Instruction


FACTA UNIVERSITATIS  
Series: Electronics and Energetics Vol. 28, N

o
 2, June 2015, pp. 177 - 192 

DOI: 10.2298/FUEE1502177S 

STUDY ON THE SPATIAL GENERATION 

OF BREAKDOWN SPOTS IN MIM CAPACITORS 

WITH DIFFERENT ASPECT RATIOS  

Xavier Saura
1
, Michele Riccio

2
, Giuseppe De Falco

2
, Jordi Suñé

1
, 

Andrea Irace
2
, Enrique Miranda

1
  

1
Departament d’Enginyeria Electrònica,  

Universitat Autònoma de Barcelona, Barcelona, Spain 
2
Department of Electrical Engineering and Information Technologies,  

University of Naples Federico II, Naples, Italy 

Abstract. Metal-insulator-metal (MIM) large area (>10
-4

 cm
2
) capacitors with different 

aspect ratios were subjected to severe stress conditions (Eox>4-5 MV/cm) with the aim 

of generating a large density of breakdown spots (from 10
5
 to 10

6
 spots/cm

2
) in the 

same device. The resulting mark pattern on the top metal electrode associated with the 

failure events was analyzed first using conventional functional estimators for two-

dimensional spatial statistics. Second, as a double check, the attention was focused on the 

same breakdown spot patterns but in relation to the probe point location. In this latter 

case, the objective was to rule out any stochastic dependence of the breakdown spot 

distribution on the position of the source of degradation and therefore to confirm whether 

or not the spots follow a complete spatial randomness (CSR) process. In order to simplify 

the mathematical treatment of the point-to-event distributions, the voltage probe was 

assumed to be located at one corner of the observation window which significantly 

reduces the number of cases to analyze. Infrared images revealed that the generation of 

the spots is associated with micro-explosions within the insulating material (HfO2) and 

with the local volatilization of the top metal electrode (Pt). 

Key words: oxide breakdown, oxide reliability, infrared thermography, HfO2 

1. INTRODUCTION 

Failure analysis of thin dielectric films in metal-insulator-metal (MIM) and metal-

insulator-semiconductor (MIS) devices consists in the application of controlled electrical 

stress and the determination of lifetime estimates using accelerated degradation tests 

[1,2]. In order to achieve good estimates a large number of devices need to be subjected to 

identical stressing conditions until a failure event is detected according to a pre-established 

criterion (current or voltage jump, progressive leakage current increase, anomalous noise 

                                                           

 Received January 5, 2015 

Corresponding author: Enrique Miranda 

Departament d’Enginyeria Electrònica, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallés, 
Barcelona, Spain  

(e-mail: Enrique.Miranda@uab.cat) 



178 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

increment in the measurements, etc.). In this way, the time-to-breakdown (BD) statistics 

using multiple units is constructed. However, the fact that each device provides a single 

data point to the sampling distributions is a limiting factor for data collection and a time-

consuming process which in general requires careful planning and execution. Fortunately, 

it has been demonstrated that in certain cases more than one failure event per device can 

be generated and detected by means of electrical methods, mainly using constant voltage 

stress (CVS) tests. The successive BD statistics has been used not only to estimate the 

lifetime of devices but also the temporal and spatial correlations among the failure events 

[3]. In [4], the successive failure event statistics was obtained assuming Poisson area 

scaling for large virtual area devices and the weakest link property of dielectric BD. 

Although this procedure allows examining large data sets (with a single BD event per 

device) and assessing their order statistics, the method cannot be considered a direct 

proof of uncorrelated events. On the other hand, in this work, the attention is particularly 

focused on the spatial correlation of multiple failure events occurring in the same device. 

The temporal correlation of the BD events was investigated in [5] and it was shown to be 

consistent with a homogeneous Poisson process. 

From the microscopic viewpoint, it is well known that during electrical stress, defects or 

traps are generated within the dielectric film which ultimately leads to the creation of a 

filamentary pathway spanning the oxide layer [6]. The formation of this path and the 

consequent leakage current increase is identified as a BD event. If the occurrence of this 

event is sufficiently energetic, the damage becomes visible on the top electrode as a mark 

associated with the local volatilization of the metal layer. Since during the application of 

CVS the degradation of the device does not stop after the occurrence of the first BD event, 

it is possible therefore to generate more than a single mark per device. At the outset, the BD 

spot generation rate is governed by the magnitude of the stress voltage but as the 

degradation proceeds, it becomes limited by the interplay of the current that flows through 

the parallel BD paths and the voltage drop across the series resistance associated with the 

device itself or with the measurement setup. When the oxide voltage reduces, the generation 

of spots ends. The final result is a visible point pattern on the top electrode material that can 

be analyzed using the methods of two-dimensional (2-D) spatial statistics [7,8]. Functional 

estimators and point-to-event distributions are used then to assess the statistical properties 

of these failure events. In addition, in order to illustrate the origin of the BD spot pattern, 

we used the transient infrared (IR) thermography characterization technique, which is based 

on a real-time mapping of the thermal activity of the device during the degradation stage. 

