Theory of transient emission current in MOS devices and the direct determination interface trap parameters

Solid-State

Electronics,

1974, Vol.

17, pp.

117-124.

Pergamon

Press.

Printed

in Great

Britain

THEORY OF TRANSIENT EMISSION CURRENT IN MOS DEVICES AND THE DIRECT DETERMINATION INTERFACE TRAP PARAMETERS* J. G. SIMMONS and L. S. WEIt Electrical Engineering Department, University of Toronto, Toronto, Canada (Received 23 February 1973; in revised form 6 July 1973)

Abstract-The physics is discussed of the emission of electrons from interface states in conditions. metal-insulator-semiconductor (MIS) systems, under isothermal, non-steady-state Generalized equations are then derived which permit the determination of the non-steady-state, emission current vs time characteristics for MOS systems containing an arbitrary distribution of surface states; the special case of a discrete surface state is also studied. More important, however, by appropriate plotting of the data, it is shown how to directly extract from the experimental data the energy distribution and the capture cross section of the interface traps in the upper-half of the band gap in the case of n-type semiconductors, and in the lower-half of the band gap in the case of p-type

N,, Nz, Nx

constants for uniform, exponential and Gaussian trap distribution, respectively, cm-* charge of an electron, C semiconductor depletion charge, C/cm’ charge at the gate electrode, C/cm* fixed charge in the insulator, C/cm’ interfacial trapped charge, C/cm’ time, set time at which (I-t) is maximum, set arbitrary time intervals, set temperature, “K arbitrary temperatures, “K thermal velocity, cm/set fixed voltage applied to the gate electrode, V width of depletion region, cm constant for exponential trap distribution, eV_’ delta function peaking at E = E, S(E - E,) 4 permittivity of silicon, F/cm difference between metal and 4 m, energy semiconductor work function, eV Y attempt-to-escape frequency, set-’ Y constant for Gaussian trap distribution, eV_’ s. electron capture cross-section cm*

NOTATION

A E C, Cl, C,, C, Ee: EC EP Et E” EO E,, Ez SE G(E,t) : k n, ME& NC

N.2 N‘

area, cm2 normalization constant depletion capacitance, F/cm* insulator capacitance, F/cm* constants electron emission probability, set-’ non-steady state Fermi energy, eV energy of the bottom of the conduction band, eV energy at which G(E,t) is maximum, eV energy level of a discrete trap, eV energy of the bottom of the valence band, eV characteristic constant of the Gaussian distribution function arbitrary energy levels of uppermost-filled trap at equilibrium, eV incremental energy function defined in (16) current, A current density, A/cm2 Boltzmann’s constant number of trapped electrons per unit area, cm-’ number of trapped electrons per unit energy, cm-’ eV-’ effective density of states in the conduction band, cm-” donor concentration, cm-’ interface trap density, cm-’ eV-’ discrete trap density, cm-*

*This study supported by Defence Research Board and National Research Council of Canada. tC!urrent address: Atomic Energy of Canada Ltd., Commercial Products, Ottawa, Canada.

1. INTRODUCTION

methods have been described in the literature for determining the energy distribution of surface states in MOS capacitors and transistors. Most of these methods are made under quasi-steady state or steady-state conditions, and involve measurement of either, or both, the a.c. conductance and capacitance[ l-81. The limitations on the useful amount of information that can be obtained from Several

117

J. G. SIMMONSand L. S. WEI

118

some of these techniques have been discussed in detail by Zaininger and Warheld [9] and Frankl[ 101. Very recently, the effect of interface traps on the properties of MOS devices under non-steady-state conditions has been discussed[ll, 121. A technique hitherto not exploited, which does not utilize a.c. monitoring techniques, is that involving the measurement of the non-steady-state d.c. current (I) vs time (t) characteristics of the device. In principle this technique is considerably simpler than any of the a.c. methods, since it consists of simply applying a step voltage to the MOS system and monitoring the d.c. relaxation current as a function of time as the system relaxes to the equilibrium state under isothermal conditions. That the method has not been used is not too surprising, because it is normally possible to obtain closedform theoretical solutions for only the very simplest of trapping solutions. Since real surface-state distributions can be quite complex, laborious numerical analysis is usually required of the experimental data, in order to identify the trap distributions. Even so, such a procedure is usually not unequivocal, particularly if the trap distribution is complex, because the bland nature of the isothermal I-t characteristic per se does not contain any distinctive structure which would, as it were, serve to directly identify salient features of the trap distribution. Here we will present generalized equations that provide for closed-form analytic solutions of the isothermal I-t decay characteristic of MOS systems whatever the interface trap distribution. More important, however, we will demonstrate, by means of an appropriate artifice, how to extract directly the trap distribution and capture cross section from the isothermal data. 2. THEORY

