Dear Narendar,

first to Yuri's suggestion. This (googling) is always a good starting point, but it can often mislead. it seems that there exist a whole number of active (!!) devices, where one talks about negative capacitance. This is not really interesting, because it does not address the real problem, namely, "...Is there a physical process in condensed phase, where the electrical response is correctly described by the negative capacitance ?" Remember that if there is an inductive process (be that standard self-induction and/or kinetic induction - ballistic transport or superconductivity) taking place in your system under test SUT), it can be misinterpreted as a negative capacitance.

The only serious attempt to attack this "problem" to my knowledge has been made by A.Jonscher (see the attached article) . We confirmed some of his results, but do not agree with others. 

Myself, I have not had time yet to write about the phenomenon, but my preliminary conclusion is that it is indeed a real physical effect. With the step voltage input (Heaviside), the usual current output from the any SUT is non-increasing, that is, it either decreases with time and/or is constant with time( for time-> "infinity").  There are though apparently situations, where, within some time interval, the responding current increases. This then causes the "negative capacitance effect". 

We have seen this type response (numerical solutions of Maxwell equations) even in the ultra pure, defect free, single crystal Silicon samples with Schottky-Schottky electrical contacts. The effect becomes stronger if the sample contains a deep defect level. It is then seen also experimentally.

I hope this will help you on the way.

With best regards

Petr 

Do your homework yourself. Google will help you! Good luck

Dear Narendar,

first to Yuri's suggestion. This (googling) is always a good starting point, but it can often mislead. it seems that there exist a whole number of active (!!) devices, where one talks about negative capacitance. This is not really interesting, because it does not address the real problem, namely, "...Is there a physical process in condensed phase, where the electrical response is correctly described by the negative capacitance ?" Remember that if there is an inductive process (be that standard self-induction and/or kinetic induction - ballistic transport or superconductivity) taking place in your system under test SUT), it can be misinterpreted as a negative capacitance.

The only serious attempt to attack this "problem" to my knowledge has been made by A.Jonscher (see the attached article) . We confirmed some of his results, but do not agree with others. 

Myself, I have not had time yet to write about the phenomenon, but my preliminary conclusion is that it is indeed a real physical effect. With the step voltage input (Heaviside), the usual current output from the any SUT is non-increasing, that is, it either decreases with time and/or is constant with time( for time-> "infinity").  There are though apparently situations, where, within some time interval, the responding current increases. This then causes the "negative capacitance effect". 

We have seen this type response (numerical solutions of Maxwell equations) even in the ultra pure, defect free, single crystal Silicon samples with Schottky-Schottky electrical contacts. The effect becomes stronger if the sample contains a deep defect level. It is then seen also experimentally.

I hope this will help you on the way.

With best regards

Petr 

Thank you so much Petr Viscor,

What you said is correct, I had gone through googlig but, unable to find which is correct one. I hope this paper will help me to understand the "Negative Capacitance".

Thanks for your concern and valuable suggestions.

Narendar -

The paper that Petr has uploaded actually tells it all, about the origin of negative capacitance, from mathematical viewpoint (Fourier transform of the transient current in response to applied voltage step). Microscopic physical mechanisms leading to a non-monotonically decreasing current waveform (for example - increasing waveform) are different in different device types.

I should say that there is a different "negative capacitance" (NC) effect that is being actively researched in the last decade or so - related to ferroelectric materials, with polarization switching. This is a quite different type of phenomena, and I am not sure if "negative capacitance" term is the right one here, since in ferroelectric switching the charge changes abruptly (discontinuously) with switching / voltage, and hence a derivative (of charge with respect to voltage = C) is not defined, strictly speaking. But NC term is used there, introducing quite a confusion, in my opinion.

Maxim Ershov

Dear Naendar,

Both by definition (*The capacitance, C is the ability of a body to store an electric charge*) and formalization (*The capacitance, C, is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors*) capacitance conceptually can’t be negative conception.

So, actually this is nothing but slang that is used by non-educated authors.

Yet, sometime it may be used (in the gauche *non-scientific* discourse) in the comparative sense, i.e. if 2 bodies have capacitance C1 and C2 while C1>C2, than the difference ΔC= C1-C2>0 may be read (but only read!) as the negative capacitance of the second body relatively the first one.

This way, negative capacitance as an entity simply doesn’t exist. Don’t be confused.

Good luck.

V. Dimitrov

PS. Well, RG isn’t area for *child’s questions*. So, following Prof. Geletii, read first any text-book (or Google).

VD.

Dear Naendar,

Both by definition (The capacitance, C is the ability of a body to store an electric charge) and formalization (The capacitance, C, is defined as the ratio of the magnitude of the charge on either conductor to the potential difference between the conductors) capacitance conceptually can’t be negative conception.

So, actually this is nothing but slang that is used by non-educated authors.

Yet, sometime it may be used (in the gauche *non-scientific* discourse) in the comparative sense, i.e. if 2 bodies have capacitance C1 and C2 while C1>C2, than the difference ΔC=C1-C2>0 may be read (but only read!) as negative capacitance of the second body relatively the first one.

This way, negative capacitance as an entity simply doesn’t exist. Don’t be confused.

Good luck.

V. Dimitrov

PS. Well, RG isn’t area for *child’s questions*. So, following Prof. Geletii, read first any text-book (or Google).

VD.

Some references on negative capacitance:

1. A. K. Jonscher, "The physical origin of negative capacitance", J. Chem. Soc. Faraday Trans. II, vol. 82, pp. 75-81, 1986.

2. M. Ershov, H. C. Liu, L. Li, M. Buchanan, Z. R. Wasilewski, A. K. Jonscher, "Negative capacitance effect in semiconductor devices", IEEE Trans. Electron Devices, vol. 45, no. 10, pp. 2196-2206, Oct. 1998.

Dear Narendar and Vasilij Ivanovich,

to Vasilij I would say that in my view, it is not fair to critisize Narendar for asking this question. Firstly, it is not childish, but relevant and intriguing and secondly, as Maxim points out, it seems to be a real effect and not "a slang, used by the un-educated authors".

It is a purpose, again in my view, of RG forum to discuss this type of non-trivial (but also trivial must be welcomed !!) questions and I am always happy to see this here.

Negative capacitance (and again, Maxim might be right that one should perhaps call it different name) is a dynamic effect when one measures electrical response in materials (non-linearity migh be playing a role). As I have already indicated a year ago, it is apparently a result of response current being non-monotonic, going through a minimum (in time) and then increasing before becoming time independent. This happens even in perfect monocrystals with residual deep defects and/or under dc bias voltage load. As an example, I will leave you with a plot of complex capacitance, measured in a perfect monocrystalline Silicon sample (Schottky-Schottky contacts) dc biased to -0.1 Volt.

with best regards

Petr

Deaf Petr Viscor.

Your remark indirectly confirms my point of view. The decrease of capacitance (minimum what are you talking about) is of relative nature and in fact is nothing but the interim capacitance drop at subsequent time t2 as compare to previous time t1. In this comparative sense it may be read as negative phenomenon. But absolute negative value C= ̶ |c| simply doesn’t exist (like negative mass or viscosity or temperature in Kelvin scale, etc.) ̶ this was the essense of my remark.

So, certainly the very conception of “negative capacitance” is nothing but mislead parlance. That is the physical point of view.

Another situation occurs in computational math. when one solve numerically Navier ̶ Stokes equation. It is well ̶ known that a negative term that may be read as “negative viscosity” appears in the computational formula. Yet, it should be absolutely clear that this term is of artificial nature providing only the very stability of the computing circuit on the whole. Physically it is meaningless.

Regards, V. Dimitrov

Dear Vasilij Ivanivich,

let me agree and disagree with you :

1. I agree that the phenomenon of "negative capacitance" is of dynamical nature and NOT of static nature , something you discuss in your remark, with which I have no problem.

2. It is a known EXPERIMENTAL fact that when measuring optical properties of metals, the real part of the dielectric response function goes negative at low frequencies. This is not an artefact of some mathematical numerical analysis.

3. For linear and causal response processes, it also a known fact that the response functions, such as dielectric response function (~capacitance of the system) is time(frequency) dependent and as one moves from high frequencies through the transition, the real part diverges to minus infinity and re-occurs from plus infinity below the given optical resonant transition (Kramers-Kronig integral relation between the real and imaginary part of the response).

4. What about if we re-name this "negative capacitance" a "part of dynamical (time/frequency dependent) dielectric response of a system" ? Then we do not use the same name for two different things, but believe me , it is real enough and it is seen in experiments, not in numerical analysis. It also has Farad as unit.

5. The problem with this dynamic phenomenon is that it is often not clearly resolved and it might be due to some measurements errors. But as Andrew Jonscher pointed out (and he certainly knew what he was talking about) , we should not be scared to discuss this effect, even if running the risici that we will be proven wrong. We, experimentalists , we see this too often and usually we measure it well above the noise level and after eliminating inductive effects.

In conclusion, I think we talk about two different things, the static and the dynamic capacitance.

With best regards

Petr

Petr -

I think that another source of confusion regarding negative capacitance is that the most of the people were exposed to the concept of capacitance in electrostatics - where the charges can accumulate only on the (surfaces of) conductors, and do not flow through the space between the conductors (dielectrics). In electrostatics, indeed, capacitance cannot be negative (excluding some special cases, like moving boundaries of the conductors with voltages etc.). Also, in electrostatics, capacitance is directly related to electric field energy.

In more general case - semiconductors, dielectrics with losses, metals, and so on - the concept of capacitance is generalized, and capacitance is defined through complex admittance:

Y=dI(omega)/dV(omega)

C(omega) = Im(Y(omega))/omega

In this case, capacitance is determined by various complex physical phenomena - carrier transport, injection, capture/emission, etc. It is not related to energy, and it is not determined by DC changes of charges in response to applied voltage step. The changes of charges may be distributed all over space, and it may be impossible and unphysical to associate them with this or that conductor (terminal, contact,...).

This more general definition of capacitance coincides with C=dQ/dV definition in the case of electrostatics (no carrier movement between conductors).

The problem is that this generalized concept of frequency-dependent capacitance (and admittance) is not discussed in the textbooks - although frequency-dependent dielectric permittivity (or susceptibility) is being discussed widely.

