Theory of electron-plasmon coupling in semiconductors
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Theory of electron-plasmon coupling in semiconductors
Fabio Caruso and Feliciano Giustino
Department of Materials, University of Oxford, Parks Road, Oxford, OX1 3PH (Dated: September 12, 2016)The ability to manipulate plasmons is driving new developments in electronics, optics, sensing,energy, and medicine. Despite the massive momentum of experimental research in this direction,a predictive quantum-mechanical framework for describing electron-plasmon interactions in realmaterials is still missing. Here, starting from a many-body Green’s function approach, we developan ab initio approach for investigating electron-plasmon coupling in solids. As a first demonstrationof this methodology, we show that electron-plasmon scattering is the primary mechanism for thecooling of hot carriers in doped silicon, it is key to explain measured electron mobilities at highdoping, and it leads to a quantum zero-point renormalization of the band gap in agreement withexperiment.
Plasmons are collective oscillations of electrons insolids that can exist even in the absence of an externaldriving field. During the last decade plasmons generatedtremendous interest owing to the rise of plasmonics, thescience of manipulating light and light-matter interac-tions using surface plasmon polaritons . Plasmonic ma-terials and devices show exceptional promise in the areasof nanoelectronics , photovoltaics , and radiation treat-ment therapy . While the electrodynamic laws gov-erning plasmonics at macroscopic length-scales are wellunderstood , little is known about the interaction ofplasmons with matter at the atomic scale. For examplequestions pertaining the interaction between plasmonsand charge carriers in semiconductors have not been ad-dressed on quantitative grounds, yet they are criticalto engineering materials for semiconductor plasmonics .Up to now microscopic quantum-mechanical theories ofelectron-plasmon interactions have been limited to ide-alised models of solids, such as the homogeneous electrongas . While these models laid the theoretical founda-tions of the theory, they are not suitable for predictivecalculations.In this work we introduce a first-principles method tostudy electron-plasmon coupling in solids. As a firstapplication we focus on doped semiconductors, wherethe manifestations of electron-plasmon coupling are mostspectacular. In contrast to metals and insulators, dopedsemiconductors can sustain ‘thermal plasmons’, that isplasmons with energies comparable to those of latticevibrations. Under these conditions electron-plasmon in-teractions can modify carrier lifetimes, mobilities, andoptical gaps in a manner similar to electron-phonon in-teractions. Using this method we find that, in the caseof degenerate n -type silicon, thermal plasmons lead toultrafast relaxation of hot carriers, provide the main bot-tleneck to carrier mobility, and induce a zero-point renor-malization of the band gap that exceeds the phonon-induced renormalization.In free-electron metals the energy of a plasmon is ~ ω P = ( ~ e n/ε m e ) , where ~ is the Planck constant, ε is the dielectric permittivity of vacuum, and e , m e , and n are the electron charge, mass, and density, respectively.At typical metallic densities, as in common plasmonic metals such as Au and Ag ( n = 3-8 · cm − ), plas-mons have characteristic energies in the range of 5-10 eV.In these cases electron-plasmon scattering is suppressedby the energy-conservation selection rule. At variancewith this scenario, in doped semiconductors the electronmass in the above expression is replaced by the band ef-fective mass, and the vacuum permittivity is replaced bythe dielectric constant. As a result the plasmon energy isconsiderably smaller, and at standard doping levels it caneasily reach the thermal range, ~ ω P =10-100 meV. Un-der these conditions electrons can exchange energy withplasmons, hence the populations of carriers and plasmonsbecome mutually coupled.In order to investigate the consequences of this cou-pling, we start by characterizing plasmonic excitationsin doped silicon from first principles. Figure 1 showsthe