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I know it's bit fundamental question but I really want to understand the relationship between the various elastic constants and phonon branches? Can anyone suggest some review literature/notes/books where I can find some lucid and simple explanation regarding the relationship between the duo.
May be the relationship holds for long wavelength limit (near to gamma point) but then why still some structures show elastic instability but dynamical stability (+ve phonon frequencies)?
Any kind advice would be acknowledged.
Thank you so much Alexander, looks really promising!Following
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I am having unstable high temperature phases and I want to calculate free energy of those structures, what is the best and efficient method to calculate it?
I know there are ways to calculate it such as:
Quasi harmonic Debye model, SCAILD method, AIMD method and fast free energy calculations, but I am bit confused which one I should chose?
Every method has some limitations for example, Debye model is kind of analytic, SCAILD and AIMD is too computationally expensive, but no idea about fast free energy. So, under such circumstance, it would be really good if anyone has some experience regarding these methods and can suggest something.
Thanks in advance.
Dear Tanmoy, we have done some work using SCAILD as well. It seems to work well but takes many iterations to converge and is therefore somewhat computationally expensive. We've been looking at the mechanical stabilities (stable phonons) but the free energy is computed as well.Following
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Why the creation of such phases is difficult?
Because they require a dense regime with at least several particles within the interaction range, which can be difficult to achieve.
Or there are other causes?
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Dear Dr Gagan,
please also send me book Of Cullity at firstname.lastname@example.orgFollowing
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Thank you very much for this answer Dr. Jardim.Following
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From my understanding, for long range interatomic interactions, not only the interactions of the unit cell with their nearest neighbors should be considered, other atoms around the unit cell should be taken into account as well. So when I tried to calculate the dispersion relation with classical potentials, I included the interactions to the 5th nearest neighbors. However the dynamical matrix is not Hermitian, i.e. some eigenvalues are negative numbers. Is it because the number of neighbors is not enough or is there any other way to calculate the dispersion relation when long range interactions are involved? Thanks.
I use a central force model, see Physica C 506 (2014) 100, available at my ResearchGate home.Following
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I know the effective mass of electron " me* "and hole " mh* " for ZnS and CdS separately and I want to know to calculate effective mass of electron and hole for solid solution ZnxCd1-xS ?
For accurate solution I'd recommend ab initio calculation of this alloy, this would provide the band structure from which effective masses can be obtained. You should contact some group capable of employing the CPA approximation for alloys.Following
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Can Fermi level go above top of the conduction band?Following
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How can we calculate the bohr excitone radius of nano-particle?
Thank you sir...Following
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It seems Lifshitz transition refers to the change of Fermi surface without symmetry breaking. But I consider it is quite common that the Fermi surface changes as a function of doping, as in Fe-based superconductors. Furthermore, in semi-metals, doping can change the Fermi energy, resulting in Fermi surface change where electron bands cross the Fermi energy and the system become metallic. So how is the 'Lifshitz transition' different from these simple Fermi surface changing phenomena?
There is no fundamental difference. In fact, the Lifshitz transition was originally considered by Lifshitz (see the attached review article by Lifshitz and Kaganov) in the light described by you here above, for a given band structure. The issue that has come into prominence in recent years is that of the Lifshitz transition in strongly-correlated systems (in doped high-Tc compounds and heavy-fermion systems), where the notion of bands, and in particular of rigid bands to which electrons are added or from which electrons are removed (by appropriate doping), is of limited applicability, if at all. Due to strong electron-electron interaction, the electronic structure, as observed experimentally (say, by means of photo-emission spectroscopy) can substantially change upon doping, possibly in conjunction with changes in other parameters relevant to the system (such as temperature, pressure, etc.). As a result of interaction, Lifshitz transitions may be observed that are absent in mean-field treatments.Following
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Both follow the same mathematical relation.
Indeed, they don't have the same status:
• Thermally activated hopping conduction is a physical phenomenon. The one-electron wave function is hopping from one site to a neighbor one, each time overcoming an energy barrier that is at first order equal to the observed activation energy, although there might be some corrections due among others to entropy factor or lattice relaxation. The prefactor is related to the eigenfrequency of the ground state.
• The Meyer-Neldel rule (1937) is an empirical law describing a correlation between prefactor and activation energy in transport experiments in disordered materials. None of the theoretical models proposed to explain the origin of the Meyer-Neldel rule is universally accepted. It seems that several effects, depending on the materials and conditions, may lead to a similar global behavior. Many authors stress out that prefactors for the conductivity provided by the Meyer-Neldel interpretation are difficult to interpret physically. Look e.g. at Widenhorn et al., JAP 89, 8179 (2001), Widenhorn et al., JAP 91, 6524 (2002).
