The opposite extreme is represented by materials with strong 𝑒-ph coupling and transport in the hop regime, such as MnO [15] and CeO2 [16]. The conventional treatment of 𝑒-ph interactions consists of breaking them up into different physical mechanisms and analyzing them in the long wavelength limit [17]. In the last decade, first-principles methods have emerged and rapidly matured to accurately calculate 𝑒-ph interactions without using empirical parameters [19].
First-Principles Electron-Phonon Interactions
The displacement of the 𝑢𝜅 ions also creates a perturbation in the KS potential that can scatter the propagating electrons. With similar notations as in the above treatment, we extend the KS potential with respect to ion displacements as. The coupling constant𝑒-ph𝑔𝑚 𝑛𝜈(k,q) is then calculated as. which is the complete beginning since all quantities involved in the integral have been obtained from DFT and DFPT calculations.
Band Transport and Electron-Phonon Scattering
Details of the interpolation schemes can be found in the Perturbo code paper [34] developed by the Bernardi group at Caltech and used in this thesis to calculate the β-ph coupling matrix elements and transport properties. Since the -ph interactions in the bandgap regime are relatively weak, electrons can be described by weakly interacting quasiparticles in Bloch states |𝑛k, and the probability that these electronic states are occupied is quantified by the distribution function 𝑓𝑛k. In addition, many materials in the intermediate and hopping regime contain transition metals with an open 𝑑 or 𝑓 shell.
Thesis Outline
In Chapter 4, we extend the first-principles methodology to the hopping regime and investigate small polaron formation in materials with strong 𝑒-ph coupling. The small polaron is a self-localized electronic state that is trapped by the local lattice distortion it causes. The formalism presented in this chapter enables predictions of small polaron formation in materials, as demonstrated by the examples of alkali halides and metal oxides.
BANDLIKE HOLE MOBILITY IN NAPHTHALENE CRYSTAL
- Introduction
- Methods
- Results
- Discussion
- Conclusion
- Supplementary Materials
The relaxation times𝜏𝑛cused in the mobility calculations are the inverse of the diffusion rates, 𝜏𝑛k = 1/Γ𝑒-ph. In panels (b)-(d), yellow is used for positive and blue for negative isosurfaces. the local𝑒-ph coupling in Eq. 2.4), namely the square of the HOMO WF,|𝑤R(r) |2, and the perturbation potential Δ𝜈q𝑉KS due to the atomic motions associated with the given state. Band structures and phonon dispersions of naphthalene crystal, for the structure used at 300 K. a) HOMO and HOMO−1 electronic bands, where black is used for the DFT bands and red for the bands with GW correction.
ELECTRON-TWO-PHONON SCATTERING FROM NEXT-TO-LEADING ORDER PERTURBATION THEORY
- Introduction
- Results
- Discussion
- Methods
- Supplementary Materials
The amplitude of the 2ph process is thus inversely proportional to its extension outside the shell, Δ𝐸 = 𝐸−𝜀𝑛. At each iteration = ℏ/2[Γ(1ph)+Γ(2ph)(𝑚−1)]. The 2 ph dissipation rate exhibits a trend as a function of energy with three plateaus near the end of the conduction band.
Remember that the 2ph velocity is inversely proportional to the square of the off-shell extent, Δ𝐸2. The 1e1a scattering rate decreases from region II to region III due to subtle reasons related to the lifetime of the intermediate state. Let us discuss the temperature dependence of the 2ph scattering processes, focusing on the ratio Γ(2ph)/Γ(1ph) between the 2ph scattering rates and the leading-order 1ph rates.
Both the absolute value of the mobility and its temperature dependence are improved when the 2ph processes are included. In summary, our calculations of the 2ph scattering rates and their contribution to the mobility pave the way to study higher order 𝑒-ph interactions and charge transport in polar materials from first principles. The computational costs of the 2ph propagation velocities are thus approximately 104−105 higher than the 1ph propagation velocities.
Temperature dependence of the ratios of the 2ph scattering processes to the leading order 𝑒-ph scattering rate.
FACILE AB INITIO APPROACH FOR SELF-LOCALIZED POLARONS FROM A CANONICAL TRANSFORMATION
- Introduction
- Small Polaron Hamiltonian
- Methods
- Results
- Discussion
- Supplementary Materials Polaron wavefunctionPolaron wavefunction
Here, we demonstrate an efficient real-space approach to calculate the small polaronic energy, starting with an experimental polaronic wave function. In this scenario, an electron or hole quasiparticle forms a small polaron and becomes self-trapped due to lattice distortion; the polaron formation energy is therefore the difference between the polaron energy 𝐸𝑚 𝑚 and the corresponding band edge. In the following, we take the maximally localized WF as the experimental wave function and calculate its polaron energy.
4.1(d)-(f), conclude that electrons in these materials do not form a self-trapped polaron, as shown by the fact that the polaron energy is above the CBM. One of these is the ability to minimize the polaron energy over the space of possible trial wave functions (here in the form of maximally localized WFs), leading to a more accurate calculation of the polaron formation energy. Equation (4.17) gives the polaron energy for the WF created by the operator 𝑎†𝑚. Yet this WF is not unique.
The true polaron wave function of the system is considered to be the most stable polaron wave function, which is the WF with the lowest polaron energy. A self-localized state is formed when the polaron energy is lower than the conduction band minimum, that is, when We plot the ratio of the polaron energy in the transformed basis to that in the untransformed basis, Δ𝐸𝑡/|Δ𝐸𝑢| in Fig.