calculated electron energy loss function, Im ǫ − ( q , ω ),which encodes information about how an electron travel-ling through a solid dissipates energy . Here ǫ − ( q , ω )denotes the head of the inverse dielectric matrix for thewavevector q and the frequency ω , evaluated within therandom phase approximation . In the case of intrin-sic silicon at zero temperature (Fig. 1a) the loss functionexhibits a continuous energy distribution (brown region)with a threshold set by the fundamental gap. This broadstructure arises from interband transitions from the filledvalence bands to the empty conduction bands, and phys-ically corresponds to the generation of electron-hole pairsby impact ionization. This is schematically indicated as‘process 1’ in Fig. 1d. The scenario changes drasticallyin the case of doped silicon. Fig. 1b and Fig. 1c showthe loss function of heavily n -doped silicon, correspond-ing to n = 2 . · cm − and n = 2 . · cm − ,respectively. As a result of the partial filling of the con-duction band valley near the X point of the Brillouinzone, new dissipation channels become available. In par-ticular, ‘process 2’ in Fig. 1b corresponds to the genera-tion of low-energy electron-hole pairs. In this case we seesharp structures which define ‘ghost’ bands as a functionof the momentum loss ~ q . These features are understoodin terms of intraband and interband transitions from oc-cupied initial states with wavevector k near the bottomof the conduction band to empty final states of wavevec- Figure 1. (a–c) Calculated electron-energy loss function of n -type silicon for momentum transfers q along the Γ X high-symmetryline. The carrier density increases from left to right, from 10 to 10 cm − . (d) LDA band structure of silicon, and Fermilevel ( E F ) for n = 2 . · cm − . The step-like structures in (b) and (c) are only a numerical artifact arising from the limitedBrillouin-zone sampling. (e) Variation of the plasmon peak in the loss function vs. carrier density, evaluated at q = 0. (f)Plasma energies extracted from peaks in (e), plotted vs. carrier concentration (blue dots). The red line corresponds to theanalytical result obtained for a homogeneous electron gas with the calculated isotropic effective mass and dielectric constantof silicon ( m eff = 0 . ǫ Si = 12). tor k + q . The intensity of these features increases withthe doping level from Fig. 1b to Fig. 1c. The peaks inthe loss function denoted by ‘process 3’ cannot be ex-plained in terms of the previous two mechanisms. In factfor q = 0 these structures are much sharper than thosedescribed above, and exist below the energy (momen-tum) threshold for the generation of electron-hole pairsvia interband (intraband) transitions. These processescorrespond to the emission of plasmons, and are charac-terised by well-defined energy resonances, as it is shownby Fig. 1e for q = 0. By mapping these plasmon peaks inthe loss function we can see in Fig. 1f that the plasmonenergy ~ ω P scales with the carrier concentration, follow-ing the same trend expected for a homogeneous electrongas. In this figure we also see that the plasmon energy ishighly tunable via doping, from thermal energies at car-rier densities around 10 cm − , to half an electronvoltat densities near 10 cm − .At large momentum transfer ~ q the distinction be-tween plasmons and electron-hole pairs is no longermeaningful, since the fluctuations of the charge density happen on length-scales approaching the size of the crys-tal unit cell. In the following we identify plasmons inthe loss function by analogy with the homogeneous elec-tron gas, where well-defined plasma excitations exist onlyfor momenta below the electron-hole continuum . For aplasmon of energy ~ ω P the critical momentum is given bythe wavevector q c = k F (cid:2) (1 + ~ ω P /ε F ) / − (cid:3) , with k F and ε F being the Fermi wavevector and the Fermi energy,respectively. The critical wavevector q c marks the onsetof Landau damping, that is, the decay of a plasmon uponexcitation of an electron-hole pair. Thus, for q < q c ther-mal plasmons are undamped collective phenomena withlifetimes set by plasmon-phonon and plasmon-plasmonscattering processes . This boundary is shown aswhite dashed lines