Consider a metal-oxide-semiconductor* device with its gate electrode positively biased, so that the device is in the accumulation mode. In this case, the interface traps in the upper-half of the band gap will be filled to an energy, say I?,, at which the Fermi level intersects the semiconductor surface (Fig. l(a)). Suppose now that the gate electrode is negatively biased so that the device is in the inversion mode. In this case, under quasi-equilibrium condi*Here we will confine ourselves to n-type semiconductors. The case of p-type semiconductors follows directly from the discussion of the n-type system, the principle difference being that in the p-type system the surface states in the lower half, rather than the upper half, of the band gap are studied.

Empty

l---l

states

-.-

-.

‘Ox’ide

(01

Electron Bulk

lb)

hole recomb,na+ion

generotm (Cl

(d)

Fig. 1. Energy diagram illustrating: (a) the accumulated mode; (b) the non-steady state mode during surface emission of trapped electrons from interface states in the upper-half of band gap; (c) the non-steady-state during surface and bulk generation of electron-hole pairs and the recombination of electrons trapped in interface states in the lower-half of the band gap with holes in the valence band and; (d) the quasi-equilibrium inversion mode. tions only those traps are filled that lie in levels below the energy, say E2, at which the Fermi level

intersects the surface in the lower-half of the band gap (Fig. l(d)). Thus, when the sample is switched from the accumulation to the inversion mode, the interface traps located between the energies El and E2 lose their electrons. This relaxation process does not occur instantaneously, and during the period it occurs, the device is in the non-steady-state. Electrons escape from the interface traps in the upper-half of the band gap by thermal excitation into the semiconductor conduction band (Fig. l(b)); when these traps have emptied, surface or bulk generation of electron-hole pairs becomes significant. In this event, electrons in the traps located in the lower-half of the band gap escape by recombining with the generated holes in the valence band (Fig. l(c)). The electrons that are emitted to the semiconductor conduction band are immediately swept out of the depletion region by the high field therein, causing a current to flow in the external circuit. Thus the current that initially flows in the circuit is due to the surface-emission process, while that which flows at later times is due to the surface-generation process;

Theory of transient emission current in MOS devices it is also possible for a bulk-generation current to succeed the surface-generation current, depending on the relative density of the bulk and surface traps. Here we will be concerned solely with the emission current.

Et

n

t

2.1 Emission current equation The current I flowing in the external circuit is proportional to the rate of emission of electrons from the interface traps. The rate of emission is determined as follows. We define a parameter n*(E,t), as the energy distribution of electrons per unit energy per unit area at the interface at any time t. The number of electrons per unit area in an incremental energy range SE at any time r after a negative pulse has been applied to the system is, therefore, equal to n,(E,t)dE. It follows then that the rate of emission per unit area of electrons from the traps located between E and E + 6E is d(bn,)/dt

= - e, 6% = - e.n,(E,t)dE

In (l), e. is the emission which is given by

coefficient

Fermi - Diroc

---

Non-steody

function

state exp (-e,t)

(3)

f =lsec t = IO set .t = 100 set

Fig. 2. Non-steady-state probability distribution as a function of energy with time as a parametric variable. Each small division of the energy axis represents l/&T of energy.

non-steady-state occupancy function. The function exp (- e,,t) is shown plotted in Fig. 2 as a function of energy with time as a parametric variable, and it is seen to have a similar dependence on energy as the Fermi-Dirac function. Thus, we may define a non-steady-state Fermi level E*, given by exp [- e,(E*)t] = l/2 or E* = EC - kT fin vt + 0.3651,

In obtaining (3) we used the boundary condition that an,= N,(E) dE at t =0, where N,(E) is the energy distribution per unit area per unit energy of the interface traps; note that this statement implies that in the process of accumulating the device the states below El are completely filled (Fig. l(a)). Substituting (3) into (1) yields d(Sn,)

-=dt

From an inspection that

-N,(E)e.

exp(-e.t)dE

(4)

of (1) and (4) it will be apparent

n,UW = N,(E) exp (- e.0,

function

(2)

where v is the thermal velocity of electrons in the silicon conduction band, cr. is the electron capture cross section for the interface states, NC is the effective density of states in the conduction band, and v(= v a. NJ is the attempt-to-escape frequency for the interface states. The solution of (1) for constant temperature is S n, = N,(E) exp (- e.t) dE

