Third generation Photovoltaics: beyond the Shockley-Queisser limit

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Nature of the protective film

Unlike a passive metallic material, copper cannot produce a passive film. But the corrosion products formed on copper and its alloys, to some extent; provide a protection against corrosion. In most cases, the protective oxide film on copper is Cu(I) oxide in aqueous environments at room temperature. This adherent and relatively impervious film acts as a diffusion barrier, but it is easily affected by change in hydrodynamic conditions.
The good corrosion resistance of Cu alloys in seawater is related to the formation of a protective film of corrosion products in the early stages of exposure. It is generally considered that the inner part of the film is made of cuprous oxide (Cu2O) with cupric oxide (CuO) appearing in the outer part of the film, and that the film contains metallic ions together with chlorides, hydroxides and carbonates [48]. Although this film will start developing during initial contact with oxygenated water, it may take several weeks for the film to be fully protective. When the film is fully developed and reaches steady-state, the corrosion rate is usually very low. In unpolluted seawater, a loosely adherent porous cupric hydroxychloride (Cu2(OH)3Cl) corrosion product scale forms over a thin, tightly adherent layer of cuprous oxide (Cu2O) that increases corrosion resistance with increasing exposure time [49, 50]. The inner film is normally reddish. The outer film may be greenish, brown or yellow brown. Although Cu2O oxide is the principal component of the film, the lattice usually includes other metallic ions, e.g. iron, nickel, aluminum, calcium, silicon, and sometimes other species. Principal anions include chlorides, hydroxides [51], carbonates and bicarbonates.
Once a protective surface film is formed, the corrosion rate will continue decreasing over a period of years, related to the classical parabolic growth rate of protective layers. For this reason, it has always been difficult to predict the life time of copper-nickel based alloys based on short-term results. Usually, general corrosion rates of 0.02-0.002 mm/yr or 20 to 2 µm/year are anticipated [52].
Formation, structure and chemical composition of the protective layer are complex and have been the subject of many investigations [53-56]. The complexity of the films on Cu-Ni alloys in marine conditions was studied by Kato et al. [55, 56]. From exposure of 90Cu-10Ni samples to air-saturated 3.4% NaCl solution and analysis of the corrosion product layers by SEM and X-ray diffraction, authors concluded that the protective films, formed under open-circuit corrosion conditions, had the following features:
a) thick outer layer, mainly cuprous hydroxy-chloride [Cu2(OH)3Cl], and inner layer containing appreciable amounts of chloride, oxygen, copper and some nickel;
b) rich in chloride throughout the film with a maximum concentration along a plane located within the inner layer near the inner layer/outer layer interface;
c) relatively poor in Ni and Fe in the inner layer, compared to levels in the outer layer,
and, in early stages of growth, outer interface surface consisting of a cuprous compound (probably Cu2O) which, with time, gives rise to carbonate and finally to cuprous hydroxy-chloride [Cu2(OH)3Cl] compounds.

GALVANIC CORROSION

An electrochemical potential almost always exists between two different metals when they are immersed in a conductive solution. If two different metals are in electrical contact with each other and immersed in a conductive solution, as seawater, a potential results that enhances the corrosion of the more electronegative metal of the couple (anode) and protects the more electropositive one (cathode).
Usually, copper alloys are more cathodic than other metals (due to their position in galvanic series) such as steel and aluminum. Copper alloys usually corrode preferentially when coupled with high-nickel alloys, titanium or graphite.
The seawater high conductivity gives rise to the possibility of formation of galvanic cells with the cathodic and the anodic areas at some meters from each other, giving rise to highly localized corrosion processes.
Accelerated damage due to galvanic effects is usually the most important near the junctions, where the electrochemical current density is the highest [57]. Another factor that affects this kind of corrosion is area ratio; it happens when the cathodic area is large and the anodic one is small.

PITTING CORROSION

Pitting is the usual form of corrosive attack at surfaces on which there are incomplete protective films, non-protective deposits, or extraneous deposits of dirty or other substances. Pitting of copper and its alloys always occurs under relatively low flow velocity, usually less than 0.6 to 0.9 m/s, and has long been associated with chloride ions [58].
Once a pit is initiated, it may propagate at a significant rate because of the development of a macro-cell, the surrounding surface and the inside of the pit. Due to the difference in electrode potential between the large passive surface area (more anodic) and the small active pit (more cathodic), the pit acts as a small anode and the external surface as a large cathode.
Copper alloys do not corrode primarily by pitting, but due to metallurgical and environmental factors that still need to be clearly understood, is a common problem detected. In order to prevent a copper alloy from pitting, the correct choice of copper alloy for the environment is necessary. For example, aluminum brass is the best choice for protection against pitting attack, while the high-copper alloys are somewhat more inclined to pitting [57].