in Fig. 1b and Fig. 1c.In order to investigate the effects of plasmons onthe electronic structure we generalise Pines’ theory ofelectron-plasmon interactions in the homogeneous elec-tron gas to ab initio calculations for crystalline solids.Our strategy consists of the following steps: (i) We iden-tify the energy vs. wavevector dispersion relations of Figure 2. (a) Calculated rates of electron scattering by plasmons, and (b) corresponding electron lifetimes in doped silicon,for several carrier concentrations. The electron energy is referred to the conduction band edge. (c–e) Comparison betweenthe imaginary part of the electron-plasmon self-energy, the electron-phonon self-energy, and the self-energy associated withelectron-hole pair generation. The carrier concentration increases from (c) to (e), and the electron energy is referred to theconduction band edge. Shaded regions indicate the dominant scattering mechanism at a given electron energy, and ‘PL’, ‘PH’,‘EH’ stand for plasmons, phonons, and electron-hole pairs, respectively. (f) Energy vs. doping map of the largest contributionto the electron self-energy. The energy is referred to the conduction band edge. (g) Diagrammatic representation of the electronplasmon scattering process. (h) Calculated plasmon-induced band gap renormalization in silicon as a function of carried density(orange squares and line), compared to the optical data from Ref. (experiment 1) and Ref. (experiment 2). The dashedhorizontal line indicates the renormalization of the band gap by electron-phonon interactions, as reported by Ref. . thermal plasmons. This is achieved by determining theplasma energies from the poles of Im ǫ − ( q , ω ) for mo-menta below the critical wavevector q c20 . (ii) We singleout the plasmonic contribution to the macroscopic dielec-tric function ǫ M via the Taylor expansion ǫ P ( q + G , ω ) = ∂ǫ M ∂ω (cid:12)(cid:12) ω = ω P ( q ) [ ω − ω P ( q )] + iη in the vicinity of the plas-mon frequency ω P ( q ). (iii) We calculate the electron-plasmon self-energy starting from many-body perturba-tion theory, and retain only the plasmonic screening.This leads to the retarded electron self-energy in Raleigh-Schr¨odinger perturbation theory :Σ eP n k = Z d q Ω BZ X m | g eP mn ( k , q ) | (cid:20) n q + f m k + q ε n k − ε m k + q + ~ ω P ( q ) + iη + n q + 1 − f m k + q ε n k − ε m k + q − ~ ω P ( q ) + iη (cid:21) . (1)In this expression k and q are Bloch wavevectors, m and n band indices, ε n k and ε m k + q Kohn-Sham eigen-values, n q and f m k + q Bose-Einstein and Fermi-Dirac oc-cupations, respectively, and η a positive infinitesimal.The summation runs over all states and the integral isover the Brillouin zone of volume Ω BZ . The quantities g eP mn ( k , q ) represent the electron-plasmon scattering ma-trix elements between the initial state ψ n k and the final state ψ m k + q , and are given by: g eP mn ( k , q ) = (cid:20) ε Ω e ~ ∂ǫ ( q , ω ) ∂ω (cid:21) − ω P ( q ) | q | h ψ m k + q | e i q · r | ψ n k i , (2)with Ω being the volume of one unit cell. Eqs. (1)and (2) are derived in the Appendix. The present ap-proach to electron-plasmon coupling in semiconductorsis formally identical to the standard theory of electron-phonon interactions . In particular, the 1 / | q | diver-gence of the electron-plasmon matrix elements at longwavelengths is reminiscent of the Fr¨ohlich interaction be-tween electrons and longitudinal-optical phonons in polarsemiconductors . This analogy is consistent with thefact that bulk plasmons are longitudinal waves. We nowanalyse the consequences of the self-energy in Eq. (1).From the imaginary part of the self-energy in Eq. (1)we obtain the rate of electron scattering by thermalplasmons, using Γ n k = 2 Im Σ n k / ~ . Physically thetwo denominators in Eq. (1) describe processes of one-plasmon absorption and emission, respectively. A dia-grammatic representation of these processes is given inFig. 