In that sense, the status of the Meyer-Neldel rule is similar to that of 1/f noise. Both are a commonly encountered behavior in many types of materials and devices, but they don't convey much physical information.
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Hopefully this can help you.
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You are welcome.Following
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Is there something about the calculations that you wish to ask more specifically?Following
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- Petr Viscor added an answer:How do holes recombine with electrons in single carrier type steady state conduction?
Suppose that e-h pairs are generated on anode surface of an insulator by surface absorption of light. some of the electrons recombine there in surface very shortly. The remaining electrons are pulled out of the insulator by the positive anode. Now the remaining holes start their journey towards cathode.
Books and papers say, the holes recombine with electrons/get trapped in defects and so their density decreases as they move forward. In a paper "High Field Effects in Photoconducitng Cadmium Sulfide" by Many, the hole density in steady state is given as
Which clearly shows that the hole density decreases exponentially with the thickness (x) of the insulator.
- The situation becomes confusing to me (probably I am not thinking correctly) when the cathode can not inject electrons into conduction band. How do the moving holes recombine with electrons in the middle of the insulator (there are no free electrons in CB to recombine with) and so they decrease with distance?
- I can understand in case of transient period before establishing the steady state, that the holes might get trapped in traps and so they decrease with distance. However, when they maintain a dynamic equilibrium of trapping and detraping in steady state, how do the holes decrease in numbers? (Seems like there are infinite traps and so the injected holes keep filling the trap without detraping.)
- Suppose that the cathode inject electrons to move through conduction band allowing holes to recombine them. But now, is it purely a hole current ? is not there electron current contribution ?
in order to hopefully help you in most efficient way, I have copied your answer below and will comment it paragraph after paragraph. In general (and at this point) I have no specific experience/knowledge about photoinduced currents and their decay, apart perhaps from my own work on persistent photoconductivity in amorphous Ge films (published in J.Phys C, together with Gerhard Fasol). Therefore take my comments as general ones, regarding the electrical response in condensed phase :
Gyanendra Bhattarai · University of Missouri - Kansas City
Thank you Petr Viscor for your time and answering my question.
Regarding my question, I was trying to understand what mechanism controls the current in steady state. Although not explicitely stated, my intention was to understand the use of steady state photocurrent for Hecht analysis.
Petr: That I understood, but what is Hecht analysis ?
In this context, I have now a bit different question. Though it is quite long, I hope you take a look.
1. In the paper I cited above, "High field effects in Photoconducting CdS" Many talks about the transient current induced (displacement current) by the photogenerated charge moving inside CdS crystal.
Petr: General comment :
The term "displacement currents" should be used for the time dependent electrical response of the bound charges, where the mobility is zero. The response of this type affects the dielectric response function eps,
The time dependent response you are talking about is due to charges with finite mobilities and I named this response "mobile charge polarisation" . This type of response does not change the dielectric function/constant eps, but rather the measured time(frequency) dependent resistance(real part of the measured electrical impedance), determined by mobile charge densitiy and mobility at place x and time t .
Clearly, they can not make steady state photocurrent as the crystal is not connected electrically. They are using mica spacers to couple the crystal capacitively. I believe the displacement current is same as the current explained by shockley-ramo theorem.
Petr: I think you are right, but in general it will depend on how much extra charge (el-hole pairs) is generated at the left electrode (anode) and how much of this extra charge dies away on the way to the cathode. Using spacers as mica is quite a dangereous think to do because the overall response can be dominated by the time dependent effects at electrode mica and mica-sample interfaces. Finally I do not know what is Schockley-Ramo theorem
2. In one case, they use the approximation of surface absorption of light to create holes (they irradiate +ve contact with light so holes gererated migrate towards negative contact).
Petr: Yes, but remember that this type generation creates extra charge many microns inside the sample, it is NOT a delta function type generation as is the case in electron beam generation (penetration depth is in Angstroms, well almost)
So in ideal case, the mechanism of decrease of carriers as they move forward is because of trapping.
Petr. No, it is because of them DISAPPEARING at the recombination centres (in electron and hole, out nothing !!). The recombination centres can be considered as true sinks and sources , charges disappear from the electrical circuit and/or are generated there.
The exponential decrease is not related to exponential attenuation of light intensity.
Petr: OK, that is good that we have clarified this point (at least in principle)
The paper assumes that the photogenerated carriers are low enough to believe the electric field inside is uniform (not affected by generated charges)
Petr: That is a fundamental fallacy !! Until you really calculate it, you can not make this assumption. The authors are apparently not aware that the depletion regions are in general of the same magnitude (the fields due to depleted regions might be (in fact they usually are !!) comparable to field changes caused by the external photo-generated charges
Below is my understanding and confusion.
