To dynamical theory of electrothermal degradation and NDT of defects in metal
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 elimination conditions. REFERENCES
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