another look at the theory of electrostatic field ionization

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Electrical & Electronic Engineering

High-Electric-Field Nanoscience

ANOTHER LOOK AT THE THEORY OF ELECTROSTATIC FIELD IONIZATION Richard G. Forbes Advanced Technology Institute & Dept. of Electrical & Electronic Engineering, University of Surrey [Permanent e-mail alias: [email protected]]

APTM16, Gyeong-ju, July 2016

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ANOTHER LOOK AT THE THEORY OF ELECTROSTATIC FIELD IONIZATION METADATA This oral presentation was made at the International Conference on Atom Probe Tomography and Microscopy (APT&M 2016), held in Gyeongju, Korea, 12 to 17 Jun, 2016. It may be cited as: "R.G. Forbes, Internat. Conf. on Atom Probe Tomography and Microscopy (APT&M 2016), Gyeongju, Korea, June 2016. [Paper O-C1-6, Abstracts p. 79]. doi:." In a few places, slight modifications have been made to the original presentation, to enhance clarity of a non-oral presentation. Some key references have been added at the end of the file. The author's permanent e-mail alias is: [email protected] . This document has been allocated a doi number by ResearchGate. At the time of writing, this number can be found by "clicking" on the image at the left of the relevant entry in my list of contributions. This will bring up a framed copy of the document with the doi at the top left of the frame.

1

Introduction: Terminology

Terminology: International System of Quantities In the 1970s, the International Standards Organization (ISO) introduced the SI units system, and recommended that the related system of quantities and equations be the preferred system for scientific communications. At that stage, the system of equations was called the metre-kilogramsecond-ampere (mksa) system, [or the rationalized metre-kilogrammesecond (rmks) system]. Since 2009, the "official" system of quantities and equations has been called the International System of Quantities (ISQ). It is now easier to make a distinction between choice of a system of equations and choice of a system of units. Field electron and ion emission tend use ISQ equations, but an ISQcompatible customary unit system based around the elementary charge e , the electronvolt (eV) and the nanometre (nm).

Terminology: Why "Electrostatic" FI ? It is well established that electrostatic fields can cause field ionization (FI). I now call this effect electrostatic field ionization (ESFI). Some scientists think that electromagnetic fields, too, can cause FI; this effect can be called electromagnetic field ionization (EMFI). Some scientists think that ESFI is the low-frequency limit of EMFI, and that ESFI theory is the low-frequency limit of EMFI theory. However, the link between ESFI and EMFI is strongly contested by Reiss, who argues that postulating a general link between EMFI theory and ESFI theory is not compatible with Einstein's theory of special relativity. Because this controversy is not resolved, I now prefer to use the term "electrostatic field ionization" when the field is not time-varying. There is nothing to stop a slowly varying electrostatic field from causing slowly varying ESFI.

Talk structure

1. 

Introduction: Terminology

2. 

Why look again at the theory of field ionization ?

3. 

Queries about molecular cluster theory

4. 

ISQ derivation of the Landau and Lifschitz formula

5. 

Comparisons with older FIM-type ESFI theory

6.

Illustrative applications

Relevance of ESFI 1. ESFI is basic theory for the field ion microscope (FIM), gas field ion source (GFIS) and helium (etc.) scanning ion microscopes. 2. ESFI is the theory of post-field-ionization in field evaporation. [Several theoretical weaknesses in Kingham's theory are known.]

There are also recent questions about the APT of complex semiconductors that appear to involve electrostatic field ionization:


There are also recent questions about the APT of complex semiconductors that appear to involve electrostatic field ionization: 3. Can O– and O2– ions escape from the apex of an operating emitter ? [Or would they be neutralized by ESFI as soon as they bounce on the surface?] 4. Can neutral emitter atoms, formed in surface or near-surface break-up

processes, reach the detector region without suffering field ionization ? 5. In the theory of APT of wide-band-gap semiconductors, can an electron escape from being trapping in a surface state, by means of field ionization into the conduction band ?


