To dynamical theory of electrothermal degradation and NDT of
defects in metal-dielectric-metal (MDM) structures Valentin M.
Bogomol’nyi
Moscow Institute of Technology, 141220, Moscow Region, Pushkinsky
raion, Cherkizovo-1, Russia ABSTRACT
On base of the solid state physics and theory of nonlinear
oscillations interpretations a development of the thermofluctuation
fatique of mechanics theory is formulated. It is shown that
electrical damage haves resonance nature. An influence of the
electron processes on the first time pre-breakdown stage with
mainly of microdefects formation is considered. The proposed theory
contents consideration of polarization of the local domains
the“cross-pieces”between neighbouring micropores, which formed
elementary electrical dipoles. Strong external constant electrical
field lead to negative differential resistance of the local
dielectric domains with N-or S-type current-voltage-characteristic
(CVC) parts and as result to current oscillations and
electromagnetic wave radiation from MDM structure (as in Gunn’s
diode). On base of A. Puankare’s limit cycles nonlinear
oscillations theory it is shown that defects formation leads to
selfexciting current oscillations and microwave radiation. This
information can be used in thermosense NDT and, that is principal,
for elimination of the defects, which arised under fabrication of
electronic devices.
Keywords: Electrothermal, resonance, degradation, dielectric,
polarization, thermosense INTRODUCTION
Dielectric thin films aging and breakdown is a phenomenon of major
technological significance in optoelectronics, micro- and
nanoelectronics. Study of the microdefects formation haves a
practical interest at first for improvement of industrial
electronic devices quality. On other hand the specially used
structure defects lead to sensors, diodes and transistors
functional parameters enhance1.
Pre-breakdown reversible effects are used in radars as sources of
high-frequency radiation (0, 1 - 103GHz), for signals amplification
in GHz diapason, where usual transistor not can be used,
thermistors, electronic switching2, low-voltaic “cold” cathodes.
Reversible breakdown used in “electrical forming”- an known
technology process fabrication of the vacuum luminescent screens
and origins of high energy electron radiation. By this in space
regular local defects (high conducting microchannels, which content
low - temperature plasm) are formed in voltage controlled -
negative -resistance MDM structures, considered in this work.
Interest to study of electrical degradation connected also with an
elaboration of high reliability optoelectronic metal - oxide -
semiconductor (MOS) devices on baseSi - SiO2, Si - SiN4, which used
in computer technique, memory and recording elements. The surface
of dielectric, contacting with metal electrode, is the most likely
domain, where destruction takes place3, 5. Local defects on surface
are seats of electrical and temperature field concentrations, where
origins of the mechanical destruction and electromagnetic radiation
are located.
The main structure defects - dislocations (linear, plane, screw)
arise as rule in all electronic devices under their fabrication.
For example, distinction between temperature - expansion -
coefficients at dielectric surface metalization lead to creation in
GaAs density of dislocations - 108cm-2. When an electrical
breakdown event occurs local evaporation of the electrode material
can be so that this regions become electrically and mechanically
disconnected from the rest structure and a complete series of
measurements can be made to renew of devices reliability, physical
background of these methods is aim of this work.
B. K. Ridley suggested a breakdown essence in SiO2 based devices,
which contents the assumption of the presence of ~100protuberance
at electrode surface. Greatly current injection enhanced at
protuberances produces high temperature filaments in which
dissociation of ion coupling takes place. The resulting positive
ions drift to the cathode, producing a positive feedback connection
on the current, which lead to its instability and
oscillations4-9.
A. K. Jonsher et al propose a breakdown statistical model,
connected with the existence of many point defects5, 6. High
electrical field lead to charge carriers injection in dielectrics
and to generation of cumulative dynamical process of formation of
the defects clusters5. Dislocations and micropores can instantly
grow by clusters generation into a highly conducting channels (with
diameter 40-80) connecting the electrodes. Accepted mechanism of
breakdown involves the further creation of gaseous channels through
the dielectric. The channels volume is ~ 10-3-10-5 from common
volume of dielectric. High conductance at breakdown is associated
just with these channels and not with conduction trough the all
rest dielectrics.
The formation of the gaseous channels before a marked change
integral conductance of MDM structures is ascribed to energy supply
from the external electric field and its storage in the solids by
polarization dielectric (“memory effect”), heating and collision
ionization, trapping charge corriers processes,
atomic“displacements”, temperature rise and as result broken
molecular bonds. At first time dielectric breakdown occurs with
very little “warning”of an increased current preceding the
breakdown. By this short pulses of current were observed5, 10, 11.
