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Breakdown First Principle Modeling of Intrinsic Breakdown

Breakdown First Principle Modeling of Intrinsic Breakdown. Ying Sun , Ghanshyam Pilania , Steven Boggs, and Rampi Ramprasad. 10/27/2011. Breakdown is the 800 lb Gorilla. Dielectric constant is intrinsic, whereas conductivity and breakdown are extrinsic for good dielectrics.

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Breakdown First Principle Modeling of Intrinsic Breakdown

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  1. Breakdown First Principle Modeling of Intrinsic Breakdown Ying Sun, GhanshyamPilania, Steven Boggs, and RampiRamprasad 10/27/2011

  2. Breakdown is the 800 lb Gorilla • Dielectric constant is intrinsic, whereas conductivity and breakdown are extrinsic for good dielectrics. • We can hope to estimate well “intrinsic breakdown”, but this is of limited interest. • However intrinsic breakdown depends on the energy loss (to phonons) per unit distance as a function of carrier energy, and the nature of this function can give us insight into the nature of important “defects”.

  3. Phenomenology of High Field Conduction • At low field, conduction tends to be Ohmic. • At medium fields (E>kBT/ql), typically ~107 V/m , conductivity increases exponentially with field as energy gained from field is greater than thermal energy (l=distance between traps). • Fields >g/ql result in high carrier mobility, progressive degradation, and eventual breakdown (g=relevant activation energy, i.e., effective trap depth). About 2x108 V/m for PE and 5x108 V/m for BOPP.

  4. Source of Thermal Activation Energy - PE • In PE, numerous impurity states are ~1 eV above the valence band and below the conduction band. • This results in an activation energy of ~1 eV, probably with contributions from both holes and electrons. For holes, we have seen the hybridized states which promote transfer between polymer chains.

  5. Thermal Activation Energy - BOPP • The measured thermal activation energy of BOPP is about 0.73 eV (Dr. Janet Ho, ARL, unpublished). • Work by Rampi and his students indicates that carbonyl causes impurity states ~0.7 eV above the valence band and ~1.5 eV below the conduction band, which suggests that the activation energy is caused by hole mobility, while breakdown is the result of electron mobility. • Unlike for PE, the activation energy does not appear to be well correlated with the breakdown field. 6 0

  6. Intrinsic Breakdown • Quantum mechanical descriptions of “intrinsic breakdown” are well over 50 years old, but until recently, the problem is that the relevant parameters could not be measured or computed. • We now have the computational tools to implement such theories. The question is what we can hope to learn by doing so • The shape of energy loss per unit distance as a function of carrier energy can give us insight into the nature and effect of relevant “defects”.

  7. Current Density, Nanocavities, Energy Loss • The shape of the loss curve determines important defects. • If the loss levels off at high energy, carrier multiplication will occur. If the loss curve levels off at high field and a carrier encounters a 10 nm cavity at 250 MV/m, it will gain 2.5 eV and may exit with sufficient energy to cause ionization, which will result in high field degradation near the cavity. Trapped

  8. Stable Transport at High Fields Leveling Off of Energy Loss at High Energy Can Define Breakdown

  9. Objectives in Study of Breakdown • Compute the energy loss curve as a function of electron energy to determine intrinsic breakdown. • Incorporate the effects of inevitable chemical impurities, such as carbonyl, through their effect on the phonon spectrum. • Incorporate the effects of amorphous and crystal-line regions to estimate the effect of morphology on energy loss as a function of carrier energy. • Relate electron-phonon coupling to fundamental parameters at the molecular level as a basis for identifying high breakdown materials (e.g., SiO2).

  10. Electron Avalanche Breakdown • Assume initial electrons are available. • Electrons gain energy from electric field and lose energy to phonons. • At sufficiently high electrical field, energy loss to phonons no longer balances energy gain from the field, and once the electron energy reaches the ionization threshold energy eI, an exciton is created, which injects a second electron at the bottom of the conduction band, and the process repeats until the electron density is sufficient to damage the material.

  11. First Principle Approach to Intrinsic Breakdown Average Electron Model • Breakdown occurs when the average rate of energy-gain from the electric field exceeds the rate of energy-loss to phonons. • > • Both the electron phonon scattering rate, gk(e),and average energy loss, ħwphgL(e), are related to W±i,j,a which is the probability of an electron in state i being scattered to state j by phonon mode a. • W±i,j,a is related to the electron-phonon interaction matrix, Mai,j • The breakdown criterion is: • E: electric field • m*: effective electron mass • tk: mean time between scattering • wph: average phonon frequency • gl: loss relaxation frequency • i,j are electron states and a is the phonon mode, • dV is the change in potential caused by ion displacement

  12. Modification of Quantum ESPRESSO Code • Quantum ESPRESSO (elphon.f90) evaluates the electron-phonon interaction matrix, Mi,j, but the program uses the matrix to calculate superconductivity of metals rather than compute the parameters of interest for dielectrics. Mod 5 Mod 4 Mod 1 Mod 2 Mod 3 • Modification 1: divide by density of states N(e). • Modification 2: sum over phonon modes and phonon wave vectors. • Modification 3: divide by phonon frequencies wqλ • Modification 4: add phonon occupation factor nqλfrom Bose Einstein distribution, T=300 K. • Modification 5: change arguments in double delta function.

  13. Estimate of Intrinsic Breakdown for Silicon k point mesh: 32x32x32 q point mesh: 4x4x4 Gaussian broadening: 27.2 meV • Maximum phonon scattering rate, gk, of200/ps results in maximum energy loss, ħwphgL(e) of 7 eV/ps for Si effective electron mass of 0.26 m0

  14. Preliminary Calculation Results • Average energy gain vs. electron energy for various electric fields. • Breakdown occurs when energy gain > energy loss over the entire range of energy. • Calculated breakdown field Ebd =7.9x107 V/m • Calculated intrinsic breakdown field of Si is 7.9x107 V/m compared to the experimental value of 5x107 V/m.

  15. Conclusions • The balance of electron energy gain with energy loss to phonons determines the intrinsic breakdown field. • In a semi-crystalline material, breakdown may occur at lower field as a result of nanoscopic imperfections, such as “cavities” or “low density areas”. • The challenge will be to develop realistic breakdown models of semicrystalline and amorphous materials, including the effects of nanoscopic imperfections, chemical impurities, etc.

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