DEALLOYING

Dealloying is a corrosion process in which the more active metal is selectively removed from an alloy, leaving behind a spongy layer of the more noble metal. Copper-Zinc alloys containing more than 15% of Zn are susceptible to a dealloying process called dezincification [57]. In the dezincification of brass, selective removal of zinc leaves a relatively porous layer of copper and copper oxide. It can be readily observed with naked eyes because the alloy develops a reddish color that contrasts with its original yellowish color.
Generally, there are two types of dealloying. Uniform or layer dealloying commonly occurs in high zinc alloys where the outer layer is de-alloyed and becomes dark while the inside is not affected; plug dealloying is typical of low zinc alloys and is characterized by the presence of de-alloyed dark plugs in the unaffected matrix of low zinc alloys [59].
Two theories have been proposed for dealloying of brass. One states that there is simultaneous anodic dissolution of both copper and zinc, while dissolved copper ions precipitate plate back from the solution on the remaining brass surface as a porous layer; the other states that the less noble alloying elements vs selectively dissolved, leaving vacancies in the brass lattice resulting in skeletal copper with poor mechanical integrity [59]. In the past decade, many alloying elements, as arsenic (As), have been used to minimize the dezincification and corrosion of brass alloys.

AMMONIA ATTACK

Ammonia can affect strongly the corrosion behavior of Cu-base alloys condenser tubes by forming soluble copper-amine complex that causes metal loss and pitting in the tubes [60]. At the condenser tube water side, ammonia compounds are produced in case of stagnant water, due to the fermentation processes occurring in presence of high organic compounds and biofilms. Nevertheless, ammonia attack may also affect the vapor side, as the water of thermal cycle of power plant is treated with hydrazine or other reducing compounds to control pH and the oxygen content.
Ammonia can also cause stress corrosion cracking in some copper alloys. In the presence of air and ammonia, aluminum brass is subject to stress corrosion cracking. Aluminum bronze is more resistant and copper nickels are highly resistant to ammonia stress corrosion cracking [61].

SULFIDE ATTACK

Sulfides are present in polluted seawater and can also be produced under static conditions due to the decomposition of organic matter. Copper-nickel alloys corrode in the presence of sulfide, sulfur, polysulfides, or combinations of these species. Thus, essentially non-corrosive or slow corrosion systems in de-aerated seawater turn into highly corrosive systems [62], and the effect of sulfur-containing compounds is to interfere with the formation of surface film, producing a black film made up of cuprous oxide and sulfide.
Sulfide ions have been shown to significantly increase the corrosion rate of Cu-Ni alloys [63-66]. In sulfide-containing environments, a porous copper sulfide layer forms on the surface of the alloys, which does not protect against corrosion [65]. This layer also prevents the formation of a protective copper oxide layer. During the enhanced corrosion in sulfide containing environments, preferential copper dissolution is observed [64, 66].
Eiselstein et al. [67] proposed two different mechanisms for Cu-Ni corrosion in sulfide-polluted seawater. One for the de-aerated seawater, where sulfide ions react with Cu(I): the decrease in the Cu(I) concentration will cause the anodic reaction to be shifted to lower potentials. In the case of aerated seawater, Eiselstein claimed that S2- reacts with both Cu(I) and oxygen. In consequence, a shift of the anodic and cathodic reactions will occur in such a way that the intersection of the two polarization curves at the corrosion potential occur at higher corrosion currents.

EROSION-CORROSION

Copper alloys are relatively sensitive to erosion-corrosion when they are exposed to water with high flow velocity, and especially when turbulences occur. It is a common water-side phenomenon that is only a problem for copper alloy condenser tubes [9]. It occurs above a critical local flow intensity (Fig. 1-9), creating local energy densities which are high enough to break down protective scales, layers or films on the metallic surface. With the increase of seawater flow rate, corrosion rates remain low due to the tenacity of the protective surface film. However, when the velocity exceeds a critical value for a given geometry, the shear stress acting on the film can lead to its breakdown resulting in high corrosion rates. A critical velocity is found at which localized corrosion occurs [68].
The flow rate of the electrolyte also affects the corrosion behavior of Cu-Ni alloys. In the presence of high turbulence, a unique corrosion morphology, the so-called horse-shoe corrosion develops [58, 69].
In order to avoid corrosion problems, some suggested maximum cooling water flow velocities are shown in Table 1-4. Minimum flow rates of more than 1 m/s are usually preferred to avoid sediment build up [70]. The effect of the water velocity on the corrosion of Cu-based alloys is discussed on section 1.3.6.

