Generic formula including many-body effects for field emission current

[hiyok]

Hello to every one. I'm presently embarking on a problem about electron emission under applied electric field. Of course, basically there are two mechanisms for electrons to escape into the vacuum from a sample: the thermionic and the tunneling. Here I'm concerned only with the latter. As usual, one assumes zero field in the sample duo to shielding effect. Often, the Nordheim-Fowler formalism is used to calculate the emission current. However, in some materials, such as one-dimensional, many-body effects are pronouncing, which cannot be accounted within the NF jargon. So, I'm wonderring if there's already in the literature a sufficiently general context to address this issue. I know there exists some, of which examples are given in the work by Bardeen and others. However, these methods involve approximations that are hardly justified. So, could anyone expose me to some extensive reviews on this problem? Thank you very much!

 

[ZapperZ]

Do you have any specific example where the "approximations" are not justified?

I can see why the FN formula may not work under certain cases because the analytical form made use of several level of simplifications (see several papers by Kevin Jensen). The analytical FN equation, for example, was derived at low temperatures (T~0K) and will show deviation when the temperature approaches 1000K. However, I haven't seen any kind of arguments or experimental results in which the "many-body" effects come into play to render the simplified description to be invalid.

Furthermore, unless you are arguing for the many-body effects inducing something like the Luttinger Liquid spin-charge separation in 1D conductors, I don't see why the renormalized Fermi Liquid would not work here. This is how we treat conventional conductors even in a quasi-1D geometry. I haven't yet seen any results from field emission that have argued for a many-body effect.

Zz.

 

[hiyok]

Dear Zz, thank you for your reply. Well, when I said the "hardly justified approximations",
I have just in mind the Luttinger liquid, which is the very topic I'm undertaking.

To clarify what I actually intended, I'd like to consider a specific example. Suppose there's a system composed of a sample and the vacuum. Let's take the vacuum into two parts: the channel (with electric field) and the far region (with no electric field). Now electrons shall tunnel to the far region from the sample through the channel. The tunneling current is determined by the rate of this occurence. This rate may be calculated via golden rule as usual. To this effect, one needs know how the sample is efectively coupled to the far region, that is, one needs the so-called transfer operator, which is often frequency dependent as typical of an open system. So, my question is, what's the transfer operator? Or, considerring the fact that this transfer operator may be found by elimination of the freedom degrees of the channel, we may ask, how to model the coupling between the sample and the channel?

Thank you!
hiyok

 

[ZapperZ]

Dear Zz, thank you for your reply. Well, when I said the "hardly justified approximations",
I have just in mind the Luttinger liquid, which is the very topic I'm undertaking.

To clarify what I actually intended, I'd like to consider a specific example. Suppose there's a system composed of a sample and the vacuum. Let's take the vacuum into two parts: the channel (with electric field) and the far region (with no electric field). Now electrons shall tunnel to the far region from the sample through the channel. The tunneling current is determined by the rate of this occurence. This rate may be calculated via golden rule as usual. To this effect, one needs know how the sample is efectively coupled to the far region, that is, one needs the so-called transfer operator, which is often frequency dependent as typical of an open system. So, my question is, what's the transfer operator? Or, considerring the fact that this transfer operator may be found by elimination of the freedom degrees of the channel, we may ask, how to model the coupling between the sample and the channel?

Thank you!
hiyok

You need to consider this important point: under what condition would this phenomenon manifest itself? In other words, it may be interesting, but is it important? When would such a thing occur?

I have never heard of a 1D field emitter, even though some of the protrusions that I've worked with in various models are pretty small in their transverse dimensions. All of them can be adequately described via the standard FN model. Even the more complicated scenario requires only a modest modification of the analytical FN model.

So why are you doing this?

Zz.

 

[hiyok]

Dear Zz,, the reason is Because I think carbon nanotubes may be a good 1D emitter, at least a quasi-1D emitter. For example, for arm-chair CNT, if the radius is small, the low-energy model may be a double-leg Hubbard ladder. So, in this regard, it may be as interesting as useful to kick the question.

regards,
hiyok

 

[ZapperZ]

Dear Zz,, the reason is Because I think carbon nanotubes may be a good 1D emitter, at least a quasi-1D emitter. For example, for arm-chair CNT, if the radius is small, the low-energy model may be a double-leg Hubbard ladder. So, in this regard, it may be as interesting as useful to kick the question.

regards,
hiyok

From all that I've read, the field current from such CNTs are adequately described by straightforward FN model. See this, for example:

http://cobweb.ecn.purdue.edu/~mdasilva/other.shtml

The deviation at high fields is also something we have seen in ordinary (non-1D) field emitters, which can be explained. So this isn't a many-body issue.

