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Figure 1. (a–c) Calculated electron-energy loss function of n-type silicon for momentum transfers q along the ΓX high-symmetry − line. The carrier density increases from left to right, from 1017 to 1020 cm 3. (d) LDA band structure of silicon, and Fermi 20 −3 level (EF) for n = 2.5 · 10 cm . The step-like structures in (b) and (c) are only a numerical artifact arising from the limited Brillouin-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 the analytical result obtained for a homogeneous electron gas with the calculated isotropic effective mass and dielectric constant of silicon (meff = 0.25, ǫSi = 12).

tor k + q. The intensity of these features increases with happen on length-scales approaching the size of the crys- the doping level from Fig. 1b to Fig. 1c. The peaks in tal unit cell. In the following we identify plasmons in the loss function denoted by ‘process 3’ cannot be ex- the loss function by analogy with the homogeneous elec- plained in terms of the previous two mechanisms. In fact tron gas, where well-defined plasma excitations exist only for q = 0 these structures are much sharper than those for momenta below the electron-hole continuum15. For a described above, and exist below the energy (momen- plasmon of energy ~ωP the critical momentum is given by 1/2 tum) threshold for the generation of electron-hole pairs the wavevector qc = kF (1 + ~ωP/εF) − 1 , with kF via interband (intraband) transitions. These processes and εF being the Fermi wavevector and the Fermi energy,   correspond to the emission of plasmons, and are charac- respectively. The critical wavevector qc marks the onset terised by well-defined energy resonances, as it is shown of Landau damping, that is, the decay of a plasmon upon by Fig. 1e for q = 0. By mapping these plasmon peaks in excitation of an electron-hole pair. Thus, for q < qc ther- the loss function we can see in Fig. 1f that the plasmon mal plasmons are undamped collective phenomena with energy ~ωP scales with the carrier concentration, follow- lifetimes set by plasmon-phonon and plasmon-plasmon ing the same trend expected for a homogeneous electron scattering processes16. This boundary is shown as gas. In this figure we also see that the plasmon energy is white dashed lines in Fig. 1b and Fig. 1c. highly tunable via doping, from thermal energies at car- In order to investigate the effects of plasmons on rier densities around 1018 cm−3, to half an electronvolt the electronic structure we generalise Pines’ theory of at densities near 1021 cm−3. electron-plasmon interactions in the homogeneous elec- At large momentum transfer ~q the distinction be- tron gas15 to ab initio calculations for crystalline solids. tween plasmons and electron-hole pairs is no longer Our strategy consists of the following steps: (i) We iden- meaningful, since the fluctuations of the charge density tify the energy vs. wavevector dispersion relations of 3

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 between the imaginary part of the electron-plasmon self-energy, the electron-phonon self-energy, and the self-energy associated with electron-hole pair generation. The carrier concentration increases from (c) to (e), and the electron energy is referred to the conduction 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 contribution to the electron self-energy. The energy is referred to the conduction band edge. (g) Diagrammatic representation of the electron plasmon 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.17 (experiment 1) and Ref.18 (experiment 2). The dashed horizontal line indicates the renormalization of the band gap by electron-phonon interactions, as reported by Ref.19.

thermal plasmons. This is achieved by determining the ∂ǫM [ω − ω (q)] + iη in the vicinity of the plas- ∂ω ω=ωP(q) P −1 q plasma energies from the poles of Im ǫ ( ,ω) for mo- mon frequency ωP(q). (iii) We calculate the electron- 20 menta below the critical wavevector qc . (ii) We single plasmon self-energy starting from many-body perturba- out the plasmonic contribution to the macroscopic dielec- tion theory, and retain only the plasmonic screening. tric function ǫM via the Taylor expansion ǫP(q + G,ω)= This leads to the retarded electron self-energy in Raleigh- Schr¨odinger perturbation theory21:

eP dq eP 2 nq + fmk+q nq +1 − fmk+q Σnk = |gmn(k, q)| + . (1) Ω ε k − ε k q + ~ω (q)+ iη ε k − ε k q − ~ω (q)+ iη BZ m n m + P n m + P Z X  