MICROBIOLOGICALLY INFLUENCED CORROSION (MIC)

The well-known toxicity of cuprous ions towards living organisms does not mean that the copper-based alloys are immune to biological effects on corrosion. The electrochemical nature of the metallic corrosion still remains present in the microbial corrosion. There is an anodic process of metallic dissolution and a complementary cathodic process that is dependent of the metal-biofilm characteristics (pH, aeration, chemical composition…), as the reduction of dissolved oxygen (in aerated environments and neutral pH) or the reduction of water (no-aerated environments). Microorganisms can change the metal/solution interface to induce, accelerate or inhibit the anodic and or cathodic reactions of the corrosion process.
The biological process is illustrated by a microbial colony growing up on the metallic substrate (Fig. 1-10). An anaerobic region is formed under the microbial colony, due to the oxygen consumption by the microbial respiration (in case of aerobic microbes) and another region, where more oxygen reaches the external part of the colony, in contact with the aerated liquid.
Any material in contact with natural waters is rapidly colonized by microbial species growing in a complex micro-environment named biofilm. Despite the good performance of copper alloys in seawater, the formation of biofilm on surfaces of heat exchangers and the subsequent settlement of macrofouling induce microbiologically-influenced corrosion (MIC), also called microbial corrosion or biocorrosion. Thus, these phenomena modify the integrity and functionality of metallic materials employed in condensers cooled with seawater [71, 72]. When immersed, copper alloys are swiftly covered by colonies of bacteria which, unlike macrofouling organisms, are not affected by the toxicity of copper, since there are protected by a mucopolysaccharide matrix [73]. The first stage in the biofilm formation is the adsorption of biomolecules, such as proteins, and other organic matter dispersed in seawater. Then, bacterial adhesion occurs and bacteria colonies are formed, together with corrosion products, algae and other microorganisms, resulting in a big complex microfouling that adheres to the metallic surface [74]. Since microfouling modifies the chemical and physical characteristics of the surface and facilitates the development of corrosion processes, an undesirable resistance to the heat exchange is introduced [75].
Biofilms are capable to modify the electrochemical characteristics of metallic surfaces [76, 77], modifying the kinetics of corrosion processes occurring at the metal-biofilm interface.
Some species of Pseudomonas, Sphingomonas, Sphingomonas paucimobilis, Rhodotorula, Flavobacterium, Acidovorax delafieldii, Cytophaga johnsonae, Micrococcus kristinae, Acidovorax and Sphingomonas have been identified in biofilms responsible for MIC in copper pipes with drinking water [78-81]. Genus of Pseudomonas have been shown to colonize copper surfaces [82] and to accelerate corrosion of copper and copper alloys [83].
The influence of marine aerobic Pseudomonas strain on the corrosion behavior of 70Cu-30Ni alloy was investigated by S.J.Yuan et al. [84]. A potential shift towards more cathodic value, in the presence of Pseudomonas bacteria was confirmed. Their results showed that, in the presence of Pseudomonas, the cathodic Tafel constant increases (dc) while Ecorr shift in the cathodic direction and the absolute corrosion current increase.
However, if Pseudomonas were not present, Ecorr shifted slightly to more anodic values and the current density was reduced. At the same time, the anodic Tafel constant (ba) shift to higher values. The impedance spectra shown that the alloy surface, in the sterile medium, was comprised of an outer organic compound conditioning layer and an inner compact and protective oxide film layer; while in the Pseudomonas inoculated medium, a duplex layer of an outer porous, heterogeneous and non-protective biofilm layer and an inner porous oxide film layer was present. Their results also demonstrated a growth of the outer film with immersion time (impedance increase).
Methods used to prevent MIC, should aim in either inhibit the growth and/or metabolic activity of microorganism, or modify the environment in which the corrosion process takes place in order to avoid adaptation of microorganisms to the existing conditions. These methods can be divided in: a) cleaning procedures, b) biocides, c) coatings and, d) cathodic protection.

