Theory of “Current-Ratio” Method For Oxide Reliability: Proposal and
Validation of a New Class Two-Dimensional Breakdown-Spot
Characterization Techniques
M. A. Alam#1, D. Monroe#2, B. E. Weir#2, and P.J. Silverman#2
ECE Department, Purdue University W. Lafayette, IN-47906 (765-494-5988/[email protected])
#2
Agere Systems, Allentown, PA18109
#1
Abstract
A theory of the Current-Ratio technique, which is
widely used to locate gate oxide breakdown spots in one
dimension (i.e., distance from source or drain), is
proposed and verified. The theory shows that the
Current-Ratio method is a special case of generalized
van der Pauw technique, and as such, can easily be
generalized to locate oxide breakdown spots in two
dimensions. We develop the theoretical framework of
this new class of breakdown-spot characterization
techniques and then validate the theory by experiments.
We conclude by discussing the implications of locating
breakdown spots in two dimensions for reliability
projections of ultra-thin gate oxides.
Introduction
The 1D “current-ratio” technique (1D-CR) has been
widely used to locate breakdown spots (BD-spots) in
thin gate dielectrics and to interpret oxide reliability [14]. Yet surprisingly, no one has justified the technique
theoretically. As shown in Fig. 1(a) the idea of 1D-CR
is simple [1]: once an oxide develops a soft breakdown
spot with only a localized increase in gate current (IG),
the device can be biased in accumulation such that IG is
balanced by source (IS) and drain (ID) currents. And the
ratio of source to drain-current locates the BD-spot, i.e.,
x/L=ID/(IS+ID) .
(1)
Fig. 1(a): Side view of an NFET biased in accumulation.
IG is injected at the BD-point x. (b): n1 is the hole density
at the BD-spot and S/D are perfect sinks. The slopes of
the carrier concentrations determine IS and ID. (c, d) Top
view of the oxide (dashed regions are S/D). Two
different BD-spot distributions are indistinguishable by
1D-CR (same x, but different y), yet would have very
different soft-BD lifetimes ([4], see Fig. 11 for reliability
implications).
Yet, the prescription does not tell us (a) If the formula is
exact, (b) If it is valid in inversion, (c) If it is limited to
1-D, and if not, if it can be generalized. This 2D
generalization is crucial because 1D-CR will lead one to
incorrectly conclude that pair distances between BDspots 1,2,3 in Figs. 1c and 1d are identical (1D-CR
measures x only!), although BD-spots in 1c is
significantly more localized (therefore has reduced
TDDB lifetime [2-4]) than those in Fig. 1d. The theory
proposed here provides the first interpretation of 1D-CR,
and helps develop several 2D techniques to determine
both x & y coordinates of SBD-spots.
Theory of the existing technique
Since J ( x, y ) = qnun ∇φ + qDn ∇n and ∇ • J ( x, y ) = 0 ,
nun ∇2φ + Dn ∇2 n = 0 . (2)
In accumulation (Fig. 1a), the electric field in small
(except near the source and drain) and minority (hole)
diffusion dominates, so that
(3)
Dn ∇2 n = 0 .
Fig. 1b shows that the solution of Eq. (3) in 1D is given
by , IS=qADnn1/x and ID=qADnn1/(L-x) and their ratio
satisfies Eq. 1. This derivation is simple, exact, and
wrong! Fig. 2(a) explains why: the 1D derivation
assumes line injection (the side-view of Fig. 1a is
misleading), while calculation of current from a BDspot requires a 2D solution (Fig. 2b). Surprisingly,
however, both the Monte-Carlo solution of Eq. 3
(a)
(b)
(d)
(c)
Fig. 2: Top view of oxide: (a) 1D analysis of CR method
assumes a line charge, while (b) actual diffusion from the
BD-point to S/D (dashed region) is 2D. However, 2Ddiffusion (c) can be mapped onto 1D diffusion with
variable delay (d) – explaining why the 1D solution of CR
(a) gives the exact result for the 2D problem.
Before we verify Eq. (5) experimentally, note that
instead of determining the VD (VS) by injecting IS (ID) at
each x, the Reciprocity Theorem allows determination
of all VD(x) in a single calculation with I injected at the
drain (source) to determine V(x)=VD(x) [red line with
filled square]. Later, we will use this theorem to
calculate the potential induced in 2D geometry.