2g. Multi-plasmon processes are not included inthe present formalism, similarly to the case of electron-phonon interactions , therefore we limit our discussionto low temperatures ( n q ≪ n -type silicon.The carrier energies are referred to the conduction bandedge. For standard doping levels ( n < cm − ) thescattering rates fall below 10 s − as a result of the lowintensity of the plasmon peaks in Fig. 1e, which is re-flected in the strength of the matrix elements in Eq. (2).However, at doping levels above 10 cm − , the strengthof the plasmon peak in the loss function increases consid-erably, and the frequency of scattering by thermal plas-mons becomes comparable to electron-phonon scatter-ing rates, 10 -10 s − . Fig. 2a shows that at evenhigher doping levels these rates keep increasing by or-ders of magnitude, and eventually dominate the coolingdynamics of excited carriers.A complementary perspective on the carrier dynamicsis provided by Fig. 2b. Here we show the electron life-times corresponding to the rates in Fig. 2a, calculated as τ n k = 1 / Γ n k . Time-resolved reflectivity measurements ofnon-degenerate silicon ( n = 10 cm − electrons photo-excited at ∼ . In the same dopingrange our calculations yield plasmon-limited carrier life-times well above 10 ps, indicating that under these condi-tions electron-plasmon scattering is ineffective. However,the scenario changes drastically for degenerate silicon,for which we calculate lifetimes in the sub-picosecondregime. In particular, for doping levels in the range 10 -10 cm − the electron-plasmon scattering reduces thecarrier lifetimes to 25-150 fs. In these conditions electron-phonon and electron-plasmon scattering become compet-ing mechanisms for hot-carrier thermalisation.In order to quantify the importance of electron-plasmon scattering we compare in Fig. 2c-e the imagi-nary part of the electron self-energy associated with (i)electron-plasmon interactions, (ii) electron-phonon inter-actions, and (iii) and electron-hole pair generation. Themethods of calculation of (ii) and (iii) are described inthe Supplemental Materials . From this comparison wededuce that plasmons become increasingly important to-wards higher doping, and their effect is most pronouncedin the vicinity of the band edge. By identifying the largest contribution for each doping level and for eachelectron energy, we can construct the ‘scattering phasediagram’ shown in Fig.2f. This diagram illustrates the re-gions in the energy vs. doping space where each scatteringmechanism dominates. Unexpectedly in degenerate sili-con electron-plasmon scattering represents the dominantmechanism for hot-carrier relaxation. This finding couldprovide new opportunities in the study of semiconductor-based plasmonics, for example by engineering the dopingconcentration so as to selectively target the ‘plasmon re-gion’ in Fig.2f.We also evaluated the impact of electron-plasmon scat-tering processes on the carrier mobility in silicon, by us-ing the lifetimes computed above as a first approximationto the carrier relaxation times. As shown in Fig. S1 ,the explicit inclusion of electron-phonon scattering is es-sential to achieve a good agreement with experiment. Onthe other hand, were we to consider only electron-phononscattering and electron-hole pair generation, we wouldoverestimate the experimental mobilities by more thanan order of magnitude.The real part of the electron self-energy in Eq. (1) al-lows us to evaluate the renormalization of the electron en-ergy levels arising from the dressing of electron quasipar-ticles by virtual plasmons. Since the renormalization ofsemiconductor band gaps induced by electron-phonon in-teractions attracted considerable interest lately , wehere concentrate on the quantum zero-point renormal-ization of the fundamental gap of silicon. Computationaldetails of the calculations and convergence tests are re-ported in the Supplemental Material . Considering fordefiniteness a carrier density of n = 2 . · cm − , wefind that the electron-plasmon coupling lowers the con-duction band edge by ∆ E c = −
37 meV at zero tem-perature, and rises the valence band edge by ∆ E v =30 meV. For carrier concentrations of 2 . · cm − and 2 . · cm − we verified that the BGN changesby less that 1 meV for temperatures up to 600 K (seeSupplemental Material ). As a result at this dop-ing concentration the band gap redshifts by ∆ E g =∆ E c − ∆ E v = −