EFFECT OF DIFFERENT PARAMETERS ON THE CORROSION BEHAVIOR OF 70CU-30NI ALLOY AND AL BRASS

A copper alloy is the combination of copper with one or more other metals to form a material that can improve the performance of pure copper. Doping with divalent or trivalent cations is an effective way of improving the corrosion resistance of copper.
Copper is the most noble metal in common use. It has excellent resistance to corrosion in the atmosphere and in fresh water. In seawater, copper and copper alloys (particularly those associated with nickel) are widely and successfully used, due to their corrosion resistance, mechanical strength and workability, excellent electrical and thermal conductivity [86] and resistance to macrofouling [87]. In practice, aluminum, zinc, tin, iron, and nickel are often used as alloying elements and reduce noticeably the corrosion rate of copper alloys.

EFFECT OF IRON AND NICKEL

Nickel and iron present in Cu-Ni alloys improve the corrosion and erosion resistance properties of the oxide film. There is beneficial effect of incorporating iron in copper-nickel alloys and it is essential to obtain adequate corrosion resistance by assisting in protective film formation [88, 89].
Copper and its alloys do not form a truly passive corrosion product film. In aqueous environments at ambient temperature, the corrosion product predominantly responsible for protection is cuprous oxide (Cu2O), a p-type semiconductor. It has been established that this is the main component of the protective film formed, in the early stages of growth. The protective properties are enhanced by the incorporation of nickel and iron that leads to a decrease in both ionic and electronic conductivities [89, 90]. The corrosion process can proceed if copper ions and electrons migrate through the Cu2O film. In order to improve the corrosion resistance of the material, the ionic and/or electronic conductivity of the film must be reduced by doping with divalent or trivalent cations [91].
Small additions of iron to copper-nickel alloys are also known to improve their resistance to erosion-corrosion [92] because iron is necessary for the occurrence of nickel enrichment in the corrosion product layer [93].
According to Popplewell et al. [95], the nature of the corrosion film depends on the iron content of the alloy: the film on samples with 0.3 % of iron in the bulk alloy or less is bright green, that on samples with 1.5 % of iron is dark green, and that on samples with 1.5 % of iron, heat treated to precipitate an iron rich phase, is nearly black.

EFFECT OF TEMPERATURE

Generally, the corrosion of copper-based alloys in de-aerated seawater flowing from 0.9 to 2.7 m.s-1 increases as the seawater is heated to 63°C. Maximum corrosion occurs at intermediate temperatures from 54 to 71°C. If seawater reaches too high temperatures (≈107°C), a significant decrease in corrosion is noted [96].
According to Ijsseling et al. the appearance, thickness and adherence of the oxide film, in aerated seawater, depends on the temperature of formation [97]. In flowing seawater (at 1.5 m.s-1), an adherent dark brown layer is formed at 10° C, a less adherent dark brown one is formed at 20-30°C, and a very thin, adherent, gold-brown film is visible at 40-50°C.

EFFECT OF PH

Depending on pH, the anodic polarization of copper may result in anodic dissolution or film formation. In acidic aqueous solutions, such as HCl or H2SO4, Cu(I) complexes are formed by bonding with Cl- or SO42-. These porous corrosion products do not prevent copper from further dissolution. As pH decreases, the corrosion rate increases. Low pH levels prevent copper-based alloys from developing or maintaining protective films and thus high corrosion rates are encountered.
Feng et al. systematically investigated copper corrosion in simulated tap water over a wide pH range [98]. The results of this study are summarized in Table 1-5. The thickness of the oxide films and the dependence of corrosion rate on pH were estimated as shown in Figure 1-11. Thus, dissolution of oxide films takes place when pH is less than 4; for a pH between 4 and 10, cubic Cu2O crystals grow while the crystal size becomes smaller and the film becomes thinner; when pH is greater than 10, monoclinic CuO films start forming.
Figure 1-11: (a) Variation of oxide film thickness with pH estimated from electrochemical methods (EC) or weight loss method (WL); (b) corrosion current density of copper in solutions of various pH calculated from polarization curves or weight loss measurements. Copper immersed during 24 h in simulated tap water at 30ºC (adapted from Feng et al. [98]).
The influence of metal-biofilm interface pH on aluminum brass corrosion in seawater was studied by Cristiani et al. [99]. According to these authors, biofilms may act on corrosion by acidification at the metal-biofilm interface. Differences have been found in corrosion current density and in surface morphology for specimens exposed to natural seawater compared to those exposed to artificial seawater (at pH 8.3).
It has been shown that at 65°C the erosion-corrosion attack is much more extensive and appears at lower velocities when the pH is decreased from 8 to 6.5 [100]. If the pH of warm (35°C) seawater flowing at about 2.7 m/s is adjusted to 3.6, no protective film forms on the surface of 90Cu-10Ni [101].