There are also recent questions about the APT of complex semiconductors that appear to involve electrostatic field ionization: 3. Can O– and O2– ions escape from the apex of an operating emitter ? [Or would they be neutralized by ESFI as soon as they bounce on the surface?] 4. Can neutral emitter atoms, formed in surface or near-surface break-up

processes, reach the detector region without suffering field ionization ? 5. In the theory of APT of wide-band-gap semiconductors, can an electron escape from being trapping in a surface state, by means of field ionization into the conduction band ? It seems likely that an updated theory of ESFI would be helpful. But developing it may be a long process. This paper reports a beginning.

3

Queries about molecular-cluster theory

Molecular clusters in high fields

In connection with the APT of oxides, there have been recent calculations on "oxide clusters in an external electrostatic field". A significant result of these calculations has been reported as "the HOMO-LUMO gap closes". [HOMO = Highest Occupied Molecular Orbital] [LUMO = Lowest Unoccupied Molecular Orbital] [HOMO-LUMO gap is like a bandgap in a semiconductor]

These calculations are a welcome addition to the literature, but I have two unanswered questions: (a) Are these calculations relevant to real emitters ? (b) What do we mean by "the gap closing" ?

Are these calculations relevant ?

field

cluster in free space

Are these calculations relevant ? HVG field field

cluster in free space

vacuum system wall

clusters on emitter surface

? Does the field actually "get at" the inside of a cluster when it is on the emitter surface ? ? Or is the field screened, in some way ? [There is a similar question in field electron emission theory.]

What does "closing the band-gap" mean ?

Ec Ev

Emitter Surface

Vacuum


Ec Ev

Ec Ev

Emitter Surface

Vacuum


Ec Ev

Ec Ev

Ec Ev

Emitter Surface

Vacuum

What does "closing the band-gap" mean ? Emitter Surface Difficult to escape ?

Ec Ev

Ec Ev

Easy to escape ?

Vacuum


ELUMO

Gap

exists EHOMO

ELUMO EHOMO

NO gap

Does "closing the band-gap" imply that a tunnelling barrier still exists ?

NO gap

ELUMO EHOMO

Tunneling barrier

ELUMO→Ec ELUMO→Ev

Electron tunnels

Tunnelling barrier implications

Tunneling barrier

ELUMO→Ec

Electron tunnels

ELUMO→Ev

If a tunnelling barrier of this kind exists at a semiconductor surface, then its theory is either: (a)

Zener-tunnelling theory.

(b)

Field ionization (ESFI) theory. [ESFI from surface electron state of some kind into the conduction band.]

4

ISQ derivation of the Landau-Lifshitz formula

4

ISQ derivation of the Landau-Lifshitz formula

An updated theory of ESFI - introduction

•  One needs to begin with a theory of hydrogen-atom ESFI in free space, even though we will want to apply theory near surfaces. •  Topic first sensibly discussed by Oppenheimer in 1928. •  First reasonably correct treatment by Landau & Lifschitz (LL) in 1958 [in first English edition of their textbook on Quantum Mechanics]. •  Many later treatments; sensible review by Silverstone & colleagues in 1977 which comments that: ESFI has proved mathematically tricky.

An updated theory of ESFI - introduction

•  Since 1958, it has been customary in the mainstream theoretical FI community to present ESFI theory using the atomic units equation system. In atomic units, the LL result for ESFI rate-constant kFI is written kFI = (4/Fau) exp[–2/3Fau] where Fau is electrostatic field as expressed in the atomic units system. This result is "valid in the limit of low fields". •  This equation form is not helpful for the APT community (and possibly other applied communities) because (a) most applied scientists do not understand this equation system; (b) for discussing ESFI near surfaces, we need to know how kFI varies with the ionization energy I (because the effective ionization energy varies with distance from the surface). It would also be helpful to have a result that is more general than the low-field limit.

An updated theory of ESFI - introduction Literature investigations showed the following. • 

There were several different opinions as to what power of I should be present in the pre-exponential, including I , I3/2 and I5/2. [However, since in atomic units the hydrogen ionization energy is 1, the atomic-units equation for kFI does not discriminate between these possibilities.]



• 

As far as I could discover, nowhere in the literature (either papers or textbooks) was there an ISQ derivation of the ESFI rate-constant formula, even though it is now 40 years since the ISO recommended that the ISQ should be the primary equation system for scientific communication (particularly between theoretical physicists and applied scientists and engineers).