-

(1)

of the traps,

e. = v G NCexp ((E - EJIkT) = Y exp ((E - E,)lkT),

119

since for energies more than about 2kT above E* the interface traps are essentially empty, while for energies more than about 2kT below E* the interface traps are essentially full. In essence, then, (6) gives the energy of the uppermost-filled trap at time t after the device has been switched into the nonsteady-state mode. The rate of emission of electrons from all the interface traps is obtained by integrating (4) over all the occupied surface states in the upper-half of the band gap to yield 4

(5)

from which it will be clear that exp (- e.t) is the

(6)

ri =

I E”

N,(E) e. exp (- e.t) dE.

(7)

120

J. G. SIMMONS and L. S. WEI

From the charge neutrally requirements

we have (8)

Qg+Qt+Qc,+Q1=0

where Q, is the charge per unit area on the gate electrode, Qt is the surface-trap charge per unit area, Qd is the charge density in the semiconductor depletion region, and Q1is the fixed charge per unit area in the insulator. Differentiating (8) yields Q~+Q+Q,f=o

Qd=qANdxd=-

Cd Q

cd+cI,

(13)

where Cd= E,/x~is the capacitance of the semiconductor depletion region. Substituting (13) into (10) gives

(9)

Now the current density J circulating in the external circuit is simply the rate of change of the charge on the gate electrode Q,; hence, substituting J = Q, in (9) yields J=-

trode, and &, is the difference in energy in between the semiconductor and gate electrode work functions. From (11) and (12) we obtain

&Qt

c, Q, cI+cd’

.l=-

(14)

and since QI = - qri,, we have from (7) and (14) exp (-e,t)

dE

(10)

N,(E)e,

(11)

2.2 Generalized solution of emission current equa-

(15)

Furthermore, Qd=qNdxd

where Nd is the donor density, and xd is the width of the semiconductor given by [ 1l]

tions

Generally

speaking, the integrand of (15) cannot exactly; however, using appropriate approximations we will now show that (15) can be solved in closed form for any trap distribution. The G(E) t be integrated

function

(12) In (12), E, is the permittivity of the semiconductor, CI is the insulator capacitance per unit area, V, is the (constant) negative bias on the gate elec-

4

defined

by

G(E,t) = e, exp (- e,t)

is shown in Fig. 3 to exhibit a sharp maximum at an

E,

I

I.0

f =100 set

t =lU4sec 0.6 w_5sec

wee

0,6

E#=I.1eV

0.4

r = 300*K

T 200°K q

02

E”

EV

Arbitrary (0) Fig.

(16)

unit (b)

3. The function G(E, t) as a function of energy with time as a parametric variable, for v = 10” see-’ and, (a) T = 3OO”K,(b) 150°K.

121

Theory of transient emission current in MOS devices energy E,, which is time dependent termined by the condition that

in a manner de-

CO. aE- .

(17)

From (16) and (17) we obtain E,=E,--kTln

vt,

where A is the area and I the isothermal current; hence the isothermal current at time t is directly proportional to the trap density N(E,) at E,, where E, is given by (18). Contrary to (15), which can be solved in closed form for only a few very limited trap distributions, (25) may be used to compute the I-t characteristics in closed form for any trap distribution.

(18)

and by comparing (6) and (18) we see that E, is practically synonymous with the non-steady-state Fermi level, the two differing only very slightly in energy (= H/3) at all temperatures. Furthermore, it is seen that G(E,t) has significant values only within an energy range 2kT of E,, which means that only electrons in interface states within about 2kT of the non-steady-state Fermi level contributed to the emission current at any time. Because G(E,t) has significant values over only an extremely narrow range of energies about E,, it may be approximated by the delta function

B S(E- E,) = e. exp (- e.t),

(19)

2.3 I-t Characteristics for special trap distributions in order to demonstrate the efficacy and accuracy of (25), we compare the exact and approximate isothermal current-time characteristics for a few special trap distributions. (i) Unifom trapping distribution. Substituting N(E,) = N, = constant into (25) yields J = q kTN,lt.