Fig. 3: Both the Monte-Carlo method (diffusion
simulated by following random walk of particles in 2D
and 3D from BD- point in 2D) and the Image Charge
method (multiple reflections on S/D from a point
charge) confirm 1D-CR (Eq. 1).
(computed by injecting particles at each point (x,y) and
counting the number of particles collected by the source
and drain) as well as the image charge solution of Eq. 3
(determined by using the equivalence of diffusion
equation and Laplace equation and then obtaining the
charge induced in source and drain by iterated images of
a point charge placed at (x,y) ) in 2D and 3D still agree
with Eq. (1) (see Fig. 3). This paradox is resolved by
realizing that a particle injected from a BD-spot and
collected, through diffusion, by two infinite parallel
plates (S/D contacts) in any dimension can be
transformed into an equivalent 1D problem (Fig. 2c): as
far as progress toward the electrode is concerned,
motion in the y-direction is indistinguishable from no
motion. Therefore, y-motion corresponds to a delay for
particle collection. Thus the 1D-CR technique is exact
in any dimension, for infinite electrodes. Fig. 2a gives
the correct result because each point of the injection line
satisfies the 1D-CR individually.
Fig. 5 shows that locations of BD spots, determined by
the standard 1D-CR technique and 1D-VR technique
(open circuit voltage ratio technique) just developed, are
identical for all gate voltages beyond threshold. The VGindependence (above threshold) reflects the fact that it is
the ratio of open-circuit voltages (in 1D-VR, Eq. 5) or
the ratio of short-circuit currents (in 1D-CR, Eq. 1),
rather than their absolute values, that determine the
location of BD-spots. Below threshold (~0.5V), the
agreement between the techniques is not as good,
Fig. 4: With BD-spot located at x and the gate held at a
certain bias, the source grounded, and the drain floating
(open-circuited), the potential profile in the channel is
given by the solution of Eq. 4 (black line with open
circles). The floating drain voltage, VD, equals that of the
BD-spot, because no current flows between drain and BDspot. The Reciprocity Theorem provides an efficient
alternate method to calculate the VD(x) [red filled squares].
Generalization to inversion
With the theory of 1D-CR established, we now
generalize the technique to inversion, where Eq. 2
simplifies to
(4)
µn n( d 2φ / dx 2 ) = 0
because inversion is drift-dominated and the diffusion
component is negligible. If one monitors Drain opencircuit voltage VD with IS flowing between BD-spot and
Source, then VD=αISx/LC because no current flows
between drain and BD-spot and the line joining them
must have the same potential (black line with circles in
Fig. 4). A similar measurement of current flow between
drain/BD-spot determines the floating source voltage,
VS=αID(LC-x)/LC. The ratio eliminates the constant α
and locates the BD-spot, i.e.,
x/L=VDID/(VDID+VSIS).
(5)
Fig. 5: Locations determined by 1D-CR and 1D-VR
techniques give identical results indicating equivalence
of the techniques.
Fig. 6: BD-spots located by 1D-CR and 1D-VR agree
within the margin of error (as predicted by the theory).
Results from 5 different devices are shown.
because subthreshold transport is diffusion-dominated
and 1D-VR can not be used. Fig. 6 summarizes the
measurements for several different devices, indicating
that in all cases the measured locations of BD-spots by
these two techniques are essentially identical.
Fig. 7: Four different test structures to locate (x, y)
positions of a BD-spot (red square). (a) Tester 7a adds
y-contacts (probes 3&4) to two standard x-contacts in
1D-CR, (b) Tester 7b simplifies Tester 7a by reducing
probes 3 & 4 to point-probes). (c) The four point-contacts
of Tester 7c are located at the corners. (d) Tester 7d
indicates, like a general 4P-VDP technique, that the BDspot can be located by arbitrarily-placed probes.
Generalization beyond 1D
Fig. 7(a-d) shows four different test structures, any of
which can be used to localize a BD-spot (indicated by
red square) in 2D and they are all conformationally
equivalent to 5-probe van der Pauw (5P-VDP) [5-7]).
The theory of each of the testers is discussed below,
because some may be easier to implement than others.
Our measurements are based on test structure 2 (Fig. 7b).
For the first test structure in Fig. 7(a), two “S/D”
contacts in the y-direction allows numbering of probes
(P) by respective current density, J3>J1>J2>J4. The BDcoordinates, x=0.5+[0.5J2/(J1+J2)]0.5 & y=[0.252(J1+J2)(0.25-x2)]0.5, are obtained by solving Eq. 4 (see
Fig. 8 for validation by Monte-Carlo and [8,9] for
special cases). One might wonder if measurement of
only two currents, J1 and J2, is sufficient to ‘triangulate’
a point in 2D. The key to this puzzle is that J1 and J2
can be specified only if J3 and J4 are also measured,
which provides the additional required information for
‘triangulation’ of the BD-spot.
