Inter Traps Tunneling

7/30/2019 Inter Traps Tunneling

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phys. stat. sol. (a) 201, No. 13, 29662979 (2004) /DOI 10.1002/pssa.200406849

2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Inter-trap tunnelling in thin SiO2 films

S. Simeonov*, I. Yourukov, E. Kafedjiiska, and A. Szekeres

Institute of Solid State Physics, Bulgarian Academy of Sciences, Sofia 1784, Bulgaria

Received 23 February 2004, revised 21 June 2004, accepted 2 July 2004

Published online 23 September 2004

PACS 71.55.Jv, 73.40.Gk, 73.40.Qv

An expression for the currentvoltage characteristics of insulators in the case of inter-trap tunnelling is

obtained. This expression gives an opportunity to estimate the energy position and concentration of traps

responsible for charge transport by inter-trap tunnelling. Tunnelling currents at 77 and 300 K are observed

in p-Si/SiO2 structures with 13 and 65 nm SiO2 films subjected to hydrogen plasma treatment at 20, 100and 300 C. It has been shown that these currents together with tunnelling currents in some other insulator

films, mainly SiO2, are carried out by inter-trap tunnelling.


1 IntroductionWith the continuous reduction of the thickness of gate insulators in contemporary advanced MOS inte-

grated circuits the problem of tunnelling currents through the gate SiO2 film acquires great importance.

Besides the indispensable tunnelling current from the metal to the Si conduction/valence band or the

tunnelling current in the opposite direction, it is established that deep levels in the SiO2 energy gap also

contribute to the tunnelling currents through SiO2 films. The indispensable tunnelling current from metal

to Si and vice versa as a function of the applied electrical field is given by the FowlerNordheim expres-

sion.

It is also established that deep levels generated by high electrical fields in SiO2 films create an addi-

tional path for charge carrier tunnelling, namely the stress-induced leakage current (SILC) (e.g. see [1]).

Similar paths for trap-assisted tunnelling (TAT) are connected with deep levels in SiO2 films generated

by ion implantation [2], X-ray exposure [3] or deep levels generated during SiO2 growth by thermal

oxidation of Si [4]. It is widely accepted that the rate-limiting step of the TAT process is the tunnelling

of charge carriers from the occupied deep levels to the conduction or valence band of SiO2 films.

Another possible tunnelling mechanism for charge carrier transport via deep levels is inter-trap tunnel-

ling. In this case the charge carrier tunnels from an occupied deep level to the next-nearest unoccupied

one. When the inter-trap distance is smaller than the charge carrier path from the occupied deep level to

the insulator conduction or valence band, this mechanism will prevail over the FowlerNordheim-type

tunnelling from deep levels. Because of the high density of deep levels in SiO2 films this inter-trap tun-

nelling should be taken into account for explaining the excess tunnelling currents in SiO2 films prepared

by different methods. By establishing the relation between the trap concentration and the tunnelling cur-

rent via these traps it becomes possible to estimate the trap density in different SiO 2 and other insulator

films. Such estimation is needed for further development and control of the growth, deposition and other

preparation methods of insulator films in the contemporary semiconductor technology.

At present, radio frequency (rf) plasma processes are widely applied in semiconductor device technol-

ogy. During plasma treatment Si/SiO2 structures are subjected to ion bombardment and UV light expo-

sure. Because of this, defects are introduced in the SiO2 film, at the Si/SiO2 interface and in the Si bulk

*Corresponding author: e-mail: [email protected], Phone: +359 2 71 44 228, Fax: +359 2 975 36 32


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phys. stat. sol. (a) 201, No. 13 (2004) / www.pss-a.com 2967


(e.g. see [5]). Therefore it is interesting to study the role of deep levels, introduced during plasma expo-

sure of Si/SiO2 structures, in the charge carrier transport through SiO2 films.

In this paper an expression for the currentvoltage characteristics in insulators for the case of inter-trap tunnelling is derived. An expression for inter-trap tunnelling current density as a function of applied

electrical field is described in Section 2. The sample preparation, plasma treatment and characterization

of p-Si/SiO2 structures are described in Section 3. The observation of tunnelling currents in these

p-Si/SiO2 structures is considered in Section 3.3, where also the analysis of these tunnelling currents is

presented. This analysis shows that the current in these SiO2 films is indeed caused by the inter-trap

tunnelling. In Section 4 it is shown that currents of tunnelling type in some other MIS structures are also

due to inter-trap tunnelling.

2 Inter-trap tunnelling conduction in insulators

As the traps in SiO2 are distributed in broad bands, in some cases with a single dominant trap, the energy

distribution,D(qt), of these traps should be taken into account. The inter-trap tunnelling is governed bythe position of the electron quasi-Fermi level, F. It is assumed that when an electrical field is applied to

the SiO2 film this level remains parallel to the energy bands of SiO2. It is a constant value below the SiO2

conduction band edge along the SiO2 film thickness. In these circumstances the electron tunnelling cur-

rent density, Jd, from occupied traps, with energy position qt (measured from the conduction band

edge), to the unoccupied ones in the direction where the electrical field decreases the energy barrier for

electron tunnelling is given by

( ) ( ) ( ) ( )( )d t t t t t t d t0

1g

J q D q f q f q wP d q

= (1)

where is the electron attempt to escape frequency in the occupied trap, w is the distance from the occu-

pied trap to the next-nearest unoccupied trap,ftis the trap occupation function and P

dis the probability

factor for this tunnelling.

