First Principles Calculations of Intrinsic Breakdown in Covalently Bonded Crystals

Ying Sun, Steven Boggs, and Ramamurthy Ramprasad Institute of Materials Science, University of Connecticut

97 North Eagleville Rd. Storrs, CT 06269-3135, USA

Abstract - A first principles quantum-mechanical method has

been developed for estimating intrinsic breakdown strength of insulating materials. The calculations are based on an average electron model which assumes breakdown occurs when the average energy gain from the electric field and phonon absorption exceeds average energy loss to phonons. Our approach is based on density functional perturbation theory (DFPT) and on the direct integration of electronic scattering probabilities over all possible final states, with no adjustable parameters. Computed intrinsic breakdown field is provided for several covalently bonded materials and compared with experimental data from the literature with good agreement. The numerical model provides physical insight into the material properties which affect breakdown.

INTRODUCTION

High breakdown strength is a critical characteristic for many insulating materials. Breakdown of most insulating materials is an “extrinsic” property, i.e., dependent on factors which are not inherent to the material but which could be described as “imperfections”. Such imperfections can range from chemical impurities, i.e., carbonyl, etc., in a polymeric insulator to statistical variations in morphology and nanocavities. Never-theless, developing a predictive theory for intrinsic breakdown is the first step down the path to predicting extrinsic break-down, which is our ultimate objective.

Breakdown theories can be classified broadly into “thermal” and “intrinsic” based on the conditions which govern energy transfer from the electric field to the lattice. Intrinsic breakdown is based on electron-avalanche theory [1] in which energy gained from the electric field causes electrons to drift toward increasing energy in the conduction band, while energy is lost to phonons. At sufficiently high electric field, electron energy gain is no longer balanced by energy loss which results in ionization and electron multiplication, i.e., an electron with sufficient energy ionizes an atom in the material, resulting in two electrons near the bottom of the conduction band. Rapid electron multiplication causes sufficient electron density to damage the material. Quantum mechanical descriptions of “intrinsic breakdown” are well over 50 years old [1-4], but until recently, the relevant parameters could not be computed so that the theories remained “academic exercises”. With the development of computational quantum mechanics, that is no longer the case.

Density functional theory (DFT) can predict many material parameters well, such bond angles and bond lengths, typically to within 1%. Band gap and dielectric constant, which are two important parameters relevant to dielectrics, can also be predicted. As yet, methods for computing intrinsic and extrinsic breakdown characteristics are lacking. Here, we present a first order method for predicting intrinsic breakdown of covalently bonded materials. We are presently extending this method to ionic materials, after which we will attack extrinsic breakdown through inclusion of chemical impurities, morphological characteristics, etc.

The first principles method for calculating intrinsic break-down field described below is based on electron phonon interactions using an average electron model. Using DFT and DFPT, the electron phonon scattering rate and average energy loss are calculated by direct integration of electronic scattering probabilities over all possible final states. The computation of intrinsic breakdown is verified for covalently bonded materials by comparing the computed data with experimental measure-ments for diamond, silicon and germanium, over which the breakdown field varies by almost two orders of magnitude.

AVERAGE ELECTRON MODEL FOR INSTRIC BREAKDOWN

Electron-avalanche theory assumes that a sufficient number of initial electrons are available in the conduction band to initiate electron avalanches. At sufficiently high electrical field, electrons gain energy until they reach a threshold energy, I, at which an electron is excited from the valence band to the conduction band, which results in two electrons near the conduction band minimum (CBM). Repetition of this process leads to avalanche breakdown, although the analytical breakdown criterion in the average electron model is that the electron energy gain from the field is greater than the energy loss to phonons for all electron energies below ionization.

In the average electron model [1,2], which affords the simplest mathematical treatment, electron energy gain from the electric field can written as

(1)

while energy loss to phonons is described by

(2)

ph LL

d

dt

e w gæ ö÷ç =÷ç ÷÷çè ø

2 2

*3k

E

d e E

dt m

e tæ ö÷ç =÷ç ÷çè ø

34978-1-4673-0487-0/12/$31.00 ©2012 IEEE

where e is the electronic charge, the electron energy, m* the effective electron mass, E the electric field, k the mean time between phonon scattering of an electron, ph the average phonon frequency, and L the energy loss relaxation frequency. Effective electron mass is the mass the electron appears to carry in the semi-classical model of transport in a crystal. The two key parameters, L and k