Maxim

Dear Maxim,

yes, I agree and this is precisely what I was trying to point out to Vasilij. In fact, it is just one possible representation of the electromagnetic (well, electrical) response to external perturbation, on the same footing as the optical spectroscopies, the only difference being the classical frequency range (from dc to ~1 THz).

In my analysis of the response, I usually relate it to the space-time changes/evolution in the local total electrical charge density ro(x,t). Interestingly enough, it kind of ressembles , the density functional method in electron band structure calculations.

With best regards

Petr

Dear Peter -

yes, agreed, this time-dependent "response function" or frequency-dependent "transfer function" (the one is being Fourier transform of the other) are very general and useful things. In different areas of physics/engineering they are called different things:

- complex admittance (complex frequency-dependent conductivity and capacitance)

- freuqnecy-dependent dielectric function

- complex frequency-dependent capacitance (real part is the capacitance, and imaginary part is G/omega)

The space-time evolution of the charges and related processes (trapping/emission, injection, extraction, etc.), different in different physical systems, can be used to explain the specifics of the transient or frequency-dependent response (like negative capacitance). But in most cases it's impossible to assign these changes to specific contact - these are things distributed all over space.

Regards,

Maxim

Dear Maxim,

it is always nice to see that one's view agree with the view of a colleague. I hope that Narendar has a benefit of our discussion also.

You write further :

"...But in most cases it's impossible to assign these changes to specific contact - these are things distributed all over space..."

Here, I do not know whether I understand. If, by contact, you mean metal electrodes - sample interface regions, then I have to say that this is not impossible, although it may be a difficult task. In fact, the numerical analysis of the electrical response IS a boundary value problem and for simple systems (like a monocrystal, glass and/or simple liquid), the modelling of boundary response is relatively easy. This is not so for more complicated systems, like polycrystals, spatially inhomogeneous systems (human tissue) etc..

So , although various physical processes may be distributed in various spatial regions, we can model these, using reasonable simplifications.

With best regards

Petr

Dear Peter -

yes, "contact" is a term usually used in semiconductor device community to denote an equipotential system (usually a highly conductive one - a conductor / metal), which voltage / potential is well defined, and which can be used to control a device - either by applied voltage or by injected current.

Regarding my comment on difficulty to attribute a change of charge to a specific contact, when a step voltage is applied to one of the contacts - what I mean is that the definition of capacitance C=dQ/dV is (strictly) valid only in the case of electrostatics, when there is no DC current between contacts. This definition should not be used in systems with non-zero DC conductivity.

In electrostatics, dQ is the change of charge on one of the contacts, does not matter on which one (in a two-terminal devices) - one contact gets charge increment +dQ, and the other -dQ. The sum of the two charge increments is zero.

In devices with non-zero conductivity, the charges (mobile charges - electrons and holes, as well as "fixed" charges - traps, impurities, defects, etc.) are spread all over the space between the contacts/terminals, and the incremental change of the charge distribution dQ(x) (x is the coordinate - 1D, 2D, or 3D) cannot be attributed, in general case, to one or the other contacts. That was my point.

Regards,

Maxim

Dear Maxim,

I think, I understand what you meant. What I mean by contact is indeed the interface plane, separating the metal electrode and the sample (System Under Test - SUT). In this way, the boundary conditions can be well defined and modelled.

The important point is though that the external charges, applied to the sample and sitting within the Debye screening length on the metal electrodes sides of the interface planes that form and keep the external applied voltage constant, do not enter at all the calculation of the space-time evolution of the total and local electrical charge density ro(x,t). This quantity is the el.charge denisty within the sample, while the external applied charges enter the calculation/simulation of the response only through the value of the applied dc voltage.

Therefore, as you point out correctly, ro(x,t) describes all the charges throughout the entire sample, be those localised and/or delocalised (finite mobility).

Once the corresponding currents j(x,t) are known(calculated), the space integral of those gives the response of the system J(t) to the applied voltage step Vo (for example) and the LaPLace transform of the J(t)/Vo is then the frequency dependent (in general) admittance of the sample. When divided by (iw), you get representation of the response in terms of the complex capacitance C [Farad] , a perfectly well defined quantity, irrespective whether the system is a pure dielectric, pure conductor and/or a mixture of both.

So, at no point one is attributing the charge ro(x,t) to charges at one or the other electrode. ro(x,t) is just response of the system to the applied voltage which, in turn, is faciliated by the charges at the electrodes.

To me it seems that we do not really disagree here. I just want to stress that the measured/transformed response in terms of complex capacitance has also very definite simple physical meaning in certain frequency regions, like geometrical capacitance of the bulk of the sample, geometrical capacitance of the interface regions of SUT and/or the strength (density) of the deep level that relaxes towards equilibrium/steady state, when the external voltage is applied.

And of course and under special circumstances, the real part of this complex capacitance can go even negative (our previous discussion).

With best regards

Petr

Dear Petr Viscor,

ok, this[1] is (among others) a "perfectly well defined quantity"; however, this defined quantity is NOT, always, a weighty "(complex) capacitance C [Farad]"[1], apart the archetypal case study, of an ideal C (model) and some related feeble models. In practice, we are exceedingly spreading this fanciful calculation[1] to some "near C" cases (R//C), e.g. the turn up of the Cimag. part, also. To sum things up, the result, from this[1] calculation, should be taken under some precautions, and a possible (but bizarre) case, with a Creal

Dear Ioannis,

let us not misunderstand each other here. :

1. One thing is a transform C(w)=Y(w)/(i*w) and/or Y(w)=1/Z(w) and so on.. and other thing is: Does this complex function has a clear simple physical meaning. Yes, it does in certain frequency ranges , but the rest is just part of the response, expressed in units of Farad.

2. From the fact that in most cases in passive systems, the response is delayed in time relative to the perturbation, Z(w) and thereby also

Y(w), C(w), L(w), M(w) ,..(and other functional forms of the same response) are complex quantities.

3. In fact, one need not talk about complex capacitance, one can just use the word "response" or complex impedance. But I think we agree here.

4. Real part of complex Capacitance (and of complex dielectric response function) in metallic Drude systems like Au, Cu or Pt are also negative at low frequencies, so the so called negative capacitance is not really something unique to semiconductors/insulators/glasses.

5. I can only agree with you that for the time being, it must be still considered as "fragile hint", mostly, as Andrew Jonscher often stressed, because peoples are often "shy" to talk about this subtle effect. My point was anything BUT an attempt to "..proof for negative C scenario...". I would be seriously misunderstood here. What I say is only that the phenomenon is seen in some experiments and I would add here that it suprisingly enough comes out also in the corresponding numerical solutions of governing set of Maxwell equations.

6.For illustration, I attach a figure. It is a measurement of the electrical response and of the corresponding theoretical calculation in ultra pure monocrystalline Silicon sample with Schottky-Schottky contacts. The other examples I believe are in the two books by A.Jonscher on Dielectric Relaxation in Materials and Universal response.

With best regards

Petr

Dear colleagues (involved persons and all, whom to it may concern).

To be open, I wasn’t going to continue discussion on the capacitance, but seems this is my turn for “childish question”. The plot that was referred to Petr Viscor (minimum at H=10−3) is nothing but local extremum (no matter how small it is and what is the ratio real/imaginary terms).

Yes, in a physics we are dealing with some counter-phenomena (acceleration/deceleration (breaking), melting/freezing, resistance/conductivity, etc.). Yet, nobody defines conductivity as a “negative resistance”.

Formally, any number (rational, irrational, no matter) is characterized by TWO features, i.e. attribute (sign) and value (modulus) |X|. Now I’m talking on the attribute. So, we read -X as a negative number -X0. Usually, for positive number, we omit sign, hence we read X (no sign) as a positive number.

Do you see that 10(−10), 10(−100), 10(−Googol) (no matter) still are>0. These are positive figures! Reread the very definition. By definition, ”The capacitance is the ABILITY…, etc.”

So, the “childish question” is as follow: “Are you enable to image a “negative ability”?

Let me to confess - I’m unable.

To cover the issue, I believe that it’s proper time and place to remember grand René Descartes (Renatus Cartesius). I cite “…Define the sense of the words, and you will clean up the World from the half of the possible delusions …” - end of the citation).

In other words - ‘define the definitions’. You don’t like definition? Are you disagree?

OK, no objection - change it promptly! But up today, for me “negative ability/capacitance” is nothing but something like “negative temperature” (in Kelvin scale) or “fried ice”.

Sorry, but I’m out of the further discussion.

Regards to all.

V. Dimitrov.

Dear Vasili Ivanovich,

It seems to me that we two still talk about different things. In the attachment, I am trying to express my view as simply and as clearly as possible, in order that we might agree on something. Otherwise, we have to agree to disagree, at least at this point.

With best regards

Petr

Dear all,

Capacitance may generally be viewed as the density of potential energy stored in a dielectric medium due to certain formation of charge. In quantum mechanics, and at certain critical densities of charge carriers, e.g. electrons, the associate capacitance may become negative. It means that at these conditions, removal of a unit of charge (dq0), therefore C=dq/dV

Dear Pouya Dianat,

this is an interesting suggestion and it would be interesting to try to design an experiment to test this.

However, the apparent negative capacitance effect I am referring to above, is a dynamical classical phenomenon (the numerical simulations are using Classical Electrodynamics equations) and the charges involved are viewed as classical particles (even the electrons - semi-classical approximation and/or quasi-particle concept of L.Landau).

With best rgards

Petr

Dear colleagues,

From a strictly electronic point of view whenever any of you said "dynamic capacitance" I could not hear anything but "impedance" . Which can of course become negative , zero, or take an arbitrary value. If we mix them up, we get a fine mess. Yes there are a number of strongly non-linear materials, yes lately we got those meta-materials with interesting (global) properties, etc. But as far as I remember, the definition for capacitance is the electrostatic one while once we talk about dynamic effects we talk about impedance.Think of the so-called dielectric constant -and the fact that it really is not so constant but a function of frequency and material electric (and quantic ) properties.For not-so-small frequency spans and a great many situations we can approximate its behavior to a constant though. So yes, in a matter of speaking, one can get a "variable capacity" if the frequencies are variable and high enough , and in some circuits the behavior of certain elements can simulate a "negative capacitance"- of course it is just a complex impedance ..being itself. But yes if you want to I guess you can stick a label on it saying "negative capacitance". But it is in fact impedance- with a large C value, that happens to be negative at that particular point, and yes, it does happen to impedances once in a while in, say, oscillators and scintillating tubes etc. From my point of view, capacitance is a static construct, while impedance is a complex-valued construct , positive or negative or indeed zero- that can contain resistance, inductance and capacitance terms and may locally become Z=-C*i for example . That does not change the definition of capacitance- a static concept related to variations of voltage with charge. Yes there are materials and devices that sometimes exhibit negative impedance and it might just be capacitive at that particular instant under those conditions- but the definition of capacitance does not allow by itself negative values.