The ratio of the polar energy in the transformed basis to that in the untransformed basis, Δ𝐸𝑡/|Δ𝐸𝑢|, for four different values of 𝑔1/𝑔0.
CONCLUSION AND FUTURE DIRECTIONS
The derivations involve Matsubara sums of two-loop Feynman diagrams, and the numerical challenges are overcome by using Monte Carlo integration together with a self-consistent update of the intermediate state lifetimes. We apply our method to GaAs, a weakly polar semiconductor with a dominant long-range𝑒-ph interaction, and find that the 2ph scattering rates are as large as nearly half the value of the one-phonon (1ph) rates . 5.1(a) that originates from the first derivative of the KS potential with respect to lattice vibrations [see the expansion in Eq.
In addition to the stretching series of the KS potential, we also introduce the harmonic approximation in the calculation of the network dynamics in Eq. A thorough consideration of all these leading-order scattering processes will provide more insights into charge carrier dynamics and place the utility of perturbation propagation on more solid ground. By heuristically choosing the lattice distortion, we construct a self-trapped electronic state that is hopping-free and isolated from all modes of oscillation.
Whether charge carriers in a material prefer a localized small polaron state over a delocalized Bloch state can be deduced by a direct comparison of the calculated polaron energy with the conduction or valence band edge. With a recent implementation that reconciles the DFPT and the DFT+U calculations, we can correctly calculate the local pH coupling constant and thus extend the applicability of the polaron energy calculations to a wider range of materials. This is one of the most promising candidates to explain the transport mechanism in materials with very low mobility.
Combining the ab initio 𝑒-ph calculation with this small polaron transport theory will provide further insight into the transport mechanism of materials in the hopping regime, and complement the usual band transport picture at the other end of the mobility spectrum.
ANALYTIC DERIVATION OF THE SCATTERING RATES OF TWO-PHONON PROCESSES
- Feynmann Rules
- Electron Self-Energy
- One-Loop Diagram I
- Two-Loop Diagram IIa
- Two-Loop Diagram IIb
- Resonance
- Summary
We will mainly be interested in the scattering rate at electron energy 𝜉𝑛k, so we will set 𝐸 = 𝜉𝑛k to get the self-energy on the shell for the state with band 𝑛and crystal momentom. Now we calculate the first two-loop diagram shown in the figure above. Fortunately, both cases give the same expression for two-phonon scattering processes, as we explicitly show below.
The other three poles are simple poles and can be handled in the usual way. Using a notation we introduced above,𝐴𝜅sums all terms independent of A useful reasonable check is that our finite temperature results should reduce to zero temperature results in the limit → 0, from which we check that the structure of poles of the finite and zero temperature expressions to be consistent with each other.
Let us denote the (𝜈q ↔ 𝜇p) term with its dummy variables replaced in the manner indicated by the arrows, 𝜈 ↔ 𝜇, q↔ p, etc. A common practice in this situation, which also appears in other quantum field theories, is to consider the full electron propagator 1/(𝐸 −𝜉+𝑖𝜂−Σ), as shown in the figure below, which introduces a finite lifetime for the intermediate electronic state. Diagrammatically, this approach is equivalent to performing a resummation of the diagrams for all tasks, as done in the well-known GW self-energy {see Eq.
For our 2h scattering rate expression, we simply add the self-energy of the intermediate state to the denominators of all the 𝛾 terms above, which removes the divergences.
BOLTZMANN TRANSPORT EQUATION WITH TWO-PHONON CONTRIBUTIONS
Summary
BOLTZMANN TRANSPORT COMPARISON WITH TWO-PHONON CONTRIBUTIONS. substitutingF𝑛k on the right-hand side of Eq. B.1) to obtain an updatedF𝑛k, until a convergedF𝑛k is reached. Our work in Chapter 3 presents calculations, both within the RTA and the ITA, that include or neglect the 2ph processes; if only 1ph processes are included, 𝜏𝑛k is set to the inverse ofΓ(1ph). The ITA with 2ph contributions is the most accurate level of theory and the one that best matches the experiment, while the ITA with only 1ph processes overestimates the experimental result.
This is a good approximation if for most (q,p) pairs, there is only one process in Eq. 3.3) that dominates, as in the case of GaAs that we have verified.
Derivation of the Boltzmann Transport Equation with Two-Phonon Scat- tering Precessestering Precesses
The function F𝜆 will be interpreted as the mean free displacement later, while for now it is just a parameterization for the first-order deviation. Substituting the expansion into Eq. B.2) and keep the terms only up to the first order of E, the left side becomes. kuv𝜆 is the band velocity and ℏ is set to unity. The right side of Eq. B.2) is the total rate of change of the distribution function due to all types of distribution processes [3]:. where𝑃 is the net rate of change of 𝑓𝜆 due to transition between states on the two sides of the arrow.
For example, 𝑃𝜆↔𝜆0𝛼 is the difference between the rate of state𝜆 making a transition to states𝜆0𝛼 and the rate of its reverse process. Note that if there is no electric field and the system reaches thermal equilibrium, 𝑃𝜆↔𝜆0𝛼 should vanish. Using the detailed balance and keeping the terms up to the first order of E, we get.
The second term of the equation is similar. Again, linearizing and keeping terms up to first order E, we get B.8), the rate of change of 𝑓𝜆 due to first-order 𝑒-ph scattering can be expressed as Let's move on to the next leading order scatter given in the second line of Eq.
If we linearize these terms of the second order and keep the terms up to the first order E, we get