67 meV. This phenomenology is en-tirely analogous to the zero-point renormalization fromelectron-phonon interactions . Our finding is consis-tent with the fact that the self-energy in Eq. (1) andthe matrix element in Eq. (2) are formally identical tothose that one encounters in the study of the Fr¨ohlichinteraction. The doping-induced band gap renormaliza-tion was also reported in a recent work on monolayerMoS , therefore we expect this feature to hold generalvalidity in doped semiconductors. In order to performa quantitative comparison with experiment, we show inFig. 2h our calculated plasmonic band gap renormaliza-tion and measurements of the indirect absorption onsetin doped silicon . We can see that there is alreadygood agreement between theory and experiment, even ifwe are considering only electron-plasmon couplings as thesole source of band gap renormalization. Surprisingly themagnitude of the renormalization, 15-70 meV, is compa-rable to the zero-point shift induced by electron-phononinteractions, 60-72 meV .In summary, we presented an ab initio approach toelectron-plasmon coupling in doped semiconductors. Weshowed that electron-plasmon interactions are strong andubiquitous in a prototypical semiconductor such as dopedsilicon, as revealed by their effect on carrier dynamics,transport, and optical properties. This finding calls fora systematic investigation of electron-plasmon couplingsin a wide class of materials. More generally, a detailedunderstanding of the interaction between electrons andthermal plasmons via predictive atomic-scale calculationscould provide a key into the design of plasmonic semi-conductors, for example by using phase diagrams suchas in Fig. 2f to tailor doping levels and excitation en-ergies to selectively target strong-coupling regimes. Fi-nally, the striking similarity between electron-plasmoncoupling and the Fr¨ohlich coupling in polar materials mayopen new avenues to probe plasmon-induced photoemis-sion kinks , polaron satellites , as well as supercon-ductivity, in analogy with the case of electron-phononinteractions. . ACKNOWLEDGMENTS
F.C. acknowledges discussions with C. Verdi andS. Ponc´e. The research leading to these results hasreceived funding from the Leverhulme Trust (GrantRL-2012-001), the Graphene Flagship (EU FP7 grantno. 604391), the UK Engineering and Physical Sci-ences Research Council (Grant No. EP/J009857/1).Supercomputing time was provided by the Univer-sity of Oxford Advanced Research Computing facil-ity (http://dx.doi.org/10.5281/zenodo.22558) and theARCHER UK National Supercomputing Service.
Appendix A: Electron self-energy for theelectron-plasmon interaction
Here we provide a derivation of the electron-plasmoncoupling strength and the self-energy [Eq. (1) and (2)] bygeneralizing the theory of electron-plasmon interactionfor the homogeneous electron gas to the case of crystallinesolids. We start from the electron self-energy in the GW approximation :Σ n k ( ω ) = i ~ π X m GG ′ Z d q Ω BZ M mn G ( k , q ) ∗ M mn G ′ ( k , q ) Z dω ′ W GG ′ ( q , ω ′ ) ~ ω + ~ ω ′ + µ − ˜ ǫ m k + q , (A1)where M mn G ( k , q ) = h ψ m k + q | e i ( q + G ) · r | ψ n k i are the optical matrix elements, µ is the chemical potential, and ˜ ǫ m k + q = ǫ m k + q + iη sign( µ − ǫ m k + q ). The matrix W GG ′ ( q , ω ′ ) = v ( q + G ) ǫ − GG ′ ( q , ω ′ ) represents the screened Coulombinteraction, and is obtained from the bare Coulomb interaction v ( q ) = e /ε Ω | q | via the inverse dielectric matrix ǫ − GG ′ ( q , ω ′ ). The spectral representation of W is given by: W GG ′ ( q , ω ) = v ( q + G ) π Z ∞ dω ′ ω ′ ω − ( ω ′ ) Im ǫ − GG ′ ( q , ω ′ ) . (A2)The dielectric matrix may be decomposed into: ǫ − GG ′ ( q , ω ) = ǫ − ( q + G , ω ) δ GG ′ + ǫ − GG ′ ( q , ω )(1 − δ GG ′ ) . (A3)where ǫ − ( q + G , ω ) is the inverse macroscopic dielectric function. Since the plasmon energy ~ ω P ( q ) is defined bythe condition ǫ M ( q + G , ω P ( q )) = 0, the plasmonic contribution to the dielectric matrix ǫ P can be singled out byTaylor-expanding ǫ M around the plasmon energy. Following Pines and Schrieffer we have: ǫ P ( q + G , ω ) = ∂ǫ M ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω P ( q ) [ ω − ω P ( q )] + iη. (A4)Making use of the identity ( a + iη ) − = P (1 /a ) + iπδ ( a ), and combining