If you want a (silly) analogy, it is like your financial balance - a construct by definition made up of all your income (necessarily positive or God forbid, zero) and subtracting all your debt. In nasty cases the balance can be negative but that does not mean that your salary ,for example, is minus (some number) euros.Impedance can be negative and made up of only a capacitive term, in some cases- but it does not make the capacity itself negative. In the cases where metals under low frequency or metamaterials or various strongly non-linear materials are discussed, capacity might be a function but if we stick to classical definitions, it is positively defined. The effect of a locally negative function involving terms resembling capacity is a complex impedance. If the material is strongly non-linear one might argue that the capacitance of the material does simply not exist (* it does but is a function of frequency and other special conditions depending on material but in a classic sense we cannot say a capacitor with that material as a dielectric will have a capacity of.. xxx farads but that we built a sensor for.. frequency, temperature or whatever that has a variable capacitance output ). In a way think of a piezoelectric material. At rest it has a capacitance. Strike it with an impact hammer and the capacity varies- sometimes decreases- while generating a very very small current by re-arranging charge carriers.Local negative slope for capacitance- negative impedance yes, negative capacitance never. That is my take on the problem.

Dear Alexandru,

you are both right and wrong:

1. You are right that the electrical response is measured as frequency dependent complex impedance. That is what I was talking about.

2. You are wrong if you claim that the term capacitance reflects static electrostatics. In any material , the response can be viewed in a number of trivial transforms, all of which convey the same information :

Z(w) [Ohm]->Y(w)=1/Z(w) [Siemens]-> C(w)= Y(w)/(i*w) [Farad]. The complex capacitance C(w) is just generalisation of the dc limit C(w->0).

3. You are wrong when you state that it is only in "active" special devices that you see "negative capacitance" (the negative real part of the measured complex capacitance C(w)).

4. The "negative capacitance" is seen in passive devices when the output measured current (Heaviside step voltage input) increases with time in a certain time interval. This is confirmed in the "exact" numerical solutions of Maxwell equations for samples with Schottky-Schottky contacts. Also presence of deep levels in semiconducting materials leads to observation of this effect.

With best regards

Petr

Thank you dear Petr for your insight in the matter- indeed whenever discussing special materials and combinations thereof there are such effects. That Schottky double junction is such a construct.

Some schools of thought consider the capacitance as a static construct only and the generalization a matter of impedance- others embrace your definition. Of course your arguments are solid and, I might add, a little more..modern than the rigid concepts we were once taught :) . Special effects in doped crystals , Schottky double junctions and advanced semiconductor physics are indeed the bleeding edge of the art and I, for one, am happy to learn new things every day. Thank you for the clarification, I will look deeper into the matter and will come back . (Buddha said you cannot see the truth unless you have no opinion, and a thing is not wrong or right, it just is. Methinks- if only things were that simple...After this I have no opinion and therefore nothing more to say- until I find one....).

Dear Alexandru,

I am happy that I might have helped you on the way. please feel free to come back to discuss further.

i think budha was a wise man, but the sentense should probably go as follows :

"You can not see the truth unless you HAVE an opinion". Then I would understand the message.

You are right "a thing just IS..", but only until you start investigating. Then, at least in science, you might have a correct or incorrect description of this "..thing.." - part of ther Objective Reality. the judge is the empirical evidence (experiment and/or observation).

with best regrads

Petr

A slightly different look at NC problem, originated from studies of electrical double layers:

Please review the story described in the following publications.

Article Influence of the metal electrode on the capacitance of the c...

Article Density functional approach to the metal-solid electrolyte i...

https://arxiv.org/ftp/physics/papers/0208/0208048.pdf

Article The question of negative capacitance and its relation to ins...

Article Limitations and strengths of uniformly charged double-layer ...

Article Relaxing gap capacitor models of electrified interfaces

Please also read the answer below

Michael -

it is indeed a different take on NC effect.

Normally, in electrostatics, and in semiconductor (or, more generally - electronic devices) devices, it is implicitly assumed that the geometry of the system (conductors, dielectrics, the boundaries, etc.) is fixed. This is a pretty good approximation in electronic / semiconductor devices.

Allowing the system to change its geometry immediately leads to new interesting effects, in particular - NC effect. It's interesting to see how easily NC appears in a simplest system - a parallel plate capacitor, with one plate fixed, and another connected to a mechanical spring.

Maxim

Also, a microphone inside an op[1]-feedback circuit might be based on the above[2] parallel (with a moving CpP) plate, electro-mechanical capacitor (C), such as a mechanical spring-CpP, composite EC-model.

1. operational preamplifier, with a positive-feedback circuit.

2. Maxim Ershov, proposal neg-C-model, above.

Maxim Ershov:

"Normally, in electrostatics, and in semiconductor (or, more generally - electronic) devices, it is implicitly assumed that the geometry of the system (conductors, dielectrics, the boundaries, etc.) is fixed. This is a pretty good approximation in electronic / semiconductor devices. "

-----------------------------------------

Response:

Thanks, Maxim. To the readers not familiar with this field, we owe an explanation. In the context of our discussion, your statement quoted above may lead to confusion. Thus, someone may erroneously assume that electro-mechanical models with variable dimensions ("geometry") have no relation to the properties of interfaces important for electrochemistry or electronics, because the geometry of experimentally studied devices is reasonably assumed to be fixed.

Let us clarify the picture using electrochemical applications as an example.

While the macroscopic dimensions of electrochemical cells are indeed fixed, there are significant changes in the "geometry" of charge distributions induced at atomic scales by the electrode charge. For this reason, the interfacial electric double layer (EDL) is theoretically described as a microscopic capacitor with movable plates ( variable effective thickness h ). In some cases, h significantly decreases with charging, and this leads to the predictions of negative differential capacitance (NC). Thus, despite the fixed dimensions ("geometry") of the macroscopic cell (electrodes, electrolyte, boundaries , etc), the "geometry" changes of EDL are profound. Exactly these changes are responsible for NC, which is the focus of our discussion.

The electroelastic models serve to illustrate the origin of this phenomenon. Their "spring" is a metaphor for all forces involved in the equilibrium of interface polarized by the field of electrode (Coulomb, entropic, exchange and correlation - you name!). Their movable "plates" are conceptually associated with the centroids of microscopic charge distributions (electrons, holes, ions, polarization charges).

In other words, these toy models mimic the variable geometry of EDL. They allowed to understand the origin of NE and analyse possibility of its observation. More importantly, they allowed to predict and analise instabilities and phase transitions in EDLs associated with the onset of NC.

The elastic capacitor (EC) was the first model demonstrating NC in isolated system with controlled surface charge density. It also demonstrates NC-related charging instability in EC connected to the potential source (battery). A more advanced model is the squishy (elastic) capacitor (SEC) with lateral flexibility of its plates, demonstrates NC-related lateral instability in the isolated system Article "Squishy capacitor" model for electrical double layers and t...

Frequency (f)-dependent response of these models is affected by "NC -related" peculiarities such as the "mode softening" in the vicinity of critical point (vertical asymptote of C) . As a result, EC behaves as a classical fixed gap capacitor at high f, and sharply deviates from this (displaying strongly non-linear response) in the low f limit (this was mentioned in

Article Self-consistent electron theory of the metal-solid electroly...

and some later publications). This result is apparently related to the properties observed in your collaborative studies Article Negative Capacitance Effect in Semiconductor Devices

______________________

Ioannis, other conventional examples of electro-elastic behaviors are the electro-compression of lipid bilayers and piezoelectricity, electrostriction, electric actuators. Thanks for your comment.

I wonder, is it possibe to stay within classical mechanics and electrodynamics to understand NC phenomenon, or one must involve exchange, correlation and etc features of quantum theory? I hope, the answer is positive.

Dear Alexandr,

I see Negative Capacitance (not inductance) both in experiments and in the corresponding Classical Electrodynamics simulations. The fit is often quite "impressive". I stil work on the precise cause, the non-linearity and boundary conditions being the most likely candidates. Not to forget the delayed response of some mobile charges relative to the majority mobile charges.

With best regards

Petr

Dear Petr,

Nonlinearity is to me quite reasonable way to get NC. Look for, please, the nonmonotonic relaxation of mechanical stress, or deformation.

Dear Alexandr,

thank you for the reference to your work on visco-elastic properties of biological tissues.

There are similarities between electrical and mechanical relaxational phenomena and in phenomena like glass transition, melting, creep ,they are even related.

I will read your work with interest, but right now I have a problem with the concept of geometry caused non-linearity (p.210 of the book).

I can not see how a combination of ideal springs(constant, time independent elasticity) and ideal dampers(constant, time independent viscosity) can lead to genuinely non-linear behaviour .

In dielectric relaxation, this type of models (variouis topological nets, comprising ideal resistors(viscous flow) and capacitors(elastic deformation)) give rise to the distribution of various Debye-like relaxation times that can be fitted to the (non-linear) experimental results. This though does not mean that they(relaxation times) have a physical interpretation. The explanation, in my view, should be looked for in the non-linear nature of the "ideal springs" themselves. In atomic systems this leads to a genuine non-linear behavior (Fermi, Pasta, Ulam paradox).

With best regards

Petr

Dear Petr,

Thanks for prompt reply to my note.

1. It's a true thing that electrical cirquit with capasitors and resistors may give much more opportunities in modeling than network of springs and damps. I'm not enough expirienced in this field.

2. Two exponential pattern in relaxing is quite common in nuclear magnetic and electron paramagnetic resonance, with spin-spin, and much more rapid spin-lattice relaxation as a physical origin of it. They are usualy quick at first and then in time scale it goes a long tail, still being monotonically decreasing.

3. In nonlinear systems, however, we can observe nonmonotonic relaxing quantity (after step-wise Heaviside excitation). The derivative is at first positive, nul in maximum, and then, negative.