EFFECT OF OXYGEN CONTENT

The oxygen content of the electrolyte has a significant effect on the corrosion resistance of copper alloys [59]. For instance, at low to moderate oxygen concentrations ([O2] ≤ 6.6 mg/L), the 70Cu-30Ni alloy is more corrosion resistant than the 90Cu-10Ni ones and in saturated seawater; both alloys have similar behavior [59].

EFFECT OF POLLUTED SEAWATER

Copper-nickel alloys corrode at increased rates in polluted waters (compared to clean waters), particularly when sulfides or other sulfur compounds are present [48, 94, 102]. Sulfides form a black corrosion product, which is less adherent and protective than the normal oxide film.
Gudas and Hack reported that 0.05 ppm of sulfide or more was necessary to cause increased corrosion of 70Cu-30Ni alloy, and even 0.01 ppm of sulfide, during one day, caused accelerated attack of 90Cu-10Ni alloy [103]. This is in agreement with the work of Alhajji and Reda who noted that sulfide was very corrosive towards the alloys with low nickel content [104].
The corrosion of Cu-Ni alloys, in flowing (1.6 m/s) seawater containing sulfides, polysulfides, or sulfur, was investigated by Anderson and Badia [105] and MacDonald et al.
[62]. Cuprous sulphide forms as the principal corrosion product causing damage to the protective film on the metal surface. The importance of proper protective film formation on tubes and pipes must be emphasized. It is stated that if during the early life of condenser tubes clean seawater passes through, good protective films will form which are likely to withstand most adverse conditions. The ideal situation, whether in a ship or a power plant, is to re-circulate aerated, clean seawater from initial start up to a time sufficient enough to form a good protective film. If, however, polluted waters are encountered during the early life, the films formed on the condenser tubes will likely not be fully protective and the risk of premature failure will be considerably increased [102].
Yuan et al. showed that when 70Cu-30Ni bare metal was in contact with sulfur ions, the surface film evolves from an oxide bi-layer (formed in the absence of S-species) to S-based layered films [84]. The authors also showed that the film was porous and not very protective in the presence of S-ions. These latter dissolve copper oxides but nickel is not affected as readily as copper. Thus, it is possible that a semi-protective film richer in nickel is formed at longer times, explaining the apparent repassivation.

EFFECT OF WATER VELOCITY

Although higher coolant flow velocities improve the efficiency of condensers and heat exchangers, the velocity cannot be increased above the endurance limit of materials.
Giuliani et al. [106] found that the sensitivity of Cu-Ni alloys corrosion rate to water flow rate was relatively low in Cl- containing solution.
Based on the work carried out by Efird [107], seawater moving over a metallic surface creates a shear stress between that surface and the layer of seawater closest to the metal. As velocity increases, the corrosion rate first increases slowly, as a result of increased oxygen supply, (cathodic depolarization) and removal of the corrosion products from the metallic surface. At higher velocities, the degree of turbulence and the shear stress are such that the protective film can be locally removed and the active underlying metal can be exposed to water; at this « breakdown » velocity, the rate of attack increases dramatically. Efird [107] studied and estimated the critical shear stress for various Cu-based alloys. His research suggested that shear stresses depended on water velocity and pipe geometry. As pipe diameter increases, copper-based alloys tolerate higher nominal velocities.
Sato and Nagat [108] showed that the shear stress at the inlet-end of a condenser tube is about double that further down the tube. This explains why inlet-end erosion corrosion is such a common occurrence, and also explains the preference for copper-nickel alloys that have been developed because of their greater velocity tolerance.
At low flow velocities, sand, debris, mud, etc. may deposit, increasing the chance for the formation of macro-corrosion cells (differential aeration and concentration cells).

