Differences between direct current and alternating current capacitance nonlinearities in high-k dielectrics and their relation to hopping conduction

Abstract

Capacitance nonlinearities were studied in atomic layer deposited HfO2 films using two types of signals: a pure ac voltage of large magnitude (ac nonlinearities) and a small ac voltage superimposed to a large dc voltage (dc nonlinearities). In theory, ac and dc nonlinearities should be of the same order of magnitude. However, in practice, ac nonlinearities are found to be an order of magnitude higher than dc nonlinearities. Besides capacitance nonlinearities, hopping conduction is studied using low-frequency impedance measurements and is discussed through the correlated barrier hopping model. The link between hopping and nonlinearity is established. The ac nonlinearities are ascribed to the polarization of isolated defect pairs, while dc nonlinearities are attributed to electrode polarization which originates from defect percolation paths. Both the ac and dc capacitance nonlinearities display an exponential variation with voltage, which results from field-induced lowering of the hopping barrier energy.

Full text

Differences between direct current and alternating current capacitance nonlinearities in
high-k dielectrics and their relation to hopping conduction
O. Khaldi, P. Gonon, C. Vallée, C. Mannequin, M. Kassmi, A. Sylvestre, and F. Jomni
Citation: Journal of Applied Physics 116, 084104 (2014); doi: 10.1063/1.4893583
View online: http://dx.doi.org/10.1063/1.4893583
View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/116/8?ver=pdfcov
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Differences between direct current and alternating current capacitance
nonlinearities in high-k dielectrics and their relation to hopping conduction
O. Khaldi,
1,2
P. Gonon,
1,3,a)
C. Vall
ee,
1,3
C. Mannequin,
1,3
M. Kassmi,
1,2
A. Sylvestre,
4,5
and F. Jomni
2
1
Univ. Grenoble Alpes, LTM, F-38000 Grenoble, France
2
El Manar University, LMOP, 2092 Tunis, Tunisia
3
CNRS, LTM, F-38000 Grenoble, France
4
Univ. Grenoble Alpes, G2ELAB, F-38000 Grenoble, France
5
CNRS, G2ELAB, F-38000 Grenoble, France
(Received 17 July 2014; accepted 9 August 2014; published online 27 August 2014)
Capacitance nonlinearities were studied in atomic layer deposited HfO
2
films using two types of
signals: a pure ac voltage of large magnitude (ac nonlinearities) and a small ac voltage
superimposed to a large dc voltage (dc nonlinearities). In theory, ac and dc nonlinearities should be
of the same order of magnitude. However, in practice, ac nonlinearities are found to be an order of
magnitude higher than dc nonlinearities. Besides capacitance nonlinearities, hopping conduction is
studied using low-frequency impedance measurements and is discussed through the correlated bar-
rier hopping model. The link between hopping and nonlinearity is established. The ac nonlinearities
are ascribed to the polarization of isolated defect pairs, while dc nonlinearities are attributed to
electrode polarization which originates from defect percolation paths. Both the ac and dc capaci-
tance nonlinearities display an exponential variation with voltage, which results from field-induced
lowering of the hopping barrier energy. V
C2014 AIP Publishing LLC.
[http://dx.doi.org/10.1063/1.4893583]
I. INTRODUCTION
Metal-Insulator-Metal (MIM) capacitors are used in
the microelectronic industry for mixed-signal and radio-
frequency (RF) applications. For such applications, it is
important to have a high level of linearity over a broad volt-
age range (low voltage coefficients). Regarding the voltage
dependence of the capacitance, it is common to describe the
capacitance-voltage (C-V) characteristic by a second-order
polynomial law,
DC
C0
¼aV2þbV;(1)
where C
0
is the capacitance at zero bias, DC¼C(V) C
0
is
the capacitance increase at bias V, and aand bare voltage
coefficients. The quadratic voltage coefficient (a) can reach
up to several 1000 ppm/V
2
when materials are not optimized.
