[Eternal Student (OP)]

Hi.

    I have a question and the Tech section hardly gets used, so I thought I might use it.   It's a question about semiconductors and that's vaguely "technology" stuff.   It may also require some background, which can be boring for others to read, so we'll break it into pieces and stop.   I'll have a check to see if anyone is following the topic so far, if they aren't, I'd be better off asking elsewhere and you would be better off not having to read more background.

Background, part I:

   We have some Gallium Arsenide.   That's an intrinsic semiconductor with this sort of crystal structure (often described as Zinc blende):



During formation you can dope the semiconductor with Silicon.    The overall crystal structure is essentially the same but some Si atoms will replace some of the Ga atoms,  that's a group III to group IV element switch,  so you have a spare electron whenever this happens.    This gives us a donor energy level in the semiconductor and we have an n-type doped semiconductor.    That's the background (part I) complete.   I'm going to guess I still have half the original audience, so I'll continue to part II.    That's almost chemistry and I suppose the post could have gone in the chemistry section - but it had to go somewhere and the tone or topic is going to shift to something more physics based very soon.

Background, part II:

    Now we have an electron we're interested in.   It's one of those "spare" or almost spare electrons from where a Si atom has replaced a Ga atom in the crystal lattice.    We're mainly interested in how it might be bound, where it is located and how available it may be to become de-localised and go off elsewehere in the crystal lattice and contribute to the electrons we would have in the conduction band.   In other words, the standard stuff that would make the doping useful for the usual purpose of creating an extrinsic n-type semiconductor.
     There is a standard model that is used to describe or estimate these things,  it's usual called the Hydrogenic model for donor impurities in a semiconductor  or just "the Hydrogenic model".
    Here's a short description summarised by (someone) of Glasgow university:
http://web.eng.gla.ac.uk/groups/sim_centre/courses/hydrogenic/hydro_1.html
    It's good because it's short and doesn't make much apology for anything that was done.  It hardly attempts to justify what was done or make many claims about the basic or fundamental physical assumptions that may have been made when proposing this model.   (It also has some minor typing or display errors - but you can't have everything).
    I'll provide a short summary of the procedure below:
(i)   We have a model for the Hydrogen atom already.
(ii)   So we have a formula for the ionisation energy of H atoms and the Bohr radius of H atoms.   Those formulae involve ε₀,  the permittivity of free space,   and also m_e , the ordinary or sometimes called "bare" mass of an electron when semiconductors are discussed.
(iii)  Just take those formulae and replace  ε₀    with  ε for the semiconductor environment   and replace  m_e   with  m_e*,  the effective mass of an electron in the semiconductor.
(iv)  That's it,  you've got your estimates.

    That's pretty much the end of part II of the background.   I'm going to stop soon and check if anyone is still following.
It's probably vital that I actually present the main question I have, so let's do that.   It still may not make perfect sense without some more background BUT it's a forum and it's all about getting some discussion and not just presenting some (possibly) factual information.

The question:
      What are the fundamental assumptions that were made or are required when proposing the Hydrogenic model?

I do already have some thoughts of my own but that will wait for a later post.  For now, I just need to check if anyone is still following or may be willing to discuss the hydrogenic model.

Best Wishes.

[alancalverd]

I hadn't come across the first equation previously (must have been asleep in one of the solid state physics lectures!) but it does look like a statement of electrostatics. The neat thing is that the answer matches the highest energy of the absorption line spectrum, so we do at least know it is correct!

We have experimental methods for estimating effective mass and permittivity so it could be said that no assumptions are required to propose the hydrogenic model, it just happens to work. Or you could argue that it is actually a circular argument because m_e* is derived from a knowledge of electron resonance and ε, which makes it a somewhat fictitious interpretation of "mass" with an apparently enormous range for 0 to 1000 m_e or more!

 

[Eternal Student (OP)]

Hi.

    Firstly, I am extremely grateful that someone has left a reply of some kind.    It has to be said that you do seem to go out of your ( @alancalverd ) way to make some reply to almost every post.    Thank you.
     I thought there may be more engineers or people who know something about semiconductors but maybe there is just the one person.     In this forum, one person is about 10% of the total regulars, so maybe I'll proceed with the post.
     At the moment I don't think there's going to be a lot of mileage for this thread if you're the only one replying - but I can at least make a reply to your reply.

