Dispersion and lineshape of plasmon satellites in one, two and three dimensions Derek Vigil-Fowler,1 Steven G. Louie,2 and Johannes Lischner3, ∗ 1National Renewable Energy Laboratory, Golden, Colorado 80401, USA. 2Department of Physics, University of California at Berkeley and Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, USA. 3Department of Physics and Department of Materials, and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London SW7 2AZ, United Kingdom Using state-of-the-art many-body Green's function calculations based on the \GW plus cumulant" approach, we analyze the properties of plasmon satellites in the electron spectral function resulting from electron-plasmon interactions in one-, two- and three-dimensional systems. Specifically, we show how their dispersion relation, lineshape and linewidth are related to the properties of the constituent electrons and plasmons. To gain insight into the many-body processes giving rise to the formation of plasmon satellites, we connect the \GW plus cumulant" approach to a many-body wavefunction picture of electron-plasmon interactions and introduce the coupling-strength weighted electron-plasmon joint-density states as a powerful concept for understanding plasmon satellites. PACS numbers: 73.22.Pr, 71.15.Qe, 71.45.Gm, 71.10.-w Introduction.|The interaction of electrons with body processes can be difficult. In this paper, we de- bosons is of fundamental importance for many phenom- velop a complementary many-body wavefunction-based ena in condensed matter physics, plasma physics and cold approach for plasmonic (and more generally, bosonic) atom physics. Recently, there has been great interest in satellites in the electron spectral function which offers the coupling of electrons and plasmons, which are collec- a clear and simple physical picture of electron-plasmon tive excitations describing quantized oscillations of the interactions and leads to new insights into the results of charge density. For example, the decay of plasmons into GW+C calculations. Specifically, this approach reveals energetic or \hot" electron-hole pairs in metallic surfaces that the concepts of satellite dispersion, satellite line- and nanoparticles, which is triggered by electron-plasmon shape and satellite linewidth are closely related, explains coupling, has led to a new generation of plasmonic devices why in three-dimensional materials the plasmon satel- for photovoltaics and photocatalysis [1{3]. lite band structure looks like a shifted copy of the quasi- Satellite features in the spectral function of elec- particle band structure and demonstrates that previous trons are another consequence of electron-plasmon in- models of plasmon satellites in three dimensions are over- teractions. Such plasmon satellites have long been simplified and cannot be applied to lower-dimensional known in core-electron photoemission spectra [4, 5]. In systems. We present results for three-dimensional [sili- recent years, valence band plasmon satellites, which con and the three-dimensional electron gas (3DEG)], two- were observed experimentally in three-dimensional met- dimensional (doped graphene) and one-dimensional [the als and semiconductors [6{9], but also in two-dimensional one-dimensional electron gas (1DEG)] systems. systems, such as doped graphene and semiconductor Green's function theory.|The electron spectral func- quantum-well electron gases [10{12], received much at- tion is related to many observables, such as the tunnel- tention. ing and photoemission spectrum, and the contribution AIP(!) (with k denoting the wave vector and we omit To analyze and design the properties of plasmon satel- k a band index) describing the removal of an electron is lites for photonics and plasmonics applications, an accu- given by [19, 20] rate, material-specific theoretical description of electron- plasmon interactions is needed. This is achieved by the X AIP(!) = jhN − 1; λjc jGSij2δ(! + E − E ); GW plus cumulant (GW+C) approach [13, 14], where the k k N−1,λ GS λ cumulant expansion of the electron Green's function G is (1) arXiv:1606.06619v1 [cond-mat.mes-hall] 21 Jun 2016 truncated at second order in the screened Coulomb inter- action W . GW+C calculations yielded good agreement where jGSi and EGS denote the ground state wave func- with experimental photoemission and tunneling spectra tion and energy of the N-electron system, respectively, in a wide range of physical systems [6{8, 15{17] and and jN − 1; λi and EN−1,λ denote the eigenstates and also with highly accurate coupled-cluster Green's func- energies of the (N − 1)-electron system. tion calculations [18]. The spectral function is related to the one-electron While Green's function methods, such as the GW+C Green's function Gk(!) via Ak(!) = 1/π × jImGk(!)j. approach, often produce highly accurate results, gain- Within the generalized GW+C approach, the retarded ing intuition and insights into the underlying many- Green's function is expressed as function of time t via 2 [21] contribution to the spectral function closely related to the coupling-strength weighted electron-plasmon joint- −iEHFt+C (t) G (t) = −iΘ(t)e k k ; (2) D P 2 k density of states Jk(!) = 1=L × q gqδ(! −Ek−q +!q) comprising only plasmon-hole pairs with total momen- HF where Ek denotes the Hartree-Fock orbital energy tum k. HF X xc xc X (given by Ek = k + Σk − Vk with k, Vk and Σk Wavefunction theory.