It can be positive (in most high-k materials such as transition
metal oxides) or negative (in silicon nitrides, silicon oxides,
or in crystalline perovskites). The linear voltage coefficient
(b), up to several 100 ppm/V, is introduced to take into
account departure from a pure quadratic dependence (elec-
trode effects). Usually, the quadratic term (aV
2
) dominates
over the linear term (bV). The origin of capacitance variation
with bias (Eq. (1)) is not yet clear, and several models have
been proposed.
1–8
They can be categorized as bulk vs. inter-
face models. Bulk models refer to field-dependent ionic,
3
electronic,
6
or dipolar
8
polarizabilities, but also to electro-
striction
6
or to more general thermodynamic arguments.
5
Interface models refer to interface traps
1
or to electrode
polarization.
4
At present, there is not a unique model which
is able to explain all the experiments. For instance, some
interface models, such as the electrode polarization model,
4
are unable to explain negative acoefficients as observed in
some dielectrics.
8
On the other hand, bulk phenomena can-
not explain the presence of the linear term (bV). Indeed,
bulk-related nonlinearities should be same whatever the sign
of the electric field, i.e., DC(V) must be an even (symmetric)
function of V and the linear term (bV) must vanish. When
DC(V) is not symmetric,
2
the linear term must be introduced
to take into account electrode/oxide interface effects, which
depend on bias polarity (V <0orV>0). Therefore, in the
same material, different effects can coexist. An example was
reported for SiO
2
where, depending on film thickness, ais
negative (thick films, bulk effects were proposed by authors)
or positive (thin films, interface effects were proposed).
7
When measuring nonlinearity, i.e., C-V characteristics,
it is usually assumed that V is a dc bias. The capacitor is sub-
jected to a small ac amplitude superimposed to a large dc
bias, and the dc bias is responsible for nonlinearity (this will
be termed "dc nonlinearity" in the following). However, in
some applications, the capacitor may be subjected to a large
ac signal (with no dc bias). The question arises of what could
be the magnitude of nonlinearity under large ac voltages
(termed "ac nonlinearity"). In theory, a given polarization
mechanism should lead to similar ac and dc nonlinearities
(this will be reminded here). In practice, it will be shown
that dc and ac nonlinearities could be very different. The
case will be evidenced for HfO
2
thin films where ac nonli-
nearities are much higher than dc ones. The origin of such a
difference will be related to bulk vs. interface polarization
a)
Author to whom correspondence should be addressed. Electronic mail:
Patrice.gonon@cea.fr.
0021-8979/2014/116(8)/084104/8/$30.00 V
C2014 AIP Publishing LLC116, 084104-1
JOURNAL OF APPLIED PHYSICS 116, 084104 (2014)
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phenomena, both being linked to hopping charge transport.
The present study follows our preceding works on the elec-
trical properties of HfO
2
in view of its application to MIM
devices.
9–11
II. EXPERIMENTS: DC VERSUS AC NONLINEARITIES
The tested MIM capacitors consist in Au/HfO
2
(10 nm)/
TiN structures. HfO
2
is deposited by Atomic Layer
Deposition (ALD) from HfCl
4
and H
2
O precursors. Films
are grown on TiN/Si substrates (bottom electrode), followed
by the evaporation of a top gold electrode (1.5 mm in diame-
ter). HfO
2
deposition is carried out at 350 C, resulting in
polycrystalline films (mainly orthorhombic). High resolution
transmission electron microscopy (HRTEM) shows grains
with random orientation, having a lateral size in the
15–20 nm range, and extending from the bottom TiN to the
top Au (see HRTEM images of similar samples in Ref. 9).
Capacitance measurements are carried out using two
kinds of test signals. The first (usual) one consists in a small
ac voltage (0.1 V
rms
, 1 kHz) superimposed to a dc bias vary-
ing from þ1.5 V to 1.5 V (dc polarity is the one applied to
the TiN electrode). It is used to probe dc nonlinearity. The
second test signal consists in a pure ac voltage (1 kHz) whose
amplitude varies from 0.05 V
rms
to 2 V
rms
. It is used to probe
ac nonlinearity (Fig. 1).
Fig. 2shows the capacitance (C) for dc and ac bias.