Dear Alexandr,

to your points :

1. wonder whether they do (R,C elements in electrical response as compared to springs and dampers in mechanical response). All these elements are based on the ideal response, that is to say R(ideal, time/frequency independent conductivity) and corresponding damper (ideal, time/frequency independent viscosity) are both purely dissipative, while C(ideal, time/frequency independent permitivity) and corresponding damper (ideal, time/frequency independent bulk modulus) are both purely non-dissipative (90 degrees out of phase to applied force). In both cases the characteristic response time is given either by conducto-permitivity relaxation time tau= permitivity/conductivity or Maxwell visco-elasticity relaxation time tau=viscosity/bulk modulus .

Now, I am not an expert in mechanical relaxation , but I believe the expressions are correct. In conclusion therefore I also believe that various topological networks should give similar type of response.

2. Not only two, in principal there can be even more of simple exponential relaxations (Debye relaxations) working in parallel/series.

3. Yes, and it is this I was referring to. You can get this type of response, if the system is spatially non-homogenoeus, but still built with ideal components. That we see in the electrical response, but in most cases, the required spatial in-homogeneities (spatial variations of Rs and Cs ->spatial variations of the corresponding conducto-permitivity relaxation times) are out of reasonable physical limits . For example the system will have to be composed of series impedances (each impedance consisting of parallel R(x) and C(x)) going from almost perfect insulator (conductivity->0) to almost perfect metal ( permittivity->0 idealisation though !!) . In conclusion here I therefore agree with your reference main point "...spatial non-homogeneity.." as being the cause of non-monotonic relaxation .

With best regards

Petr

Dear Petr,

Its a great pleasure to hear from you, very interesting for me, but i was engaged in that some years ago.

The idea was that some (even 2D topology) construction composed of linear Hook springs with different stiffness may have nonlinear behavior in stress-strain curve. Rhombic model with more soft transversal diagonal spring demonstrates 'toughening' effect as if its Young's modulus increase with stretching. This effect is quite common in biological tissues and play important role in natural behavior. Btw, coiled spring, as a prominent human investigation, has similar toughening after 'unfolding' of coils when stretching modulus works, while it is rather complinable in initial coiled state when shear modulus works.

As to nonlinear capacitor, i remember Misha Paryenskii's (my school mate) spring - in parallel - capacitor model which he envolved early in NC problem for electrolytes.

Good wishes in new year,

Respect,

A Kobelev

[email protected]

Dear Alexandr,

it is interesting to envisage a single block with an internal structure (Rhombic model), encompassing a number of different ideal Hook's springs with varying Bulk (Young) modulus. I have to think of the electrical response analogy.

However, in the linear response regime (all elements are stil ideal) it seems to me that the response is again that of a distribution of characteristic relaxation times (Maxwell visco-elastic relaxation times) . And morover, in order to get time/frequency dependent behaviour, dampers (dash-pots) have to be present as well.

The mechanical response has always intrigued me, perhaps it is time to get more involved in the matter. the Glass Transition phenomenon I have mentioned, does indeed involve both (inclusive NMR).

All the best wishes for the year 2019 to you as well !!

With best regards

Petr

Dear Petr,

If i worry you with my replies, please, tell me to have a break, don't hesitate. l'm pleased conversating weekendless.

1. What do you mean of Young's modulus being not constant? By hands? As to my mind, each nonlinearity is due to internal topological structure, at each scale, and it may be fractal.

2. If my memory serves me, linear Newton's damper can be attached to Hook's spring in two ways: in parallel (Kelvin-Foigt model), and then it has no relaxation at all, only hysteresis loop, and in series (Maxwell model), which fits well to mimic relaxation, but it has no equilibrium state, with infinite creep.

3. It's quite easy to programm both purely elastic, or posessing viscosity, complex (2D, or 3D) graphs to get response to uniaxial stretching, or perhaps, shear loading. I hope, i'm ready. Dependent on the goal, something may be easy in mechanics, smth in electrical cirquits.

Good luck,

A Kobelev

Alexander: is it possible to stay within classical mechanics and electrodynamics to understand NC phenomenon or one must involve exchange, correlation and etc features of quantum theory? .

Alexander, the answer depends on what do you mean by understanding the " NC phenomenon ". And primary, on to which phenomenon are you referring to? Please, describe the observations, and we will try to discuss their possible interpretation.

With best regards.

M

Dear Michael,

People believe the NC phenomenon exists, i can hardly imagine that its mechanism in one case differs from the other. Let's take the electrolyte boundary, for instanse.

Best wishes,

AK

Dear Michael,

to your comment to Alexandr :

1. By understanding I believe Alexandr means a sufficiently precise description of the phenomenon of Negative (dynamic) Capacitance using Classical Electrodynamcis and Classical Mechasnics. I believe this to be indeed the case, apart from the energies of the electrically active particles . These might have to be calculated quantum mechanically.

2. As to the observations, please have a look at the enclosed WD. I have chosen two cases, both for the "perfect" system, namely defect free monocrystalline Silicon, where almost everything is known to a high degree of accuracy.

With best regards

Petr

Thank you, Petr, for sending the real stuff for case study. Sorry if my questions are immature.

(1) Do you apply a constant component of voltage (should I say "non-zero bias"?) in you experiments?

(2) What is the equivalent circuit of your setting?

(3) Could you please reflect on the assumptions made to derive the sign and value of C from the measured imaginary component of impedance ? ( I understand the relation for regular RC (RCL) circuit, but your case is different).

(4) In your response, you asked for the classical example of the NC, and added "(dynamic)". Why?

PS You can contact me at [email protected] with anything that does not fit in the frames of public forum. Otherwise, we can use some collaborative tool (say GDocs) for more private discussion.

Thank you. Alexander. I am not aware of any unambiguous measurement of NC at the electrochemical interfaces which does not mean that they do not exist. If you find an example, similar to one sent by Peter, we can include it in our case study.

With best regards. M

PS If the private setting suits you better, you are welcome to use my email or a collaborative tool.

Dear Michael,

1) Please, have a glance at Oleg's old paper.

2) May i scetch down my general view. With my poor awareness, i left all high-level activities in the field to you, and to much yonger Petr. As to my abilities, i hope i can model NC analogue in massless viscomechanics. The capacitor equals spring, resistor - dashpot, (and induction coil works as inertion mass). The fish is, to me, in friction, which needs dinamics, and in nonlinearity. They both lead to nonmonotonic relaxation (see, pls. prviously attached file). According to cited by Petr paper, which i'v read a weak later, it may serve as an origin (they mentioned there 'positive derivative', but it's completely unstable case).

3) Memorizing one of ancient Kourovka's, and Kurkin's question to your report, the problem arises with NC prohibition by some conservation law. The answer was, i remember, in periods in time domain, with NC, while further the right things restore.

Good wishes,

Alexa

Sasha, evolution of our views on NC is reflected in the following publications.

Article Influence of the metal electrode on the capacitance of the c...

Article Self-consistent electron theory of the metal-solid electroly...

Article The admissible sign of the differential capacity, instabilit...

Article Limitations and strengths of uniformly charged double-layer ...

Article "Squishy capacitor" model for electrical double layers and t...

Article Relaxing gap capacitor models of electrified interfaces

I'm am sorry for self-references, but this is the most concise way to respond to your questions, explicit and implied.

My comments and questions to your paper are too technical, and do not satisfy the Q&A form of this public forum. So, it is better to discuss them in private setting.

Misha, i've send you reply at your private address [email protected].

Petr may also contact me privately, if needed.

A Kobelev

Dear Michael(Misha),

your questions are definitely NOT immature . Her I try to answer them :

(1) Do you apply a constant component of voltage (should I say "non-zero bias"?) in you experiments?

Not always, but the data I have sent you actually were measured under dcBias Voltage (~0.1 to 1.0 Volt). I know though that I have somewhere the data without dcBias and still showing Neg. Cap. effects.

(2) What is the equivalent circuit of your setting?

I do not have a good approximate linear equivalent R,C,L circuit for these effects, but I am sure that one can be made, although ideal inductance element would have to be involved and that would have no physical interpretation (simple). The response is not inductive.

In my simulations I solve the complete set of Maxwell equations in real space-time and then La Place transform them to get frequency dependent Admittance/Impedanca/Capacitance etc.. The agreement between the data and the numerical simulations are quite good in the case of one or two deep levels in perfect semiconductor (monocrystalline Silicon).

(3) Could you please reflect on the assumptions made to derive the sign and value of C from the measured imaginary component of impedance ? ( I understand the relation for regular RC (RCL) circuit, but your case is different).

As is clear from the above, I just solve the equations and get the impedance.

The convention that I follow when working with R,C.L networks is

Z(w)=Z1(w) + i*Z2(w). That gives -90 degrees for the purely capacitive response and +90 degrees for purely inductive response. I hope I have understood your question correctly.

(4) In your response, you asked for the classical example of the NC, and added "(dynamic)". Why?

1.NC-A classical phenomenon

I have not finished the analysis of this phenomenon ( there is also a truly negative phase, inductive response in some of these samples, which is due to ballistic transport - kinetic inductance effects that complicate things), but for the time being I can explain the data at hand just through classical electrodynamics and elements of classical mechanics ´(forces on the particles ). Therefore I do not believe that it is a pure Quantum Mechanical effect.

2. NC - A dynamical effect

Here I want to make a distinction between a dc concept of a capacitance and a dynamic measurement of a capacitance in order to avoid some critical comments like ...there is absolutely no NC in Nature... This kind of comments have occurred here.

When things settle in time (time->infinity) I stil believe that Q(charge)/V(voltage) = C(capacitance of the system). T least in the linear response regime.

With best regards

Petr

Dear Petr, your response is very interesting, and provokes many questions.Here is just a few.

1. In studies of electrochemical double layers with no Faradic current across the interface, the charge Q is essentially the electron charge localized at atomic scales near the surface of a (metallic) electrode. What is the composition of Q in your experiments: its components, origin, spacial distribution? There are obvious differences due to differences in Debye lengths of semiconductors , but are there other, dynamics -related differences?

2. The differential capacitance dQ/dV , not the integral capacitance Q/V , is usually used in Electrochemical studies of interfaces. This is similar to using instantaneous rather than average velocity describing the mechanical motion. Why does your response mention only the integral capacitance?

