t~, a new and well defined transition region region will be established. The value of A V is measured as shown, and )t is calculated using eqn (3). (ii) thermal relaxation: in this measurement the steady state transition region width is established before the short circuit is applied by holding the diode under bias VR for a time long compared with the thermal relaxation time re. In this case Ao is calculated using eqn (5) with the measured value of % In the calculations of A and Ao it is necessary to use values of c, and no. The capture coefficient, c,, must be obtained from a separate measurement, and the capture cross section value used here for the gold acceptor level was 9 × 10 ,7 cm 2 (7). The no can be obtained from the value of reverse bias measured immediately after the short circuit application when all the traps are still filled with majority carriers and the space charge density is No - NT through the space charge region, i.e.

0.4

no= ND-- NT =2EsE°(VR- A V + Vo) qL 2

0

I

10 °

I

|

I

I

10 4

10 s

I

I

I

1 0 ~2

t

"¢c

Fig. 3. Comparison of the time dependent transition layer width obtained from numerical and analytical calculations. (No = 1.04× 1014cm 3, NT =4.0× 1012cm-3, T = 100 K, c, = 1.0× 10-8 cm3/s).

Vo is the built-in junction potential. The results of the measurements are given in Table 1 and the values of A V varied with the measurement conditions. As expected, the smallest value of A V was obtained from the thermal relaxation measurement, in which the transition region width is anticipated to be the greatest. It will be noted that the value of AV obtained

S.D. BROTHERTON

990 Experimental measurement

Voltage woveform

~3o~ (1)

k

~ Non -steady state

LOw temDeroture short circuit (t m <<~, )

|&v

(2) Thermal relaxation

Steady state

Fig. 4. Measurement procedures used to estimate the concentration of deep level impurity.

Table 1. Gold concentration measurements on p+n diode (Al11/1/375) [(a)--this value of ,~ is the steady state value calculated at 209.7 K] Measurement Technique

(I)

Low Temperature Short Circuit

[2)

Thermal Relaxation

Temperature

VR

tX

AV

(K)

(v)

(s)

(v)

(um)

77.0

8.046

O.517

0.328

209.7

8.005

0.448

0.545

IO.O

X

NT

(cm-3)

i.O3x2014

(a)

from the low temperature short-circuit measurement is 15.4% larger than obtained in the thermal relaxation measurement, and yet the values of NT calculated are within 6% of each other, with the larger AV giving the slightly smaller NT.

(x-18.1)

l.lOxlO 14

plicable to the DLTS measurement of trap concentrations using the quasi-constant capacitance procedure published by Whight[8], in which the value of t used in eqn (3) is the pulse width at times tl and t2 in Fig. I, Ref. [8].

4. CONCLUSION

REFERENCES

A simple analytical expression has been developed for the width of the non-steady state transition region in junction space charge regions under constant space charge width conditions. Numerical analysis has demonstrated the validity of this expression between certain well defined limits. The analytical expression is particularly suitable for the analysis of deep level parameter data measured under constant capacitance conditions as illustrated by the experimental results. It is equally ap-

l. D. Pans, Appl. Phys. Letts. 37, 413 (1980). 2. A. Zylbersztejn, Appl. Phys. Lens. 33, 200 (1978). 3. S. D. Brotherton, P. Bradley and J. Bicknell, J. Appl. Phys. 50, 3396 (1979). 4. H. G. Grimmeiss, L-/~ Ledebo and E. Meijer, Appl. Phys. Letts. 36, 307 (1980). 5. J. M. Noras. Solid-St. Commun. 39, 1225, (1981). 6. C. T. Sah, L. L. Rosier and L. Forbes, Appl. Phys. Lett. 15, 161 (1969). 7. S. D. Brotherton and J. Bicknell, I. Appl. Phys. 49, 667 (1978). 8. K. R. Whight, Solid-St. Electron. 25, 893 (1982).