For the dc bias case (Fig. 1(a)), C varies from 18.88 nF at
0 V, to 19.11 nF at 1.5 V. Measurements are performed using
a high precision impedance analyzer (Alpha model from
Novocontrol Technologies). For the above capacitance
range, the absolute error of the impedance analyzer is 0.1%.
This corresponds to an absolute error of around 0.02 nF on
the measured capacitance. Relative error is 0.01%, which
means that error between two consecutive measurements
are around 0.002 nF, well below the variation observed in
Fig. 1(a). C-V characteristics are asymmetric, nonlinearities
being higher for positive dc bias (Au as a cathode). For posi-
tive bias, it is found that a¼4840 ppm/V
2
,b¼290 ppm/V,
and for negative bias a¼2150 ppm/V
2
,b¼1140 ppm/V.
Fig. 2(b) shows C for ac bias. Remarkably, when com-
pared to dc nonlinearities, ac nonlinearities are much higher.
For instance, for a dc bias of þ1.5 V we get DC/C
0
1.2%,
while for an ac voltage of 1.5 V
rms
we have DC/C
0
25%,
more than a decade above. Like dc nonlinearity, ac nonli-
nearity can be described by a second-order polynomial law.
It is measured that a¼67 450 ppm/V
2
,b¼62 310 ppm/V,
values which are much larger than the dc case.
To confirm the above results, measurements were
repeated using a different probe station connected to a differ-
ent impedancemeter (Agilent analyzer). For another sample,
when the ac voltage increases from 0.1 V
rms
to 0.7 V
rms
(the
maximum ac voltage delivered by the Agilent analyzer), ca-
pacitance increases by 11%, which is of the same order of
magnitude as the C increase in Fig. 2(b). This definitely rules
out any artifact which could have been related to the mea-
surement equipment, and confirms the large magnitude of ac
nonlinearity.
III. DISCUSSION
A. dc versus ac nonlinearities in theory
In theory, nonlinearities in ac and dc can be related to
each other. Let us consider the electric displacement
D¼e
0
EþP, where E is the applied electric field and P is the
dielectric polarization. The electric field is composed of dc
and ac parts, E ¼E
dc
þE
ac
sin xt, then
D¼e0Edc þe0Eac sin xtþP½Edc þEac sin xt;(2)
P[E] can be expressed as a power series (Maclaurin series),
FIG. 1. Test signals used to probe dc nonlinearity (Vdc þsmall Vac) and ac
nonlinearity (pure Vac).
FIG. 2. Capacitance (a) vs. dc bias and (b) vs. ac bias (at 1 kHz).
084104-2 Khaldi et al. J. Appl. Phys. 116, 084104 (2014)
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PE
½
¼X
1
0
Pn
ðÞ 0
ðÞ
n!En:(3)
Using Eqs. (2) and (3), D can be written as
D¼e0Edc þe0Eac sin xtþP1ðEdc þEac sin xtÞ
þP3ðEdc þEac sin xtÞ3þ;(4)
where P
n
¼P
(n)
(0)/n!. Since D must reverse when the polar-
ity of E is reversed, only odd powers of E are present in
Eq. (4). In the following, P
5
(and higher order nonlinearity
terms) are neglected. This is justified by the fact that C(V)
can reasonably be approximated by a 2nd order polynomial
function (see Fig. 2). This is to say that P is approximated by
a 3rd order polynomial function, and P
n
0 as soon as n >3.
Expanding (E
dc
þE
ac
sin xt)
3
in Eq. (4), and consider-
ing only the terms at x(the ones that are measured by the
impedance meter), one gets the dielectric constant at x,
eðxÞ¼DðxÞ=ðEac sin xtÞ
¼e0þP1þð3=4ÞP3E2
ac þ3P3E2
dc:(5)
Note that for linear dielectrics, P
3
¼0, and since
P
1
¼dP/dE ¼e
0
v, one gets the familiar expression e¼e
0
(1 þv), where vis the dielectric susceptibility.