Table of contents :

Introduction 
1 The Hot-Carrier Solar Cell 
1.1 Third generation Photovoltaics: beyond the Shockley-Queisser limit
1.1.1 The last decade objective
1.1.2 « There’s plenty of room above »
1.1.3 Promising already existing technologies
1.2 Carrier cooling, the hot topic
1.2.1 Carrier-carrier scattering
1.2.2 Intravalley versus intervalley scattering
1.2.3 Auger scattering and carrier multiplication
1.2.4 The role of phonons in carrier cooling
1.3 Hot Carrier Solar Cell: the ultimate PV device
1.3.1 Principle and comparison with a single p-n junction solar cell
1.3.2 Maximum efficiencies expected
1.3.3 Recent experimental achievements on hot-carrier solar cell issues
1.4 Materials for hot-carrier solar cell absorber
1.4.1 Carrier cooling and atomic scale engineering
1.4.2 Formal requirements
1.4.3 Physical challenges for materials science
1.5 Open questions
2 Phonons: density of final states 
2.1 The Physics of phonons
2.1.1 Interatomic force constants
2.1.2 Equation of motion, dynamical matrix and secular equation
2.1.3 Phonon dispersion and one-phonon density of states
2.2 Phonon decay
2.2.1 Conservation rules
2.2.2 Decay channels nomenclature
2.2.3 Engineering phonon band structure for hot-carrier solar cells
2.2.4 Need for a complete picture
2.3 Phonons within Density Functional Perturbation Theory
2.3.1 Principle and formalism
2.3.2 Successes and limitations
2.4 Two-phonon final states in bulk semiconductors
2.4.1 Practical implementation
2.4.2 Detailed two-phonon states analysis
2.4.3 Discussion
2.5 Gaps in the density of states of nanostructured materials
2.5.1 Practical implementation of PhDOS calculation on large systems
2.5.2 Application to superlattices
2.5.3 Application to quantum dots
2.6 Gaps in the density of states, a lost cause ?
2.6.1 Superlattice: how gaps are filled
2.6.2 Quantum dot: how gaps vanish because dots are not round
2.6.3 How it turns out that gaps are sufficient but not necessary
2.7 Conclusion on the two-phonon final states investigation
3 Phonons: decay rate 
3.1 LO-phonon lifetime
3.1.1 Phonon decay from experiment
3.1.2 Lifetime in bulk materials
3.2 Formalism – Beyond the harmonic approximation
3.2.1 Derivation
3.2.2 State of the art of phonon lifetime calculation
3.2.3 Drawbacks of tantalizing approximations
3.2.4 Misleading success in fitting experimental data
3.3 Third order anharmonic tensor within DFPT
3.4 Practical implementation
3.4.1 Phonons eigenvectors and eigenfrequencies
3.4.2 q-space mesh and energy smearing
3.4.3 The two-phonon density of final states
3.4.4 Validation and commentary on DFPT for HCSC
3.5 Application to SiSn
3.5.1 Two-phonon density of states
3.5.2 Lifetime of the LO-phonon: beyond the zone centre approximation
3.5.3 Lifetime dependence on the LO-phonon reciprocal position
3.5.4 Eventual photovoltaic efficiency
3.6 Towards higher orders-phonon processes
3.6.1 Previous discussions on four-phonon processes
3.6.2 Estimation of the four-phonon processes contribution
3.6.3 Consecutive three-phonon processes
3.6.4 Crystal symmetry and higher-order anharmonic tensors
3.7 Conclusions on LO phonon lifetime calculations for HCSC
3.7.1 SiSn features as references
3.7.2 Limitations
3.7.3 Towards other ways to hinder carrier cooling
4 Electron-phonon interaction 
4.1 Dimensionality issue
4.1.1 Electron cooling in superlattices
4.1.2 Self-induced electric field and intraband scattering
4.1.3 [InAs]n-[GaAs]n
4.2 Directionally dependent electron-phonon interaction model
4.2.1 Electron-phonon coupling strength
4.2.2 Electron wave function
4.2.3 Electron evolution equations
4.3 Full-band cascade: practical implementation
4.3.1 Electronic band
4.3.2 Phonon polar field
4.3.3 Electronic cascade
4.3.4 Validation: bulk case
4.4 Effects of the superlattice size
4.4.1 Effect on the electric field
4.4.2 Effect on the electron-phonon interaction dimensionality
4.5 Conclusions on the electron-phonon interaction in superlattices
4.5.1 Approach and results
4.5.2 Approximations and subsequent limitations
4.5.3 Perspective on the hot-carrier effect in superlattices
Conclusion

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