Microwave radiationtakes place on the first growing part of the
current - voltage - characteristic (CVC) andmight be not connected
with integral volume negative differential resistance
mechanism.
Simultaneously with microwave radiation in In Sb crystals, for
example, the low-frequency current oscillations (105 - 108c-1) were
observed. In electrical circuit is not oscillations arised the same
as radiated from dielectrics. From this it is follows,
thatmicrowave radiation is not depends on integral volume
dielectric properties and defines eigenfrequency properties of
defects.
Usually theoretical treatments of the electrical breakdown were
concerned with “quasistatic” behavior of high energy electrons12
and ionization avalanches in isolators4, 5. The aim of this work a
study of coupled dynamic thermoelectron emission currents3, 14-19,
polarization5, 6 and heating processes13in dielectrics at the first
time pre-breakdown stage before of the gaseous channels
formation.
Following model of the pre-breakdown stage it is supposed, taking
into account the electron injection in dielectric from cathode. If
a first injecting electron start from cathode towards anode
a“polarization echo” channel (“memory track”) remains in
dielectric. The second and following electrons move in the same
“polarization-echo” channel. As result at first time constant
current “filaments”arise. Electrical current in dielectrics with
high electrical resistance lead to its heating. Owing to
temperature rise then the electrical conductivity increases. This
selfexciting nonlinear periodic process lead to high heating to and
nonlinear N-or S-type CVC. The“polarization echo” - “memory
track”collects charges from a relatively large catchment of
environment rest dielectric volume, this constitutes the basis of
temporal energy effectiveness of breakdown coupled with secondary
electron emission from volume of dielectrics into“memory
track”.
Once the increase of the electrical field and temperature shorten
time interval between moving in“memory track” electrons to the
“point” and the “memory track” retains sufficient strength
channeling becomes positive feedback, which lead to rapid increase
of the micropores formation in direction of the force lines of the
external electrical field and further conducting channels. The
energy dissipation resulting from environment dielectric
polarization fluctuations caused by narrowly focused “beam”of
charged particles rapidly leads to electrothermoelastic destruction
of dielectrics5, 6, 13.
As result of secondary electron emission from environment
dielectrics the charges are stored on the surfaces of the cavities.
External electrical field separates the negative and positive
charges on inner surface of each pore. In each“cross-piece”between
neighbouring pores the additional strong electrical field appears,
which amplitude might be sufficiently more than magnitude of the
initial external electrical field. In dielectric cross-piece the
negative differential resistance and N-or S-type CVC can be
realized. By this each cross-piece under constant external
electrical field“works”as acoustoelectronic generator or tunnel
diode (origin of stable oscillations or radiation). THEORY
Thin layer dielectrics, contacting with metal electrodes, shown
semiconductor properties. Thermoelectron injection currents can be
drawn through defects in insulator (mainly through dislocations and
other defects). Space - charge - limited currents (SCLC) become a
the most simple tool for measuring the imperfection in ionic
crystals3. Presence of traps lead to distort the shape of the
current-voltage-characteristic (CVC) from square law to a much
higher power dependence current from voltage14-19.
SCLC offer resonance mechanism of pre-breakdown stage taking into
account following CVC cube-law.
In A. Rose’s work it is shown that in CdS crystals at concentration
of traps ~ 1010 traps/cm3, h =10-3cm (h - thickness of dielectric)
at 105V/cm a thermal breakdown observed. By this CVC have form14 ,
(n >1) (1) where n is constant, U - voltage.
The cube - law regime of the CVC in condition of double injection
from electrodes (the simultaneously injection electrons from
cathode and holes from anode) was considered in M. A. Lampert’s
work, where was given following formula15 , (2)
where is dielectric permittivity, is a life time of charge
carriers, and are mobilities of electrons and holes, L is a
thickness of dielectric diode. The negative resistance part on the
CVC has its origin in the unequal capture cross section of defects
(unclude traps) and corresponds to acceptor - like behavior of
various defects. Exhibiting a current-controlled negative
resistance might be revealed experimentally through either of two
effects: spontaneous oscillations under application of an
appropriate dc voltageor an breakdown at some critical voltage.
Both types of these phenomena have been observed in
high-resistivelyGe15. The expressions analogous to formulas (1) and
(2) were given also in works16-19. Thin layer SiO2films for example
at all kinds their fabrication have micropores and cracks. Averidge
magnitude of through pores inSiO2 coating with thickness 0, 1 is 10
cm-2. For thin Ni-Mo-SiO2-Si films (30-600) the current- voltage
characteristic have form18 , (3)
where Ez is electrical field strength, A and k are constants. From
experimental data it is follows that under EZ =105 - 3 ·105V/cm k
have magnitudes in interval 2, 3 , (4) where are known from
experiment constants.