(26)

In Fig. 4, curves (a) illustrate the exact (15) and approximate I-t characteristics for this case. the correlation is seen to be very good. (ii) Exponential trapping distribution. The trapping distribution is assumed to be given by N(E) = N2 exp [- cw(E, - E)],

where B is a normalization constant that is evaluated by integrating both sides of (19) with respect to energy:

(27)

Ec

B =

I E”

e, exp (- e,t) dE,

cm

or B

Substituting

=

!$(I _ e-“‘).

(19) and (21) into (15) yields =<

4 CI (c*+ C,)

m)=

I

N,(E) F( EU

1 -em”?S(E - E,) dE (22)

or

AICL N,(E,) Z(f)=(cs+c3

g(l

-em”)

(23)

Since v is usually of the order 10” set-‘, vt is generally much greater than unity, so that (23) simplifies to N (E,). J(t)=c,+ Cd lf_T t -

Normally

(24)

Cd Q C, so that (23) further reduces to I(t)=qAkTN(E) t

P9

(25)

I

IO-

I

1o-4

10-3

I

1

10-2 10-l f. set

I

IO0

la

lOI

102

Fig. 4. Approximate and exact J - t characteristics for: (a) a uniform trap distribution, N,(E) = 5 x lo’* cm-’ (eV)-‘; (b) an exponential trap distribution, N,(E) = 10” exp [- 2(E, -E)] cm-* (eV)-’ and; (c) a Gaussian trap 5 x 10” exp [- 25(E - 0*38)‘] cm-* (eV)-‘. distribution, Other parameters used; Y = 10” set and T = 200°K.

J. G. SIMMONS and L. S. WEI

122

where N2 and a! are characteristic constants of the distribution. Substituting (27) into (25) and using (18) we obtain J = q N&T y-a’= t-‘“‘T+”

or

(28)

Correlation between the exact and approximate characteristics is again very good, as illustrated in Fig. 4. (iii) Gaussian trapping distribution. We assume a trapping distribution given by N(E) = NJ exp[- y(E - Ed’l,

(2%

where N,, y, and & are characteristic constants. Substituting (29) into (25) and using (18) yields In J= Cj + C2 In t + G(ln t)‘,

density at E,. Also, we know that E, is related to time by E,-- E, = kT In vt

(30)

where C, = ln(2q kT N, exp[- Y(E~- E0 - kT In u)‘]} C,=2kTy(E,-Eo-kTln v)-’ c3 = - y(kT)*.

EC-E, = 2.303 kT(log,ot +log,ov);

(32)

hence, using (32) the log& axis can be transformed into energy measured with respect to the bottom of the conduction band. Therefore, it will be apparent from an inspection of (31) and (32), that plotting It as a function of loglot provides a direct image of the energy distribution of the surface traps in the upper half of the band gap. In other words, using the transformations provided by (31) and (32), the It log t characteristic may directly converted into a N(E,) vs E, - E, characteristic, that is, into an actual plot of the surface trap distribution. To demonstrate the applicability and efficacy of the method, the I-t curves shown in Fig. 4 are shown replotted in Fig. 5 as It vs log& characteristics; the axes have been transformed to N,(E,) and EC-E, axes to obtain the energy distributions.

The exact and approximate characteristics are illustrated in Fig. 4 by curves (c), and, again, the correlation between the two characteristics is seen to be extremely good. 3. DIRECT

DETERMINATION

OF TRAP

DISTRIRUTION

For the reasons mentioned in the Introduction, the isothermal I-t characteristics are not particularly fruitful from the point of view of yielding data on the surface trap distribution. That this is so is particularly well demonstrated by the three sets of curves in Fig. 4, for although the trap distributions responsible for the curves are quite disparate, the curves themselves are very similar in nature, and might well have been thought to have been generated by almost identical trap distributions. Although the I-t characteristics per se do not provide much information on the trap distributions responsible for them, we will now show that the product of the isothermal current and time (It) plotted as a function of log& actually yields directly the surface trap distribution. 3.1 Distributed traps Equation (25) may be expressed NO%)

=-j&y

in the form (31)

which relationship shows that the product of current and time is linearly proportional to the trap

t,

set

Fig. 5. The curves of Fig. 4 replotted in the form It vs loglOt. The It and Iog,,t axes are also expressed in terms of N, (Em) and E, - E,,,, respectively.