In many cases the change of fixed oxide charge in SiO2 films during electrical conduction can be ne-

glected. Then from the time-dependent continuity equation (the Esaki equation)

( )t in t out td

1d

fJ f J f

t= ,

whereJin andJout are, respectively, the ingoing and outgoing electron fluxes, in the steady state it follows

thatJin =Jout andft = 1/2. With

t

t F

eff

1

1 exp

fq

kT

=

+

,

where Teff is the effective temperature of tunnelling electrons in the SiO2 film, this mean that the tunnel-

ling current contribution is a maximum when the trap energy coincides with the quasi-Fermi level F.

Taking into account that

( ) ( )tt F

eff

1 11

21 cosh

f f tq

kT

=

+

(2)

it is clear that electron tunnelling from traps with other energies decreases sharply.


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2968 S. Simeonov et al.: Inter-trap tunnelling in thin SiO2 films


Equation (1) can then be transformed to

( )

=

+

t t d up t

t F0

eff

( )1

21 cosh

g D q P P dqJ q w

q

kT

(1a)

where Pup is the probability factor for an electron tunnelling from an occupied trap to the next-nearest

unoccupied one in the direction where the electrical field increases the energy barrier.

Taking into account that the factor 1 + cosh [(qt F)/kTeff] as a function ofqt increases sharply

when qt is moving away from F, in comparison with Pd and Pup also as functions ofqt one may

transform Eq. (1a) to

( )( )t t t

d up

t F0

eff

1

2

1 cosh

g D q dqJ q w P P

q

kT

=

+

(1b)

where Pd and Pup are calculated for qt* = F. As will be shown later the errors introduced by this ap-

proximation for calculation of inter-trap tunnelling are practically negligible.

Taking into account that

t t Feff

efft F

eff

dtanh

21 cosh

q qkT

kTq

kT

=

+

.

Equation (1b) can be transformed to

J= qwNt(Pd Pup) (1c)

where trap concentrationNt is equal toD(F)kTeff.

For the calculation of Pd in accordance with [4] it is assumed that under an electrical field, E, the

shape of the energy barrier between two adjacent deep levels with energy depth qt and distance w is

trapezoidal (see Fig. 1). For simplicity it is assumed that the distances from the occupied deep level to

next-nearest unoccupied ones along or against the electrical field are equal. It will be shown later that

this assumption does not put any constraint for practical use on the expression obtained for tunnelling

current. If the electrical field, E, decreases the electron energy barrier for electron tunnelling from an

occupied trap to the next-nearest unoccupied one, the probability factor, Pd, for this tunnelling in the

q(t-E.w)q(t+E.w) qt

ww

Fig. 1 Schematic of electron potential barriers at

nearest traps in an insulator under an electrical field.


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WKB approximation is given by the integral

1/ 2 1/ 2

d t

0

2exp (2 ) ( ) dw

P m q Ex x =

. (3)

After an integration of Eq. (3) Pd is transformed to

1/ 2 3/ 2 3/ 2

t td

4(2 ) [( ) ]exp

3

m q EwP

E

=

. (4)

WhenEwt and taking into account that

1/ 2 23/ 2 3/2 t

t t 1/2

t

3 3( )( )

2 4

Ew EwEw

+ +

2

1/ 2

3( ).

4 t

Ew(5)

Equation (4) can be transformed to

=

1/ 2 1/ 2 1/ 2 2

td 1/2

t

2(2 ) (2 )exp exp .

m q w m q w E P (6)

When the electrical field increases the energy barrier for an electron tunnelling in the opposite

direction from the same occupied deep level to the next-nearest unoccupied one the derivation of the

probability factor for this tunnelling, Pup, is similar to that for Pd. In this case the probability factor is

given by

1/ 2 1/ 2 1/ 2 2

tup 1/2

t

2(2 ) (2 )exp exp

m q w m q w E P

=

. (7)

The concentration of traps,Nt, near to the electron quasi-Fermi level is approximately 1/w3

; taking intoaccount Eqs. (1c), (6) and (7) the net electron current density from the occupied deep levels to unoccu-

pied ones is expressed by

d up3( )

2

q wJ P P

w

= . (8)

When Pd and Pup in Eq. (8) are replaced by Eqs. (6) and (7), respectively, the current density is given by

1/ 2 1/ 2 1/ 2 2

t

2 1/2

t

1 2(2 ) (2 )2 exp sinh

m q w m q w E J q

w

=

. (9)

Both the field-dependent and field-independent terms in Eq. (9) depend on the energy position in the

insulator energy gap, q t, and the inter-trap distance, w. Therefore, measuring theIVcharacteristics ofa metalinsulatorsilicon structure it is possible to determine the energy position and concentration of

traps in the insulator if inter-trap tunnelling takes place.

3 Characterization of hydrogen plasma-treated p-Si/SiO2 structures

3.1 Sample preparation

Float-zone grown boron-doped Si(100) wafers with a specific resistivity of 3.64.5 cm were thermally

oxidized in dry oxygen (H2O < 3 ppm) at 1050 C. Two sets of Si/SiO2 structures were prepared. One

set, D1 samples, with 13.5 nm thick SiO2, and the other set, D6, with 65 nm thick SiO2. The thickness of

the SiO2 layers was determined from the ellipsometric measurements.