Equation (5) shows that ac nonlinearity magnitude (given
by (3/2)P
3
E
rms
2
, where E
rms
¼E
ac
/冑2) should be 1/2 of dc
nonlinearity magnitude (given by 3P
3
E
dc
2
). This is derived on
the basis that the same polarization mechanism governs the dc
and ac response. Thus, in principle, a measure of dc nonlinear-
ity (using a small ac signal superimposed to a dc bias) allows
to predict ac nonlinearity (under a pure ac signal).
On the contrary, it is observed that ac nonlinearity is
much higher than the dc one. Therefore, it is immediately
apparent that P
3
in the ac term is not the same as P
3
in the dc
term. In other words, two different polarization mechanisms
govern the ac and dc behaviors.
B. ac nonlinearity
1. Low-frequency admittance
To get a deeper insight into physical mechanisms, the
low-frequency response of the admittance (Y ¼GþjCx)
was measured as a function of ac amplitude (Fig. 3).
At low V
ac
(0.01, 0.1, and 0.5 V
rms
) and at low frequency
(<100 Hz), the conductance varies as x
s
,withs¼0.8.
Gxsðs<1Þ:(6)
This is typical of hopping conduction.
12
Exponent s increases
as V
ac
increases (>0.5 V
rms
), to reach values close to s ¼1at
high V
ac
.ThisisshowninFig.4(a), where s was calculated
from dG/df in the 0.1–10 Hz range, see Fig. 4(b). The increase
in s with V
ac
(Fig. 4) and the ac nonlinearity (Fig. 2(b))appear
to be correlated, as they both rise above 0.5 V
rms
.
2. Classical theory of hopping at low ac fields
The Correlated Barrier Hopping (CBH) model
12
will be
considered. Under the influence of an ac field, electrons
FIG. 3. Frequency dependence of (a) capacitance and (b) conductance as a
function of ac amplitude (pure ac signal).
FIG. 4. (a) Hopping exponent s as a function of Vac. (b) (dG/df) vs. f, from
which s was calculated.
084104-3 Khaldi et al. J. Appl. Phys. 116, 084104 (2014)
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oscillate between trap 1 and trap 2 (pair model) by hopping
over a barrier W,
W¼WMe2
pER;(7)
where W
M
is the trap depth and R is the distance between
traps. The second term in Eq. (7) is the barrier lowering due
to traps overlapping (Fig. 5).
12
Electrons hop back and forth between traps and their
movement is equivalent to oscillating dipoles. Calculation of
the ac conductivity requires to establish the polarization
response to a step field (E ¼E
0
H(t), where H(t) is the
Heaviside function) and then to operate a Fourier transform.
13
When a field E
0
is applied to the pair (direction 2 !1) the
electron transition rate from trap 1 to trap 2 (1!2hopping
probability per unit of time) is
w12 ¼w0expðW=kTÞexpðeE0R=2kTÞ;(8)
while the 2 !1 transition rate is
w21 ¼w0expðW=kTÞexpðeE0R=2kTÞ;(9)
where w
0
is the attempt to escape frequency (phonon
frequency).
The related characteristic time s(relaxation time) of the
equivalent oscillating dipole is given by
13
s¼1=ðw12 þw21Þ¼s0expðW=kT Þ=coshðeE0R=2kTÞ;
(10)
where s
0
¼1/2w
0
.
At low fields, i.e., eE
0
R2kT, cosh(eE
0
R/2kT) 1
and ss
0
exp(W/kT). From Eq. (7), one gets
ss0expðWM=kTÞexpðe2=peRkTÞ:(11)
A pair of traps, with a given separation distance R, lead to a
simple Debye response in the frequency domain. However,
the separation distance R varies from pairs to pairs, so does s
(see Eq. (11)). Thus, calculation has to be performed by inte-
grating over the R distribution.