In this expression k and q are Bloch wavevectors, m state ψmk+q, and are given by: and n band indices, εnk and εmk+q Kohn-Sham eigen- 1 values, nq and fmk+q Bose-Einstein and Fermi-Dirac oc- − 2 eP ε0Ω ∂ǫ(q,ω) 1 iq·r cupations, respectively, and η a positive infinitesimal. g (k, q)= hψ k q|e |ψ ki, mn e2~ ∂ω |q| m + n The summation runs over all states and the integral is  ωP(q) over the Brillouin zone of volume ΩBZ. The quantities (2) eP gmn(k, q) represent the electron-plasmon scattering ma- with Ω being the volume of one unit cell. Eqs. (1) trix elements between the initial state ψnk and the final and (2) are derived in the Appendix. The present ap- proach to electron-plasmon coupling in semiconductors is formally identical to the standard theory of electron- 4 phonon interactions22. In particular, the 1/|q| diver- largest contribution for each doping level and for each gence of the electron-plasmon matrix elements at long electron energy, we can construct the ‘scattering phase wavelengths is reminiscent of the Fr¨ohlich interaction be- diagram’ shown in Fig.2f. This diagram illustrates the re- tween electrons and longitudinal-optical phonons in polar gions in the energy vs. doping space where each scattering semiconductors23,24. This analogy is consistent with the mechanism dominates. Unexpectedly in degenerate sili- fact that bulk plasmons are longitudinal waves. We now con electron-plasmon scattering represents the dominant analyse the consequences of the self-energy in Eq. (1). mechanism for hot-carrier relaxation. This finding could From the imaginary part of the self-energy in Eq. (1) provide new opportunities in the study of semiconductor- we obtain the rate of electron scattering by thermal based plasmonics, for example by engineering the doping plasmons, using Γnk = 2ImΣnk/~. Physically the concentration so as to selectively target the ‘plasmon re- two denominators in Eq. (1) describe processes of one- gion’ in Fig.2f. plasmon absorption and emission, respectively. A dia- We also evaluated the impact of electron-plasmon scat- grammatic representation of these processes is given in tering processes on the carrier mobility in silicon, by us- Fig. 2g. Multi-plasmon processes are not included in ing the lifetimes computed above as a first approximation the present formalism, similarly to the case of electron- to the carrier relaxation times. As shown in Fig. S116, phonon interactions22, therefore we limit our discussion the explicit inclusion of electron-phonon scattering is es- to low temperatures (nq ≪ 1). Fig. 2a shows the calcu- sential to achieve a good agreement with experiment. On lated electron-plasmon scattering rates in n-type silicon. the other hand, were we to consider only electron-phonon The carrier energies are referred to the conduction band scattering and electron-hole pair generation, we would edge. For standard doping levels (n < 1018 cm−3) the overestimate the experimental mobilities by more than scattering rates fall below 1011 s−1 as a result of the low an order of magnitude. intensity of the plasmon peaks in Fig. 1e, which is re- The real part of the electron self-energy in Eq. (1) al- flected in the strength of the matrix elements in Eq. (2). 18 −3 lows us to evaluate the renormalization of the electron en- However, at doping levels above 10 cm , the strength ergy levels arising from the dressing of electron quasipar- of the plasmon peak in the loss function increases consid- ticles by virtual plasmons. Since the renormalization of erably, and the frequency of scattering by thermal plas- semiconductor band gaps induced by electron-phonon in- mons becomes comparable to electron-phonon scatter- 28–33 12 14 −125,26 teractions attracted considerable interest lately , we ing rates, 10 -10 s . Fig. 2a shows that at even here concentrate on the quantum zero-point renormal- higher doping levels these rates keep increasing by or- ization of the fundamental gap of silicon. Computational ders of magnitude, and eventually dominate the cooling details of the calculations and convergence tests are re- dynamics of excited carriers. ported in the Supplemental Material16. Considering for A complementary perspective on the carrier dynamics definiteness a carrier density of n = 2.5 · 1020 cm−3, we is provided by Fig. 2b. Here we show the electron life- find that the electron-plasmon coupling lowers the con- times corresponding to the rates in Fig. 2a, calculated as duction band edge by ∆Ec = −37 meV at zero tem- τnk =1/Γnk. Time-resolved reflectivity measurements of perature, and rises the valence band edge by ∆Ev = 17 −3 non-degenerate silicon (n = 10 cm electrons photo- 30 meV. For carrier concentrations of 2.5 · 1019 cm−3 excited at ∼0.8 eV above the band edge) indicate ther- and 2.5 · 1020 cm−3 we verified that the BGN changes 27 malisation rates around 350 fs . In the same doping by less that 1 meV for temperatures up to 600 K (see range our calculations yield plasmon-limited carrier life- Supplemental Material16). As a result at this dop- times well above 10 ps, indicating that under these condi- ing concentration the band gap redshifts by ∆Eg = tions electron-plasmon scattering is ineffective. However, ∆Ec − ∆Ev = −67 meV. This phenomenology is en- the scenario changes drastically for degenerate silicon, tirely analogous to the zero-point renormalization from for which we calculate lifetimes in the sub-picosecond electron-phonon interactions29. Our finding is consis- 19 regime. In particular, for doping levels in the range 10 - tent with the fact that the self-energy in Eq. (1) and 20 −3 10 cm the electron-plasmon scattering reduces the the matrix element in Eq. (2) are formally identical to carrier lifetimes to 25-150 fs. In these conditions electron- those that one encounters in the study of the Fr¨ohlich phonon and electron-plasmon scattering become compet- interaction. The doping-induced band gap renormaliza- ing mechanisms for hot-carrier thermalisation. tion was also reported in a recent work on monolayer 34 In order to quantify the importance of electron- MoS2 , therefore we expect this feature to hold general plasmon scattering we compare in Fig. 2c-e the imagi- validity in doped semiconductors. In order to perform nary part of the electron self-energy associated with (i) a quantitative comparison with experiment, we show in electron-plasmon interactions, (ii) electron-phonon inter- Fig. 2h our calculated plasmonic band gap renormaliza- actions, and (iii) and electron-hole pair generation. The tion and measurements of the indirect absorption onset methods of calculation of (ii) and (iii) are described in in doped silicon17,18. We can see that there is already the Supplemental Materials16. From this comparison we good agreement between theory and experiment, even if deduce that plasmons become increasingly important to- we are considering only electron-plasmon couplings as the wards higher doping, and their effect is most pronounced sole source of band gap renormalization. Surprisingly the in the vicinity of the band edge. By identifying the magnitude of the renormalization, 15-70 meV, is compa- 5 rable to the zero-point shift induced by electron-phonon ACKNOWLEDGMENTS interactions, 60-72 meV19. F.C. acknowledges discussions with C. Verdi and In summary, we presented an ab initio approach to S. Ponc´e. The research leading to these results has electron-plasmon coupling in doped semiconductors. We received funding from the Leverhulme Trust (Grant showed that electron-plasmon interactions are strong and RL-2012-001), the Graphene Flagship (EU FP7 grant ubiquitous in a prototypical semiconductor such as doped no. 604391), the UK Engineering and Physical Sci- silicon, as revealed by their effect on carrier dynamics, ences Research Council (Grant No. EP/J009857/1). transport, and optical properties. This finding calls for Supercomputing time was provided by the Univer- a systematic investigation of electron-plasmon couplings sity of Oxford Advanced Research Computing facil- in a wide class of materials. More generally, a detailed ity (http://dx.doi.org/10.5281/zenodo.22558) and the understanding of the interaction between electrons and ARCHER UK National Supercomputing Service. thermal plasmons via predictive atomic-scale calculations could provide a key into the design of plasmonic semi- conductors, for example by using phase diagrams such Appendix A: Electron self-energy for the as in Fig. 2f to tailor doping levels and excitation en- electron-plasmon interaction ergies to selectively target strong-coupling regimes. Fi- nally, the striking similarity between electron-plasmon Here we provide a derivation of the electron-plasmon coupling and the Fr¨ohlich coupling in polar materials may coupling strength and the self-energy [Eq. (1) and (2)] by open new avenues to probe plasmon-induced photoemis- generalizing the theory of electron-plasmon interaction sion kinks35, polaron satellites36–38, as well as supercon- for the homogeneous electron gas to the case of crystalline ductivity, in analogy with the case of electron-phonon solids. We start from the electron self-energy in the GW interactions.39–44. approximation45–47:

~ ′ ′ i dq mn ∗ mn ′ WGG (q,ω ) k q ′ k q Σnk(ω)= MG ( , ) MG ( , ) dω ′ , (A1) 2π Ω ~ω + ~ω + µ − ǫ˜ k q GG′ BZ m + mX Z Z mn i(q+G)·r where MG (k, q)= hψmk+q|e |ψnki are the optical matrix elements, µ is the chemical potential, andǫ ˜mk+q = ′ ′ −1 ′ ǫmk+q + iη sign(µ − ǫmk+q). The matrix WGG (q,ω ) = v(q + G)ǫGG′ (q,ω ) represents the screened Coulomb 2 2 interaction, and is obtained from the bare Coulomb interaction v(q) = e /ε0Ω|q| via the inverse dielectric matrix −1 ′ ǫGG′ (q,ω ). The spectral representation of W is given by: ∞ ′ v(q + G) ′ 2ω −1 ′ WGG′ (q,ω)= dω Im ǫ ′ (q,ω ). (A2) π ω2 − (ω′)2 GG Z0 The dielectric matrix may be decomposed into:

−1 −1 ′ −1 ′ ǫGG′ (q,ω)= ǫM (q + G,ω)δGG + ǫGG′ (q,ω)(1 − δGG ). (A3) −1 ~ where ǫM (q + G,ω) is the inverse macroscopic dielectric function. Since the plasmon energy ωP(q) is defined by the condition ǫM(q + G,ωP(q)) = 0, the plasmonic contribution to the dielectric matrix ǫP can be singled out by 11 Taylor-expanding ǫM around the plasmon energy. Following Pines and Schrieffer we have: ∂ǫ ǫ (q + G,ω)= M [ω − ω (q)] + iη. (A4) P ∂ω P ω=ωP(q)

−1 Making use of the identity (a + iη) = P (1/a)+ iπδ (a), and combining Eqs. (A1), (A2), and (A4) yields the electron-plasmon self-energy: −1 i~ dq 2ω (q) ∂ǫ v(q + G) eP nm k q 2 ′ P M Σnk(ω)= |MG ( , )| dω ′2 2 ′ . (A5) 2π ΩBZ ω − [ωP(q)] ∂ω ω + ω + µ − ǫ˜mk+q mG Z Z " ω=ωP(q)# X

This expression may be recast into the form of a self-energy describing the interaction between electrons and bosons in the Migdal approximation 21,48–51: ~ eP i dq ′ eP 2 ′ ′ Σnk(ω)= dω |gmn(k, q)| Dq(ω ) Gmk+q(ω + ω ). (A6) 2π m ΩBZ X Z Z

Since for doped semiconductors qc is typically within the reciprocal lattice vectors G. The matrix elements ap- first Brillouin zone, we dropped the dependence on the 6 pearing in this expression are defined in Eq. (2); G rep- the electron-plasmon coupling coefficients can be evalu- resents the standard non-interacting (Kohn-Sham) elec- ated analytically, giving the results of Ref.11, geP(q) = 1 ~ −1 2 2 tron Green’s function, Gnk(ω) = [ ω − (˜εnk − µ)] , (2πe ~ωP(q)/ǫ0q ) 2 . and we introduced the ‘plasmon propagator’: Dq(ω) = Finally, we emphasize that the structure of Eq. (A5) ~ 2 2 2ωP(q)/[ (ω − ωP(q))]. Equation (A6) represents the stems directly from the identification of the plasmonic prototypical electron self-energy arising from electron- contribution to the dielectric function through the lin- boson interactions. From this expression the result in earization of Eq. (A4), and it is reflected in the inclu- Eq. (1) follows by standard integration in the complex sion of the plasmon oscillator strength ∂ǫM in ∂ω ω=ωP(q) plane21. the coupling coefficients [Eq. (2)]. This procedure distin- For completeness we note that Eq. (A6) can also guishes the electron-plasmon self-energy from the conven- be derived from the electron-boson coupling Hamilto- tional GW self-energy in the plasmon-pole approxima- ˆ eP −2 eP † ˆ nian H = ΩBZ nm dkdq gnm(k, q)ˆcmk+qcˆnk(bq + tion, and justifies its application to the study of thermal ˆ† ˆ† ˆ † plasmons in doped semiconductors. b−q), where b−q P(bq)R andc ˆmk+q (ˆcnk) are the bo- son and fermion creation (destruction) operators, re- spectively. As a consistency check, we note that the electron-plasmon coupling coefficients Eq. (2) reduce Appendix B: Plasmon damping to the results of Pines and Schrieffer for homogeneous systems11. In particular, for an homogeneous electron To investigate the effects of extrinsic carriers on ther- nm 2 gas we have MG (k, q) = δnm and ǫM = 1 − ωP(q)/ω . mal plasmons, we consider the Fermi golden rule for the In this case, the partial derivative in the definition of rate of change of the plasmon distribution function11:

BZ 2π dk eP 2 Rq = |g (k, q)| [(nq + 1)f k q(1 − f k) − nqf k(1 − f k q)]δ(ǫ k + ~ω (q) − ǫ k q) (B1) ~ Ω mn n + m m n + m P n + k BZ nm X Z X

eP where ~ωP(q) are plasmon energies, g electron-plasmon coupling coefficients, and n/f are Bose/Fermi occupation factors for plasmons/electrons. In practice, the first term accounts for the increase of the plasmon population in- duced by the absorption of an electron-hole pair, whereas the inverse process is described by the second term. Ther- mal plasmons are well defined for momenta smaller that the critical momentum cutoff given by the wavevector: 1/2 qc = kF (1 + ~ωP/εF) − 1 , with kF and εF being the Fermi wavevector and the Fermi energy, respectively.  15  By definition (see, e.g., ) qc is the smallest momentum satisfying the condition ~ωP(q) = ǫnk+q − ǫmk. Thus for q < qc, the Dirac δ in Eq. (B1) vanishes, indicat- ing that, while excited carriers may decay upon plasmon emission, the inverse processes, whereby a thermal plas- mon decays upon emission of an electron-hole pair, is for- Figure 3. Momentum dependence of plasmon peak in the bidden. Therefore, thermal plasmons are undamped by loss function of silicon at a doping concentration of 1.25 · 20 −3 other electronic processes, and their decay for q < qc may 10 cm , corresponding to a critical momentum cutoff qc = be ascribed exclusively to plasmon-phonon and plasmon- 0.05 in 2π/a units. plasmon scattering. To exemplify the effect of Landau damping on the plas- mon dispersion, we illustrate in Fig. 3 the plasmon peak in the loss function of silicon at a doping concentration of 1.25 · 1020 cm−3. At these carrier concentration, we obtain a momentum cutoff qc = 0.05 in units of 2π/a, with a being the lattice constant. For q < qc, the loss intensity is reduced as a consequence of the lifetime ef- function exhibit well defined plasmon peak with a peak fects introduced by Landau damping, and its intensity intensity larger than the continuum of electron-hole ex- becomes essentially indistinguishable from the spectral citations. For q > qc, on the other hand, the plasmon signatures of electron-hole pairs. 7

Figure 4. (a) Calculated electron mobility in n-type silicon, as a function of carrier density and energy relative to the chemical potential. (b) Comparison between calculated and measured electron mobilities in silicon as a function of doping. The black circles indicate experimental low-temperature mobility data from Ref.52. The orange squares and line represent our complete calculation including electron-plasmon (pl), electron-phonon (ph), electron-hole (eh), and impurity scattering. (c) Partial contributions to the mobility are shown as red (ph), and yellow (ph+eh).

Appendix C: Plasmon-limited mobility 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. We now evaluate the impact of electron-plasmon scat- Electrical measurements at high doping52 yield mobili- tering processes on the carrier mobility in silicon. In the ties in the range of 100-300 cm2 V−1s−1 for carrier den- relaxation-time approximation the mobility is given by 18 20 −3 tot ∗ ∗ sities between 10 and 10 cm ; these data are shown µ = eτ /mem , where m is the conductivity effective as black circles in Fig. 4b-c. Were we to consider only mass, that is the harmonic average of the longitudinal tot electron-phonon scattering and electron-hole pair gener- and transverse masses, and τ is the scattering time ation, we would overestimate the experimental mobilities arising from processes involving plasmons (eP), phonons by more than an order of magnitude (red and yellow lines (ep), electron-hole pairs (eh), and impurities (i). Not- in Fig. 4c). Impurity scattering reduces this discrepancy ing that scattering time and relaxation time differ by to some extent, but there remains a residual difference less that 10% at low carrier concentrations53, we follow tot ~ ep eP at the highest doping levels. It is only upon accounting Matthiessen’s rule to calculate τnk = /2 Im(Σnk+Σnk+ eh i ep eh i for electron-plasmon scattering that the calculations ex- Σnk +Σnk), where Σnk, Σnk, and Σnk are the electron hibit a trend in qualitative and even semi-quantitative self-energies associated with each interaction. agreement with experiments throughout the entire dop- Strictly speaking the mobility µ is an average prop- ing range. In particular the scattering by plasmons is key erty of all the carriers in a semiconductor; however, for to explain the anomalous low mobility of 100 cm2 V−1s−1 illustration purposes, it is useful to consider a ‘single- above n = 1019 cm−3. Even through the inclusion of tot ∗ electron’ mobility obtained as µnk = eτnk /mem . This electron-plasmon scattering a residual discrepancy be- quantity is shown in Fig. 4a. In this figure we see that tween theory and experiment is still observed, which we the mobility decreases as one moves higher up in the ascribe to the simplified models adopted in the descrip- conduction band; this behavior relates to the increased tion of electronic scattering with electron-hole pairs and phase-space availability for electronic transitions. In ad- impurities. This observation leads us to suggest that the dition we see that the mobility decreases with increas- origin of the mobility overestimation in earlier calcula- ing carrier concentration. In order to analyse this trend tions could be connected with the neglect of electron- we give a breakdown of the various sources of scattering plasmon scattering54,55.

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