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The p-Si/SiO2 structures were subjected to rf hydrogen plasma exposure at temperatures of 20, 100

and 300 C for 15 min in a home-built planar reactor similar to the one described in [6]. The plasma was

excited between two parallel electrodes by a 13.56 MHz rf generator with a power density of 75 mW cm 2delivered to the upper electrode. The hydrogen gas pressure was kept at 13.3 mbar. The metal body of

the reactor and the lower electrode were grounded. The samples were placed on this grounded electrode.

During plasma exposure the SiO2 surface was charged up to the floating potential in the reactor (2 V)

measured by a Langmuir probe.

3.2 CVcharacteristics of p-Si/SiO2 structures

To study electrically active defects in these p-Si/SiO2 structures capacitancevoltage (CV) measure-

ments using a 1 MHz E7 10 LCR meter and currentvoltage (IV) measurements were performed at

77 K and room temperature. The CVcharacteristics of the hydrogen plasma-treated p-Si/SiO2 struc-

tures were compared with those of p-Si/SiO2 structures formed in the same oxidation runs but without

plasma treatment. Such unexposed structures will be further denoted as reference samples.Using the expression for the capacitance,

ox s

1 1 w

C C = ,

and for the electrical potential

2

A Afb

s ox2

qN w qN wd V V

= +

of a MOS structure, where Vfb and Vare the flat-band and applied voltages, respectively, w and dare the

thickness of the space charge layer and the oxide film, respectively, s and ox are dielectric constants of

the same layer and film, respectively, and all other symbols have their usual meaning, one may obtain theexpression

2

fb

A S ox ox ox

2( ) 1 1 2 1 1V V

qN C C C C C

= +

for the capacitance in the depletion mode of a MOS structure. Using this expression, from CVmeas-

urements in the depletion mode of MOS structures, the flat-band voltage, expfb ,V and the doping concen-

tration in the Si substrate,NA, are determined by plotting the dependence of

2

ox ox ox

1 1 2 1 1

C C C C C

+

Table 1 Density of traps,Nox, in the SiO2 film.

Nox for the D1 set (cm2) Nox for the D6 set (cm

2)sample

measurement temperature 300 K 89 K 300 K 77 K

reference 5.80 1011 1.40 1011 4.30 1011 4.70 1011

treated in plasma at 20 C 3.71 1012 5.95 1012 1.19 1012 1.87 1012

treated in plasma at 100 C 3.07 1012 4.85 1012 1.07 1012 1.47 1012

treated in plasma at 300 C 6.90 1011 1.06 10

12 3.70 1011 5.30 10

11


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-12 -10 -8 -6 -4 -2 0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0300K

reference

treated at 1000C77K

reference

treated at 1000C

C/C

ox

Voltage (V)

0 5 10 15 20 25 3010

-5

10-4

10-3

10-2

10-1

100

101

102

300K

77K

J(Acm-2)

Voltage (V)

on the applied voltage. The slope of this plot gives theNA value and the intersection with the voltage axis

of the extrapolated curve determines the expfbV value. With this value ofNA the ideal flat-band voltagei

fbV for the corresponding Si/SiO2 structure is determined byideal

fbqV = m Eg + kTln[Nv(T)/NA],

where m is the metal work function, is the electron affinity andNv is the effective density of states inthe Si valence band.

The CV characteristics of all samples subjected to hydrogen plasma are shifted towards negative

voltages. This is evidence that hydrogen plasma exposure generates positive charge in the SiO 2. As an

illustration, in Fig. 2 the CVcharacteristics, measured at 77 and 300 K, of the reference and the sample

treated in hydrogen plasma at 100 C from the D6 set are shown.

The density of fixed oxide charge in the SiO2 film, qNox, is given by qNox = Cox(exp

fbV i

fb ).V The val-

ues of trap density,Nox, for different samples are given in Table 1.

For the references and the samples treated in plasma at 300 C the trap density at room temperature is

in the range of 1011 cm2. The trap density values of reference samples are close to the same values of

corresponding samples treated at 300 C. The trap density for both D1 and D6 samples treated in plasma

at 20 and 100 C is in the range of 1012 cm2. This observation shows that the majority of plasma-induced

defects are annealed out at 300 C. The Nox values for the D1 set of samples are approximately three

times higher than the corresponding values for the D6 samples.The CVcharacteristics of all samples measured at 77 K are shifted towards more negative voltages

in relation of the corresponding characteristics measured at 300 K. This is evidence that donor-like local-

ized traps exist at the Si/SiO2 interface [7]. In accordance with [7] their energy density, Nit, in the Si

energy gap can by estimated by the expression

exp exp

fb fbit ox

F

(77) (300)V VN C

=

,

where F is the shift of the Fermi level towards the Si valence band when the temperature decreases

from 300 to 77 K. The values of the energy densityNit are given in Table 2.

Fig. 2 Normalized CV characteristics of reference

and hydrogen plasma-treated at 100 C D6 p-Si/SiO2

structures measured at 77 and 300 K.