Integral characteristics of MDM structures: capacity, induction and
resistance with N-type’s CVC with negative differential resistance
part correspond to analogous parameters of the lamp (valve tube)
dynatron oscillator20. In accordance with Kirchoff’s law we have ,
(5)
where IR, IL, IC are currents in resistor, induction coil and
condenser, t is time (6)
Set (4) in (5) and take into account (6) we have following
differential equation , (7) is time derivative. From equation (7)
we have Van-der Pol’s equation , .
From theory of nonlinear parametric oscillations in considered case
the selfsynchronous oscillations arise with one limit closed
trajectory in phase plane.
CVC (4) is in accordance with given in work19, where current
magnitude was determined from electrical field strength, electrons
mobility and geometrical parameters of the roughness on the metal
electrode’s surface. By this CVC was estimated in condition of the
nonlinear dependence of the electrons mobility on electrical field
strength18, 19. Considered N-type of the CVC correspond to case,
when temperature of dielectrics is not distinct itself from
environment medium temperature. By N-type CVC nonhomogeneities of
space-charge occurs in dielectric. S-type CVC corresponds to
nonuniform current distribution in dielectric with formation of the
conducting micro- channels (or“current filaments”)and then as
result micropores. In this case the influence of temperature rise
on CVC is sufficiently great in comparison with case of N-type’s
CVC. Thermal effect of space-charge-limit-currents (SCLC) was
investigated in E. Gray’s work21, where he gives the most correctly
physical theory of Joule’s heating of dielectric diode in condition
of traps-filled-limit-voltage. E. Gray’s model corresponds to
square-current-voltage-law, however this theory can be used also
for estimation of the lower boundary valuation for the cube-law,
see formula2, 14, 15. E. Gray writes the CVC and temperature
equations in the form (8)
T0 is environment medium temperature, Et is a “depth”of traps (or
activation energy), which mark off the zone conductivity boundary,
whereI is a current density, U is a voltage applied to electrodes
MDM structure, Et is a “depth” of traps, Nt - their
concentration;
where h is a thickness of dielectric, is a mobility of electrons,
is a dielectric permittivity, NC is the density of electron states
in conductivity energy zone, R is a heat resistance of dielectric
diode, T is absolute temperature, R is the Boltzman’s constant.
Used nondimensional variables we have , (9) .
From (9) we determine current-voltage characteristic in parametric
form , . (10)
From (10) we determine the temperature magnitudes, in which
differential resistance changes the derivative siqnum, as the
squares of equation, . (11)
From expression (11) at condition we determine the critical
temperature point, in which fulfilled condition ,
where 1 and 2are the first and second magnitudes of temperatures
which are conform to with principal turning - points of the S-type
CVC. From experiment it is follows, that main part of external
electrical field energy is transformed into heating and
polarization losses in dielectrics13, 22.
Temperature measurements with remote sensing methods can be used in
NDT of the microelectronic devices quality and also for remote
estimation of the inner electrical fields in thin films structures.
For example we estimate a valuation of the electrical field
strength, which may be used for elimination of the microdefects in
MDM structures23, 24. The temperature increase T in MDM structure
can be obtain from formula , (12)
where jc is current of electrons, Ez is an electrical field
strength, t is electrons “fly” time, C0 is a thermic capacity, is a
dielectric density; , (13)
where e is electrons charge, its mobility, nis a concentration of
the free electrons, which determined with following formula, ,
(14)
where is a capacity on init of dielectric area, is a dielectric
permittivity, h is dielectric thickness, U is a voltage on
electrodes of the MDM structure. The “capture” of electrons
coefficient is determined with following expression3, 15 , (15)
The magnitudes of Ez and can be approximately determined with
formulae . (16) From (12) taking into account (13)-(16) we have .
(17)
At first turning - point on S-type part of CVC the sharply (almost
vertical) temperature increase and differential electrical
resistance make start and electrical aging begins therefore. The
electrical field strength lower boundary valuation we determine
from (17) taking condition of equalityTminwith one degree over
initial environment temperature (which can be exactly measured with
remote thermosense methods) . (18)
From (18) for thin ferroelectric film with @ 103 and C @ 0, 1
[calory/gram. degree], @ 2 [gram/cm3] and Tmin=1oC we have Ez =
3*104 [V/cm]. At the second turning - point on the S-type part of
CVC the current “filaments” and micropores make start3, 14, 15, 18.