I hadn't come across the first equation previously

    You almost certainly have, you just aren't recognising it.    It's the energy of the n=1 state of the Hydrogen atom solution obtained from the usual Quantum Mechanical model for the Hydrogen atom.
     It's not "some statement from electrostatics".    Electrostatics - well just the Coulomb potential - was only used to obtain the potential U(r, t)  for the electron in the Hydrogen atom,   after that it's all put into the Schrodinger equation.   1 to 2 pages later in a QM textbook you see it broken down into separable equations involving a Centre of Mass co-ordinate R  for the atom (which you aren't going to be too interested in,  for example we can assume the atom is stationary) and a relative position co-ordinate r  for a fictitious particle.   The "reduced mass" of this fictitious particle is so close to be being the electron mass and the centre of mass is so close to be being the location of the nucelus,  that overall the fictitious particle can be regarded as though it is just the electron with position r from an origin located at the nucleus.
    A few more pages later and you're onto separating the wave function for the electron into  a radial, polar angle and azimuthal angle component.   The angle components for the wave function will be based on the Spherical Harmonic functions,   the radial component function is based on Laguerre polynomials.    Hopefully the words are sounding familiar even if the procedure was something you did a few years ago.   You couldn't have slept through ALL of it.
    Anyway, the final solutions for the wave functions you will end up with are known and from that the Energy of each state (with given quantum numbers  n,l,m of which only n will be important for the Energy) can be determined.   It's this:



where a₀ is the Bohr radius.
 

[Equation 1]

The neat thing is that the answer matches the highest energy of the absorption line spectrum

   The answer?  -13.6 eV  I suppose.    It's not coincidence that it matches the energy to take an electron from n=1 to infinity.   That formula was the energy of the electron in the n=1 orbit where  n=∞ was taken as reference zero energy as determined by the usual QM model of the Hydrogen atom.

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

    Anyway, hopefully some of the original post will now make a bit more sense:

(i)  We have a formula for the Bohr radius and ionisation energy of the Hydrogen atom and this involves ε₀   and  m  (the bare electron mass).
(ii)   Rather than solving the Schrodinger equation all over again for the spare electron in the crystal lattice,  we say  "mmmm.... looks tricky,  tell you what let's assume it's like the solution for the Hydrogen atom".    We'll just change a few things.    Replace  ε₀  with  ε  for the semiconductor  and  m  for  m*.

What assumptions may have been made?   Well that's the main point of the thread, that's what I'm not sure of and what I'm asking about.    The texts I've seen don't agree on what the assumptions may be.  Many are quite "hand wavy".   On the face of it,  this spare electron from the Silicon dopant is in the semiconductor,  so using ε  for the semiconductor instead of ε₀ is superficially understandable.   I say "superficially" because that's all it is.   
    If the spare electron had stayed close to the Silicon atom then it is just moving in what is free space  (the space between atoms) and we should have described the Coulomb force attracting it to the Silicon nucleus with ε₀.   It's not until you know that the (modified) Bohr radius for this system ( Silicon ion + its spare electron )  is going to be large and span over several atoms in the lattice that it's really sensible to start replacing ε₀  with  ε.    Unfortunately,  the modified Bohr radius won't be all that big unless you have already made the substitution  ε → ε₀.   Take a look at the equation for the Bohr radius  ("Equation 1").   It has ε₀ in the numerator,   the relative permittivity of a semiconductor is typically a large number > 1,   e.g.  ε_r = 13 for Gallium Arsenide.   So  ε = the permittivity in the semiconductor =  ε_r . ε₀   =  13 times bigger than ε₀.   That results in the (modified) Bohr radius for the  (Silicon + spare electron ) system which is 13 times bigger than the usual Bohr radius for the Hydrogen atom.   That ordinary Hydrogen Bohr radius ~  order of size of inter-atomic spacing in the lattice.    Since this is also a spherical region that we assume the spare electron roams and volume ~  r³,  we see that the spare electron in the (Silicon + spare electron) system roams over a region of space that spans over hundreds if not thousands of atoms in the crystal lattice.    However,  that all happens BECAUSE you changed ε₀  to  ε (for the semiconductor environment).
       Now, hopefully you can then see the dilema:    You need some justification to make the change ε₀  →  ε   to begin with.   It's nonsense to argue that the large radius somehow  justifies using ε for the semiconductor instead of ε₀ when the radius could only be that big because you have already made the exchange of ε₀  →  ε.     However, this "hand wavy" argument is used in many texts.