Suppose we have direct electrical contact say blocking so we can measure steady state photocurrent. In steady state, I think the displacement current contribution vanishes and the continuity equation is explained only by drift current. If there are traps, I think two possibilities:
Petr: No, Although the displacement currents vanish, there will be a steady state current (electron plus hole) that is determined by the rerspectrive drift AND concentration gradients (it is actually given at all times by the gradient of the respective quasi-electrochemical potential(s) at a given quantum energy level(s) that contributes to the steady state electrical current.
a. The traps are filled uniformly spatially and so the electric field remains uniform. The drifting carrier concentration remains uniform and so continuity equation is hold.
Petr : The traps are not filled uniformly in space due to the difference between the respective electro-chemical potential and the energy level in question. The electric field is never uniform in usual Schottky contacts. Only when the electrochemical potential of the metal electrode is identical/equal to electrochemical potential of the sample, the electric field inside the sample can be considered as uniform. This is of course never the case, it would require that the metal electrode and the sample are identical materials !
The carrier concentration is not spatially uniform in general. In general, when the sinks and sources are present, majority carrier concentration in x decreases and minority carrier concentration increases correspondingly. With no sinks and sources (deep levels - recombination centres) and in strong one carrier system (n or p), it is though correct that the majority carrier current is almost equal to the steady state total current through the sample. THIS DOES NOT MEAN that both the concentration and the field are uniform. On contrary, they are not, otherwise you would never get into steady state condition (constant, time independent current through the sample) !!!
Continuity equation is an integral part of one of Maxwell equations and when sinks/sources are present, continuity equation IS NOT satisfied for each relevant quantum energy level (each transport "channel").
b. Since carriers are generated near anode, they are trapped more in anode region and so the carrier density drops exponentially along distance. In this case, for the continuity equation to hold, the space charge is created and the electric field will be non uniform (like in space charge limited current.)
Petr: Well, this is more of a speculation. Although I do this type of numerical calculations, I will not come up with such a strong argument/statement. The result will depend on the concentration of excess charges, their mobility and first of all on the capture cross sections (the probability that a moving charge will be captured by a trap and/or deep level(recombination centre). With almost zero capture cross sections, the carrier may be first trapped at the anode or not al all. Lastly, I do not see, why the x dependence of the excess charge in steady state should be exponentially decreasing ?!
However in both cases, the total current is because of drift current and the total current is given by J = e*p(C)*mu*E(C). where p(C) is density of holes collected at cathode and E(C) is electric field at cathode. I don't think we need to integrate the charge carriers entire sample to get the current.
Petr: Again, this is a strong approximation, not valid in general. As you just pointed out, the excess charges will create a space charge (non-zero field) and therefore there will (and must be !) always a term in the expression for the total current that is given by concentration gradient. For time->infinity(steady state),
J(x)= e*p(x)*mu(x)*E(x) - (e/kT)*D*dp(x))/dx
There are many papers describing Hecht analysis. Most of them cite Many's paper. Though Many's paper describes transient current, other authors use the same mathematics for steady state.
Petr: If the equations to be solved are properly defined, then this is possible, because steady state=transient state(time->infinity)
Though they find p(z), E(z) for every z, they integrate e*p(z)*mu*E(z) over the entire length and divide by the volume of sample to get the current. I am not convinced by this integration because once one knows p(z), E(z), the current is just J = e* p(z) *mu* E(z) = e* p(C)*mu* E(C).
Petr: The proper sequence is: Start of the time->calculation of local, space-time dependent current density i(x,t) [Amp/M*2] ->calculation of the total current density through the sample at time t I(t)= integral(I(x,t)dx - > Next time tnew=timeold plus delta time ->back to the start, where time=time new. After sufficient time interval (in general time->infinity), I(t)->const and also (in fact the definition of the steady state) i(x,t)->i(x) = constant for all x. So,in the steady state, i(x) is the (must be) the same at each x, right from the anode to the cathode.
In short, I can say that, I don't understand how Shockley-Ramo theorem is used in steady state.
Petr: As I already mentioned, I do not know what Chockley-Ramo theorem states and therefore I can not give an useful comment.
I don't see its applicability as there is no displacement current unless there are electrons inside the material so the charge carriers disappear
Petr: You contradict yourself here, with trappnig hypothesis, you will get a loss of excess charges (disappearing into traps->no further movement) in space and time. But you are right that in the limit time->infinity(steady state), there will be (principle of detailed balance) no further time change in the current.
because of recombination
which then requires S-R theorem to calculate total current. But then, the current will not be entirely due to holes. There must be some way that electrons move within the crystal in CB to recombine with holes.
Petr: The electrons do not have to move, they just have to be there ! As pointed by A.Kumar, there will always be a finite concentration of thermally excited electrons and ,in case of presence of deep levels(recombination centres) much large concentration of electrons there (only~half band gap activation energy).
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