This stage of electrical aging with cube - law of the CVC
considered in this work. The upper boundary valuation of the
critical electrical field strengthEz(max) we determine from (17)
taking into account the condition T=2T02... .
In consequence with thermo-electrical nonstabilities arising in
dielectrics at pre-breakdown stage their polarization and as result
under external constant electrical field the polar dielectrics
domains oscillations of various physical nature were experimentally
observed4-9, 11, 19.
The evidently, principal positive feedback in auto - wave
electrical aging processes realizes as result of energy dissipation
of the polar dielectric domains at their harmonical oscillations13.
Controlled with external electrical field heating can be used in
NDT and various technology processes: electroadhesion“welding” of
multilayer structures, polarization et al. The condition of the
even temperature distribution through MDM structure thickness can
be used in thermosense control of their quality and reliability.
Temperature control theory we briefly consider in this work. The
thermal - conductivity equation have form13 , (19)
where T is temperature, C0 is a thermic capacity, t is time, Z is a
space coordinate, which mark off the middle surface of dielectric
layer, is a frequency, are components of energy dissipation
function, which are determined with following expressions, is a
thermo - conductivity coefficient; , (20) , (21) , (22)
where Cel is an dielectric capacity, U0 is a voltage, is volume; ,
, are experimentally determined constants, which characterize
dielectric, mechanical and piezoelectric (coupled
electromechanical) losses in polar dielectrics at harmonical
excitations, Ee is a elasticity module, is Puasson’s coefficient, i
(i = 1, 2, 3) are elastic relative deformations (or strains) of
dielectrics, is an electrical field strength; are the piezoelectric
constants. Boundary and initial conditions we take in form , ,
(23)
where T0 is initial temperature, is thermointention coefficient. At
constant temperature distributionthrough dielectric layer we obtain
solution of boundary problem (22), (23). Afterwards of the
integration of (22) taking into account (23) we have , (24) The
solution of the equation (24) have form , (25) , , Tmax is a
maximum initial temperature T0 increase.
Under parameters of piezoelectric transducer on base piezoceramic
PZT-4: h=2mm, U0=100V and =6, 28*103s-1temperature of polar
dielectric was calculated with computer use. From this numerical
calculation13 it is follows, that after 32 minutes the stationary
regime realized with Tmax=1, 72 oC. From experiment in considered
case it is Tmax is 2, 04 oC (with error of measurements 0, 4 oC).
The conditions of uniform temperature distribution trough thickness
of MDM structure and its stationary state can be used at
thermosense control of the microelectronics devices under
industrial technology processes. CONCLUSION
As distinct from known theory in this work the main criteria of the
electromechanical fatique: critical electrical field strength of
electrical degradation and heating breakdown temperature point were
determined. From experiments it is follows, that in first - time
electrical pre-breakdown the mainly part of external electrical
energy (80-96%) transforms in the heat and polarization
fluctuations5, 6, 13 in the local dielectric domains of MDM
structures (~10-3-10-5from common dielectric volume). By this the
conductivity channels (with diameter ~40-80) and as result of
electromechanical degradation the micropores formation appears at
first pre-breakdown stage18.
Each pore becomes of electrical dipole as result of the secondary
electron emission from rest dielectric volume. The additional
electrical field strength in the“cross-pieces”between micropores
can be sufficiently more than initial external electrical field
strength. By this in dielectric“cross-pieces”between pores
polarization in direction of the forces of the external electrical
field and negative differential resistance arised, which lead to
selfexciting oscillations of local dielectric domains. Energy
dissipation under resonance oscillations of the polar dielectrics
taking into account dielectric, mechanical and piezoelectric losses
lead to increase temperature MDM structures. Numerical and
analytical methods of temperature calculation for polar dielectrics
by harmonic oscillation are given in work13. The temperature rise
lead in one’s turn to increase of amplitude oscillations. This auto
wave processes with positive inverse coupling haves selfexciting
character20.
Amplitude-frequency characteristics wave processes of various
physical nature, can be used for identification of types and
dimensions of the structure defects3-11. If the MDM structure
surface temperature is in twice more as the first time initial
environment temperature the formation of current“filaments” starts,
as result micropores arised18 . This regime correspond to
space-charge-limited currents (SCLC) with,
current-voltage-characteristic (CVC) of the cube-law3, 14-18as
result at constant external electrical field in condition of
sufficiently temperature increase the S-type CVC and oscillations
arised14-19. The temperature on MDM structure surface can be
measured with standard thermosense methods, compare its with
environment initial temperature can be obtain information on
microdefects state at electrothermal degradation and theirs
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