Interestingly, the methods described here are particularly relevant for devices exhibiting the 

resistive switching (RS) effect, which basically consists in the formation and rupture of a 

single or multiple conduction channels spanning the dielectric layer in a MIM structure [9]. 

In this regard, several papers have reported the appearance of degradation patterns on the 

top metal electrode of the structures during the switching process arising from the out-

diffusion of oxygen ions from the oxide layer [9,10]. Although this subject is out of the 

scope of this paper, it is worth mentioning that multifilamentary RS is currently being 

considered for multilevel memory devices and could represent a breakthrough in the field of 

large capacity information storage systems [11].  

This paper is organized as follows: in Section II, the samples under investigation and 

the experimental setup used to characterize the IR emission during the multiple dielectric 

breakdown process are described. In Section III, the generation of the BD spot pattern 

and the associated thermal effects are discussed. In Section IV, the spatial distribution of 



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 179 

the failure events is investigated using conventional functional estimators for 2-D spatial 

point processes. In all the cases, the estimated curves are accompanied by the corresponding 

95% confidence bands. In Section V, the same BD spot patterns are examined using 

distance and angular point-to-event probability distributions for rectangular area 

observation windows. In this case, the probe point location is considered as the reference 

point from which the distances and angles to the BD events are computed. For the sake of 

mathematical simplicity, the point probe is considered to be located at one of the vertices of 

the observation window. Finally, in Section VI, the conclusions of this work are presented.  

2. SAMPLES AND EXPERIMENTAL SETUP 

The devices investigated were MIM capacitors fabricated according to the following 
steps: a 200 nm-thick thermal SiO2 layer was grown on a n-type Si (100) substrate, after 
which a Pt (200 nm-thick) layer was deposited by electron-beam (E-beam) evaporation. 
The samples were then placed in a Cambridge NanoTech Fiji atomic layer deposition 
(ALD) system where HfO2 (30 nm-thick) was deposited using TEMAHf precursor and 
H2O. The samples were then returned to the E-beam evaporator and a Pt layer (200 nm-
thick) was deposited on top of the HfO2 film. Lithography and a lift-off process were 

used to form arrays of rectangular capacitors of different sizes (400 x 400 m
2
, 400 x 100 

m
2
, 1600 x 100 m

2
). Access to the bottom Pt metal was enabled via a dry etching 

technique using a mask/resist process that removed the HfO2 to the bottom Pt metal while 
at the same time protected the top Pt metal of the patterned devices. The oxide extends 25 
µm beyond the perimeter edge of the top metal electrode. The devices were stressed 
using a HP4155 Semiconductor Parameter Analyzer with the bottom electrode grounded. 

A schematic of the experimental setup used to carry out the IR mappings discussed in 
Section III is illustrated in Fig. 1. The core of the system is the Merlin-MID IR camera 
together with an external synchronization logic block implemented on a FPGA digital 
circuit. The IR camera presents a focal plane array (FPA) of 320×256 pixels InSb sensor 
(30×30 µm

2
 pixel area) that guarantees a minimum temperature resolution of 25mK. The 

observation field is about 30 µm with a 1:1 optical magnification lens. The system is 
capable of detecting temperature distributions both in steady-state and transient conditions. 
In transient operation it is possible to use real-time and equivalent-time measurement 
modes. In the former case, the maximum sampling-rate is limited by the thermo-camera 
framerate to 50Hz. In the latter case, if the experiment can be repeated periodically, the 
system can reach an equivalent bandwidth of 100 kHz full-frame limited only by the 
minimum integration time of the FPA sensors (10µs). In this work, a direct method for 
transient characterization of the device temperature was used. The information provided by 
the system is subsequently edited using image processing software. More details about the 
IR mapping setup can be found in [12].  

To conclude this Section, it is worth pointing out that the statistical study reported in 
Sections IV and V was carried out using the Spatstat package for the R language [13]. 
This package supports creation, manipulation and plotting of point patterns, exploratory 
data analysis, simulation of point process models, parametric model fitting, as well as 
many other statistical tools (including over 1500 user-level functions). Spatstat can be 
downloaded for free from the R website [14]. 



180 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

 

Fig. 1 Schematic of the experimental IR setup with detail  

of connections between different system parts. 