Fig. 3 IV characteristics of D6 p-Si/SiO2 struc-

ture treated in hydrogen plasma at 100 C measured

at 77 and 300 K.


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Table 2 Energy density of interface traps,Nit, in p-Si/SiO2 structures.

sample Nit for the D1 set(eV1 cm2)

Nit for the D6 set(eV1 cm2)

reference 3.85 1012 1.76 1011




11

3.3 Tunnelling currents in p-Si/SiO2 structures

The IVcharacteristics of the structures were measured at both forward and reverse bias applied volt-

ages. Here only the forward current at negative voltages applied to the dot electrode on the oxide is con-

sidered. The current densities of different samples are close to each other, but the lowest densities are

observed in the references and in the samples treated in plasma at 300 C, while the highest densities arefor samples treated at 100 C. In Fig. 3 theIVcharacteristics, measured at 77 and 300 K, of D6 sample

treated in plasma at 100 C are shown. The CVcurves show that at applied voltages larger than 10 V

this p-Si/SiO2 structure is in the accumulation mode and, therefore, the whole electrical field is applied

across the SiO2 film. With decreasing the temperature from 300 to 77 K the electrical conduction

changes a little. Even at voltages larger than 10 V the conductivity at 77 K is slightly higher than that at

300 K. Such behaviour of the conductivity, i.e. a small increase with decrease of temperature, is ob-

served in Ge tunnelling diodes [8].

The small change in the conductivity with temperature suggests that under an accumulation the con-

duction through the oxide is of tunnelling type. As the oxide thickness in the studied structures is 13.5 or

65 nm, FowlerNordheim tunnelling through the SiO2 film is excluded. Therefore, the observed conduc-

tivity is due to trap-assisted tunnelling of charge carriers in the SiO2 film. The electron and hole effective

masses in SiO2 are 0.5m0 and 0.46m0, respectively [4]. One can expect that the observed tunnelling cur-

rents through the oxide of the plasma-treated structures may be carried either by electrons or holes. In the

first case the electrons tunnel from the Al dot electrode through the SiO2 film towards the p-Si substrate.

In the second case the holes tunnel from the hole-accumulation layer at the p-Si/SiO 2 interface through

the SiO2 film towards the Al dot electrode. It has been established that in the case of hole accumulation

at the Si/SiO2 interface the current through the 45 nm SiO2 films on p-Si substrate is carried by elec-

trons injected from the metal [9]. Because of this, only the electron current will be considered further for

an explanation of tunnelling currents in our 13.5 and 65 nm thick SiO 2 films. However, possible inter-

trap tunnelling currents by holes in other insulator films can be treated by a simple replacement of the

electron effective mass with that of holes in Eq. (9).

If one assumes that FowlerNordheim trap emission is responsible for tunnelling currents, then for

the tunnelling current analysis two expressions are applied. In some cases such current density is ex-

pressed by

1/ 2 3/ 22 t4(2 )exp

3

m qJ AE

E

=

(10)

whereA = q3/[162(qt)] (e.g. see [10, 11]), or by

1/ 2 3/ 2

t0

4(2 )exp

3

m qJ J

E

=

(11)

whereJ0 is unspecified [12]. Equation (8) is correct for the tunnelling of electrons from the gate metal to

the semiconductor or vice versa through the SiO2 layer with an energy barrier at the metal/SiO2 interface,

qt.


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Taking into account that the fixed oxide charge in these Si/SiO2 structures is positive, the electrical

field in the SiO2 layer,E, in the accumulation mode is expressed as

exp

fbV VEd

=

where V is the applied voltage, expfbV is the flat-band voltage and d is the SiO2 film thickness. Using

Eq. (10) from the plot of ln(JE2) versus (V exp 1fb )V it is possible to determine qt either by the slope of

this dependence or by the pre-exponential term. From theIVcharacteristics, measured at 300 K, of the

sample from the D6 set, treated in plasma at 300 C, the qt values are calculated from both the slope and

pre-exponential term and are 0.077 and 1.83 107 eV, respectively. These values calculated from the

IVcharacteristics, measured at 77 K, are 0.16 and 6.29 105 eV, respectively. For the D6 reference, the

corresponding values at 300 K are 0.21 and 3.8 106 eV, respectively, while at 77 K they are 0.15 and

4.1 106 eV, respectively. The large discrepancies in the values of qt determined from the plot of

ln(JE2) versus (V exp 1fb )V show that Eq. (10) is not appropriate to account for electron tunnelling via

traps in the SiO2 layer.As is known, the tunnelling current is given by a product of the probability factor for tunnelling with

the supply function of tunnelling charge carriers and subsequent integration of this product. The electron

supply function depends on the character and density of electron states. The character and density of

electron states in the SiO2 conduction band are very different from the character and density of localized

electron states at traps in SiO2. Therefore the supply function for electrons that tunnel from traps in the

SiO2 is different from the supply function for electrons in metal or SiO2 conduction band. Because of this

the current density due to electrons tunnelling from traps to the SiO 2 conduction band cannot be ac-

counted for by Eq. (10) as is shown above.

