Condensed Matter Theory

Condensed Matter Theory

  • Vadim Adamyan added an answer:
    Is there any concept of quasi-particle in classical systems?

    By quasi-particle I mean in the sense of particles dressed with their interactions/correlations? If yes, any references would be helpful. 

    Vadim Adamyan

    The normal modes in the classical theory of small oscillation typify quasiparticles.

  • Pius Augustine added an answer:
    Why, whilst depositing PMN-PT on LSCO buffered platinised silicon substrate, on and off, is the film conducting?

    What could be the possible reason? 

    Pius Augustine

    Thank you Tarun.

    Sometimes between two points on the surface of the film shows very low resistance. 

    Also between the bottom platinum electrode and top surface of PMN-PT film. 

    Is it because of the O2 partial pressure ? 

  • Anton Nikonov added an answer:
    How to solve the problem with Nb-Ti-Al potential in LAMMPS?

    I'm trying to simulate the structure of B2 alloy Ti-Nb using the potential

    Created structure collapses with a large release of energy during relaxation. The lattice parameters were taken from the test report. The LAMMPS script is attached.

    Anton Nikonov

    Thank you.

    This structure is shown in the table with the test results of the interatomic potential.

    The same result is obtained when using the NVT.

  • Konrad Gruszka added an answer:
    "relax" or "vc-relax", which one for optimization of primary cell in QE (or generally other DFT codes)?


    Let's suppose I have some unit cell for example Hexagonal Yttrium. I would like now to introduce a defect in the form of substitution of one of the atoms by let's say bigger one. Before doing further calculations I know, that first I should lead to minimize the forces acting on the atoms inside the cell. Quantum Espresso lets me do this by two ways: one using older 'relax' optimization where cell parameters don't change and a second one where I can optimize not only positions of atoms inside cell but also other things like eg. cell size or angles, possibly leading to a lower total energy.

    Which option should I use, to ensure that my calculations would be physicaly right? Does forcing the system to remain in a particular unit cell is appropriate? What if I do not know the true unit cell due to the lack of experimental data?

    Konrad Gruszka

    Thank You all for your answers.

    I can see now, that this case  is more of my assumptions than 'only one good path' that I should follow..

    Ang Feng: I can imagine that when only isolated defect is present, the latter will "arrange" to fit this inclusion, so relaxation of whole cell is needed .  I think that also a much bigger supercell is appropriate. 

  • Carlos Paz de Araujo added an answer:
    The averaged energy of two-electrons repulsion U - What is it?

    As is well-known, the so-called averaged energy of two-electrons Coulomb repulsion U has been introduced both in quantum theory of atoms / molecules and of condensed matter, which is typically defined, as is shown in the attached figure (a), - where the integral written there is taken over the whole 6-dimensional configurational space (r1=(x1, y1, z1) and r2 = (x2, y2, z2)).

    For instance, the on-site two-electrons repulsion energy U appears in superexchange theory, where the antiferromagnetic contribution to exchange integral is obtained as: Jaf ~ b2/U (where b is a hopping integral), it appears in LDA+U approach intended to reproduce the band structure of strongly correlated crystalline systems correctly, and so on.

    As far, as I can judge, the U energy is introduced as was shown (or in equivalent way) in manifold textbooks and papers.

    But it is absolutely evident, that the integral defined so diverges, in other words, is equal to infinity, except the trivial and physically insignificant case, if at least one of one-electron orbitals is identically zero at the whole space. Actually, most probably no other physically reasonable form of one-electron orbitals can be proposed to eliminate the singularity in the denominator at ANY point of "line" r1 = r2 in 6-dimensional space (see also attached figure (b), where this point is symbolically depicted for the case of "one-dimensional" electrons).

    Note, that the approach like "Let`s deviate from "line" r1 = r2, next, take the integral over the whole space except the deviation vicinity (see also figure (b)), and finally calculate the limit of the result approaching the measure of deviation to zero".. so, something like that evidently is not valid - because it also does not eliminate the singularity, actually, the integral over the whole space except the deviation vicinity might be arbitrarily large (keeping its sign to be invariable), as depends on the deviation measure value.

    Sorry for a long text, but it is related to my question directly. On the one hand - I cannot find the logical errors in argumentation given above, as well, as cannot find the explanation in textbooks and publications I have ever seen. On the other hand, manifold sources deal with the definitely FINITE values of U (typically, some eV).

    Can someone explain me, how this contradiction could be solved?

