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SEMI MF1771-0304 © SEMI 2003, 2004 7 all corrections indicated in Equation 3 must be take n into account. Figure 2 Determination of Noise Threshold Current Level, Slope Ratio Applicability, and Voltage Step Time Variatio…
SEMI MF1771-0304 © SEMI 2003, 2004 6
5. Divide the current value measured at the previous
point by the capacitor area, and record this value as the
hard failure current density.
9.10 After hard failure has been detected by one of the
criteria in Section 9.9 or the upper test voltage limit has
been reached, perform a post-test using the same
criteria defined for the pretest in Section 9.4.1. This is
to be done whether a pretest was elected in Section 9.4
or not. If the unit fails the post-test conditions, modify
the failure category in some way to reflect this
observation. For example, one may change the sign of
the failure category number, or add an asterisk.
9.11 Proceed to the next device to be tested, and repeat
Sections 9.3–9.10 until all devices are tested.
10 Calculations
10.1 Current and Current Density — To calculate
current (I) from a current density (J), multiply the
current density by the area of gate contact (A) as
follows:
AJI ×= (1)
NOTE 3: Example: Given a current density (J) of 1 µA/cm
2
and a gate area (A) of 0.08 cm
2
, the current would be 80 nA.
10.1.1 Similarly, compute current density (J in A/cm
2
)
from measured current (I in A) and area (A in cm
2
)
using
A
I
J = (2)
10.2 Oxide Voltage and Electric Field Strength as a
Function of Applied Voltage — Since ramp voltage test
increments and ramp rates as well as some breakdown
voltage data analyses are specified in terms of oxide
electric field strength, it is important to consider the
conversion of applied voltages to oxide voltage and
electric field values for these two cases. Historically,
oxide electric field strength has been approximated
simply by dividing the applied voltage by the thickness
of the oxide. This approximation is in error because it
neglects the offset in the zero values of applied and
oxide voltage brought about by the work function
difference between gate electrode and silicon substrate,
and the voltages dropped across the substrate (and the
gate electrode, if it is non-metallic) when the capacitor
is strongly biased during breakdown testing. These
voltage drops are made up of two components; one
associated with band bending at the interfaces with the
oxide that establishes the high fields required for
testing, and another involving additional voltage drops
due to series resistances that become significant at very
high currents. These corrections, which together
normally range up to 1 to 2 V in magnitude, might
reasonably be neglected for samples with oxides greater
than 20-nm thick, as the correction amounted to only a
few percent of the breakdown voltage values. For
thinner oxides, these additional voltage components
must be taken into account.
10.2.1 The voltage V
app
applied across an MOS
capacitor with a given gate-substrate work function
difference Φ
ms
and oxide fixed charge Q
f
may be
expressed as follows:
gatesubms
ox
f
oxapp
VV
C
Q
VV ++
Φ+−= (3)
where:
V
ox
= voltage across the oxide, V,
C
ox
= oxide capacitance, F/cm
2
,
V
sub
= voltage across the substrate, V,
V
gate
= voltage across the gate electrode (arising
from polysilicon depletion or series
resistance, V.
10.2.2 At zero volts applied, the offset due to the work
function difference and oxide charge appears
predominantly across the oxide. For setting a voltage
ramp rate, the incremental change in V
ox
with changing
V
app
is not affected by this offset, but may be decreased
by voltage increases across the silicon substrate or the
gate electrode due to band bending and series resistance
drops. For a sample in which the applied bias voltage
polarity accumulates the substrate, and the gate
electrode is metallic or of the opposite conductivity
type from the substrate (for example, p-type silicon
substrate and n+ polysilicon gate), these effects are
appreciable only for very small and very large applied
voltages. At low bias, silicon bands bend until the
surface becomes degenerate, after which the rate of
band bending becomes very low. At high biases,
significant resistive voltage drops may develop if the
test structure design is not optimized. Over the largest
portion of the test in which neither of these effects are
large, ramp rates may be computed assuming that
voltage increments applied to the device under test
appear completely across the oxide, and ∆E
ox
, the
electric field increment across the oxide, is given as
follows:
ox
app
ox
W
V
E
∆
=∆ (4)
where:
W
ox
= oxide thickness, cm.
10.2.3 For computations of oxide field strength
associated with the various hard and soft failure criteria,
SEMI MF1771-0304 © SEMI 2003, 2004 7
all corrections indicated in Equation 3 must be taken
into account.
Figure 2
Determination of Noise Threshold Current Level,
Slope Ratio Applicability, and Voltage Step Time
Variations
10.3 Determination of Noise Threshold Current Level
and ln J-V Slope Ratio Specification — Useable values
of noise threshold current level and ln J-V slope ratio
depend upon properties of the test system as well as the
samples to be evaluated, and a preliminary test may be
required in order to specify them properly. To do this,
it is necessary to store all the current-voltage data pairs
as well as the incremental voltage step time readings in
data arrays. This is not required for performance of the
bulk of the testing, but is sometimes useful for more
thorough analysis of the test results. When such data is
stored, it is possible to construct a figure like that
shown in Figure 2, which is a semi-logarithmic plot of
sample current, step time, and slope ratio versus applied
voltage. Data given here was taken on a 3-nm oxide on
a p-type silicon substrate, so the current and voltage
values are magnitudes of negative readings. Both direct
and Fowler-Nordheim tunneling components are seen
in the I-V data.