12,13
Assuming a random dis-
tribution of traps (density N), it is found
12
that
G¼1
24 p3eN2R6x:(12)
At low fields, the sub-linear dependence on frequency
(Eq. (6),s<1) is embedded in the R
6
term. Indeed, for a
given frequency of measurement x, major contribution to G
comes from traps with a relaxation time s¼1/x(reso-
nance). However, sdepends on R (Eq. (11)). Therefore, the
resonance condition s¼1/xcorresponds to a specific R
value. From Eq. (11), one gets
12
RðxÞ¼ ðe2=pÞ=½WMkTLnð1=xs0Þ:(13)
As xdecreases, Eq. (13) shows that the hopping dis-
tance R increases. As a consequence, the R
6
term in Eq. (12)
increases as xdecreases, and this leads to the sub-linear de-
pendence of G on x(Eq. (6)).
3. Hopping at high ac fields
The case of high fields will now be discussed.
Derivation of Eq. (12), established for low fields, is based on
the Fourier transform of the step response function.
12,13
It is
valid for linear systems, i.e., at low fields for which the
superposition principle applies (the time response to the field
E
1
þE
2
is the sum of the separate time responses to fields E
1
and E
2
). However, at high fields the superposition principle
does not apply. This is due to the field dependence of the
relaxation time, Eq. (10), that leads to a Debye response for
which s¼s(E). Therefore, the Debye response at (E
1
þE
2
)
is not the sum of separate Debye responses at E
1
and E
2
.Itis
no longer possible to express the polarization response to an
arbitrary (high) field as the convolution of the impulse
response and the field. The classical analysis (Fourier trans-
form of the step response) leading to Eq. (12) fails. Thus,
from a strict point of view, at high ac fields, Eq. (12) is not
valid.
Calculating the frequency response of nonlinear systems
from their time response is a complex mathematical task.
14
Such a calculation was not attempted here. Only a qualitative
discussion will be given. As an extreme case, when eE
0
R
WkT (very high fields), Eq. (10) shows that s!s
0
(the
minimum possible value for s). Thus, sis independent of R.
Previously, at low fields, the pairs with a separation distances
R¼R(s¼1/x) had a major contribution to ac hopping (lead-
ing to s <1). Now, at very high fields, since sis independent
of R, all pairs equally contribute to ac hopping. This is the
source of the linear variation of G with x(s ¼1). It is inter-
esting to note that Eq. (12), even though it was derived at
low fields, is able to predict that if R becomes independent
of x, then G x(s ¼1).
In short, the source of ac nonlinearities is the following.
As the ac field increases, more and more pairs are involved
in ac hopping (not only the pairs satisfying Eq. (13)), and
s!1. The increase of the number of pairs participating in ac
hopping leads to an increase in bulk polarization, and C rises
(Fig. 2(b)).
Up to now, a second-order polynomial law (Eq. (1)) was
used to describe C(V
ac
). It is a matter of usual practice
and this polynomial law has no specific physical signifi-
cance. It is possible to describe C(V) by an exponential law,
see Fig. 6,
FIG. 5. Pair model (correlated barrier hopping): Trap levels 1 and 2, sepa-
rated by a distance R. Trap depth is W
M
, barrier between traps is W.
Electron transition rate from trap 1 (2) to trap 2 (1) is w
12
(w
21
).
084104-4 Khaldi et al. J. Appl. Phys. 116, 084104 (2014)
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DC
C0
¼a½exp bV
ðÞ
1;(14)
where for the ac case (Fig. 6(b)), a¼0.13 and b¼0.69 V
1
.
The exponential law acquires physical significance because
it is rooted in Eqs. (8)–(10), i.e., in barrier lowering by the
field. The 2nd order polynomial law simply appears as a
Taylor series expansion of the exponential law. At present, it
is difficult to give analytical expressions for the parameters a
and b. It would require an analytical model for ac hopping at
high fields.
4. Generalization to ALD-deposited high-k oxides
To check if the above results are specific to our HfO
2
films, we also carried out measurements on ZrO
2
films which
were grown in a different reactor and using different condi-
tions (10 nm thick ZrO
2
films, deposited by Plasma-
Enhanced ALD, using (C
5
H
5
)Zr[N(CH
3
)
2
]
3
as precursor and
direct O
2
plasma as reactant).