The probability factor for electron tunnelling from traps to conduction band in SiO2 is given by

exp [(4(2m q)1/2t3/2)/3E]. In this case one has to use Eq. (11) which takes into account only the

probability factor for tunnelling from traps andJ0 remains unspecified [12]. In this case it is possible to

determine only qt from the slope of the dependence of ln(J) on (Vexp 1

fb ) .V The qt values for the

samples of the D1 set, treated in plasma at 20 and 100 C, are in the range of 0.110.26 eV. From the D 1set the qt values for the reference and the sample treated in plasma at 300 C are in the range

0.260.36 eV. The corresponding values of qt for the D6 set are in the range 0.0830.15 eV and

0.140.29 eV, respectively. Such low values ofqt are not compatible with the WKB approach to the

tunnelling by which approach Eqs. (10) and (11) are obtained. The WKB approximation is valid when

1/ 2 3/ 2

t

12(2 ) ( )

E

m q Ex

=

.

The mean values ofare 0.15 and 0.18 only for the D6 samples at 100 and 300 C. In all other cases

ranges from 0.76 to 6.42. Therefore the use of both Eqs. (10) and (11) is not justified at least for these

latter cases. Moreover these values ofqt are considerably lower than the values of trap energy positions

in the SiO2 film established by measurements other than IVones [1315]. For these other measure-

ments qt values are in the range 1.82.8 eV. Therefore, one can reach to a conclusion that charge carrier

tunnelling via deep levels in these SiO2 films is not the FowlerNordheim-type emission described by

Eqs. (10) and (11).Another possibility for trap-assisted tunnelling is inter-trap tunnelling. In this case one has to use

Eq. (9) for analysis of the tunnelling current. The plots of the natural logarithm of the current density,

measured at 77 K and room temperature, as a function of (Vexp

fb )V for the D1 sample treated in plasma

at 300 C are shown in Fig. 4. As is seen, at applied voltages larger than expfbV the slope of the plot of ln(J)

versus (Vexp

fb )V is constant. This slope is 0.538 and 0.398 V1 for 77 and 300 K measurements, respec-

tively. Also, extrapolating the curve to the intersection with the ln(J) axis at (V expfb )V = 0 the ln(J0)

value is obtained. The values ofJ0 are 0.51 and 0.413 A cm2 for 77 and 300 K measurements, respec-

tively. Then one may evaluate the distance between the nearest traps, w, and the trap position in the SiO2


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-2 -1 0 1 2 3 4 5 6 7 8 9 10-16

-14

-12

-10

-8

-6

-4

-2

0

2

4 300K77K

ln[J(Acm-2)]

(V - VFB

)(V)

energy gap, qt, by comparing the value of this slope and that ofJ0 with the corresponding values of the

slope andJ0 in Eq. (9) for inter-trap tunnelling, which are equal to

1/ 2 2

1/ 2

t

(2 )m q w E

and

1/ 2 1/ 2

t

2

1 2(2 )exp

m q wq

w

,

respectively.

For an estimation of the order of magnitude of the attempt to escape frequency, , Mott proposed a

relation h = kT where T is the Debye temperature [16]. With a Debye temperature of 552 K [17] the

estimated value for in SiO2 is 1.15 1013 s1. The frequency of dominant phonons in SiO2 at 100 K

estimated from Fig. 9 of [17] is = 1013 s1. In our calculations the value of= 1013 s1 will be used. The

inter-trap tunnelling current density given by Eq. (9) depends exponentially on qt and w and it is pro-

portional to . Because of this if= 10

13

s

1

is replaced by = 10

12

s

1

the calculated value for qt de-creases by 15%, while the value for w decreases by 4%. If = 1013 s1 is replaced by = 1014 s1 the

calculated value for qt increases by 15%, while the value for w increases by 4%. This means that one

needs only the order of magnitude of the attempt to escape frequency to evaluate the values ofw and qt.

The effective electron mass in SiO2 is taken as m* = 0.5me. From theIVmeasurements at 77 K of the

D1 sample treated at 300 C the values ofw and qt are 1.73 107 cm and 2.18 eV, respectively. From

the room temperature measurements of the same sample these values are w = 1.57 107 cm and

qt = 2.74 eV, respectively.

Using these values ofqt and w one may estimate the error of replacement ofPd(qt) in Eq. (1b) with

Pd(qt*) where qt* = F. It is smaller than 12% for qt in the range F 4kTeff


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the errors are similar but for D6 samples they are smaller. Therefore, one can use Eq. (9) for analysis of

the tunnelling currents in D1 and D6 samples.

The values of the slope in these plots for applied voltages corresponding to the accumulationmode show that the hyperbolic sine function can be replaced by the exponential function. This means

that practically all electrons trapped at deep levels always tunnel in the direction where the electrical

field decreases the electron energy barriers. Therefore in these circumstances one can neglect the

Pup term in relation to the Pd one for these values of applied voltage. The distance w has the meaning

of the mean value of the inter-trap distance and it does not depend on the electrical field direction. Be-

cause of this w values used for calculation of Pd and Pup as shown in Fig. 1 are taken as equal to one

another.