• 

As far as I could discover, nowhere in the literature (either papers or textbooks) was there an ISQ derivation of the ESFI rate-constant formula, even though it is now 40 years since the ISO recommended that the ISQ should be the primary equation system for scientific communication (particularly between theoretical physicists and applied scientists and engineers).

•  In the English 3rd edition, LL do give a version of their formula that is valid in the Gaussian equation system [but many applied scientists are not familiar with this equation system either]. I have now created an ISQ derivation of the ESFI rate-constant formula. This may be found on arXiv, but the paper needs tidying up a little.

An updated theory of ESFI Since the hydrogen ionization energy is a universal constant in its own right, the proof appears to be more definitive if one creates it for a oneelectron atom with a nuclear charge of Ze . The outcome can be written in the alternative forms: kFI = CFI (Z5IH5/2/F) exp[–bZ3IH3/2/F] , kFI = CFI (I5/2/F) exp[–bI3/2/F] , where IH is the hydrogen-atom ionization energy, and I [=Z2IH] is the ionization energy of an atom with nuclear charge Ze. The second FowlerNordheim constant b and the (new) field ionization constant CFI are given by

b = (4/3)(2me)1/2/e! ≈ 6.830 890 eV–3/2 V nm–1 , CFI = 29/2me1/2/e!2 ≈ 1.245 354×1017 eV–5/2 V nm–1 s–1 ,

where e, me and ! have their usual ISQ meanings. The rate-constant formulae above are consistent with the LL Gaussiansystem formula, and with two other formulae in field ionization literature.

New conclusions from derivation

(1) 

If this derivation is strictly valid, it seems to imply that the usual JKWB approach to evaluating tunneling probabilities is valid ONLY IF the Schrödinger equation separates in Cartesian coordinates. If correct, this conclusion could have wide implications in scientific applications involving tunnelling.

(2) 

In their derivation, LL make a mathematical approximation that is known from modern FE theory NOT to be particularly good. This seems not to affect the low field result, but might enable a better formula at medium fields.

(3) 

Since this is a 3D formulation, it should enable us to explore the relationshp between 1D and 3D formulations of tunnelling problems [which is not well established at present].

Publication problem

There is a publication problem with this material. (1) 

Techically, a re-derivation of an existing formula in a different equation system is not new research.

(2) 

The proof is too long for the normal education journals.

(3) 

This problem remains to be solved, but the material can be found on arXiv.

5

Comparisons

An updated theory of ESFI We have deduced (by a 3-dimensional argument) the formula kFI = CFI (I5/2/F) exp[–bI3/2/F] , with &

b ≈ 6.83 eV–3/2 V nm–1 , CFI ≈ 1.25 ×1017 eV–5/2 V nm–1 s–1 .

I call this the ISQ version of the Landau & Lifshitz formula (ISQ-LL formula). For a hydrogen atom I = 13.6 eV. In a field F = 20 V/nm, the ISQ-LL-formula yields kFI ≈ 1.5 × 1011 s–1. This can be compared with (1): Gomer's 1960 formula––based on an approximate 1-dimensional treatment––which (put in ISQ form) is, for H : kFI ≈ νatt D = νatt exp[{1–2cF1/2/ I } × {–bI3/2/F}] , where D is the barrier transmission probability, νatt is the attempt frequency, and c is the Schottky constant [≈ 1.20 eV (V/nm)–1]. For the classical Bohr vibration frequency νatt= 6.57 ×1015 Hz, this yields kFI ≈ 1.8 × 1014 s–1 .

An updated theory of ESFI This also can be compared with (2): a 1-dimensional treatment derived from modern field electron emission (FE) theory, which yields kFI ≈ νatt D = νatt exp[v(µ) × {–bI3/2/F}] , where v(x) is a special mathematical function well known in FE, and µ is a barrier-modelling variable given (for H-atom ESFI) by µ = e3F/πε0I2. For values used earlier, this formula yields kFI ≈ 2.4 × 1013 s–1 .