15
As for HfO
2
films, increasing
the ac voltage in the volt range leads to C increase in the
10% range (Fig. 7(a)).
Fig. 7(a) displays the capacitance at 0.1 V
rms
after the
C-f at 3 V
rms
has been recorded. The C-f characteristic is
similar to the one recorded for the virgin sample. This shows
that measurements up to 3 V
rms
(3 MV
rms
/cm), which last
several minutes, have no degradation effect on the samples.
Thus, any possible oxide degradation as the source of ac
nonlinearities is ruled out. In other words, ac nonlinearity is
fully reversible.
As for HfO
2
, ac nonlinearities in ALD-deposited ZrO
2
are linked to a change in the s exponent, which saturates to
s¼1 at high ac voltages (Fig. 7(b)). These results tend to
show that ac nonlinearities of high magnitude (a few 10%)
and their relation to ac hopping are general features of ALD-
grown high-k oxides.
C. dc nonlinearity
Instead of a pure ac field, we now consider the case of
a large dc field to which a small ac field is superimposed.
Under the influence of the large dc field, w
12
w
21
(Eqs. (8) and (9)), all carriers are stored in trap 2 and the
superimposed small ac field is not able to bring back car-
riers to trap 1. Thus, the mechanism described in Sec. III B
under pure ac field (charges oscillating between trap 1 and
trap 2) vanishes.
1. Electrode polarization related to hopping
conduction
Up to now, isolated and disconnected pairs were consid-
ered. However, percolation paths may exist (connected
pairs). The case of conduction paths extending across the ox-
ide thickness is now considered. Electronic charges are
drifted by the dc electric field along these paths. Depending
FIG. 6. Second-order polynomial law and exponential law for (a) dc nonli-
nearity and (b) ac nonlinearity. FIG. 7. (a) ac nonlinearities in ZrO
2
films and (b) s parameter extracted
from the conductance.
084104-5 Khaldi et al. J. Appl. Phys. 116, 084104 (2014)
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on the efficiency of charge transfer at electrodes, space
charge layers build-up at electrodes. "Blocking" electrodes
correspond to bad charge transfer at the oxide/metal interface
and lead to accumulation and depletion layers at the anode
and at the cathode ("bad" charge transfer means that charge
mobility across the interface is lower than charge mobility
along hopping conduction paths). At the opposite, "Ohmic"
electrodes correspond to free electronic transport across the
oxide/metal interface and no space charge is present. The
effect of the ac field, which is superimposed to the dc field,
is to modulate the electrode space charges, giving rise to a
supplementary capacitance. This is often termed "electrode
polarization." It is equivalent to a macroscopic dipole whose
charges are the depletion and accumulation layers and whose
length is equal to the oxide thickness.
Modeling of electrode polarization was the subject of
several works in the past.
16–18
In a previous publication,
4
we
demonstrated that electrode polarization can account for dc
nonlinearity in MIM capacitors. Briefly, we considered that
the dc bias leads to a field-enhanced conductivity,
r¼r0expðeEeff R=2kTÞ;(15)
where r
0
is the low-field dc conductivity and E
eff
is the
effective field which governs hopping along chains of
defects (meaning of "effective" field is discussed below).
Field-dependent conductivity, of the form given by Eq. (15),
is commonly observed for electronic hopping in transition
metal oxides.
19
It is a consequence of barrier lowering by the
electric field, see Eq. (8).
By inserting Eq. (15) in the model of Beaumont and
Jacobs
17
for electrode polarization in solids, we showed
that
4
DC
C0
¼2
e2n
L
LD

12n1
qþ2
ðÞ
21n
ðÞ
1
x2n
r2n
0exp neEef f R=kT

1

;(16)
where L is the oxide thickness, L
D
is the Debye length of
accumulation/depletion layers, n is an empirical parameter
which accounts for the variation of C with f (for hopping
transport, 2n ¼1s, where s is the exponent appearing in
Eq. (6)). Parameter qis the "blocking parameter" which
measures charge exchange at the electrode. For "ohmic" con-
tacts, q!1, there are not any space charge layers and elec-
trode polarization vanishes. For "blocking" contacts, q!0,
large space charge layers exist and electrode polarization
reaches a maximum value.