In Table 3 the values of the energy depth, qt, and the distance, w, for samples from D1 and D6 sets are

summarized. As is seen, the values of qt are in the range 1.53.9 eV for both D1 and D6 sets. With

these values ofqt and w the mean values of the parameter

1/ 2 3/ 2

t2(2 ) ( )E

m q Ex

=

are from 3.8 103 to 0.044. As all values of are many times smaller than 1 this means that the

WKB approximation is valid for analysis of the inter-trap tunnelling in these MOS structures. For a

given sample, the observed difference of the values ofqt by several tenths of eV is due to the uncertain-

ties ofIVmeasurements and the change of the position of the quasi-Fermi level in the SiO2 layer with

changing temperature. These values of qt are in the same range as the broad bands of defects at

1.82 and 2.25 eV observed by electroluminescence in SiO2 layers [13]. They are also in the range of

deep levels at 2.7 2.8 eV by which two-step TAT takes place in 5 nm thick SiO2 layers [14] and

deep levels at 2.1 eV, established by noise measurements of SILC currents in 5 nm thick SiO2 layers

[15].

Table 3 The energy depth, qt, and the distance, w, of traps in the SiO2 energy gap, obtained with

values ofm = 0.5me and = 1013 s1 for samples of the D1 and D6 sets.

D1 set

at 77 K at 300 K

sample

w (cm) qt (eV) Nt (cm3) w (cm) qt (eV) Nt (cm

3)

reference 1.45 107 3.53 3.29 10

20 1.38 107 3.71 3.79 10

20

treated in plasma

at 20 C1.78 10

7 1.95 1.77 1020 1.81 10

7 2.17 1.69 1020

treated in plasma

at 100 C1.76 10

7 2.35 1.83 1020 1.82 10

7 2.37 1.66 1020

treated in plasma

at 300 C1.73 107 2.18 1.93 1020 1.57 107 2.74 2.57 1020

sample D6 set

reference 2.08 107 1.55 1.11 1020 1.99 107 1.59 1.26 1020

treated in plasma

at 20 C1.75 10

7 1.84 1.86 1020 1.63 10

7 2.35 2.30 1020

treated in plasma

at 100 C1.28 107 3.02 4.8 1020 1.19 107 3.9 5.93 1020

treated in plasma

at 300 C1.82 107 1.90 1.65 1020 1.80 107 1.9 1.71 1020


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Using the values of inter-trap distance, w, one may evaluate the density of traps, Nt, by which the

tunnelling takes place (Nt 1/w3). The calculated values ofNt are in the range 1.1 10

202.45 1020 cm3

and are also presented in Table 3. The values ofNt for the D6 reference sample are the smallest. Observa-tion of tunnelling currents even in the reference samples, without any plasma treatment, means that traps

are initially present in the oxides and they are responsible for the appearance of inter-trap tunnelling.

However, the concentration of these traps in reference samples is lower in relation to the trap concentra-

tion in plasma-treated samples. This result is analogous to the observation of TAT even in virgin SiO 2

layers without electrical stress [4].

The values ofNt obtained for D1 and D6 samples plasma treated at 300 C are higher than the corre-

sponding reference sample values. The comparison of the Nt values with the Nox values of the corre-

sponding samples shows that the lowest values ofNt and Nox are observed in the reference samples.

The values ofNt and Nox in samples treated in plasma at 300 C are higher than in the reference samples

but smaller than those for samples treated in plasma at 20 and 100 C. The highest values ofNox are

observed in samples treated at 20 C, while the highest values ofNt are observed in samples treated at

100 C. This shows that traps different from traps connected to the fixed oxide charges are responsiblefor inter-trap tunnelling. The concentration ofNox is in the range 6.6 10164.4 1018 cm3 in these SiO2

layers. This concentration is smaller than the Nt concentration, which is in the range 1.1 10202.45

1020 cm3. It is known that Nox is a superposition of positively and negatively charged defects in SiO2

layers [18]. Tunnelling via neutral traps unoccupied by electrons is also possible, as is suggested in

Ref. [1]. Therefore, the concentration ofNt can exceed the concentration ofNox as these concentrations

are related to different properties of SiO2 layers. Trap concentrations of the order of 3 10191020 cm3

are considered for analysis of the temperature-dependent SILC in 10 nm SiO2 layers [19]. Trap con-

centrations of the order of 3 10197 1019 cm3 are used for modelling of trap-assisted inelastic tunnel-

ling in MOS structures [15]. These trap concentrations are of the same order of magnitude as the values

ofNt obtained from the analysis of the tunnelling-type IV characteristics of the studied p-Si/SiO2

structures. Therefore, both quantities, the trap energy positions in SiO2 energy gap qt and the trap con-

centrations Nt obtained from the inter-trap tunnelling dependence ofIVcharacteristics, are consistent

with the corresponding investigations of other Si/SiO2 structures. This consistency leads to the conclu-sion that the inter-trap tunnelling is the real conduction mechanism in the investigated p-Si/SiO2 struc-

tures.

One may also notice that Eq. (9) is similar to the so-called Poole law for inter-trap conduction in the

case of thermally activated electron emission from insulator traps (see Ref. [19] and references therein).

In this law ln(J) is also a linear function of applied electrical field E, as it is in Eq. (9). However, the

relationships connecting ln(J) with w and qt and Eq. (7) are different because the charge carrier transfer

from an occupied trap to next-nearest unoccupied one is thermally activated in the Poole law and it is of

tunnelling type in the case of Eq. (9).