    I would be VERY grateful.

    Carlos Paz de Araujo

    U is the screened potential. So, you should pick RPA or Thomas-fermi to described the dressed quasi-particle and then, the integral will not diverge. There is no sense for the integral if the potential energy does not get screened within a few lattice sites, or is in the case of the Hubbard model, screening within a distance slightly larger than the Bohr radius. Thus, when you see U, understand that it is really Vexternal/dielectric function.

  • Henry Tregillus added an answer:
    How to define a theoretical room-temperature quantum state storage device?

    The coherence time of quantum dots is largely linked to their electromagnetic susceptibility (environmental interaction strength). Topological quantum computing is based upon the long coherence time of anyon pseudoparticle states.

    Is there some known material which exhibits stable quasiparticle behavior in response to photonic excitation?

    Or if not, what might such a material look like?

    Henry Tregillus

    Arijit, Franz:

    I am attempting to build such a statistical description of quantum mechanics within a solid, such that the movement and behavior of large-scale quantized behavior can be understood and applied. As it stands, the best we have in this regard is NMR, but it relies on something different than what I have in mind.

    NMR uses electromagnetic pulses to target specific magnetic moments of complex molecules in a strong magnetic field; given the known resonance of a particular section, it can be manipulated, and the molecule can be used as a circuit of some sort. Operations result due to inner-molecular interactions, and as these are probabilistic, the final measurement has to be done many times - or just over a lot of samples.

    Like Arijit is saying, NMR is NOT scalable. We just don't have the resolution necessary, nor the magnetic field strengths desired. It's also slow. The principles behind it however, of using molecules themselves as storage and logic gate structures, is a big part of what I'm trying to learn, just from the perspective of condensed matter, rather than molecular dynamics.

    I'll be sure to check out more NMR papers though, it is certainly a good building block. Thank you both :)

  • Tanmoy Chakraborty added an answer:
    What is the equation that describes the directional dependent shear modulus of orthorhombic crystals in polar co-ordinates?

    May be the question is bit specific, but I want to know if anyone has any idea about it, just in case. I know the relations for bulk modulus and Young's modulus from the paper (J. Appl. Phys. 109, 023507 (2011), Eq. 16 & 17), but they didn't calculate directional dependent shear modulus.

    So, in case if anyone knows, kindly let me know. Thanks in advance.

    Tanmoy Chakraborty

    Alright, I found the answer by myself from a paper:

    AIP Advances 5, 087102 (2015)

    The equation is given on pg. no.-13 (Eq.-10)

    So, in case anybody is interested, can follow the above reference.

  • Motahare Mohseni added an answer:
    How to calculate lattice constant from XRD spectrum?
    Lattice Constant formula.
    Motahare Mohseni

    Dear Ranjana Varma thanks for your answer

  • Manuel Morales added an answer:
    Is the Standard Model incomplete in light of String-Net theory?

    I have a doubt regarding String-Net theory. Prof. Xiao-Gang Wen stresses upon the point that all fermions must carry gauge charges. The Standard Model contain composite fermions that are neutral for U(1) × SU(2) × SU(3) gauge theory. So, according to string-net theory the Standard Model of particle physics is incomplete, and the correct/complete model should contain extra gauge theory, such as a Z_2 gauge theory. But, Coleman-Mandula theorem states, more or less, that space time symmetries (which determine spin) cannot mix with gauge symmetries in anyway. The Haag-Lopuszanski-Sohnius (HLS) extension of this theorem states that the only possible loophole to the Coleman-Mandula theorem is SUSY, as far as I understood. So, is the Standard Model incomplete from the point of view of string-net theory or is string-net theory radically inconsistent with what we see in nature, and must therefore be wrong in its present form? Also, is it meaningful to call something a Z_2 gauge theory (because, as far as I understand, discrete symmetries can at best act as "large" gauge transformations)? PS: To get an understanding/gist of what Prof. Wen is saying (as I mentioned in the first half of my question) please refer to this paper:

    Manuel Morales

    "Experiments are the judges."

    Oh really? Think again...