Do you have any references for any experiment that suggests otherwise?

Zz.

 

[hiyok]

Hello, Dear Zz, I now don't have direct evidences that suggest otherwise than FN model. Nevertheless, there is work by Watanabe and his collabrators who are seriously concerned with those correlation effects in field emission. They employed Density functional theory to study this issue. Please look at

1) Phys.Rev.Lett., 85:1750
2) ibid, 88:127601
3) Phys.Rev.B, 51:5278

Regards,
hiyok

 

[ZapperZ]

Hello, Dear Zz, I now don't have direct evidences that suggest otherwise than FN model. Nevertheless, there is work by Watanabe and his collabrators who are seriously concerned with those correlation effects in field emission. They employed Density functional theory to study this issue. Please look at

1) Phys.Rev.Lett., 85:1750
2) ibid, 88:127601
3) Phys.Rev.B, 51:5278

Regards,
hiyok

Unless I missed something, none of those have anything to do with 1D conductors or any Luttinger liquid effects.

Zz.

 

[hiyok]

Dear Zz, I never claim those work have anything to do with Luttinger liquid properties, rather, what I have been all the time stressing is that there are indeed theoretical reasons to go beyond FN jargon. But I do believe that, if one could fabricate perfect single-wall CNT, Littinger liquid effects will eventually come into play. By 'perfect', I mean long, thin and single.

hiyok

 

[ZapperZ]

Dear Zz, I never claim those work have anything to do with Luttinger liquid properties, rather, what I have been all the time stressing is that there are indeed theoretical reasons to go beyond FN jargon. But I do believe that, if one could fabricate perfect single-wall CNT, Littinger liquid effects will eventually come into play. By 'perfect', I mean long, thin and single.

hiyok

It depends on what you mean by FN model. If you mean the analytical form of it, then we already know that. I mentioned Kevin Jensen papers where he went beyond the standard FN model to include a larger range of parameters, and to include temperature effects that can significantly alter the Fermi distribution.

However, if you mean that you can actually DETECT difference between one model versus another, then that's another issue. That's why I asked you if there are any experimental observations that would necessitate that. You need to understand a little bit what is involved in experimentally measuring the field-emisson current, especially on the accuracy of what is being measured to see if such an effect and such a difference can be detectable. You have to make a decision on the effective area of the emitter, which is not trivial and highly model-dependent. You also have to consider if other factors do not overpower any many-body effects. This includes the Nottingham effect that can greatly influence the field emission current.

This point that I've been making from the very beginning. Is there any physical reason from the many field emission observation that we have to indicate the presence of the influence of many-body effects? This isn't meant as a put-down. I work in this field as well, and it would be interesting if I can actually detect such a thing. From all the field-emission data that I've looked at, I haven't seen any such indication.

Zz.

 

[hiyok]

Dear Zz, thanks for your advice. Ok, I'll collect more experimental data to gain better feel for field emission. For the moment, I have an idea. As known, usually the field emission energy distribution (FEED) as a function of energy should have a sharp peak about the Fermi energy as a result of finite quasi-particle residue, which measures the discontinuity of occupation at this energy. However, such a peak should disappear for a Luttinger liquid, which doesn't bear the mentioned discontinuity at all. How do think about this?

hiyok

 

[ZapperZ]

Dear Zz, thanks for your advice. Ok, I'll collect more experimental data to gain better feel for field emission. For the moment, I have an idea. As known, usually the field emission energy distribution (FEED) as a function of energy should have a sharp peak about the Fermi energy as a result of finite quasi-particle residue, which measures the discontinuity of occupation at this energy. However, such a peak should disappear for a Luttinger liquid, which doesn't bear the mentioned discontinuity at all. How do think about this?

hiyok

How do you plan on collecting and measuring such energy? You can't just simply collect the emitted electrons, because they have been altered due to the external field, which isn't uniform especially in the vicinity of the tip. It gives you a very wrong value of any energy measurement since these aren't the energy that the electrons had while they were in the material, unlike, say, photoemission spectroscopy.

Zz.