-1= k, are described by

(3)

(4)

where ± refers to phonon absorption or emission by an elec-tron, Wi,j, is the probability of an electron in state i being scattered to state j by phonon mode and N( ) is the electron density of states. Sum over electron wave vector k and phonon wave vector q is not shown in the above equation for simplicity. The electron scattering probability, Wi,j, is derived by Fermi’s Golden rule and is expressed as

(5)

where m is the total mass of atoms in a unit cell, n the phonon occupation number which is given by the Bose-Einstein distribution, and εi the single particle energy of initial state i and εj the single particle energy of final state j. Thus the key to implementing the model is computation of Mi,j

the electron-phonon scattering matrix, which is characterized by

, (6)

where i,j are electron states and is the phonon mode, V is the change in potential caused by atomic displacement based on phonon mode and “q” is the phonon wave vector that connects states i and j .

Based on the average electron model, a reasonable zero order approximation for intrinsic breakdown strength is provided by the criterion that the energy gain from the field (eq. 1) is greater than the energy loss (eq. 2) for all electron energies below ionization. This is also known as the Frohlich high energy criterion, and leads to the following expression for the breakdown field, Ebd [2,3,5].

. (7)

In eq(7), ħ ph L provides the energy loss rate. Based on the above discussion, the key to implementing this approach is computation of the electron-phonon coupling matrix Mi,j

which can be computed efficiently using density

functional perturbation theory (DFPT). The electron phonon matrix elements have been studied widely to calculate the transition temperature of superconducting metals.

Quantum ESPRESSO [6] is an open source integrated suite of computer codes for electronic-structure calculations based on DFT. Quantum ESPRESSO employs plane waves for the expansion of wave functions and charge density, and pseudo-potentials (both norm-conserving and ultrasoft) to represent the core electrons. Quantum ESPRESSO calculates phonon frequencies efficiently using DFPT and includes an algorithm to calculate the electron-phonon coupling matrix elements. Modifications of the Quantum ESPRESSO code were made to adapt the code to calculate the scattering rate and energy loss rate of insulating materials. Phonon linewidth [7] is an auxil-iary quantity computed by Quantum ESPRESSO as related to electron phonon coupling in metals, the mathematical expres-sion for which includes the square of the modulus of the elec-tron-phonon matrix elements Mi,j

and the double delta func-tion in eq(3) and eq(4). Using this quantity as a starting point, we have modified the code to sum over the phonon modes and phonon wave vectors, added the phonon occupation factor from Bose Einstein distribution, changed the arguments in the double delta function, divided the resulting quantity by density of states and phonon frequencies, etc.

COMPUTATIONAL RESULTS Covalently Bonded Crystals

The computational approach described above was tested on well characterized, covalently bonded systems, including diamond, silicon and germanium, prior to addressing ionic and polymeric materials. The calculation of band structure and phonon frequencies was performed using the local density approximation in DFT and using norm-conserving pseudo-potentials, as implemented in Quantum-ESPRESSO. For all three materials, a Monkhorst-pack k point mesh of 16x16x16 and q point mesh of 4x4x4 were adopted. The function in equations (3) to (5) was replaced by a sharp Gaussian function, with a broadening value of 27.2 meV. Figure 1 shows the electron-phonon scattering rate k of silicon. The green line represents the scattering rate from phonon emission, which is much greater than energy gain from phonon absorption. The calculated electron-phonon scattering rate as a function of electron energy is in good agreement with reported full band calculations [8]. Figure 2 shows the average energy loss rate ħ ph L of silicon.

The intrinsic breakdown field based on the average electron model is the electric field at which the electron energy gain exceeds energy loss over a wide range of electron energy, normally from the CBM to about 5 eV above CBM. Figure 3 shows average electron energy gain and energy loss as a function of electron energy for several electric fields. We can see that the red line represents the energy gain at the critical electric field above which electron energy gain is greater than energy loss over the entire electron range, in which case

,

1 1( ) ( )

( )k ij ijk

W dN a

ag e d e e e

t e

= = -åååò

,

1( ) ( ) ( )

( )ph L ij ij

w W dN a a

aw g e d e e e

e

= -åååò

2

, ,

1 1

2 2( )i j ij j inW M

m aa

a aa

p d e e ww

+æ ö÷ç= - ÷ç ÷çè ø

, , ,i j i k j k qM Vaaj d j +=

*3bd ph L k

m

eE w g g

é ùê ú= ê úê úë û

35

electrons continue to drift up in the conduction band until ionization occurs which results in electron multiplication, increasing “hot” carrier density and eventual damage to the material which leads to breakdown. The calculated breakdown strength of silicon is 7.9x107 V/m compared with experimental intrinsic breakdown strength of 5x107 V/m [9].