|We will now demonstrate that denoting the mean-field orbital energy, the mean-field the expression for the satellite contribution to the spec- exchange-correlation potential and the bare exchange self tral function from GW+C [Eq. (6)] can also be derived by energy, respectively). Also, Ck(t) is the cumulant func- considering the effective electron-plasmon Hamiltonian tion given by X y X y 1 Z e−i!t + i!t − 1 Hel−pl = Ekckck + !qaqaq (7) C (t) = d!jImΣ (! + E )j ; (3) q k π k k !2 k X gq y y + p c ck(aq + a ); (8) where Σ (!) denotes the GW self energy [22, 23] and E D k−q −q k k q;k L is the GW quasiparticle energy. To gain physical understanding, it is useful sepa- where ck and aq are destruction operators for quasipar- rate the cumulant function into a satellite contribution ticles and plasmons, respectively. In this Hamiltonian, sat −i!t Ck (t), which contains the e term in Eq. (3), and a the first term describes a set of non-interacting quasipar- quasiparticle contribution, which contains the (i!t − 1) ticles, the second term describes a set of non-interacting sat term. Expanding the Green's function in powers of Ck plasmons (or more generally, bosons) and the third term leads to a representation of the spectral function as the captures the interaction between quasiparticles and plas- qp sum of a quasiparticle contribution Ak and an infinite mons. (m) This electron-boson Hamiltonian plays a fundamen- series of plasmon satellite contributions Ak (with m denoting the number of plasmons that are created in the tal role in the theory of electron-phonon interactions, shake-up process). Specifically, the first satellite contri- but computing accurate spectral functions is difficult bution can be expressed as [19, 24, 25]. At intermediate coupling strengths, differ- Z ent types of perturbation theory give significantly dif- (1) 0 sat 0 qp 0 ferent results: When compared to highly accurate path- Ak (!) = d! Ck (! − ! )Ak (! ): (4) integral calculations, the self-consistent Brillouin-Wigner qp perturbation theory yields substantially worse results Approximating A (!) ≈ Z δ(Γk)(! − E ) with Z k k k k than standard Rayleigh-Schr¨odingerperturbation theory denoting the renormalization factor and δ(Γ) being a [19]. Lorentzian of width Γ, we find that A(1)(!) ≈ Z /π × k k For electron-plasmon interactions, Lundqvist demon- ImΣ (!)=(! − E )2. k k strated [24] that the application of Brillouin-Wigner per- Evaluating Eq. (4) requires the calculation of the imag- turbation theory to H results in the Dyson equation inary part of the GW self energy. To clarify the physical el−pl of the GW approach. Solving this equation, he found picture, we use the self energy of a homogeneous elec- two solutions. While the first solution corresponds to tron system in D dimensions with a plasmon-pole model a standard quasiparticle excitation, he assigned the sec- for the dielectric response. With these assumptions, the ond solution to a novel particle, the plasmaron, a strongly electron-removal part of the self energy is given by [23] coupled, coherent hole-plasmon state. Despite several re- π X ports claiming the observation of the plasmaron [10, 11], ImΣIP(!) = λ v δ(! − E + ! ); (5) k LD q q k−q q it has become clear recently that no such excitation ex- q ists in known materials and that its spurious prediction where vq denotes the Coulomb interaction in D dimen- signals the inability of the GW method (and, equiva- sions and !q and λq are the plasmon dispersion relation lently, the Brillouin-Wigner perturbation theory) to de- and the plasmon strength, respectively. Also, L is the scribe plasmon satellites [6, 7, 15, 16]. linear extension of the system and k − q corresponds to Motivated by its accurate description of electron- a hole state. phonon interactions, we now apply Rayleigh-Schr¨odinger (1) Inserting Eq. (5) into the expression for Ak yields perturbation theory to Hel−pl. Without electron- plasmon interactions, i.e. for gq = 0, the eigenstates 2 (1) Zk X gq A (!) = δ(! − E + ! ); of the (N − 1)-electron system are simply ckjGSi (with k LD (E − E − ! )2 k−q q energy E − E ) and ay c jGSi (with energy E − q k k−q q GS k q k−q GS (6) Ek−q + !q) [26].