The measuredIVcharacteristics at 77 and 300 K do not depend on the temperature. This means that

the effective temperature of electrons in traps in SiO2 under these electrical fields is not equal to the

lattice temperature and it is at least equal or greater than 300 K. Because of this it is not possible to esti-

mate the energy distribution of traps responsible for inter-trap tunnelling, D(F), from their concentra-tion,Nt, by the relationNt =D(F)kTeffused in Eq. (1c).

Defects in SiO2 films generate broad trap bands and because of this the values of the trap energy posi-

tion qt discussed so far have a meaning of the energy position of traps which dominate in the particular

tunnelling process. Because of this it is interesting to estimate the energy distribution of traps, N, re-

sponsible for tunnelling in D1 and D6 SiO2 films. In these circumstances N is equal to 1/w3qEw as

qEw eV is the energy position of the next-nearest trap to which the electrical field makes possible

the electron tunnelling. With the values of w and E in these D1 and D6 SiO2 films N is in the range

2.2 10207.15 1020 cm3 eV1. Similar or even higher energy densities of states are reported for amor-

phous materials (e.g. see [20] where the values ofN are in the range 6.37 10211.28 1022 cm3 eV1

for V2O5 B2O3BaO glasses).


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4 Tunnelling currents in other insulator films

For a further check of the inter-trap approach to the tunnelling currents in some SiO 2, SiO and otherinsulator films, theirIVcharacteristics given in Refs. [2, 2127] have been analysed with Eq. (9) in the

same way as is described in Section 3.3 for the tunnelling currents in p-Si/SiO2 structures treated in hy-

drogen plasma.

In [21] the IVcharacteristics of 380 nm SiO layer measured at 4.2 and 77 K have been reported.

These characteristics are very close to each other suggesting that in this SiO film at these temperatures a

conduction of tunnelling type takes place. Assuming that the inter-trap electron tunnelling is the real

charge transport mechanism, from the 4.2 K IVmeasurements by Eq. (9) one may obtain an energy

position ofqt = 1.8 eV and a concentration ofNt = 6.7 1018 cm3 for traps responsible for tunnelling

in this SiO film. The same values obtained from 77 K measurements are 1.79 eV and 7.26 1018 cm3,

respectively. Similar IVcharacteristics of 410 nm SiO film have also been reported [22]. In this case

from the 4.2 KIVmeasurements the values ofqt = 1.9 eV andNt = 1.38 1019 cm3 are calculated.

These values obtained from the 77 K measurements are 1.83 eV and 1.22 1019

cm3

, respectively.In [2] it has been shown that in 300 nm SiO2 films after a 100200 keV Ge implantation with a dose

of (1.51.8) 1016 cm2 the current does not depend on temperature in the range 25150 C for applied

electrical field in the range (24.5) MV cm1. Using Eq. (7) the calculated qt and w values for this

tunnelling-type current are 3.16 eV and 2.91 107 cm, respectively. The corresponding value ofNt is

4 1019 cm3.

In [23] a weak temperature dependence of the gate leakage currents of 11.5 nm thick nitrided oxide

and re-oxidized nitrided oxide in the 300400 K temperature range is also reported. For electrical fields

above 4.5 MV cm1, applying Eq. (7) the estimated values ofqt and w are 3.11 eV and 2.17 107 cm

for nitrided oxide, 3.17 eV and 2.18 107 cm for re-oxidized nitrided oxide and 3.32 eV and 2.33 107 cm

for nitrided oxides annealed in nitrogen. The corresponding trap concentrations are 9.78 1019, 9.64 1019

and 7.89 1019 cm3, respectively.

In many present day MOS structures the thickness of the SiO 2 layer is of the order of several nano-

metres and, therefore, trap-assisted tunnelling exists in these SiO2 layers. In these cases the charge carri-ers from the metal or the semiconductor tunnel firstly to a trap in the SiO 2 layer and then to the semicon-

ductor or the metal. Although Eq. (9) is obtained for the case of inter-trap tunnelling, the probability

factors (exponential terms) for the inter-trap tunnelling in an insulator and the tunnelling from an insula-

tor trap to either metal or semiconductor bands are essentially the same. The corresponding IVexpres-

sions differ only by the pre-exponential terms in them. Therefore, Eq. (9), valid for inter-trap tunnelling,

can be also applied for evaluation of energy position and concentration of traps in such MOS structures.A weak temperature dependence of the gate leakage currents in 3 nm SiO2 films of an MOS capacitor

in the accumulation mode in the temperature range 50300 K has been observed [24]. From these IV

characteristics at gate voltages greater than 2 V using Eq. (9) the values of qt = 2.8 eV, w = 2.42 107 cm

andNt 7 1019 cm3 are estimated.