An updated theory of ESFI This also can be compared with (2): a 1-dimensional treatment derived from modern field electron emission (FE) theory, which yields kFI ≈ νatt D = νatt exp[v(µ) × {–bI3/2/F}] , where v(x) is a special mathematical function well known in FE, and µ is a barrier-modelling variable given (for H-atom ESFI) by µ = e3F/πε0I2. For values used earlier, this formula yields kFI ≈ 2.4 × 1013 s–1 . The field ionization ESFI) time-constant τFI = 1/ kFI . Summarising : ISQ-LL gives

kFI ≈ 1.5 × 1011 s–1

τFI ≈ 6.5 ps

Gomer gives

kFI ≈ 1.8 × 1014 s–1

τFI ≈ 0.0056 ps

FE-type theory gives

kFI ≈ 2.4 × 1013 s–1

τFI ≈ 0.041 ps

So old theories of gas field ionization, found in old field ion microscopy textbooks (etc.), are probably not particularly good quantitatively. Kingham theory aimed to be similar to LL theory in free space, but has some difficulties of its own, too complicated to discuss here.

An updated theory of ESFI

Reasons for the differences between LL treatment and others seem to be: (a) The LL treatment is 3-D, but the others are 1-D. (b) The quantum mechanics of the LL treatment is different (and presumably better).

6

Illustrative applications

ESFI into conduction band Tunneling barrier

ELUMO→Ec

Electron tunnels

ELUMO→Ev

With significant hesitations as to applicability, let’s see what happens if we we apply the formula to MgO cluster data. Assume bandgap = 7.8 eV . Try various fields (since internal field may be/probably will be much less than external field).

ESFI time-constants Table: ESFI time-constants for parameters shown Gap

ESFI field

Time-constant

eV

V/nm

s

7.8

1

1045

7.8

2.5

10−6

7.8

5

10−6

7.8

10

10−12

7.8

25

5 × 10−16

ESFI time-constants Table: ESFI time-constants for parameters shown Gap

ESFI field

Time-constant

eV

V/nm

s

7.8

1

1045

7.8

2.5

10−6

7.8

5

10−6

7.8

10

10−12

7.8

25

5 × 10−16

Best conclusion is that ESFI formula is not obviously useful for this problem, at this time.

ESFI time-constants for oxygen-entities As heard earlier, there is a problem with measured oxide compositions. To add to the debate, here are some ESFI time-constants for oxygen-entities. Table: ESFI time-constants for species and parameters shown Species

Electron affinitty

Ionization energy

ESFI field

Time-constant

eV

eV

V/nm

s

O−

1.5

20

1 × 10−16

O−

1.5

40

2 × 10−16

O2−

0.45

20

1 × 10−15

O2−

0.45

40

3 × 10−15

ESFI time-constants for oxygen-entities As heard earlier, there is a problem with measured oxide compositions. To add to the debate, here are some ESFI time-constants for oxygen-entities. Table: ESFI time-constants for species and parameters shown Species

Electron affinitty

Ionization energy

ESFI field

Time-constant

eV

eV

V/nm

s

O−

1.5

20

1 × 10−16

O−

1.5

40

2 × 10−16

O2−

0.45

20

1 × 10−15

O2−

0.45

40

3 × 10−15

O

8.13

20

2 × 10−15

O

8.13

40

9 × 10−17

O2

12.07

20

5 × 10−13

O2

12.07

40

8 × 10−16

Conclusion: In high fields, remember that reaction pathways involving ESFI may be available.

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Thanks for your attention Citations and abstract follow

Some key references [slide number and origin

5. H.R. Reiss, J. Phys. B: At. Mol. Opt. Phys. 47 (2014) 204006. 23. J.R. Oppenheimer, Phys. rev. 13 (1928) 66. 23. L.D. Landau & E.F. Lifschitz, Quantum Mechanics, (1st English edition, Pergamon, Oxford, 1958, p. 257; 3rd English edition (corrected), Butterworth, Oxford p. 296. 23. T. Yamabe, A. Tachibana & H.J. Silverstone, Phys. Rev. A 16 (1977) 877. 26. R.G. Forbes, arXiv: 1412.1821v4 31. R. Gomer, Field Emission and Field Ionization (Harvard Univ. Press, Cambridge, Mass., 1960, p. 70; reprinted by AIP, New York, 1993). 39. M. Karahka et al., Appl. Phys. Lett. 107 (2015) 062105.

Abstract

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