3. GENERATION OF THE BD SPOTS AND THERMAL EFFECTS 

The images shown in Fig. 2 illustrate the local temperature increase caused by the 

occurrence of a single and multiple BD events in the MIM devices described in Section II. 

The failure events were generated using CVS with stress voltages of about 8-9 V applied for 

periods not longer than 1 minute. In many cases the shorts between the top and bottom 

electrodes are permanent as revealed by the IR images. In other cases just a flash is observed, 

which indicates that after the metal evaporation, the electrical contact between the electrodes 

is locally lost. The voltage probe is clearly visible in the IR images too, which points out a 

temperature increase of the contact between the tip of the probe and the top electrode (see Fig. 

2). A difference of up to eight degrees Celsius is detected between the metal electrode and the 

center of the leakage paths. Notice that this temperature can be radically different from the 

actual temperature reached inside the dielectric film during the micro-explosions. The IR 

images also show that the leakage current can be localized in a few sites (see Figs. 2.a and 2.b) 

or, if the damage is extremely severe, it can almost flow uniformly distributed over the whole 

device area (see Fig. 2.c). A thorough analysis of these experiments reveals that although the 

density of marks associated with the generation of BD spots on the top electrode can be quite 

large, the actual operating leakage current pathways spanning the dielectric film can be 

considerably less numerous. A second interesting question that arises is to what extent the 

local temperature increase associated with the formation of the current pathways does not 

affect the spatial generation of subsequent failure events. This issue will be analyzed in 

Sections IV and V in terms of the statistical distribution of the events. The results obtained by 

other authors using electrical methods indicate the absence of BD spot spatial correlation in 

MIS structures [15]. However, contrary to ours, that kind of study was limited to just a few 

BD events per device. In the next Section, the BD spot distribution in a 2-D space will be 

analyzed using a variety of functional estimators. 



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 181 

  
a) 

  
b) 

  
c) 

Fig. 2 Thermal images of different MIM structures in which a a) single and b) and 

c) multiple BD spots are generated during electrical stress. 

4. ANALYSIS OF THE BD SPOT DISTRIBUTION USING 2-D FUNCTIONAL ESTIMATORS 

In order to characterize the BD spots distribution in a 2-D region, it is necessary first 

to provide a brief introduction to the different functional estimators that are considered in 

this study. Notice that the standard reference model of a point process in the plane is the 



182 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

homogeneous Poisson point process, also referred to as CSR for Complete Spatial 

Randomness, which is commonly regarded as the null hypothesis model in spatial 

statistics [7]. The nearest neighbor distance distribution G is the cumulative distribution 

of the distance from a typical random point to the nearest other point (see Fig. 3). For a 

CSR process, G is given by the expression: 

     ( )       (    
 )   (1) 

where λ is the average intensity of the process, i.e. the number of events registered 

divided by the area of the observation window. While G>GCSR suggests that nearest 

neighbor distances are shorter than for a CSR process (clustered pattern), G<GCSR 

suggests a regular pattern, or simply the absence of neighbor points. The empty space 

function F is the cumulative distribution of the distance from a fixed point in space (but 

arbitrary) to the nearest point. F is a useful statistic summarizing the sizes of gaps in the 

pattern. For a CSR process with intensity , F reads: 

     ( )       (    
 ) (2) 

which coincides with the expression for the nearest neighbor distance distribution GCSR. 

While F>FCSR suggests that empty space distances are shorter than for a CSR process 

(regularly space pattern), F<FCSR suggests a clustered pattern. The Ripley’s K function is 

defined so that K(r) equals the expected number of random points within a radius r of a 
typical random point. For a CSR process, K is given by the parabolic relationship:  

     ( )    
  (3) 

In this case, K>KCSR suggests clustering, while K<KCSR suggests a regular pattern. 

Perhaps the most important functional summary statistics available for point patterns is 

the pair correlation function g. This function expresses the probability of observing a pair 

of points separated by a distance r divided by the corresponding probability for a CSR 

process and can be found as: 

  ( )  
 

   

  ( )

  
  (4) 

where dK(r)/dr is the derivative of K with respect to r. In this case, g>1 suggests 

clustering or attraction, while g<1 suggests inhibition or regularity. For a CSR process 

gCSR(r)=1. The estimator fails for r values close to 0 [7]. 

 

Fig. 3 Distribution of the points and circular region of generic radius r used to calculate 

the functional estimators. 



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 183 

In Figs. 4 to 6, three particular cases of MIM devices with BD spot patterns are shown. 