      On the positive side.  In this modern era we do actually have some Scanning Tunneling Microscope images that show the electron density around an impurity donor in a semiconductor lattice.   It actually looks like it's true that there is some cloud of higher than average electron density around the donor impurity which does spread out over many hundreds of neighbouring atoms in the lattice.   Armed with this experimental observation and working backwards, we would then have some justification for arguing that the orbit of the spare electron spans over several atoms in the lattice and we need to start changing ε₀  to  ε for the semiconductor.    However, we did not have this information when the hydrogenic model was first developed or proposed.   Someone or some group of people made some other sort of assumption or else a lucky guess.

    There's more I could say   BUT   I don't think the thread is going to go anywhere if you ( @alancalverd ) are the only one making a reply.   I very much appreciate your time and your expertise but if there's no other replies in (i.d.k. a few days) I'll assume the thread is not going to be productive and you ( @alancaverd ) were just being very polite and attempting to engage with the thread  (thank you).

Best Wishes.

[alancalverd]

Always a pleasure to blow half a century of dust off the textbooks, my friend. Hopefully I'll have time to read the multi page proof before they nail the lid down.

You need some justification to make the change ε0  →  ε   to begin with.

The Bohr atom assumes (as do all other sensible models) that a hydrogen atom consists of charged particles in a vacuum, so whatever is happening between them is normalised to practical units by introducing a factor of ε₀ (hence my reference to electrostatics).

Once you have delocalised the electron into a real material, its behavior is going to be modified by the bulk properties of that material, so ε_vacuum must be replaced by ε_material - electrostatics again.

It's very satisfying that an ab initio calculation aligns closely with experiment*, but a mathematician or theoretical physicist might argue that whilst ε₀ is a legitimate scaling factor, ε_r is always an experimental fudge factor to make the numbers fit the theory!

* Or does it? Suddenly we have to introduce another concept m_e*, with orders of magnitude variance between materials, to make it fit.  A number that can vary between 0 and 1000 "depending on the circumstances" exceeds even the wildest dreams of aether/phlogiston/pseudoscience/politics and looks more like religion to me. But then so does the Bohr atom. 

[Petrochemicals]

I read it.

How is this making explanation of the low temperature state? I am assuming that the free electron is loosed at low temperature?

[paul cotter]

Hi ES, I read your post yesterday around midday but I was rather busy at the time so I did not reply. I certainly agree that Alan is an excellent contributor. However I doubt if I can be of any value as my semiconductor theory(50 years ago) was extremely rudimentary as opposed to semiconductor application. I have a query peripheral to yours and I hope I am not digressing too much: the permittivity of a material is a bulk property that is due to polarisation of charges within that material--does this have a real meaning at the atomic scale?,  ie  is it not just a case of vacuum permittivity+ whatever electrostatic effects are occurring locally?  Another conundrum for me is why electron and hole mobility varies when a hole is effectively an electron moving in the opposite direction, step by step. Apologies if this is too much off topic. 

[Eternal Student (OP)]

Hi.

How is this making explanation of the low temperature state? I am assuming that the free electron is loosed at low temperature?

    That one I can answer fairy quickly.

In the earlier post  ( # 3 )   the equation just above the one I labelled  "Equation 1",  is the one for the energy of electron in the state with principal quantum number n   in the ordinary Hdorgen atom.



Focus on the n=1 or ground state,  which is the second equation on the right for E₁.
At very low temperatures all electrons in the lattice should be in their ground state, so the  (Silicon + spare electron) should be in that n=1 state with energy E₁.   However, under the Hydrogenic model for donor atoms in a semiconductor we make the assumption that we replace ε₀ with ε   and  m  with  m^* (the effective mass of the electron).  As already mentioned this typically makes the Bohr radius a₀ at least 10 times bigger.
Actually it's about 100 times bigger at low temperatures because there is also the change to using m^*  (the effective mass of the electron) that we should consider.   If you briefly take another look at the Bohr radius equation  (equation 1 in the earlier post) you'll see it has the term m in the denominator.   At very low temperatures the conduction band for the semiconductor is essentially completely empty and the effective mass of electrns is about 0.1  (ten times smaller) than the bare electron mass.  So the modified Bohr radius, a₀ for the (Silicion + spare electron) system is really about 100 times bigger than the Bohr radius for the ordinary Hydorgen atom.
    For the E₁ equation, we are dividing by  a₀².   Putting in all the numbers we see that the binding energy of the spare electron to the Silicon nucleus is of the order of 13.6  meV  ~  milli- electron Volts  instead of  electron Volts.
      The thermal energy available is of the order  K_B .  T   ,  with K_B  = Boltzmann constant and T = tempertaure in Kelvin.     At  room temperature, T ≈ 300 K   and  K_B  T ≈ 26 meV.
      In essenece then, the binding energy of the spare electrons to the Silicon impurity atom is small  ~ meV  and there is enough thermal energy available to completely remove the spare electron from the effect of the Silicon nucleus.   So, at anything resembling the temperatures we have anywhere on our planet,  all the spare electrons from the Silicon dopant atoms have been removed from the Silicion nucleus, they are de-localised and enter the conduction band for the semiconductor crystal lattice.
     You can make another calculation using the Boltzmann distribution to estimate the probabilities of the system being in the fully de-localised state rather than the state with the spare electron bound to the Silicion nucleus.   The number of available states in the conduction band is then ~  the number of atoms in the lattice,  meanwhile the number of states where the electron could be bound to the Silicon atom ~ the number of Silicon dopant atoms.   Since dopant atom concentrations are tiny (e.g.  1  Silicon atom in every 1 million atoms of the undoped semiconductor), the number of available states in the conduction band is utterly overwhelming and even at very low temperatures,  almost all the spare electrons from the dopant atoms will be found in the conduction band of the semiconductor.