3.2 Discrete trap levels The use of the delta function [see (19)] approximation for the function G(E,t) limits the application of the method described above to trap distributions greater than 3kT wide; thus, the results of method as described cannot be applied to discrete traps. Since discrete traps are an important consideration in any defect system, it is necessary that we

Theory of transient emission current in MOS devices

study their effect on the If -log t characteristic for the system. The I-t characteristic for a discrete trap level of density N, positioned at an energy E, is obtained from (16) by substituting N&E - EJ for N,(E) and integrating the resulting expression to obtain I = q N, e.(EJ expt-

e.(EJt);

(33)

in arriving at (33) we have assumed C+ Cd. Multiplying both sides of (33) by t yields It = q N, t e.(EJ exp(-

e,(EJt)

(34)

Equation (34) expressed in terms of an It--log& plot exhibits a distinct peak (Fig. 6), and the time t,

3-

log t,

see

Fig. 6. The It - log,,,t characteristic for a discrete trap at 300 and 250°K (v = lo’* set-‘, N, = 5 x 10” cm-‘, E, = O-4eV).

at which the maximum occurs is obtained from the condition d(W _ 0 dt ’

(35)

Thus, from (34) and (35) we obtain e,(&T)

= G’,

(36)

or EC- El = kT In vt,. Substituting

(37)

(36) into (33) yields (It)m.x = qN, e-‘.

(38)

Therefore, the depth of the trap below the bottom of the conduction band is obtained directly through (37) from the time at which the maximum in the If -log t characteristic occurs, and the trap density

123

through (38) from the height of the maximum in the characteristic. The effect of discrete traps can be distinguished from that of distributed traps simply by applying (33) to the experimental data: if the data and theory correlate, the traps are discrete, otherwise they are distributed. A second test is provided by the fact that the size of the If -log t characteristic is independent of temperature in the case of discrete traps (Fig. 6), whereas in the case of distributed traps the size of the characteristic is temperature dependent. 4. DISCUSSION

A knowledge of the attempt-to-escape frequency is necessary in order to fully analyze the experimental data in the manner described above. This parameter may be ascertained by obtaining the If log,,t characteristic at two different temperatures T, and T2. Then by measuring the time t, and tf on the T, and T, characteristics, respectively, at which some prominent point of the characteristic occurs, corresponding to an energy, say, E,, we obtain the two simultaneous equations El = EC- kT, In vt,

(39)

El = EC- kT2 In vtt

(40)

From (39) and (40), v and also El may be determined. Furthermore, a knowledge of v(= vu,, NC) permits a determination of the capture crosssection of the interface states. How close to the conduction band edge a study of the interface states may be made using the technique described here depends on the timeresolution of the measuring apparatus: the faster its time response the closer to the bottom of conduction band the interface states can be studied. It will be apparent from (18) that the effective energy resolution of the measuring equipment can be enhanced by cooling the sample to low temperatures. For example, if the maximum response time of the apparatus is 10e6set and v = 10” set-‘, then at 300°K one is limited to studying surface states within about 0.28 eV of the bottom of the conduction band. However, at liquid-nitrogen temperature (77X), surface states within about O-07 eV of the conduction band edge can be studied, and at liquidhelium temperature (4°K) within about 0.0035 eV. Although we have been concerned with the interface states density throughout the upper half of the band gap, by judicious initial and final biasing, various energy ranges may be investigated. The whole of the upper half may be investigated in this manner

124

J. G.

and the energy distribution from

the accumulated

constructed

SIMMONS

piecemeal

data.

REFERENCES

L. M. Terman, Solid-St. Electron. 5, 285 (1963). K. Lehovec and A. Slobodsky, Phys. Status. Solidi 3, 447 (1963). E. H. Nicollian and A. Goetzberger, Appl. Phys. Letts 7, 216 (1965). P. V. Gray and D. M. Brown, Appl. Phys. Letts 8, 31 (1966). C. N. Berglund, IEEE Trans. Electron. Devices ED13, 701 (1966).

and L. S.

WEI

6. B. E. Deal, M. Sklar, A. S. Grove and E. H. Snow, J. Electrochem. Sot. 144, 266 (1967). 7. E. H. Nicollian and a. Goetzbereer. _ Bell Svst. Tech. J. 46, 1055 (1967). 8. A. Goetzberger, V. Heine and E. H. Nicollian, Appl. Phys. Letts 12, 95 (1%68). 9. K. H. Zaininger and G. Warfield, IEEE Trans. Electron. Devices, ED-12, 179 (1965). 10. D. R. Frankl. J. am/. Phvs. 38. 1966 (1%7). 11. J. G. Simmons and -L. We;, Solid-St. Electron. 16, 43, (1973). 12. J. G. Simmons and L. Wei, Solid-St. Electron. 16, 53 (1973).