J1 ≅
(1 − y 2 )
(1 + x − x 2 )
(1 − x 2 )
J2 ≅
(1 − y 2 )
(1 − x + x 2 )
(1 − x 2 )
The test-structure in Fig. 7b simplifies 7a by reducing
the top/bottom “S/D” contacts to point-probes. The xposition is obtained by 1D-CR or 1D-VR (with P3 & P4
floating), and R3,4=y/W=(V3-V4)/2(V3+V4). Fig. 9
provides numerical (Image Charge) and experimental
verification.
The test structure in Fig. 7c reduces all contacts to
point-probes. Measurement of V1,3 (P2 & P4 sinking
current from BD-spot) and V1,2 (P3 & P4 current sinks)
locates the 2D-BD spot (Fig. 10).
Finally in Fig. 7d, k=V2,3/ρ I. ρ is measured, before
breakdown, by 4P-VPD. After breakdown, V2,3 is
Fig. 8: J1 and J2 for test structure in Fig. 7a are plotted
along y for fixed x (symbols: Monte Carlo, solid lines:
analytical solution (Eq. in inset). Note that by
symmetry, the solution needs to be plotted in only the
triangular region of Fig. 7a. Therefore, in the plots,
initial y-values increase with increasing x.
two BD-spots are uncorrelated in 2D, their distance
PDF p(l) is shown in Fig. 11 [10,11]. Any deviation
between theoretical and measured p(l), by any of the
techniques discussed in Fig. 7, would establish the
degree of 2D-correlation of BD-spots, with
corresponding modification of post-SBD oxide lifetime.
p (l ) = 4l [1 − (0.5 + 0.25π )l + l 2 / 3]
Fig. 9: 2D-position measured by test structure 7(b).
Solution of left half of the test-structure is plotted.
1D-VR based on P1, P2, & BD-point determines
BD-spot (dotted lines: V2=VX=0:0.05:0.5. The
induced voltage between P3 & P4 determines the ylocation
(solid
lines:
R3,4=-0.5:0.033:0.5.
Measurement of locations of three BD-spots by two
methods (open circles: VR, closed-box: CR)
coincide, which validates the technique.
0< l <1
p ( l ) 2.5( 2 − l ) 3
for 1 < l < 2
Fig. 11: The approximate analytical PDF of pair
distance l2=(xi-xj)2+(yi-yj)2 is supported by exact
numerical calculation. (xi,yi) are locations of i-th BDspot (see Fig. 1c and 1d). The 2D-location
determination techniques discussed in Figs. 7-10 can
be used to explore deviation from this PDF to establish
2D-correlation.
Summary
Fig. 10: 2D-position determination by test structure
7(c). (solid-lines): P1 & P3 are V-probes and P2 & P4
are I-sinks from BD-point. (dashed-line): P1 & P3 are
V-probes and P2 & P4 I-probe provides 2nd
measurement. Contour values indicate the location of
the BD-spot for measured V13 and V24.
measured between P2 & P3 and I between the BD-spot
and P1. Since k is a known function of the locations of
P1, P2 & P3 [5-7], the BD-spot is uniquely determined.
Reliability implications
The lifetime after SBD depends on the spatial and
temporal correlation of the BD-spots [3,4]. We have
previously shown that BD-spots are temporally
uncorrelated [3,4], but spatial decorrelation could be
proved, by 1D-CR, only in 1D. This does not exclude
the possibility of 2D correlation (Fig. 1c vs. 1d). The
“5P-VPD” methods [7a-d] offer a new class of
techniques to detect such correlation. Theoretically, if
We have provided the first theory of position
measurement by the current-ratio method, which
resolves paradoxes and establishes its validity on a par
with SILC, CP, etc. Our theory allows generalization of
the 1D-CR to inversion and to 2D (four techniques, all
conformationally equivalent to 5P-VPD, are proposed).
This new class of techniques will help interpret BD-spot
distributions in 2D to better predict gate oxide reliability.
Acknowledgement: M. Alam gratefully acknowledges
financial support from NCN (#671-1211-4533) and
NSF/SRC (#2004-HJ-1238).
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x2 )