3. I have doubts (likely, because of the lack of understanding and education in this field) that Z2(w) can be safely named in general "the purely capacitive response". This title is indeed adequate for RCL circuits , and actually originates from this local and linear limit. But is it true in general? (sic: I do not mean the trivial issue of filtering out the induction ) ...

4 (On your response 4.2 regarding the dynamic measurement of a capacitance). What exactly is measured dynamically? Is it some quantity that exists at rest?

5. Describing the model, you mention deep electron states in Si. Can you still consider your model "classical"? (I believe that this only sounds contradictory, and you certainly have a good answer).

This is already getting complicated. I have to stop here avoid further mess. :)

Certainly, there are more questions, but, as I mentioned in response to Alexander, they are technical and suggest further discussion. This will be a detraction from the Q&A format of this forum. So, I would appreciate if we can continue this in a collaborative setting, such as Google docs. You are welcome to reach me at [email protected].

Dear Misha,

I have just completed an answer to your questions and just before sending it off, it bloody disappeared from my screen.. Sorry, I will try again and will send it as suggested, to your e.mail. it might though take a week or so, because I am abroad.

With best regards

Petr

This is an interesting and fruitful discussion, please post your exchange here :)

One more thought on negative capacitance -

the analogy between electrical and mechanical effects says that capacitance in electricity is equivalent to stiffness coefficient (k) in mechanics.

Energy in capacitor: E=CV^2/2

Energy in mechanical spring: E=kx^2/2

Voltage across a capacitor: V=Q/C

displacement of a spring: x=F/k (Hooke's law)

and so on.

Interestingly, negative stiffness in mechanical world has been engineered and used for a long time now. Once, at an exhibit of a conference (I don't remember which one), I came across a company called Minus-K Technology, that makes products based on negative stiffness effect (tables to reduce mechanical vibrations - for example, for probe stations for microelectronics):

https://www.minusk.com/

Are there similar examples of using negative capacitance in engineering / technology?

There are a lot of papers and information on negative impedance circuits, for example:

https://en.wikipedia.org/wiki/Negative_impedance_converter

However, I have not heard much about their practical applications (for example - to reduce RC delays in ICs).

Dear Maxim,

OK, I will do it, I just have to remember to save the text not to loose it again.

Your example of negative Young modulus is interesting.

And you are right, there is a lot of parallelism/analogy between visco-elasticity and conducto-permittivity. Probably the "same type" of equations describing the dynamics.

With best regards

Petr

Thanks to Maxim for referring, but i can't realy imagine the mecamizm of negative stiffness (NS). That anti-noise tablet with NS must be completely unstable: it produses force negative in sign with deformation, at stretching it gives force which increases stretching, and at compression we go to collapse. The same thing with capasitance always lead to prohibition of it in stable case, and it may work only after shunting with "normal" elements.

Coming back to initialy asked question about devices based on NC phenomenon, the list in wiki Negative Impedance article gives hundred of such examples in electrocirquits. In electrochemistry, biochemistry and electrolyte field i am unaware of devices, only traces of NC in electrocapillarity of melted copper/ oxide melt interface with adsorption, and examples, given in Micael Partenski's contribution to this discussion.

Good luck to all of us,

A Kobelev

To Maxim, Alexandr and Michael :

• I believe that the examples given in Wiki on NC "devices" are related to active devices, not the passive ones that we discuss here (?!).
• From the Electrical Impedance Spectroscopy measurements on various systems andfrom the following following analysis, my impression is that one deals with what I call "delayed response".
• An example is again a deep level in Silicon monocrystalline sample with good metal electrodes. Here, if the emmission and capture rates for electrons between the deep level and conduction band and the deep level and valence band are appreciably different, the equilibrium/steady state is established first there, where the rates are largest and at a later stage follows the "slower channel". This leads to an increase of the the total measured current through the sample within a characteristic time window. But it then settles to a constant value and stays there to time->infinity !
• Interestingly enough , the same phenomenon happens (much weaker though) just with Schottky-Schottky system without deep level. Non-linearity and assymetry are probably the main cause.

With best regards

Petr

Dear Alexander - I think negative stiffness is a negative component, that is added to a positive (normal) stiffness, to produce effectively very small positive stiffness (small capacitance in electricity) - to minimize vibrations.

This page explains the basics, very briefly (I did not try to understand it fully):

https://www.minusk.com/content/technology/how-it-works_passive_vibration_isolator.html

Apparently, the stiffness (or a part of it) can be negative only under certain circumstances (frequency range, etc.) - similar to capacitance, that can be negative only within certain frequency range and within other limits (temperatures, applied voltages, etc.).

A constant negative capacitance would lead to an (unphysical) exponential voltage increase on the capacitor.

Maxim

By the way, a very quick search on "negative stiffness" reveals a volume of material on this effect, for example:

https://www.comsol.com/blogs/can-a-stiffness-be-negative/

https://www.me.utexas.edu/~ppmdlab/files/Kashdan_Paper11_FINAL.pdf

We may learn something useful from mechanical world, to apply and use this information in the electrical world! :)

Dear Petr -

yes, circuits reproducing effective negative capacitance (or, more generally, negative impedance) are based on opamps, that contain transistors.

In the context of electronic devices and their negative capacitances - correct, we are talking about what we can call "passive" devices, although I do not see a fundamental difference between transistors and simpler devices - p-n junctions, Schottky diodes, etc. - when these "simpler" devices display a complex dynamics, that leads to a delayed current response to applied voltage excitations (a fundamental requirement for the onset of negative capacitance, as first explained by Andrew Jonscher).

Referring to an earlier question/discussion - I believe that, in general, a phenomenon of negative capacitance has nothing to do with whether there are classical or quantum effects involved in the dynamics of the electrical system. Fundamental microscopic mechanisms can be quite different, either quantum or classical (or combination of thereof), and the main requirement is the delay between current and voltage transient signals.

Maxim

Dear Maxim,

negative additive to in general positive stiffness means decrease of Young's module with deformation. As Petr said, it may come from 'active' component, which means nonlinear feedback effect. We may imagine of course, computer-based, or from special electric scheme component with opposite phase. It looks like well known method of noise reduction by immidiate adding to the signal one with it's inverted phase. The thing is how to do it mechanically, and i know how to get increase of stiffness with stretching in this way.

With good wishes,

A Kobelev

Dear Maxim,

thanks a lot for the comsol site. Just reading, it's interesting.

Is it the 'Femlab' designer?

A Kobelev

Thanks, Maxim, for the analogy , and Sasha, Petr for further insightful comments. Apparently, this kind of analogies reflect similar nature of the properties such as dielectric and magnetic susceptibilities, various elastic "constants”, compressibility, capacitance… you name! (I can not find it now, but Petr, I belive, has already mentioned this relation in our forum).They are all related to thermodynamic response functions, and hence to stability of matter , second derivatives of different energy functions (by thy way, negative Poisson ratio, which is not directly related to stability and not thermodynamically forbidden, is still counter-intuitive and fascinating !). Their divergence indicates "criticality". Negativity is a conundrum raised in practically all such cases (see for example Kirzhnits et al in respect to static dielectric response eps(k)). Probably the most famous is the issue of negative compressibility in van der Waals phase diagrams. Thrilled by its analogy with NC, we discussed it in

Article Relaxing gap capacitor models of electrified interfaces

using "squishy capacitor model" for illustration.

The Comsol forum example looks like a nice version of Euler buckling instability. Thanks, Maxim, for the pointer.

This response is edited. I removed some confusing and lengthy parts

of the original post. MP

------------------------------------------------------------------------------------

Maxim: "I think negative stiffness (NS) is a negative component, that is added to a positive (normal) stiffness, to produce effectively very small positive stiffness (small capacitance in electricity) - to minimize vibrations".

Dear Maxim, how do you add (negative) stiffness to stiffness? If we target the analogy with capacitance, it would be appropriate to consider two basic connections: parallel and serial. Using your own k C analogy , the serial connection of two elastic elements would result in the gross elastic constant K = k1x k2/(k1 + k2). Did you mean to make it smaller by choosing proper k1< 0? Otherwise, did you mean parallel

connection?

Naturally, stiffness is not exactly k, but it is unlikely to make a big difference.

But is it really possible to add k10. It was shown, however, that this assumption is false: as soon as c1 approaches critical point (right before it becomes negative), system becomes unstable and transitions to a new state with c1>0.

Returning now back to stiffness and using your analogy, we may expect that just before k1 becomes negative, the system looses stability and transits to a new stable state with positive k1. In other words, it seems unlikely that one can control stiffness by connecting positive and negative components. "Unlikely" does not mean "impossible" - I simply do not know how, and I can be wrong.

PS: Surely there are many other ways of controlling stiffness. My doubts only concern the addition of NS element for this purpose.

Dear Maxim and Misha,

i, being not so smart enough in electrical engineering, reading article by Henrik Soennerlind in Comsol blog, reveal at last the block, composed of normal elastic Hook's springs, posessing NS in the dependence of force F(l) prodused by elongation .The nul point in the middle is the begining of instability domain. It looks like a variety of 3D networks springs - rigid rods with rapid flip-flop effect in rod's orientation. Who can give electric analogue of such mechanics?

Best wishes,

A Kobelev

Sasha: Who can give electric analogue of such mechanics?

Dear Sasha. This sounds like an interesting challenge. However, I do not clearly understand its goals.

This is a wonderful article on the role of models and their purpose.

https://nemenmanlab.org/~ilya/images/9/99/Rosenblueth-wiener-1945.pdf I remember reading a while ago that its authors had a very popular seminar on interdisciplinary modeling (started by Rosenblueth). At some point, they decided to stop this project because (I am quoting from memory) " anything can be a model of anything else, but the best model of one cat is another cat".

The question is what is the mechanical problem that you want to solve with the help of electric analogs?

Dear Misha,

Thanks a lot for poniting to Weiner (that same, sic!) notes on modeling philosophy, i agree of course, that it is mechanism of our (and i believe other animals) way of brain-governed behavior, except the modeling for modeling and couple of cats, which is added for conspiracy, i wonder.

The purpose of my effort i believe is to imagine NC 'in mechnical way'. It's not by chance you and Jordan use spring (mechanical mirror of capacitor) in your squiszy model.

If capavitor's gap varies with charging, it may not only decrease natutaly via Coulomb attraction, but some times increase due to some nonlinear nonstable understandable 'on fingers' way. Other possibility is unstable decreasing charge with voltage.

Goods, not nessesary 'material',

for you, relatives and neibours.