10.3.1 To determine the noise threshold current level,
examine the ln J-V slope ratio, shown as open triangles
in Figure 2. It is seen to be quite noisy for this test
system and sample up to a current level just below 1
nA. This value, 1 nA, is thus a good choice for noise
threshold current level for this test. Further analysis of
the slope ratio data shows that its maximum value in the
range from 1 nA up to catastrophic failure is 1.21, so
the standard specified ratio value of three (3) is
adequate to avoid significant noise interference.
10.3.2 The step time-voltage dependence plotted as
light squares in Figure 2 illustrates shortcomings in the
step time control for this configuration of the
measurement system. A voltage step time of 0.1 s is
specified for this test method in Sections 5.3 and 9.7.
Because of the high, time-dependent currents measured
below 0.5 V, there is a regime below 1 V where this is
observed, but most readings from there up to a current
value of 0.1 µA fall in the range 0.2 to 0.4 s. Also, for
higher current values, there are singular points at
electrometer range changes where 0.2-s delays are
encountered. It has been shown (Klema
4
) that
deviations of this amount do not have large effects on
breakdown voltage distributions, but these extended
delays affect total measurement time, and it would be
worthwhile to eliminate them if possible. Approaches
depend upon measurement hardware being used, but
trading off low current resolution for electrometer
autoranging time and look-ahead range changing code
are possibilities.
10.4 ln J-V Slope Hard Failure Criterion — Hard
breakdown failure of oxides is increasingly difficult to
detect as oxide thickness decreases below 10 nm.
Fowler-Nordheim emission, the dominant current
transport mechanism for SiO
2
films in this thickness
range, predicts that at a given field strength, the
logarithmic slope of the J-V characteristic increases for
decreasing film thickness, but decreases with increasing
field strength for a given oxide thickness. Thus, a
change in ln J-V slope may be a more sensitive detector
of failure for these very thin films, where high failure
current density, low oxide impedance at failure, and
high voltage drops in series resistances might lead to
very small current changes when the oxide ruptures.
10.4.1 Experience with this failure criterion for oxide
thicknesses ranging down to 3 nm indicates that a
change by a factor of 3 provides good detection of
failure while remaining above the noise level in the
data. Users may verify this condition for their
particular sample and test conditions, and change the
failure factor accordingly. Any such change must be
agreed upon by the parties to the test, and clearly
identified in the report of the data.
10.4.2 In order to minimize noise in the calculated
values and optimize the sensitivity of the failure
detection, it has been found advisable to use a set of
five data points to calculate the established and new
values of the ln J-V slope. This illustrated in Figure 3,
which shows the last few data points in the ramp
voltage test of a 50-nm oxide. Data points are spaced at
0.1 MV/cm increments, as dictated by this test method.
10.4.3 In particular, the last five data points are labeled
(V(n), I(n)) through (V(n – 5), I(n – 5), respectively.
The established logarithmic slope is as follows:
4 Klema, J., “Ramp Rate Effect on Dielectric Breakdown,” Final
Report, IEEE International Integrated Reliability Workshop (IRW),
IEEE Electron Devices Society, 1989, p. 87.
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()
()
()( )()
−−−
−
−
51
5
1
absln
abs
nVnV
nI
nI
(5)
while the new slope is computed from the last two
points,
()
()
() ( )()
−−
−
1
1
absln
abs
nVnV
nI
nI
(6)
10.4.4 Testing for the failure criterion is done by taking
the ratio of Equation 6 to Equation 5. As can be seen
from Figure 3, this failure would be identified by failure
criteria Sections 9.9.2 and 9.9.3, as well as the slope
criterion Section 9.9.5.
Figure 3
Calculation of In J-V Slope Ratio Failure
Criterion for a p-Type Sample with 50-nm Gate
Oxide
10.5 Defect Density — To calculate a defect density, a
minimum criterion must be chosen by either the user or
negotiated with the user' s customer. This criterion can
take the form of a minimum breakdown voltage or
electric field strength, or a discontinuity in the
breakdown voltage distribution of the sample. Given
the fraction of devices reaching this criterion, the defect
density calculation may be based on a Poisson
relationship (see Standard 35) using the following
equation:
)exp( ADY −= (7)
where:
Y = yield of good units in terms of the defined
failure criterion,
A = area of sample, cm
2
, and
D = defect density, defects/cm
2
.
NOTE 4: Example: Given a total of 100 devices tested with
87 devices passing the minimum criterion for success and a
gate area of 0.08 cm
2
, the defect density would be as follows:
2
defects/cm7.1
08.0
)100/87ln(
==D
(8)
10.5.1 An undefined condition results if the number of
successes is zero. It may be necessary to change the
area of the test capacitor chosen for testing in order to
resolve meaningful defect densities. Figure 4 shows the
relationship between defect density and test capacitor
area required for resolution in terms of a minimum of
10% good or defective units in the sample.
NOTE 5: For example, a test capacitor with an area of 0.1
cm
2
can resolve defect densities between 1 and 25
defects/cm
2
, with 10 and 90% defective samples.
Figure 4
Test Capacitor Area Required to Resolve Various
Oxide Defect Densities, Assuming Poisson Statistics
10.6 Weibull Distributions — To convert cumulative
percentages to Weibull format (sometimes referred to
as “smallest extreme value probability distribution III”),
use the following equation:
(
)
)1ln(ln F
−
−
(9)
where ln is the natural log operator and F is the fraction
of accumulated failures. Care should be taken so that F
is never exactly 1 since this results in an undefined
situation.