    • Source
      [Show abstract] [Hide abstract]
      ABSTRACT: The current scientific methodology is based on the paradigm of second cause logic by placing cause second to an observable physical effect, i.e., the existence of an effect causing the existence of an effect. The methodology of successfully predicting effects and the repetition of the predicted effects to serve as an indication of causality necessitates that causality is a singular mechanism in order for such a method to be valid. Conversely, the Tempt Destiny experimental model placed causality first by distinguishing which mutually exclusive selection variable caused the effects observed. The 'unambiguous' empirical evidence has shown that the second cause logic used as the foundation of scientific methodology cannot 'correctly' distinguish which first cause variables caused the effects observed.
  • Tanmoy Chakraborty added an answer:
    How can I calculate elastic constants of SQS structures using VASP ?

    I am using vasp 5.2 version where the elastic constants for a symmetric structure is automatically calculated by IBRION = 6 tag.

    Since, I am interested in SQS structure which is basically having very low symmetry. Now, my question is does vasp still keep symmetry of the structure (let say, bcc) while calculating elastic constants?

    What if I want to calculate elastic constants of a SQS-54 structure (bcc) which is basically way to expensive to calculate from computational point of view and if by a bain path transformation I reduce it to let say SQS-36 having tetragonal structure, then in that case will the elastic constants of the SQS-36 (tetragonal) structure be equivalent to that of SQS-54 (bcc) structure?

    Basically, all I want to reduce my giant SQS-54 bcc structure to some tetragonal structure of having small atoms and in that case can I consider the elastic constants of the tetragonal structure be equivalent to SQS-54 bcc structure?

    Any idea would be greatly acknowledged.

    Tanmoy Chakraborty

    Nice to hear from you Per. Yes, this is kind of by-hand method which should work. I am doing that right now, as the automatic vasp method (IBRION =6) does not work here. I was initially thinking, if somehow I can reduce my large bcc SQS-54 structure to some different structure and then calculate the elastic constants by vasp automatically (this is stupidity at its best I was thinking). Thanks for your valuable comments.

  • Boris I. Sedunov added an answer:
    What is the equivalent description of Van der Waals forces for a gas or supercritical fluid?

    In chemistry, a common method of calculating solute and solvent reaction rates in a liquid is through using the quantum mechanical descriptions of inter and intramolecular bonds. One of the intermolecular descriptions is that of Van der Waals interactions - that of the sum of electrical, quantized, yet weak bonding forces. These may include permanent dipole-dipole interactions (Keesom force), dipole-induced dipole interactions (Debye force), and spontaneous dipole interactions (London dispersion forces). Generally Van der Waals forces omit that of ionic bonds between molecules.

    As far as I know though, there isn't a model describing supercritical liquid/gas phases, such as that of CO2 which is often used to decaffeinate coffee beans. I know there are some models built to describe plasmas, but these are generally models designed for cross-section analysis used in fusion/fission reactors - they don't describe allowed energy levels in the same manner as say, a solid state semiconductor would.

    I'm not quite sure how to tackle this kind of problem. In a field theoretic condensed matter picture (or even many-body statistical Schrodinger equation) solids are generally described by phonon modes; quasiparticle states are evaluated with ladder operators, after setting up the problem with electron & ion density. Sometimes metallic conductors can be described by an electron gas - at sufficiently low temperatures, this is a Fermi liquid.

    Fermi liquids have energy levels described by momentum degeneracy and the Pauli exclusion principle. I assume something similar must apply to a gas, but there would be an absurd number of tightly packed available energy levels, and in terms of the Schrodinger equation, most particles would have a Hamiltonian of that similar to a free particle; bumping around other gases though, on the large scale, it's almost a classical description - and in fact, classical descriptions work pretty well. Is it just because the energy levels involved are so high that it's in the classical limit?

    Anyways. I can't seem to find any literature on this - all of the above is just my thinking on it. It's mostly a curiosity of mine :)

    Boris I. Sedunov

    Henry, in a pure real gas the dimension of a cluster is the number of particles bound together in the cluster. In galactics statistic this definition does not work, because masses of particles differ.

    I wish you success in your investigations!

  • Burhan Ullah added an answer:
    How does the formation of oxygen vacancies affect the quality factor?

    As we know that the reduction of Ti+4 to Ti+3 ion, which is the consequence of the formation of oxygen vacancy. The Ti+3 ion can be viewed as a Ti+4 ion that trap an electron(T+4.e) which mean that  Ti+4.e-Voo - T+4.e  possible bond will be formed. The electron will be bound by the fully ionized oxygen vacancies(Voo). So what about the impact of quality factor in such conditions? How we can view the quality factor in terms of oxygen vacancies and the trapped electron?