Energy (eV)7 8 9 10 11

Ele

ctro

n-ph

onon

Sca

tterin

g R

ate

(1014

/sec

)

0.0

0.5

1.0

1.5

2.0

2.5

Total Scattering RateAbsorption Scattering RateEmission Scattreing Rate

CBM

Figure 1. Electron-phonon scattering rate of silicon.

CBM 7 8 9 10 11

Ave

rage

Ene

rgy

Loss

( 1

014eV

/ sec

)

0.00

0.02

0.04

0.06

0.08

0.10

Energy Gain From Phonon AbsorptionEnergy Loss From Phonon EmissionTotal Average Energy Loss

Energy (eV) Figure 2. Average energy loss rate of silicon.

Energy (eV)

7 8 9 10 11

Ave

rage

Ene

rgy

Gai

n an

d L

oss

(1014

eV/s

ec)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Energy Gain at 9x107 V/m

Energy Gain at 8x107 V/m

Energy Gain at 7x107 V/mAverage Energy Loss

CBM

Figure 3. Silicon average energy gain and energy loss

The breakdown fields of diamond and germanium were also calculated using the same approach. All the computed breakdown field values are in good agreement with experi-mental data [9,10], as shown in Figure 4. Effective electron masses of 0.39m0, 0.26 m0, and 0.12 m0 were used for diamond, silicon and germanium, respectively which is the effective electron mass at conduction band minimum for these materials [11].

Experimental Breakdown Field (V/m)1e+7 1e+8 1e+9 1e+10

Cal

cula

ted

Bre

akdo

wn

Fie

ld (

V/m

)1e+7

1e+8

1e+9

1e+10

GermaniumSiliconDiamondExperimental Breakdown Field

Figure 4. Experimental breakdown field and calculation breakdown field of diamond, silicon and germanium

In order to develop an intuition for the fundamental chemical and physical factors that control intrinsic breakdown, we have attempted to correlate several attributes, such as the phonon cutoff frequency, scattering rates, density, etc., to the intrinsic breakdown field. From Frohlich’s breakdown criterion, eq(7), we see that increased electron-phonon scattering rate increases the breakdown field. Indeed, the calculated maximum scattering rates of diamond, silicon and germanium, 22.7x1014/s, 1.98x1014/s and 2.47x1014/s, respectively, correlate well with breakdown field. The lattice constants of these systems, 0.357, 0.543, and 0.566 nm, respectively, indicate that larger scattering rate may be caused by greater atomic density, as greater atomic density increases the likelihood of the electron-lattice collisions during carrier transport. While these concepts should be tested through calculations of the scattering rates and breakdown fields for a given system as a function of lattice parameters (in order to decouple the physical and chemical factors), these considera-tions are consistent with increased breakdown strength of polymeric material with increasing crystallinity.

From eq(4), we see that average energy loss increases with phonon frequency. Calculated phonon dispersion curves for diamond, silicon and germanium have similar shape with phonon cutoff frequencies of 37.9, 15.3 and 8.73 THz, respec-tively, so that the phonon cutoff frequency is correlated with breakdown, as it should be.

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Application to Crystalline Polyethylene

We have applied the average electron method to pure crystalline polyethylene, which results in an estimated break-down field of 11x109 V/m based on average electron model as shown in Figure 5. The effective electron mass was 1 m0 for simplicity. The calculated breakdown strength of crystalline PE is very high compared with experimental breakdown strength of technical PE, which is in the range of 1.6 x108 V/m [12]. No intrinsic breakdown data exist for pure crystalline PE, and the material used in the experimental studies has about 50% crystallinity and a density in the range of 0.95 g/cm3 which implies ~10% free volume in the amorphous regions. As well, technical PE contains appreciable chemical impurities, such as carbonyl, which create impurity states in the bandgap [13]. We are in the process of computing the intrinsic breakdown of PE with the inclusion of carbonyl chemical impurities and will pursue the effect of density fluctuations (morphology) and nanocavities in the future.