The images were obtained after the application of constant voltage stress. In the first two 

cases (Figs. 4 and 5), the voltage probe tip is clearly visible close to one corner of the 

device. The position of the probe tip defines the considered observation window (red solid 

line) within which the point pattern is analyzed. However, notice that for the analysis that 

follows, the specific location of the voltage probe is irrelevant, since r refers to a generic 

distance within the spatial point process. In the case of Fig. 4.a, the area of the device is 400 

x 400 m
2
, the area of the observation window is 370 x 370 m

2
 and the number of spots 

registered is 321. This corresponds to a density of 2.35x10
-3

 points/m
2
. The dashed lines in 

Figs. 4.b to 4.e were calculated using expressions (1)-(4) for a CSR process. Notice that the 

estimated curves (black solid lines) are all confined within the 95% confidence bands 

(shaded area) except for some deviation detected in G for the short distance range. This is 

merely a consequence of considering the BD spots as mathematical points (recall that 

typical lateral sizes are about 2 to 3 m) [16]. In addition, notice that g fluctuates around the 

unity mark, which is indicative of a CSR process at different scales. In summary, the 

process illustrated in Fig. 4 can be considered a CSR process. No interaction is detected 

among the BD spots, i.e. they are spatially uncorrelated. Something similar occurs for the 

device shown in Fig. 5.a. In this case the area of the device is 400 x 100 m
2
, the area of the 

observation window is 392.5 x 94.5 m
2
 and the number of spots is 168. This corresponds to 

a density of 4.53x10-3 points/m2, which almost doubles the intensity of the previous 

example. No interpoint distance below 2 m is registered but this is again a matter related 

to the spot sizes. No sign of deviation from a CSR process is detected. The case illustrated 

in Fig. 6.a is rather different since one of the sides of the device is significantly larger than 

the other. The device under consideration has lateral sizes of 1600 x 100 m
2
 and the region 

of interest (observation window) coincides with the device area. In this case, 132 points 

were registered with an average density of 8.25x10
-4

 points/m
2
. Notice the important 

deviations that take place with respect to the corresponding CSR process (see for example 

the pair correlation function gin Fig. 6.e). Particularly important is the absence of points 

in the short distance range (<5 m), but this can be related to the low density of points 

(~10
-4

) investigated. For the example illustrated in Fig. 6.a, the information about the 

location of the voltage probe is unavailable. The estimators seem to indicate that the data 

points follow a CSR process for distances larger than 5-10 m. The deviation in G for the 

long distance range (>30 m) can be attributed to edge effects associated with the low 

aspect ratio (100/1600) of the investigated device (see Fig. 6.b). In Section V, the same BD 

spot patterns explored in this Section are assessed again but in connection with the probe 

point location. Since no spatial correlation effects were detected using 2-D functional 

estimators, it is expected these results to be confirmed using alternative methods such as the 

point-to-event distance and angular probability distributions. This analysis is accomplished 

in the next Section. 



184 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

  

a) 

    
 b) c) 

     
 d) e) 

Fig. 4 a) Distribution of the BD spots in a MIM device with area 400  400 m
2
.  

2-D functional estimators: b) G , c) F , d) K and e) g.   



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 185 

 

a) 

    
 b) c) 

    
 d) d) 

Fig. 5 a) Distribution of the BD spots in a MIM device with area 400  100 m
2
.  

2-D functional estimators: b) G , c) F , d) K and e) g.   

 



186 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

 

a) 

   
 b) c) 

   
 c) d) 

Fig. 6  a) Distribution of the BD spots in a MIM device with area 1600  100 m2.  

2-D functional estimators: b) G, c) F, d) K and e) g.   



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 187 

5. ANALYSIS OF THE BD SPOT DISTRIBUTION USING POINT-TO-EVENT DISTRIBUTIONS 

In order to perform the study on the spatial distribution of the BD spots with respect 

to a particular point in the plane, it is necessary to characterize the locations of the failure 

events by two random variables (see Fig. 7): i) the distance 0x(a
2
+b

2
)

1/2
 between the 

reference point P and the event E and, ii) the angle 0α/2 subtended by the line 

connecting P with E measured with respect to the horizontal side of the rectangle a X b. 

In particular, for our analysis, we take P as the location of the voltage probe, which is 

chosen to coincide with one vertex of the observation window. This selection remarkably 

simplifies the mathematical treatment. The variable x defines the point-to-event distance 

probability distribution function (PDF) f(x), whereas α defines the point-to-event angular 

probability distribution function f(α). F(x) and F(α) denote the corresponding cumulative 

distribution functions (CDF). For the sake of simplicity, we use the same notation f and F 

for both variables though their explicit mathematical expressions are different. These 

distributions can be easily found by combining the areas of geometrical figures (see 

Ref.[17]).  