I'll get to @alancalverd and @paul cotter  replies later.

Best Wishes.

[alancalverd]

In full pedant mode, I now wonder if the question is correct?

You take a mathematical (not physical) model of a hydrogen atom (though it happens that solving the Schrodinger equations gives you the same answer), plug it into a non-vacuum environment, and add appropriate experimental values to get an answer for electron mobility that once again gives you a result close to experiment . I don't see any assumptions there, just a few inspired Ave Marias and the mystical transubstantiation of m to m*. 

More articles of faith than assumptions, I think. But at least the dictatus ex cathedra survives experimental test! 

[Eternal Student (OP)]

Hi.

Another conundrum for me is why electron and hole mobility varies when a hole is effectively an electron moving in the opposite direction, step by step.

     There's two ways sensible ways to go about answering that.

1.    Don't try to understand what electrical mobility might represent in any classical sense, just use the mathematics.  There's a formula for the mobility of a charge carrier and it depends on the effective mass of the charge carrier.   The effective mass of holes is different to the effective mass of electrons.   This only moves the question from "why is the mobility different" to  "why is effective mass different?" but we'll deal with that later.   If you were already willing to accept that holes and electrons have different effective masses then you'd be finished and have your answer for why the mobility is different. 

2.   I suppose you want some classical understanding, don't you?   We would define the electrical mobility of a charge carrier as   V_d / E   =  the ratio of the average drift velocity of the carriers that persists in the steady state to the applied electrical field.  So a charge carrier is very mobile if a small applied E field results in a large drift velocity.   
    Now we can assume a classical model like the Drude model    (  https://en.wikipedia.org/wiki/Drude_model ) for electrical conduction.
    The drift velocity can be estimated by using Newtons laws.   The force acting on a charge, e, in the electrical field, E,  is  F = Ee.    The charge carrier is accelerated until there's a scattering event, e.g. it hits something like an impurity in the lattice or a phonon (a vibration in the lattice which we can consider as a quasi-particle we call a phonon).    For a fixed E field you apply, the drift velocity of the carriers will then be governed by two things:   (i)  The effective mass of the carrier, since we're going to take the Force F and divide by m to obtain the acceleration;      (ii)  The mean free time, τ,  between scattering or collision events because that's how long you can keep the acceleration F=Ee acting on them and governs the maximum velocity reached before collisions occur.
    The mean free time will be much the same for an electron travelling in the conduction band or the valence band.   Phonons and impurities don't care about the energy levels or bands for electrons, they just are things which exist in the space in the crystal lattice.   However, the effective mass of electrons in the conduction is going to be very different to the electrons in the valence band.  Typically the effective mass for electrons in the conduction are very small.   For the Gallium Arsenide which I mentioend in this post,   the effective electron mass = 0.063 x bare electron mass.   The effective mass for electrons in the valence band is  -0.51 x bare electron mass although we would usually neglect the -ve sign and talk of holes instead,  the effective mass of holes is +0.51 m_e.
   Anyway, 0.063 is an order of magnitude smaller than 0.51,  so in the time between collisions, the electrons (in the conduction band) achieve 10 times the drift velocity as the holes (in the valence band).   You're absolutely right that the holes are just due to electrons moving the other way but the holes are due to electrons moving in the valence band  and not in the conduction band.   In the valence band, electrons are heavy and therefore drift slowly in an applied E field.