CBM 9 10 11 12 13 14 15

0.1

1

10

100

1000

10000Average Energy LossEnergy Gain at 80 x 108 V/mEnergy Gain at 115 x 108 V/mEnergy Gain at 130 x 108 V/m

Ave

rage

Ene

rgy

Gai

n an

d L

oss

(1014

eV/s

ec)

Energy (eV) Figure 5. Crystalline polyethylene average energy gain and energy loss.

Once we can estimate intrinsic breakdown based on the above approach for both covalently bonded and ionic materials, the next step is to incorporate “defects” into the model, including chemical impurities (carbonyl, etc.), density fluctuations based on the statistical mechanics of an amor-phous polymer, and “gross” defects such as nanocavities. At the electric fields of interest, a carrier can acquire ~0.5 eV/nm from the electric field, so that in going through a 10 nm cavity, a carrier can acquire in the range of 5 eV. Thus such cavities, as well as other forms of defects, may be important to extrinsic breakdown and electrical aging.

CONCLUSON

A first principles method of modeling intrinsic breakdown has been developed to estimate the intrinsic breakdown field of covalently bonded dielectric materials. Our calculations adopt an average electron model which assumes breakdown occurs when the average energy gain exceeds average energy loss to the phonons over a wide range of electron energy. We have implemented this formalism to calculate the average

electron energy gain and average electron loss, from which the intrinsic breakdown field can be estimated. The calculated breakdown fields for diamond, silicon and germanium are in good agreement with experimental data. Application of this method to crystalline PE leads to a much larger computed breakdown strength compared to that for technical PE, which suggests the need to consider extrinsic factors such as physical disorder and chemical defects in the case of such systems. We are also extending our theory and computations to ionic materials, after which we will address the problem of estimating extrinsic breakdown for engineering materials through inclusion of physical and chemical “defects”.

REFERENCES

1. A. Von Hippel, “Electric breakdown of solid and liquid insulators”, Journal of Applied Physics, 8, 815-832 (1937).

2. H. Frohlich, “Theory of electrical breakdown in Ionic Crystals”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 160, No. 901, 230-241 (1937).

3. H. Frohlich, “On the Theory of Dielectric Breakdown in Solids”, Proc. R. Soc. Lond. A 188, 521-532 (1947).

4. R. Stratton, “The Theory of Dielectric Breakdown in Solids”, Progress in Dielectrics, Vol 3, 235-292 (1961).

5. M. Spark, D. L. Mills, R. Warren, T. Holstein, A. A. Maradudin, L, J. Sham, E. Loh, Jr., and D. F. King, “Theory of electron-avalanche breakdown in solids”, Physical Review B, 24, 3519-3536 (1981).

6. Quantum ESPRESSO, URL: http://www.quantum-espresso.org .

7. P. B. Allen, “Neutron Spectroscopy of Superconductors”, Phys. Rev. B 6, 2577-2579 (1972).

8. O. D. Restrepo, K. Varga, and S. T. Pantelides, “First-principles calculations of electron nobilities in silicon: Phonon and Coulomb scattering”, Applied Physics Letters 94, 212103 (2009).

9. J. N. Park, K. Rose, and K. E. Mortenson, “Avalanche Breakdown Effects in Near-Intrinsic Silicon and Germanium”, Journal of Applied Physics, Vol 38, 5343-5351 (1967).

10. P. Liu, R. Yen, and N. Bloembergen, “Dielectric Breakdown Threshold, Two-Photon Absorption, and Other Optical Damage Mechanisms in Diamond”, IEEE Journal of Quantum Electronics, Vol. QE-14, No. 8, 574-576 (1978).

11. Martin A. Green, “Intrinsic concentration, effective densities of states, and effective mass in silicon”, Journal of Applied Physics, 67, 2944-2954 (1990).

12. S. O. Pillai, Solid State Physics, New Age International, New Delhi, 2005.

13. A. Huzayyin, S. Boggs and R. Ramprasad, “Quantum Mechanical Studies of Carbonyl Impurities in Dielectric Polyethylene”, IEEE Trans. on Dielectrics and Electrical Insulation, Vol. 17, No. 3, pp.920-925, 2010.

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