The PDF and CDF for the distance x are given by the expressions: 

  ( )  

{
 
 

 
 

  

   
     

 

  
      (

 

 
)      

 

  
*      (

 

 
)       (

 

 
)+     √     

      (5) 

and 

 ( )  

{
  
 

  
 

   

   
     

 

   
[  √  

  

  
         (

 

 
)]      

 

   
[  √  

  

  
         (

 

 
)   √  

  

  
         (

 

 
)]     √     

 (6) 

respectively. The PDF and CDF for the angle α are given by the expressions: 

  ( )  {

 

  

 

    ( )
  )        (

 

 
)

 

  

 

    ( )
      (

 

 
)    

 

 

 (7) 

and 

  ( )  {

 

  
   ( )           (

 

 
)

  
 

  
   (

 

 
  )       (

 

 
)    

 

 

  (8) 

respectively.  



188 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

 

Fig. 7 Definition of the variables used to localize the breakdown spots (E) with respect to 

the voltage probe position (P). x is the distance from P to E and  is the 

corresponding angle. 

Figures 8 to 10 show the three cases discussed in the previous Section but now 

emphasizing the distribution of the points with respect to the bottom-left vertex of the 

observation window. As expected, the experimental results (histograms for the PDFs 

and black solid lines for the CDFs) agree well with the theoretical results (red  solid 

lines) calculated using expressions (5)-(8). Notice how the mode of the distributions 

shifts to the left (shorter distances and smaller angles) as the length of the rectangle 

increases. In all the analyzed cases, it can be concluded that the data po ints are CSR 

distributed in total consistency with the observations reported in Section IV. Notice 

that a fundamental difference between the method presented here and that of the 

functional estimators is that the former one makes explicit reference to the shape of the 

observation while the second does not (no edge correction has been considered). On the 

other hand, methods for calculating the confidence bands are ready available for the 

functional estimators, whereas these methods for the point-to-event distributions have 

not been developed yet. 



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 189 

 
a) 

  
 b) c) 

  
 d) e) 

Fig. 8 a) Map of the BD spots in a MIM device with area 400 x 100 m2. The lines indicate 

the distances from the event to the bottom-left vertice of the observation window. 

PDFs and CDFs for the distances x and the angles : b) f(x) , c) F(x) , d) f() and e) 

F().   



190 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

 

a) 

  
 b) c) 

  
 d) e) 

Fig. 9 a) Map of the BD spots in a MIM device with area 400 x 100 m
2
.  

The lines indicate the distances from the event to the bottom-left vertice of the 

observation window. PDFs and CDFs for the distances x and the angles :  

b) f(x) , c) F(x) , d) f() and e) F().  



 Study on the Spatial Generation of Breakdown Spots in Mim Capacitors with Different Aspect Ratios 191 

 

a) 

b) c) 

d) e) 

Fig. 10 a) Map of the BD spots in a MIM device with area 1600  100 m
2
. The lines 

indicate the distances from the event to the bottom-left vertice of the observation 

window. PDFs and CDFs for the distances x and the angles : b) f(x) , c) F(x) , 

d) f() and e) F().   



192 X. SAURA, M. RICCIO, G. DE FALCO, J. SUÑÉ, A. IRACE, E. MIRANDA 

6. CONCLUSIONS 

The spatial distribution of breakdown spots in HfO2-based large area MIM capacitors 

generated by severe electrical stress was investigated. The analysis was performed on 

devices with different aspect ratios. Using real-time IR mapping it was demonstrated that 

the occurrence of these shorts between the top and bottom metal electrodes is associated 

with a notable local temperature increase. However, not all the generated spots remain 

conducting after their creation because of the volatilization of the top metal electrode. 

From the statistical analysis carried out in this work, it can be concluded that the 

breakdown spots in MIM devices with high-K dielectric are spatially uncorrelated. This is 

consistent with previous observations carried out on MIS devices. 

Acknowledgement: This work is funded in part by the Spanish Ministry of Science and Technology 

under contract TEC2012-32305 and the DURSI of the Generalitat de Catalunya under contract 

2009SGR783.E. Miranda thanks the funding from the Visiting Professor Program of the University of 

Naples Federico II. The authors also acknowledge I. M Povey, E. O’Connor, K. Cherkaoui and P.K. 

Hurley (Tyndall National Institute, Cork, Ireland) for device provision and assistance in the electrical 

characterization of the samples.We give our special thanks to Prof. Ninoslav Stojadinović and Prof. 

Danijel Danković from University of Niš, Republic of Serbia for their invitation to write this report.. 

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