3.   So we've delayed discussing effective mass for as long as possible but ultimately this is what it's going to come down to.
     Effective mass is something that a textbook may take several pages to introduce.   That's something which is probably worth spending the time looking at it but we're going to introduce it here in about 3 minutes just by trimming a few corners.
     In Newtonian mechanics  you have  kinteic energy  =   p² / 2m      where  p = momentum, m = mass.
Assuming an electron behaves broadly like a free electron, i.e. it's free of being in any potential,  then it's energy is just its kinetic energy.    Change the momentum (the motion) of an electron and you change its energy.   This is the standard free electron behaviour.
    Moreover, this sort of behaviour for an electron is all that we're really interested in when we're talking about electrical conduction.   We apply a voltage, that's an E field set up over the semiconductor, and we expect some electrons to move from one end of the semiconductor to the other, a current should exist through the conductor.    There were some electrons in the crystal lattice that were so tightly bound to an atom that they just won't move - and they are not important and do not influence the conductivity of the material at all.   If all the electrons were tightly bound like this, then the free electron model behaviour isn't there at all and the material is essentially an insulator rather than a conductor.
    So, the behaviour we're interested in is where the energy of an electron is related to the momentum or movement by
                      E = p² / 2m                                    [Eqn 1]

    Usually we're taking a Quantum mechanical description of an electron by its wave function,  in that situation the Energy would be given  by     
                     E = ħ² k²  / 2m.             [Eqn 2]
  You can get to this equation through the multiple pages of the textbooks but we're going to take a short-cut.
    The de Broglie wavelength for a particle of momentum p   is   λ = h/p.   
=>    p =  h/λ      =>        p = (h/2π) . (2π/λ)   =   ħk         where  ħ is the reduced plank constant and k is the wavenumber k = 2π/λ.         So just substitute  p = ħk  into    [Eqn 1]    to obtain   [eqn 2]
   
    Where we treat electrons as though they are free electrons (i.e. not bound to some atom but able to move through the lattice),  then    E = ħ² k²  / 2m    [as in Eqn 2]    is the relationship between their wavenumber and their energy.    ħ is a constant, that's great and you'd normally assume that the mass of the electron is a constant.   Under that assumption differentiate  E   w.r.t.  k  twice.    From   [Eqn 2] we obtain  (d² E / dk² ) = ħ² / m   and we'll re-arrange that to obtain:
     m* =   ħ²   /  (d² E / dk² ).                                       [Eqn 3]

I put an asterisk on the m on the LHS you may have noticed.   If the electron had genuinely been completely free then [Eqn 2] would have correctly described its energy, our differentiation would have been totally fine and the RHS of [Eqn 3] would have given us back  m,  the bare mass of an electron.  In a semiconductor, we can still determine everything on the RHS of  [Eqn 3] and we just declare that whatever that is, that is the effective mass of the electron.
    So to paraphrase this a bit,  the effective mass, m*, is what we would need to assume the mass of the electron is if we want to model the electron as though it was free to move rather than actually moving in some complicated potential that exists throughout the crystal lattice of the semiconductor.   When an E field is applied to the semiconductor, there will be a net movement of electrons and they they will move as if they have an effective mass given by [Eqn 3].  It's not that their mass is actually changed somewhow, it's really that they aren't seeing ONLY the potential from the E field or voltage you've applied across the semiconductor.  They are moving in a complicated potential determined by all the atoms in the crystal lattice but the net effect of that is to put some drag or hinderance on the electrons (if the effective mass is greater than the bare mass) or else boost the speed of movement of the electrons through the space (if the effective mass is less than the bare mass).

    We're nearly done.  All we need is to determine the relationship between the Energy of an electron and its wavenumber, k.   That's a few pages of a textbook that we will just skip because this post is already long.   Here's a diagram of what you get:


   
Take [Eqn 3] which describes the effective mass of an electron and imagine an electron that is at or near the top of the red curve,  i.e.  has an energy near the top limit of the valence band.     We're considering the second derivative of E with k,  so that's the rate of change of the gradient  (rather than just the gradient).   The red curve has a small magnitude for this, it changes gradient only slowly.   Electrons here have an effective mass ~  1 / E''  ,  so that's a large mass.    It's also negative if you notice.   The gradient becomes negative as k increases.   Rather than treating the electrons that may be able to move in the valence band as if they have negative mass, we just consider the holes as if they are the carrier particles of interest and they will have a positive mass (of the same size).
   Similarly, we can consider an electron near the bottom of the blue curve (at the bottom of the conduction band),  here the gradient changes rapidly as k increases   => E'' is a big number =>   effective mass is small,   Also E''  is a positive number => the effective mass is positive.
- - - - - - - - - - - - - - -

Sorry, that was a long post.   I'll leave your other question / comment for later.