Eqs. (A1), (A2), and (A4) yields theelectron-plasmon self-energy:Σ eP n k ( ω ) = i ~ π X m G Z d q Ω BZ | M nm G ( k , q ) | Z dω ′ ω P ( q ) ω ′ − [ ω P ( q )] " ∂ǫ M ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω P ( q ) − v ( q + G ) ω + ω ′ + µ − ˜ ǫ m k + q . (A5)This expression may be recast into the form of a self-energy describing the interaction between electrons and bosonsin the Migdal approximation :Σ eP n k ( ω ) = i ~ π X m Z d q Ω BZ Z dω ′ | g eP mn ( k , q ) | D q ( ω ′ ) G m k + q ( ω + ω ′ ) . (A6)Since for doped semiconductors q c is typically within thefirst Brillouin zone, we dropped the dependence on the reciprocal lattice vectors G . The matrix elements ap-pearing in this expression are defined in Eq. (2); G rep-resents the standard non-interacting (Kohn-Sham) elec-tron Green’s function, G n k ( ω ) = [ ~ ω − (˜ ε n k − µ )] − ,and we introduced the ‘plasmon propagator’: D q ( ω ) =2 ω P ( q ) / [ ~ ( ω − ω ( q ))]. Equation (A6) represents theprototypical electron self-energy arising from electron-boson interactions. From this expression the result inEq. (1) follows by standard integration in the complexplane .For completeness we note that Eq. (A6) can alsobe derived from the electron-boson coupling Hamilto-nian ˆ H eP = Ω − P nm R d k d q g eP nm ( k , q )ˆ c † m k + q ˆ c n k (ˆ b q +ˆ b †− q ), where ˆ b †− q (ˆ b q ) and ˆ c † m k + q (ˆ c n k ) are the bo-son and fermion creation (destruction) operators, re-spectively. As a consistency check, we note that theelectron-plasmon coupling coefficients Eq. (2) reduceto the results of Pines and Schrieffer for homogeneoussystems . In particular, for an homogeneous electrongas we have M nm G ( k , q ) = δ nm and ǫ M = 1 − ω P ( q ) /ω .In this case, the partial derivative in the definition of the electron-plasmon coupling coefficients can be evalu-ated analytically, giving the results of Ref. , g eP ( q ) =(2 πe ~ ω P ( q ) /ǫ q ) .Finally, we emphasize that the structure of Eq. (A5)stems directly from the identification of the plasmoniccontribution to the dielectric function through the lin-earization of Eq. (A4), and it is reflected in the inclu-sion of the plasmon oscillator strength ∂ǫ M ∂ω (cid:12)(cid:12) ω = ω P ( q ) inthe coupling coefficients [Eq. (2)]. This procedure distin-guishes the electron-plasmon self-energy from the conven-tional GW self-energy in the plasmon-pole approxima-tion, and justifies its application to the study of thermalplasmons in doped semiconductors. Appendix B: Plasmon damping
To investigate the effects of extrinsic carriers on ther-mal plasmons, we consider the Fermi golden rule for therate of change of the plasmon distribution function : R q = 2 π ~ BZ X k Z d k Ω BZ X nm | g eP mn ( k , q ) | [( n q + 1) f n k + q (1 − f m k ) − n q f m k (1 − f n k + q )] δ ( ǫ m k + ~ ω P ( q ) − ǫ n k + q ) (B1)where ~ ω P ( q ) are plasmon energies, g eP electron-plasmoncoupling coefficients, and n / f are Bose/Fermi occupationfactors for plasmons/electrons. In practice, the first termaccounts for the increase of the plasmon population in-duced by the absorption of an electron-hole pair, whereasthe inverse process is described by the second term. Ther-mal plasmons are well defined for momenta smaller thatthe critical momentum cutoff given by the wavevector: q c = k F (cid:2) (1 + ~ ω P /ε F ) / − (cid:3) , with k F and ε F beingthe Fermi wavevector and the Fermi energy, respectively.By definition (see, e.g., ) q c is the smallest momentumsatisfying the condition ~ ω P ( q ) = ǫ n k + q − ǫ m k . Thusfor q < q c , the Dirac δ in Eq. (B1) vanishes, indicat-ing that, while excited carriers may decay upon plasmonemission, the inverse processes, whereby a thermal plas-mon decays upon emission of an electron-hole pair, is for-bidden. Therefore, thermal plasmons are undamped byother electronic processes, and their decay for q < q c maybe ascribed exclusively to plasmon-phonon and plasmon-plasmon scattering.To exemplify the effect of Landau damping on the plas-mon dispersion, we illustrate in Fig. 3 the plasmon peakin the loss function of silicon at a doping concentrationof 1 . · cm − . At these carrier concentration, weobtain a momentum cutoff q c = 0 .