Sasha

A.K. : "The purpose of my effort i believe is to imagine NC 'in mechnical way'. It's not by chance you and Jordan use spring (mechanical mirror of capacitor) in your squiszy model. "

Sasha, initially (with Vitaly Feldman and Misha Voroibjev) we used electro-elastic models to understand the counter-intuitive and puzzling outcomes of semi-microscopic models of Double Layer (EDL) where electrons were studied in spirit of Density Functional approach. Elastic capacitor helped us to elucidate the nature of NC (which microscopically was already associated with the displacement of the "plates" - centroids of charge distributions on both "sides" of EDL) , grasp its relation to (in)stability of interface, and, later, to introduce other models addressing some extra NC-related features and possibilities.

The elastic spring was used as a component of capacitor, responsible for the charge-induced gap variation, because we needed to demonstrate that NC is compatible with some (restricted) type of equilibrium; the restriction was later termed "sigma-control"*. The spring itself (or the stabilizing force) may contain a lot of possible behaviors. For instance, it can be multi-stable, and cause instabilities even under sigma control. Such instabilities naturally enrich the DL behavior patterns under q- or Phi-control. An example of such behavior was demonstrated by "Amper - Hook" capacitor model

Article On the non-linear response to charging of a relaxing capacitor

But even this toy model was aimed to demonstrate some microscopically-discovered behaviors (where the distance of the solvent molecules' closest approach to electrode was a variable parameter determined from equilibrium , additionally contributing to gap variation).

Much more behaviors may be introduced by playing with springs (and the plate's gap-dependent potential energy in general), but we always avoided making it an end in itself.

* By the way, we considered a similar combination (and mutual effect) of two types of phase transitions in super-ionic conductors, a field-induced transition in the electrolyte and NC -related instability in DL, in our work with Yuri Kharkats.

AK: " If capacitor's gap varies with charging, it may not only decrease naturally via Coulomb attraction, but some times increase due to some nonlinear unstable understandable 'on fingers' way. "

It is well established that the gap is non-monotonous function of charge (check, for instance, "lattice saturation" entropy effects in ionic layer, leading to expansion of the gap with charging). It can contract or expand depending on electrode charge and other parameters [temperature, concentrations, etc]). There is vast literature on this subject, from old ages to recent studies of, say, ionic liquids. A variety of behaviors are also related to the "dipolar" component of DL. They all are "non-linear" and not necessary unstable.

We were not interested in modeling all these behaviors, focusing only on NC.

AK: "Other possibility is unstable decreasing charge with voltage".

Possibility of what? This statement seems vague. Can you explain clearly what does it mean. Please, try to be specific. Describe the phenomenon, so that we would be able to discuss it productively.

Dear Misha, may be i'm wrong.

1) Explain me pls simple thing (i read that early articles quite breafly and long time ago, so poorly remember the details) about DL: charge at capacitor planes exists, external voltage does not. If we apply voltage opposite in sign to make charge nul, and then restore it, can we name it NC?

2) So, equivalent scheme of transistors, resistors, etc, for DL electric behavior with NC can't be seen? And i wish it. Much simpler previous case of two springs also?

3) At last, may be i misunderstand smth, NC of flat capasitor (and cilinder one) can be obtained by a) varying gap width in "unnatural" way keeping normal charge course, or b) keeping permanent gap by opposite course of charge at external electric polirizing. This can be done via electrotech cirquits with feedback as in impedance meters.

Sorry for my nonspecific abstractal flow.

Good luck,

Sasha

Sasha, I can not grasp all the details of your questions, but you can surely Google the answers. I believe that some of this is related to the "zero charge point", but in case of ideal surface-inactive electrolytes you can safely assume (at least for the sake of discussion and mutual understanding) that V(Q=0) =0.

At this point, to avoid unnecessary complexity, I would avoid references to impedance and feedback- just deal with basic physics - charging C with fixed portions of Q, or applying fixed V.

It would be fair, if in addition to new questions you added some answers to my previous question as well. :)

Dear Misha,

Good question MAY value much.

OK, settled.

Best wishes, Sasha

Thank you, Sasha. I am trying :)

AK: 1) Explain me pls simple thing (i read that early articles quite breafly and long time ago, so poorly remember the details) about DL: charge at capacitor planes exists, external voltage does not. If we apply voltage opposite in sign to make charge nul, and then restore it, can we name it NC?

Sasha, I believe that I understand now your question (except for the early articles that you read. Are they still available? Can you send a link?)

In respect to the essence of the question, consider a thought experiment, so that all physics is on the table. Assume that you have a polarized film between the plates of capacitor, with P the surface density of polarization (dipole moment per unit area). Then if the plates are grounded (V=0), they are charged . Let's name this charge density S0. Assume for simplicity that P is frozen -does not depend on the field from the plates. ( You can additionally introduce dielectric constants eps inside the gap but the physics will stay the same). If you apply a voltage dV, the charge will change by dS exactly like in a classical high school capacitor. The capacitance C = dS/dV = constant(V). If you now apply voltage -dV, the charge decrement is - dS, and C stays the same. In other words, in this simple picture the equilibrium charge at V=0 has no effect on capacitance. Naturally, C is strictly positive. In your question, "and then restore it" corresponds directly to the second step of this experiment, no matter how exactly the charge was changed at the first step.

Best wishes. M

Thank you Misha, after some time I've just get myself this unswer. My hope was that exsistance of unpolarized DL may be the reason of NC. I've lost in mind that for lipid membrains in instance it's due to dipolar molecules arranged in a DL. The porige boil in my pot irrespective to anything, it's my fault, what can I do? :(

Take it easy as given with me, your old and true mate.

Sasha

PS if it is saved, please, sent me back privately address of YouTube clip of paradoxical English sillables usage (my reply to your 'chastushki', in verses, by some guy, I've lost)

AK: My hope was that existence of unpolarized DL may be the reason of NC. I've lost in mind that for lipid membranes in instance it's due to dipolar molecules arranged in a DL.

Hi, Sasha. Please clarify. Seemingly, I understand each word separately, but not jointly. How "existence of unpolarized DL may be the reason of NC",

and what is the relation to lipids. Just provide a simple model. You can do it in "mechanical way" if you like. Btw, there is no deficit of polar molecules at the interfaces. For instance, so called "dipolar models" of the compact layers are focused on the electric response of the polar water molecules in contact with electrode. So, if you have same polaritity- related ideas, we can discuss them. We have some modest experience in this field : Article Electron and molecular effects in the double layer for the m...

Article Model for a metal—electrolyte interface: elastically bonded ...

Dear Misha,

I guess it was my unfruitful idea. Due to my lack of education i often produce paradoxial things believing that they may lead to sucsess.

In the surface of metal DL model of TF the internal Coulomb potential which give expon tail of el distribution can not be deminished by any external voltage (may be not?). In lipid membrane DL the same?

I do this conversation believing that my fulish activity may help someone to revise his position ones more, that is vry important as to me, or perhaps to get profit to us all.

With love,

Sasha

PS that finnish gue is funny in all his clips. Till now i do not find lost English paradoxial verses about spelling and pronounsiation

OK, Sasha. Thanks for your time.

Petr: "I have just completed an answer to your questions and just before sending it off, it bloody disappeared from my screen.. Sorry, I will try again and will send it as suggested, to your e.mail. it might though take a week or so, because I am abroad."

Dear Petr. Sorry I missed this note. I am waiting for you response and feel grateful for our productive dialogue. Wherever you post your response, please send also a copy to my brandeis address as you planned.

Thank you.

Misha

Dear Misha,

I have just returned from abroad and will shortly send the promised answer to your questions.

With best regards

Petr

Thank you, Petr.

Dear Misha,

here is the promised reply to your questions :

Dear Petr, your response is very interesting, and provokes many questions.Here is just a few.

1. In studies of electrochemical double layers with no Faradic current across the interface, the charge Q is essentially the electron charge localized at atomic scales near the surface of a (metallic) electrode. What is the composition of Q in your experiments: its components, origin, spacial distribution? There are obvious differences due to differences in Debye lengths of semiconductors , but are there other, dynamics -related differences?

Petr: if there is no Faradaic current across the electrolyte-metal electrode interface, it means that there is no electron transfer (no REDOX reactions exist) and the electrical charges are only those, present in the electrolyte. In most cases these are not electrons ,but rather ions of various kinds. In my case n.1 (Monocrystalline Silicon), these charges are electrons(conduction band) and missing electrons/holes (valence band). If some other electrically active energy levels are present (like a shallow or deep level) , then there is also a contribution from these localised charges (P+ or B- for example). There is no double layer present at the interface in this case. In my case n.2 (aqueous chloride solutions), apart from Na+ and Cl- ions for example, there are also polar water molecules and it is these that form the “Helmholtz first layer” (physisorption). Their strong dipole moment get aligned so that at the positively charged electrode, it is slightly negatively charged oxygen that is “stuck” to the surface of the metal electrode and the whole molecule becomes imobile. As a consequence the dielectric constant of these water molecules falls from epsr~80 to a value characteristic of ice ~3.0 Also in this case there is no double layer. The spatial variations of the total, local electrical charge density is given by one of the Maxwell equations (Poison equation).

2. The differential capacitance dQ/dV , not the integral capacitance Q/V , is usually used in Electrochemical studies of interfaces. This is similar to using instantaneous rather than average velocity describing the mechanical motion. Why does your response mention only the integral capacitance?

Petr: There is only one response. To an applied time varying voltage V(w)=Vo*cos(wt) - input, the system responds through the measured electrical current I(w)=Io(w)*cos(wt+phi(w)) – output. Your dQ=I(w)*dt=C(w)*dV(w). No mystery here, just that the response, the capacitance C(w) is frequency/time dependent. At time->infinity, this quantity becomes frequency/time independent and you get your “integral” capacitance. I prefer to call it static capacitance.

3. I have doubts (likely, because of the lack of understanding and education in this field) that Z2(w) can be safely named in general "the purely capacitive response". This title is indeed adequate for RCL circuits , and actually originates from this local and linear limit. But is it true in general? (sic: I do not mean the trivial issue of filtering out the induction ) ...

Petr: it was only in the example given. The impedance of purely capacitive system is Z(w)=0+1/(i*wC). It could be of course also purely inductive Z(w)= 0+i*wL, or it could be a mixture of both. I am sorry if I did not express this more clearly. Purely capacitive response is not a title adequate only for RCL circuits, it is fully legitimate assignment for a purely capacitive response of a real system, like for example a piece of sapphire within the frequency range DC to some GHz .