If the stress-induced leakage current in MOS structures with 4.5, 7 and 12 nm SiO2 layers in Fig. 11

of [25] is re-plotted using Eq. (9) one may obtain qt and w values. These are 5.9, 4.5 and 3.1 eV and

2.25 107, 2.9 107 and 4.09 107 cm for 4.5, 7 and 12 nm SiO2 layers, respectively.Further we show the applicability of Eq. (9) for TAT currents in the case when it is possible to esti-

mate the inter-trap distance, w, independently. From the observed TAT given in Fig. 11 of [26] it is de-

duced that w is less than 5 nm but greater than 3.5 nm. Using Eq. (9) one may re-plot TAT for 5, 7

and 10 nm SiO2 films. The values obtained for qt and w are 4.58 eV and 2.7 107 cm, 4.55 eV and

3.02 107 cm and 4.9 eV and 3.06 107 cm, respectively. The most reliable values are for 10 nm SiO2

films because in this case the inter-trap tunnelling is an indispensable step in the whole tunnelling charge

transfer. Therefore, the value obtained of w = 3.06 107 cm is close to the independently estimated

value ofw, 3.5 nm


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In AlN/GaN heterostructures with 6 nm AlN insulating film the observed currents in accumulation

mode are 10131017 times higher than the tunnelling currents calculated without taking into account the

trap-assisted tunnelling [27]. If one assumes that the observed current is of tunnelling type then usingEq. (9) the values ofqt and w are 0.973 eV and 3.144 10

7 cm, respectively, and the corresponding

trap density is 3.21 1019 cm3.

From the above considerations the usefulness of Eq. (9) is clearly demonstrated. The possibility of

using Eq. (9) will give a quick and straightforward opportunity for an estimation of the concentration of

traps in insulator films under the conditions where tunnelling currents prevail. The estimated trap densi-

ties could be beneficial for the comparison of different preparation methods of new insulator layers, for

example high-permittivity insulator films on SiO2.

5 Conclusion

An expression for the currentvoltage characteristics in insulators for the case of inter-trap tunnelling,

Eq. (9), has been presented. This expression gives an opportunity to estimate the energy position andconcentration of traps by which the inter-trap tunnelling takes place. It is used to analyse TAT currents in

p-Si/SiO2 structures subjected to hydrogen plasma treatment, and in other SiO2 and other insulator films.

The thickness of these films and current densities in them are very different. Nevertheless the values of

energy position in SiO2 can be estimated using this expression for currentvoltage dependence for inter-

trap tunnelling. The energy gap and concentration of traps in the SiO2 films are in the range 1.53.9 eV

and 6 10182.14 1020 cm3, respectively, for oxide thickness changing from 3 to 410 nm in D1 and D6

samples and some other SiO2 layers where TAT is observed. These results show that the inter-trap tun-

nelling can account for tunnelling-type currents in SiO2 and some other insulator films. The obtained

values of energy position and concentration of traps are consistent with the same values estimated by

other methods in similar SiO2 films. Therefore the inter-trap tunnelling is a more suitable mechanism

than the FowlerNordheim trap emission for explaining TAT in SiO2 and some other insulator films. The

obtained values of energy position and concentration of traps in these films, especially the concentration

of traps responsible for inter-trap tunnelling, can be used in the development of preparation methods forcontemporary semiconductor technology.

References

[1] D. J. Di Maria and J. Cartier, J. Appl. Phys. 78, 3883 (1995).[2] J. Zhao et al., Solid State Electron. 46, 66 (2002).

[3] B. J. Cho et al., Solid State Electron. 44, 1289 (2000).

[4] A. Ghetti, Microelectron. Eng. 59, 127 (2001).

[5] S. Simeonov et al., Vacuum61, 199 (2001).

[6] A. R. Reinberg, Electrochemical Society Meeting, San Francisco, US patent 3 757 733 (1974).

[7] P. V. Gray and D. M. Brown, Appl. Phys. Lett. 8, 31 (1966).

[8] L. Esaki, Phys. Rev.109, 603 (1958).

[9] J. Maserjian and N. Zamani, J. Appl. Phys. 53, 559 (1982).

[10] D. Goguenheim et al., Solid State Electron.45, 1355 (2001).

[11] Y. L. Chiou et al., Solid State Electron. 45,1787 (2001).

[12] W. J. Chang et al., Semicond. Sci. Technol. 16, 961 (2001).

[13] C. G. Quin et al., J. Phys.: Condens. Matter13, 11751 (2001).

[14] F. Jimenes-Molinos et al. J. Appl. Phys90, 3396 (2001).

[15] K. Komiya and O. Yasuhisa, J. Appl. Phys. 92, 2593 (2002).

[16] N. F. Mott and R. S. Allgaier, phys. stat. sol. 21, 343 (1967).

[17] R. C. Zeller and R. O. Pohe, Phys. Rev. B 4, 2029 (1971).

[18] M. Pepper, Proc. R. Soc. A 353, 225 (1977).

[19] B. De Salvo et al., Solid State Electron.44, 895 (2000).

[20] M. M. El-Desoky, phys. stat. sol. (a) 195, 422 (2003).

[21] A. Servini and A. K. Jonscher, Thin Solid Films 3, 341 (1969).


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[22] N. Klein and Z. Lisak, Proc. IEEE 54, 979 (1966).

[23] S. Fleisher et al., J. Appl. Phys. 73, 8354 (1993).

[24] A. P. Jacobs et al., Semicond. Sci. Technol. 17, 942 (2002).[25] D. Vuillaune et al., Microelectron. Reliab. 38, 7 (1998).

[26] G. Ghibaudo et al., C. R. Acad. Sci. Paris 1, Serie 4, 911 (2000).

[27] S. Imanaga et al., Jpn. J. Appl. Phys. 40, 1194 (2001).

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