The exponential dependence of DC/C
0
on V, as pre-
dicted by Eq. (16), is verified experimentally (see Fig. 6(a)).
Using the general form as given by Eq. (14), we find
b¼1.7 V
1
(and a¼9.4 10
4
). If the electric field is taken
as the macroscopic field, E
eff
¼V
dc
/L, then b¼(neR)/(LkT).
Since s ¼0.80 (see Fig. 4(a), low ac fields), then n ¼(1 s)/
2¼0.10. Then, the hopping distance is calculated to be
R¼44 A
˚. Such a value is very large when compared to
interatomic distances, and therefore it appears unreasonable.
This problem was underlined and discussed by Austin and
Sayer
19
who found hopping distances of the same order of
magnitude in several transition metal oxides. According to
these authors, large values of R stem from under-estimating
the electric field. They postulated that the field appearing in
Eq. (15) is not the Maxwell macroscopic field (V/L), but an
"effective" field which results from fluctuations in energy
barriers along the percolation path. Indeed, small fluctuations
in local order lead to fluctuations in energy barrier. Pairs
with highest energy barriers are those offering nodes with
maximum resistance, i.e., nodes where the voltage drop is
the highest, so does the field. Therefore, the electric field is
not constant along the percolation path, being enhanced at
pairs of highest energy barrier. For instance, in transition
metal oxides, Austin and Sayer calculated field enhancement
factors (p) varying between 4 and 17.
19
It means that the
"effective" (local) field is ptimes higher than the average
field,
Eeff pxVdc=L;(17)
where pis around 10. Using p10, it is found R 4A
˚,
which is now consistent with interatomic distances.
2. Oxygen vacancies as traps
The nature of the traps will now be discussed. For such
a purpose, we refer to our previous study
11
where the activa-
tion energy of dc conductivity was measured at low fields
(V ¼0.1 V). The activation energy of the dc conduction was
found to be non-constant with temperature (the slope of rvs.
1/T is non-linear). This is consistent with a distribution of
energy barriers. At room temperature, the activation energy
was around 0.3 eV.
11
An activation energy around 0.3 eV
was independently reported by other groups.
20,21
Since this
activation energy was reported for low fields, it corresponds
to R
min
(the shortest hopping distance is the one offering the
lowest barrier, see Eq. (7), i.e., the favored path for dc con-
duction at low fields). Oxygen vacancies are common defects
in oxides. The distance between nearest-neighbored oxygen
vacancies is around 3 A
˚in HfO
2
.
22
From Eq. (7), taking
W¼0.3 eV and R
min
¼3A
˚, one gets W
M
2 eV. Such a trap
depth is in agreement with levels related to oxygen vacancies
(quoted around 2 eV for V
O
0
[Ref. 23], and in the
1.7 eV–2.7 eV range for V
Oþ
[Ref. 24]). Therefore, oxygen
vacancies could be the traps that are involved in hopping
conduction.
3. Electrode polarization vs. isolated pair polarization
Electrode polarization must also be considered for ac
nonlinearity (pure ac voltage, Sec. III B). However, it
involves conduction paths extending across the oxide
thickness. Such percolation paths are expected to involve a
small fraction of the total amount of defect pairs.
Therefore, under pure ac fields the polarization at isolated
pairs dominates (mechanism described in Sec. III B). As
reminded at the beginning of this section, when a large dc
voltage is superimposed to the ac voltage, polarization at
isolated pairs vanishes and the electrode polarization (of
much smaller magnitude) emerges. This is schematized in
Fig. 8.
084104-6 Khaldi et al. J. Appl. Phys. 116, 084104 (2014)
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IV. CONCLUSIONS
Capacitance nonlinearities, i.e., the variation of C with
V, were studied in MIM structures fabricated from ALD de-
posited HfO
2
thin films (10 nm). Usually, nonlinearities are
tested with a dc bias.