4 (On your response 4.2 regarding the dynamic measurement of a capacitance). What exactly is measured dynamically? Is it some quantity that exists at rest?

Petr: By dynamic I mean that the measurement is done as a function of time/frequency. When the measured quantity dQ/dV is time/frequency independent, then C becomes time/frequency independent constant and it therefore exists at rest also.

5. Describing the model, you mention deep electron states in Si. Can you still consider your model "classical"? (I believe that this only sounds contradictory, and you certainly have a good answer).

Petr: You are right, it does sound contradictory. My point here has been that in general, the energies of the various electrically charged particles should be known if we want to describe the electrical response correctly, because the electro-magnetic response process takes place in energy-space-time. And in general, you can define the relevant energies only through quantum mechanical calculation. However, in electrolytes, like aqueous chloride solutions, I can approximate these energies (for example energy of Na+ ion respective to the vacuum level) through some thermodynamical arguments/measurements.

I will send this also to your e-mail, but Maxim asked us to continue here.

With best regards

Petr

Dear Maxim,

you write:

...yes, circuits reproducing effective negative capacitance (or, more generally, negative impedance) are based on opamps, that contain transistors.

In the context of electronic devices and their negative capacitances - correct, we are talking about what we can call "passive" devices, although I do not see a fundamental difference between transistors and simpler devices - p-n junctions, Schottky diodes, etc. - when these "simpler" devices display a complex dynamics, that leads to a delayed current response to applied voltage excitations (a fundamental requirement for the onset of negative capacitance, as first explained by Andrew Jonscher).

The "fundamental" difference is that in active devices there are internal energy sources, at least as I understand this type of classification. But I guess one could argue that putting a field across a sample and forcing the sample out of equilibrium into a steady state could be considered as energy source, though external.

Andrew Jonsher in the article that you I believe, were a co-author (see my comment here a couple of years back), did not explain, but suggested that if the response current for some reason starts to increase within some time interval, this could lead to negative capacitance. In my example of a deep level in Si, this is what happens for some values of capture/emission rates between a deep level and the conduction and valence bands. The other example is a perfect Schottky-Schottky samle under moderate dc voltage bias, the third are simple aqueous chloride solutions and the fourth is ultra pure Silicon Schottky-Ohmic system, again under moderate dc bias voltage, although here it is not the negative capacitance, but rather kinetic inductance effect (ballistic transport).

...Referring to an earlier question/discussion - I believe that, in general, a phenomenon of negative capacitance has nothing to do with whether there are classical or quantum effects involved in the dynamics of the electrical system. Fundamental microscopic mechanisms can be quite different, either quantum or classical (or combination of thereof), and the main requirement is the delay between current and voltage transient signals.

Yes, under certain set of bulk electrical material parameters and for given boundary conditions , defining the system under observation, the numerical solution of Maxwelll equations (Classical Electrodynamics !) gives the negative capacitance effect in agreement with experiment. One has to be careful though to first exclude all other possible causes.

With best regards

Petr

Dear Petr, thank you! There is a lot to digest Unfortunately, I did not receive the email to which I can privately respond with silly and technical questions, and have and ongoing dialogue.

For now, one general question (I put it here, but it is also addressed to Maxim, and other experts in this field).

The capacitance of electrical double layer (EDL) is a conventionally defined equilibrium characteristic. Majority of its theoretical studies are focused on equilibrium charge distributions at the interface. NC (in equilibrium) is prohibited by thermodynamics*. However , NC can legally appear in the models implying restricted equilibrium- uniform controllable surface charge density distribution ( overwhelming majority of EDL theories belong to this class) . Prediction of NC in such models indicates charging instabilities/phase transitions under the physically accessible conditions (when the potential V or the total charged Q is controlled). These transitions lead to discontinuous behavior of C and other critical peculiarities. These properties are well illustrated by electroelastic toy models. For our discussion it is only important that their impedance has a zero- frequency limit if studied far from a critical point, and displays complex nonlinear behavior in critical region.

According to the response, there is no EDL present in cases of your interest. So (please correct if I am wrong), C is a purely dynamic property related to the phase shift between AC voltage and current. In other words, if I may, it does not have static equilibrium limit.

Question: is there any prohibition/restriction on the sign of "your" C ? If yes, please formulate it. If not, and NC is not forbidden, why is it an issue at all? Why is NC surprising and deserves special attention?

(I believe that it does, and the question is not rhetorical. Still, if the sign of C is not related to stability, why +2 is less peculiar than - 2?)

-------------------------------

Now the correction to my question that you have addressed before.

The following part was very poorly crafted:

" 1. In studies of electrochemical double layers ... the charge Q is essentially the electron charge localized at atomic scales near the surface of a (metallic) electrode. "

Here I meant the following. In conventional picture of EDL, there is a "charge Q" on the electrode and a "counter-charge" -Q distributed in electrolyte. I was trying to say that in this picture Q, the "electron excess charge" is localized at atomic distances near the surface of electrode (in well-defined region), and asked you how the "charge of electrode" is defined and distributed in your systems. Apparently, you have answered by telling that there is no EDL in your case.

* Because C is a second derivative of a corresponding thermodynamic potential

Dear Misha,

I will send a copy to your e-mail address.

Now to your question:

1. There is no restriction as to the space-time evolution of the total local electrical charge density in the neighborhood of the interface as a response to Heaviside voltage step input, apart from the interface itself (blocking ,Schottky contacts).

2. NC is not a static effect, it is a dynamic effect (time dependent currents flowing through the sample) of bulk electrical charge density responding to step voltage input under certain conditions of material parameters and boundaries.

3. My view at present is that NC has nothing to do with some kind of various surface processes, but it is a simple result of the solution to Maxwell equations, given the conditions I have just mentioned above.

4. It is misleading to talk about "capacitance of the double layer", the only thing one talk about really is the total and local electrical charge density changing in time and space. Due to finite temperature fluctuations, this el.charge is not discontinuous across the interface.

5. Negative Capacitance effect is a "problem" since its cause (see the paper by A.Jonscher and Maxim and others that I have attached to my comment for 2 years back) implies that the response is not Kramers-Kronig compatible.

With best regards

Petr

Sorry, Petr. I was re-editing my question, while you responded to the original version. Please, read my update (it comes from the heart :) ), while I am reading your original response. Thanks!

----------------

10 minutes later

After reading your response I see that it already deals with some of my updates. Kramers-Kronig reference immediately caught my eye. So, here NC contradicts the law of Entropy. Makes perfect sense - I have to read the references. First thing to check is if C in your approach a well-defined response function. Thanks again.

Mi

PS:

PV: "Due to finite temperature fluctuations, this el.charge is not discontinuous across the interface."

I agree. It is not discontinuous already due to quantum nature of electrons - the charge distributions overlap. And you are right that charge fluctuations in electrolyte induce "image" currents in electrode. Still, I believe that in time-averaged picture (in static regime) the EDL is a nice working model. Regretfully, there is no molecular dynamic studies that equally treat metal electrons and species of electrolyte, to account for fluctuations of charge distribution on both sides of interface.

Dear Maxim

In your IEEE article, is this a typo (p.2): "On the other hand, it should not be large enough that the transient current is properly resolved." ? Otherwise, what does it mean?

Dear Misha,

just a couple more comments so that we do not misunderstand each other :

1. First thing to check is if C in your approach a well-defined response function. Thanks again.

Well, it is as well defined as the linear response theory defines it (ratio of output/input). Complex capacitance C(w) is obtained as a trivial transform of the measured complex impedance Z(w) (well defined response function):

C(w)=Y(w)/i*w [Farad] and Y(w)=1/Z(w) [Siemens].

2. Just to make sure, if we take the external voltage source into account, the system (a piece of Silicon for example) can be considered thermodynamically as an open system, allowing the current to increase (for a while).

3.The solutions for the space-time evolution of the total, local electrical charge density I get by solving Maxwell equations are continuous up to the interface (from the sample side). But I assume a perfect metal electrodes surface charge that creates a constant voltage drop across the sample. The experiment is a constant voltage experiment. There are no image charge or forces (those I actually consider as a fallacy in the discussions concerning the charge distributions across the interfaces). My view is that the fluctuations "smooth out" the distributions. Also important is the fact that we deal with macroscopic Maxwell equations (in Landau sense - Electrodynamics of Continuous Media).

With best regards

Petr

Thanks, Petr!

Unfortunately it happened to me too: I wrote a response, and then lost it.

In general, I wanted to say, that reading your answers helped to understand much better your field (also represented here by Maxim), and some questions just disappeared. The most important was realization that you are not focused on the interfacial phenomena/properties (including double layers), but consider macroscopic "bulk" phenomena. The borders are accounted for by "classical" boundary conditions, and instead of surface charge distributions you calculate transient currents generated by voltage step, or the harmonic response.

Another difference is that all this is typically done in presence of conductive current (while we consider "ideally polarized electrode").

I still need your guidance:

PV: 5. Negative Capacitance effect is a "problem" since its cause (see the paper by A.Jonscher and Maxim and others that I have attached to my comment for 2 years back) implies that the response is not Kramers-Kronig compatible.

The paper is very interesting and insightful, but I did not find anything about KK - NC contradiction . Please let me know where in the paper it was is implied.

Thank you.

Misha

Dear Misha,

I am happy that at least some of my comments/observations/views have helper to clarify some of your questions.

1. Boundary conditions. You are right, they are essential, but need not be calculated quantum mechanically. I use electrochemical potential of the metal electrode and of the bulk sample to define them. Alternatively one can use the current boundary conditions. The microscopic description comes later in the analysis, when interpreting Helmholtz layer capacitance, Faraday electron transfer current etc.

2. KK contradiction is not in Andrew Jonscher paper as far as I know. It is a consequence of response current increasing in time. In linear esponse theory this current should be a non-increasing function of frequency. You can see this when you consider the real and imaginary parts of the measured complex capacitance. The real part of the capacitance at a given frequency (dielectric response function) is an integral over the imaginary part from infinite frequency to that given frequency and therefore it must be a non-decreasing function when you go from infinity to zero frequency. NC does not follow this.