05 in units of 2 π/a ,with a being the lattice constant. For q < q c , the lossfunction exhibit well defined plasmon peak with a peakintensity larger than the continuum of electron-hole ex-citations. For q > q c , on the other hand, the plasmon Figure 3. Momentum dependence of plasmon peak in theloss function of silicon at a doping concentration of 1 . · cm − , corresponding to a critical momentum cutoff q c =0 .
05 in 2 π/a units. intensity is reduced as a consequence of the lifetime ef-fects introduced by Landau damping, and its intensitybecomes essentially indistinguishable from the spectralsignatures of electron-hole pairs.
Figure 4. (a) Calculated electron mobility in n -type silicon, as a function of carrier density and energy relative to the chemicalpotential. (b) Comparison between calculated and measured electron mobilities in silicon as a function of doping. The blackcircles indicate experimental low-temperature mobility data from Ref. . The orange squares and line represent our completecalculation including electron-plasmon (pl), electron-phonon (ph), electron-hole (eh), and impurity scattering. (c) Partialcontributions to the mobility are shown as red (ph), and yellow (ph+eh). Appendix C: Plasmon-limited mobility
We now evaluate the impact of electron-plasmon scat-tering processes on the carrier mobility in silicon. In therelaxation-time approximation the mobility is given by µ = eτ tot /m e m ∗ , where m ∗ is the conductivity effectivemass, that is the harmonic average of the longitudinaland transverse masses, and τ tot is the scattering timearising from processes involving plasmons (eP), phonons(ep), electron-hole pairs (eh), and impurities (i). Not-ing that scattering time and relaxation time differ byless that 10% at low carrier concentrations , we followMatthiessen’s rule to calculate τ tot n k = ~ / ep n k +Σ eP n k +Σ eh n k + Σ i n k ), where Σ ep n k , Σ eh n k , and Σ i n k are the electronself-energies associated with each interaction.Strictly speaking the mobility µ is an average prop-erty of all the carriers in a semiconductor; however, forillustration purposes, it is useful to consider a ‘single-electron’ mobility obtained as µ n k = eτ tot n k /m e m ∗ . Thisquantity is shown in Fig. 4a. In this figure we see thatthe mobility decreases as one moves higher up in theconduction band; this behavior relates to the increasedphase-space availability for electronic transitions. In ad-dition we see that the mobility decreases with increas-ing carrier concentration. In order to analyse this trendwe give a breakdown of the various sources of scattering in Fig. 4c, and we compare our calculations to experi-ment. Here we show the carrier mobility at 300 K av-eraged on the Fermi surface defined by the doping level.Electrical measurements at high doping yield mobili-ties in the range of 100-300 cm V − s − for carrier den-sities between 10 and 10 cm − ; these data are shownas black circles in Fig. 4b-c. Were we to consider onlyelectron-phonon scattering and electron-hole pair gener-ation, we would overestimate the experimental mobilitiesby more than an order of magnitude (red and yellow linesin Fig. 4c). Impurity scattering reduces this discrepancyto some extent, but there remains a residual differenceat the highest doping levels. It is only upon accountingfor electron-plasmon scattering that the calculations ex-hibit a trend in qualitative and even semi-quantitativeagreement with experiments throughout the entire dop-ing range. In particular the scattering by plasmons is keyto explain the anomalous low mobility of 100 cm V − s − above n = 10 cm − . Even through the inclusion ofelectron-plasmon scattering a residual discrepancy be-tween theory and experiment is still observed, which weascribe to the simplified models adopted in the descrip-tion of electronic scattering with electron-hole pairs andimpurities. This observation leads us to suggest that theorigin of the mobility overestimation in earlier calcula-tions could be connected with the neglect of electron-plasmon scattering . S. Lal, S. Link, and N. J. Halas, Nat. Photonics , 641(2007). R. J. Walters, R. V. A. van Loon, I. Brunets, J. Schmitz,and A. Polman, Nat. Mater. , 21 (2010). H. A. Atwater and A. Polman, Nat. Mater. , 205 (2010). M. L. Brongersma, N. J. Halas, and P. Nordlander,Nat. Nanotechnol. , 25 (2015). E. Y. Lukianova-Hleb, X. Ren, R. R. Sawant, X. Wu, V. P.
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