1–8
Here, we used two types of signals:
a small ac voltage superimposed to a large dc bias (this is
the usual configuration, used for the purpose of testing “dc
nonlinearities”), and a pure ac voltage of large magnitude
(for the purpose of testing “ac nonlinearities”). The goal
was to compare the MIM capacitor response under dc and
ac conditions. This is an important reliability issue for
applications for which MIM devices can be subjected to
both types of signals. In theory, ac and dc nonlinearities
should be of the same order of magnitude, with the ac one
being 1
=
2of the dc one. This is true if ac and dc nonlinear-
ities originate from the same polarization mechanism. In
practice, ac nonlinearities are found to be an order of mag-
nitude higher than dc nonlinearities (artefacts which would
have been related to the electrical equipment have been
ruled out).
Large differences between ac and dc nonlinearities
point out that different polarization mechanisms coexist. In
order to identify microscopic mechanisms, low-frequency
impedance spectroscopy was carried out. The study was
focusedontheG-xcharacteristics. At low ac electric
fields, we got G x
s
,wheres¼0.8. This is typical of hop-
ping conduction.
12
Within the CBH model,
12
electron hop-
ping proceeds through isolated defect pairs with a random
distribution of defect separation distance (R). A given x
corresponds to a given value of R, i.e., only a fraction of
the pairs participate in hopping (this is the origin of the
sub-linear dependence of G on x).Whentheacfieldis
increased in the MV/cm range, the hopping exponent (s)
progressively increases to a value close to 1 (s 1). The
departure from the value s ¼0.8 is clearly correlated with
the onset of ac nonlinearities, suggesting that both phe-
nomena have the same origin. The rise of the s parameter
is thought to originate from an increase in the number of
pairs which participate in ac hopping. Since each (isolated)
pair behaves as a microscopic oscillating dipole, the over-
all bulk polarizability increases, and C rises (source of ac
nonlinearities).
The source of dc nonlinearities is electrode polarization.
Such a mechanism refers to the modulation of electrode
space charge regions (by the small ac test signal which is
superimposed to the dc bias). This process requires dc con-
duction paths from one electrode to the other, i.e., connected
pairs that lead to percolation paths through the film thickness
(grain boundaries for instance). It differs from the preceding
ac polarization mechanism (CBH) which involved isolated
pairs. When applying a large dc field, ac polarization at iso-
lated pairs vanishes (because dipole orientation is "locked-
on" by the dc field). As a consequence, only the electrode
polarization mechanism remains and can be observed.
Modelling of dc nonlinearities is carried out using theories
of electrode polarization,
17
taking into account a field-
dependent dc hopping conduction, which includes a local
field correction.
19
The capacitance-voltage relations can be
described by an exponential law (in both the dc and ac
modes). This is physically related to a lowering of the hop-
ping barrier at large electric fields. Such an exponential law
has to be contrasted with a second-order polynomial law,
which is often used to describe C-V characteristics.
Though different polarization mechanisms are at the ori-
gin of ac and dc nonlinearities, they are both related to elec-
tronic hopping at oxygen vacancy defects. The related trap
depth is around 2 eV, which is consistent with levels reported
for oxygen vacancies in HfO
2
.
23,24
Large differences
between ac and dc nonlinearities are also reported for ZrO
2
thin films (which have been grown in a different ALD equip-
ment). This suggests that the behavior is general to high-k
transition metal oxides, which are slightly oxygen deficient.
Finally, from the application point of view, the present
study demonstrates that it is important to perform ac tests for
evaluating the voltage linearity of high-k MIM capacitors.
Indeed, the usual dc bias test may largely underestimate non-
linearities which originate from the ac part of the voltage.
This could become a serious reliability issue for applications
where harmonics of large magnitude are present.
ACKNOWLEDGMENTS
R
egion Rh^
one-Alpes is gratefully acknowledged for
financial support ("CMIRA" France-Tunisia mobility
program and "Micro-Nano" research program). This work
was partly supported by "Laboratoire d’Excellence" MINOS
(project ANR-10-LABX-55-01).
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132.168.89.68 On: Tue, 21 Oct 2014 10:55:34