With best regards

Petr

Dear Petr, and Michael,

if you address a pure non-faradaic process[1] taking place on a (one side flat) WE[2] and two (the same) REs, but in a side to side topology (with respect to the WE), then

if one RE1 help to record a Zc1, we have a common, normal, Cap.1 (C1>0) case;

however, the other side (RE2) based Z, will help to record a -Zc1 (Zc2=-Zc1) value, e.g. a nominally "abnormal" Cap.2 (C2=-C1

Dear Ioannis,

Just in time you flow in a new stream into NC discussion. In electrocapillary studies of melted Cu / oxide melt interface we considered some Red-Ox reactions at the interface in that case. Adsorption/desorption processes result in potential (voltage) regions with NC.

Returning to initial point of our discussion, how long do you evaluate the way we get the NC devices?

Sincerely,

A Kobelev

Dear Alexandr Vladimirovich Kobelev,

your electro-capillary studies could be referred as a good, non-thought experiment, (among others) example(s).

Also, a certain state of your (and other similar) interface(s) might be used as a quite stable impedance (Z-)memory, as a memristive device. The general (devices' family) brand name is, already here, between us, the memristor[1,2].

A further discussion, about the give out on "how long, or how strong will be supported, do you evaluate... etc." is, rather, a marketing issue[3], also.

1. Memristive Devices for Computing: Mechanisms, Applications and Challenges http://www.hpl.hp.com/techreports/2013/HPL-2013-48.pdf

2. Memristors - How it works! see frames after 20m https://www.youtube.com/watch?v=lsLJyijsA2A

3. https://www.memristor.org/electronics/flash-storage/284/ocz-1-tb-terabyte-ssd-solid-state-disk-drive-colossus

Dear Ioannis,

Thank you very much for this new for me 'memristor' item. I'd like to study the subject, grateful to your advise.

Aleksandr

PV: In linear response theory this current should be a non-increasing function of frequency.

Dear Peter,

May be this is why I was always interested in situations where the non-linearity is important. For example, mode softening and strong low-frequency dispersion in Elastic Capacitor in the vicinity of NC range is essentially non-linear effect. Are you dealing with linear regime in your analysis? If you do, then how do you explain the breach of K-K?

Dear Ioannis and Misha,

I do not follow the usual approach to electro-chemistry of the electrolyte-metal electrode interfaces. I feel that the "normal" 3 electrode configuration for electrochemical impedance measurements are not only un-necessary, but often can be source of errors. Why ? Because one introduces extra "unknown" interfaces into the problem and for a given solution, one does not really know, where the "zero" potential exactly is.

That is also reason why I do not really understand Ioannis's argument about Negative Capacitance effect. Could you clarify for me what is the geometry of the electrodes and what are the quantities Zc1 and Zc ?. If they are the measured complex electrical impedances between Working Electrode and Reference Electrode 1 (let us say to the left from WE) and between WE and RE 2 (to the right from WE), then you have two , almost unrelated impedance systems.

In my approach I use the two electrode configuration with a very well and precisely defined geometry - a small glass container (diameter of some ~2cm and the length ~0.5 cm) that is Pt sealed on both sides. Two outlets enable the solution to be put in and taken out and the cell properly cleaned between various measurement.. In this type of cell, the two interfaces are assumed to be identical, namely Platinum-solution interface. Under dc voltage Bias there comes an assymetry, but this is taken care of by Maxwell equations automatically. What we still do not have at present is explicit incorporation of the possible REDOX reactions into the system of equations. However, the impedance data suggest that down to low frequencies where the interface is really felt, the mozt systems behave as the interfaces would be prefectly blocking - no electron charge transfer across the interfaces.

The Negative Capacitance effect does not occur in these systems as a rule.

With best regards

Petr

Dear Misha,

1. Almost all the systems where one measures electrical impedance are non-linear. However, one can measure the response (the complex electrical impedance Z(w)) within linear response regime by applying a sufficiently small perturbation (ideally zero, but then there is no measurable response). The compromise is e.V~kT.

2. Is my analysis linear ? No, I do not linearise the set of Maxwell equations that I am solving numerically. In other words, the whole problem is posed as as a set of non-linear differential equations of parabolic type with well defined boundary conditions.

With best regards

Petr

Dear Petr, thanks for your timely response. As your system is non-linear*, there is no K-K (or any other restrictions) on the sign of the "C-component" of impedance. If this is correct, them "NC" is not prohibited. And it is even experimentally observed. So, could you please elucidate the goal of your study. Do you want to build a transparent (and consistently treated) example of how NC happens, or there is still something general to be proven or disproved? Or, can we say that, being permissible, the appearance of NC is still surprising? If so, then why?

I understood you notion, that the perturbation in your study is small [but still large enough for response to be observable :)]. I am just saying that (as you well aware) the nolinear systems are full of surprises, and they may "over-react" in response to small signals. Apparently, you feel safe from such disastrous events (or have a good insurance policy :) ), while we, dealing with EDLs that you "despise" (just kidding), feel always at the edge of catastrophe (both in conventional and mathematical senses).

Now a technical question from reading your earlier response:

PV: But I assume a perfect metal electrodes surface charge that creates a constant voltage drop across the sample. The experiment is a constant voltage experiment.

Petr, when you say "constant voltage drop", what does it mean ?

Sorry, I am sure that this is purely semantic misunderstanding on my part, and your answer will help me to understand your setting.

PV: There are no image charge or forces (those I actually consider as a fallacy in the discussions concerning the charge distributions across the interfaces).

The term "Image charges" was used in my comment as a metaphor for the shielding of the field of external charges by metal electrons. It is not a fallacy and it is quite accurate at distances > 2-3 AA from the surface. However, I agree that one should be cautious in using this terminology and approach. The conundrum lies in the danger of double- counting, because in equilibrium the surface charge on electrode is actually built of the same "images". Hence, it would be wrong to consider images as an addition to the "surface charge density".

Thank you.

Misha

* In our study of EDLs the non-linearity is crucial. For example, linear treatment of electric response (say to the field of charged ( Q) electrode) always keeps centroids of induced charge distributions frozen in space and independent of Q. In our area of study this kills the NC appearance even in theories based on Sigma - control (while NC in this artificially restricted equilibrium is not generally forbidden). It's worth noting, however, that the electric field in EDLs at metal/electrolyte interface can locally reach enormous values (up to 10 V/nm !) , and the response is essentially non-linear. I am not sure that this is an issue in your study where (as you've mentioned) the role of EDLs is not important and the interfaces are accounted for via the classical boundary conditions .

Dear Petr, I just reiterated some of my previous questions (see the modified text above.

Best

M

Dear Misha,

1. ".. So, could you please elucidate the goal of your study. .."

My goal is to put the analysis of the electromagnetic response in condensed phase within classical range of frequencies (dc to THz) on a new qualitative level ("first principles approach"), using Quantum Mechanics, Thermodynamics and Classical Electrodynamics as the three essential ingrediens. Negative Capacitance phenomenon is just one of the somewhat exotic results of this analysis, consistent with experimental evidence.

2. "..Do you want to build a transparent (and consistently treated) example of how NC happens, or there is still something general to be proven or disproved?

The cause (causes) of NG needs a more extensive investigation which I have not done so far.

3.".. Petr, when you say "constant voltage drop", what does it mean ? .."

A constant voltage experiment is when you apply a Heaviside step voltage across the sample at time zero and measure /calculate the time response (the current).

4." The term "Image charges..."

The response of the system to external electrical field is to "shield" (mobile charge screening polarisation) it so that inside the system, the field becomes zero and current eventually stops. I do not know why peoples call this "image" charges. When an electron in vacuum approaches a metal surface, The so called "image force" is completely negligible (density of electrons inside is 10*27 per m*-3 !) and the apparent decrease of the potential has nothing to do with this.

5.".. I am just saying that (as you well aware) the nonlinear systems are full of surprises, and they may "over-react" in response to small signals..."

I agree and for the time being, I would use this term "over-react" rather then "delayed response" (A.Jonscher). Your description is more precise.

With best regards

Peter

Thanks, Petr.

First, I must tell that this setting is very inconvenient for more-or-less intense discussion, and awfully time-consuming . Instead of working in the same document, we have to create a new one every time we sneeze, and then copy-paste phrases from the previous... For some reason my suggestion to work in GDocs was not supported.

My Response

1. Clarified. Thanks.

2. OK. I thought that your target is NC.

3. So, "constant voltage drop" = "voltage step". Done.

4. Images is one of the most beautiful concepts in electrostatics. I completely misunderstood your objection. Btw, high electron density in a metal is exactly the reason why its boundary is equipotential, which makes images to work so simply and nicely.

5. I thought that "delayed response" is about non-locality, phase shift, etc. But, with your motivation, I see that this is a part of what happens in critical region: reaction to a small perturbation is a long-lasting story (process). Would be interesting to know how the author defines this term. I guess, it will require some extra reading

:(.

All the best.

M

Dear Misha,

is your e-mail address : [email protected] ?

I will try to send my answer(s) also there

Petr

Dear Misha,

to your points :

4. I am aware of this way (image force approach) to calculate potentials and charge distributions in electrostatics, but I am referring to "..Image force lowering of the potential barrier at metal-semiconductor interface.." It is here I use the word "Fallacy" because there is no empirical evidence of this effect. Peoples, when they do not know precisely how the interface look in their experiment, refer to the measured barrier height as being affected by this force. The quantum mechanical calculations (charge density functional method for example) reveal that is force is not affecting anything that could be identified in the corresponding experiment (internal photo-effect) or as in my case, the Electrical Impedance Spectroscopy experiment.

5. It looks interesting, I can not comment because I do not know what system it refers to. A word of caution, oscillations are typically result of numerical errors encountered in the numerical analysis of non-linear response. But this is probably an experiment ?!

With best regards

Petr

Thanks, Petr.

4. Well, then we were talking about different things. I meant very ordinary electron shielding (screening) of external charges at conductive surface.

Speaking about self-consistent calculations of surface barrier, I am not sure what r u referring to. The "image" energy is due to electron's interaction with the surrounding "hole" (which lags behind e when it moves out) and it is implicitly present in any QM treatment consistently accounting for exchange-correlation.

The picture showed very accurate solution of a non-linear toy model. I remove it because it is not self-explanatory and provokes more questions than gives answers.

The address is correct. Best.

M

PS I am going to be a poor communicator for a while, with strongly non-linear delays :(

PPS