    Burhan Ullah

    Thanks so much Respected Marcos Augusto Lima Nobre, i become very happy to see your comments on my question,  thanks for your kind suggestion and useful information. This give me a positive feedback to handle my problem.

    -------------------thanks once again to give me your time.--------------

  • Tanmoy Chakraborty added an answer:
    What is the relationship between elastic constants and phonon spectra?


    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.

    Tanmoy Chakraborty

    Thanks Luis for your suggestions

  • Per Söderlind added an answer:
    Any advice on free energy calculations for unstable high temperature phase ?


    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.

    Per Söderlind

    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.

  • Avaneesh Kumar added an answer:
    Does anyone know why band structure and other properties of nano structure is different in spin and non-spin calculation of same structure?

    Hi, I'm new in the magnetic calculation. During simulation of Doped (single atom) bulk structure (Zinc-Blende) with and without spin polarized, i found band and other properties are different, Why is this so? Also help me to which atom (dopant or hosts atom) i will assign spin for spin calculation?    

    Avaneesh Kumar

    Thanks to both of you.

  • Yulia E. Shchadilova added an answer:
    Are the superglass and the supersolid states observed experimentally? Or they still under debate?

    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?

    Yulia E. Shchadilova

    Beyond experiments with liquid helium, the superfluid-to-supersolid phase transition was observed with the cold atomic gas in a cavity, e.g. Science 336, 1570 (2012). 

  • Shielo Namuco added an answer:
    What will happen if you dope a magnetic material like manganese in a GdBCO bulk superconductor?
    Manganese has magnetic moment so I think it's magnetic moment might affect the superconductivity of the sample.
    Shielo Namuco

    Thank you very much for this answer Dr. Jardim. 

  • Xueheng Zheng added an answer:
    How can I calculate phonon dispersion relations of structures with long range interactions using classical potentials?

    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.

    Xueheng Zheng

    I use a central force model, see Physica C 506 (2014) 100, available at my ResearchGate home.

  • Karel Carva added an answer:
    How can i calculate the effective mass of electron and hole for solid solution?

    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 ?

    Karel Carva

    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.

  • Remi Cornwall added an answer:
    Can Fermi level go above top of the conduction band?

    Can Fermi level go above top of the conduction band?

  • Swapnali Dhanayat added an answer:
    What is quantum confinement? How can I calculate bohr excitone radius of nanoparticle?

    How can we calculate the bohr excitone radius of nano-particle?

    Swapnali Dhanayat

    Thank you sir...

  • Behnam Farid added an answer:
    What is Lifshitz transition and how are they different from Fermi level crossing?

    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?

    Behnam Farid

    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.

  • Vincent Mosser added an answer:
    Is there any difference between "Meyer-Neldel rule” of conduction and thermally activated hopping conduction?

    Both follow the same mathematical relation.

    Vincent Mosser


    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.

    There is now some agreement that 1/f noise originates from the incoherent superposition of many discrete lorentzian fluctuators, with time constants ranging from values shorter than the time constant of the experimental equipment to values longer than the duration of the experiment. In a certain sense, this is smearing brought to its limit. For an enlightening insight of 1/f noise in semiconductor devices, see  Kirton and Uren, Advances in Physics 38, 367-468, 1989.

    Hopefully this can help you.

    Best regards,

    Vincent Mosser

  • Behnam Farid added an answer:
    How can I calculate the spin exchange interaction parameters J for any magnetic material?
    I am trying to calculate the spin exchange interaction of a material using DFT method. I have calculated the total energy in different spin configuration in relax structure. I am getting values 5 times less than previous reported values. What could be the possible error.
    Behnam Farid

    You are welcome.

  • Vaibhav Kaware added an answer:
    Does anyone have experience in calculating vibrational properties and geometry of zeolites using Quantum ESPRESSO?

    I've interest to use Quantum ESPRESSO suite in order to obtain the properties of nanoporous materials (zeolites) under critical conditions of pressure and temperature.

    Vaibhav Kaware

    Is there something about the calculations that you wish to ask more specifically?

  • Moulay Tahar Sougrati added an answer:
    Does anyone know anything about the settings of a Mössbauer spectrometer?

    Is there a special formula for exact distance between source & detector or target?

    Moulay Tahar Sougrati

    Here are 4 references for the optimisation of the mossbauer setting :

  • 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.

    1. 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?
    2. 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.)
    3. 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 ?
    Petr Viscor

    Dear Gyanendra,

    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

    Petr: Yes

    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). 

    Best regards


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