Best Wishes.

[Eternal Student (OP)]

Hi.

You take a mathematical (not physical) model of a hydrogen atom

   What is a physical model of a Hydrogen atom?   Is it an actual Hydrogen atom?   I'll admit an actual Hydrogen atom would show the behaviour of a Hydrogen atom better than a mathematical model of a Hydrogen atom.

In full pedant mode, I now wonder if the question is correct?

   What you mean is - since the final prediction seems to agree with the observations, that's all we're interested in.  That's the engineer approach. 

    Having the final predictions useful is a bonus but it doesn't further your (human society's) understanding does it?   We've got to know how the model was put together and what assumptions were made in the process.   That's the theorists approach.

Once you have delocalised the electron into a real material, its behavior is going to be modified by the bulk properties of that material, so ε_vacuum must be replaced by ε_material - electrostatics again.

    That's pretty much the argument that some existing texts would use.  It's hand wavy as discussed in an earlier post.
A theorist is always working from the basic models and assumptions upwards toward the final prediction.  We do not know that the spare electron from the Silicon dopant would occupy a large volume of space through the conductor until we just go ahead and make the change ε₀ → ε anyway.    When we make that change then the Bohr radius is seen to be large,  which does imply the electron might see a potential from the Silicion atom as if the bulk material value of ε applies  - but it's too late then.    The large answer cannot be used in some retrospective manner to justify the change we made that did make the final answer large.
   When I say "it cannot be used" - obviously it can, you just do it.  I just mean "it shouldn't be used".   We ought to be able to build the Hydrogenic donor model from the ground up.    With the STM images we now have, you could actually claim to know the final answer already and then have justification for using ε of the bulk material.   But that isn't what the original proposers for the Hydrogenic donor model would have had.   In some text somewhere we should be able to find what their basic assumptions were.   More importantly we can reverse engineer the final result to ascertain what those fundamental assumptions would be - and thereby hope to gain some insight into how things may work on a more fundamental level.
    Simple example:   An assumption you may have to make is that the electron would occupy a large volume of space and the bulk properties of the material would be such that the electron experiences Coulomb forces involving ε_{bulk material}.   Since the relative permittivity is a consequence of polarisability of dipoles and movement of charges in the bulk material, we were lucky that the overall crystal lattice wasn't significantly affected by the presence of the Silicon dopant atoms.   For example, the available states for electrons in the crystal lattice  (e.g. the band structure for electrons) wasn't significantly shifted up or down in energy or some band structure wasn't completely destroyed when the Silicon dopant atoms are added to the crystal lattice.   Instead what we seem to have is the majority of the physical structure and electron energy band level structure left intact and there is just an extra donor energy level inserted into that structure.    I will probably talk about some assumptions that I think were made in a later post but this post is already too long.   I'm just hoping you can see just how far reaching the consequences might be of assuming the spare electron saw electrostatic fields as if ε_{bulk material} applies  - it means that charges and dipoles probably have responded and altered through the rest of the lattice (but the lattice hasn't fallen apart).
   

but a mathematician or theoretical physicist might argue that whilst ε0 is a legitimate scaling factor, εr is always an experimental fudge factor

   Now you're getting somewhere and have elevated yourself up from the level of an engineer.   Less empasis on fudge factors that just make things work and more emphasis on an underlying understanding.

Suddenly we have to introduce another concept m_e*   (effective mass) 

    I've already said something about this in the previous post replying to @paul cotter .   There's more but this post is already long.

and also @paul cotter said this:

the permittivity of a material is a bulk property that is due to polarisation of charges within that material--does this have a real meaning at the atomic scale?

    I've already started to incorporate a discussion of using bulk permittivity into the (two) replies so far.   It's a huge issue and one I would leave for another post.    The main thing is,  I have some ideas already.   I just don't know if they're right and there's limited point in leading the discussion in some way as if I know.
    I do not know what the fundamental assumptions would have been when the Hydrogenic model was proposed and exactly what that implies for the fundamental behaviour of atoms on this atomic scale.   That's what I'm asking about or hoping to discuss.   If I make another post it will probably be tomorrow and (if I remember) I'll be flagging a lot of it with   "I don't know, this isn't textbook, this is what I think"   warnings.

Best Wishes.