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(ω) (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(ω) = |hN − 1, λ|c |GSi|2δ(ω + 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 |GSi 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 |N − 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(ω) via Ak(ω) = 1/π × |ImGk(ω)|. 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(ω) = 1/L × q gqδ(ω −Ek−q +ωq) 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 † X † 1 Z e−iωt + iωt − 1 Hel−pl = Ekckck + ωqaqaq (7) C (t) = dω|ImΣ (ω + E )| , (3) q k π k k ω2 k X gq † † + √ c ck(aq + a ), (8) where Σ (ω) 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 −iωt 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 (iωt − 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 (ω) = dω Ck (ω − ω )Ak (ω ). (4) integral calculations, the self-consistent Brillouin-Wigner qp perturbation theory yields substantially worse results Approximating A (ω) ≈ Z δ(Γk)(ω − 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)(ω) ≈ Z /π × k k For electron-plasmon interactions, Lundqvist demon- ImΣ (ω)/(ω − 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(ω) = λ v δ(ω − E + ω ), (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 ωq 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 (ω) = δ(ω − E + ω ), of the (N − 1)-electron system are simply ck|GSi (with k LD (E − E − ω )2 k−q q energy E − E ) and a† c |GSi (with energy E − q k k−q q GS k q k−q GS (6) Ek−q + ωq) [26]. Only the state ck|GSi gives a contribu- tion to Eq. (1) and the resulting spectral function has a where we introduced the electron-plasmon coupling single delta-function peak and no satellite features. 2 strength gq = λqvq. Eq. (6) shows that the satellite Next, we include electron-plasmon interactions us- 3
a) b) c)
0 > 0.3 0.3 > 0.10 − 2 0 0.2 − 5 0.08 0.2 − 4
F − 10 0.06 (ω)ε − 6 − 15
k 0.1 A 0.1 Energy (eV) − 20 − 8 Energy (eV) 0.04 − 25 0.0 − 30 − 10 0.02 0.0 -1.0 -0.5 0.0 0.0 0.5 1.0 L Γ X U,K Γ (ω-E )/ω Wave vector k 0 k/kF
FIG. 1. a) GW plus cumulant spectral functions of the three-dimensional electron gas at k = 0 for rs = 1.0 (red curve) and rs = 3.0 (blue curve). b) Spectral functions (in 1/eV) of the three-dimensional electron gas at rs = 4.0 (corresponding to metallic sodium) from GW plus cumulant theory. c) Spectral functions (in 1/eV) of silicon from ab initio GW plus cumulant theory calculations. ing first-order Rayleigh-Schr¨odingerperturbation theory. which depends sensitively on the relative magnitudes of The non-interacting states that lie in the energy region of the plasmon and quasiparticle effective masses [given by † ∗ ∗ the first satellite are the hole-plasmon pairs aqck−q|GSi. mpl = 1/(2β) and mqp = 1/(2α), respectively]: if β is Including the hole-plasmon coupling yields larger α, the satellite peak has a tail towards higher bind- ing energies (i.e. away from the Fermi energy). If α is † aqck−q|GSi → greater than β, the tail is towards lower binding energies. † 1 gq If the effective masses are equal, the satellite structure is aqck−q + √ ck + ... |GSi, (9) symmetric. LD Ek − Ek−q + ωq Equation (10) predicts the occurrence of a drastic i.e. the hole-plasmon pair state acquires a quasiparticle change in the satellite lineshape as function of the component, which makes this state “visible” in the spec- Wigner-Seitz radius rs. While the quasiparticle effective tral function as a satellite structure. Inserting Eq. (9) mass only has a weak dependence on rs and may be ap- into Eq. (1), we recover Eq. (6) for the plasmon satellite proximated by its non-interacting value, i.e. α = 0.5 (in contribution to the electron spectral function. This anal- atomic units) [23], the plasmon effective mass depends ysis shows clearly that no single, coherent hole-plasmon sensitively on rs. Within the random-phase approxima- state is formed, but instead the satellite is comprised of √ tion (RPA), we find β ≈ 0.64/ rs [20]. At small rs, a large number of incoherent, weakly interacting hole- β is large and the tail of the satellite extends to higher plasmon pairs. binding energies. For rs & 1.6, β is smaller than α and Plasmon satellites in three dimensions.—We first the tail of the satellite extends to lower binding ener- study the properties of plasmon satellites in the 3DEG. gies. Figure 1(a) shows the GW+C spectral functions In this system, the plasmon dispersion is parabolic at for the 3DEG with r = 1.0 and r = 3.0 obtained with a 2 s s small wave vectors, i.e. ωq = ω0 + βq [20] and we as- plasmon-pole model. It can clearly be seen that the tails sume that also the quasiparticle dispersion is parabolic, of the satellites extend into different directions. i.e. E = αk2. k In combination with angle-resolved photoemission With these assumptions, we can analytically compute spectroscopy (ARPES), the above analysis imposes use- the coupling-constant weighted electron-plasmon joint ful limits on the value of the plasmon effective mass. density of states, which is closely related to the satellite While ARPES experiments do not measure the plasmon contribution A(1)(ω) to the spectral function [27]. For k dispersion, they can determine both the quasiparticle ef- k = 0, we find fective mass and the satellite lineshape. Depending on the direction of the satellite tail [see Fig. 1(a)], the plas- ω0 Θ (sgn(α − β)[ω + ω0]) Jk=0(ω) = . (10) mon effective mass must be either smaller or larger than 2π p|(α − β)(ω + ω )| 0 the quasiparticle effective mass. This approach is par- This result shows that the satellite feature is peaked at ticularly useful in multiband systems, where each band ω = −ω0, i.e. the satellite is shifted from the quasi- leads to an additional constraint on the plasmon effective particle energy by the lowest plasmon energy ω0. More- mass. over, the satellite exhibits a highly asymmetric lineshape, The satellite lineshape from full GW+C calculations 4 is more symmetric than predicted by Eq. (10) since a) b) additional broadening mechanisms arising from finite 0.0 0 > 500 quasiparticle linewidths [see Eq. (4)] and finite plasmon linewidths (for example, caused by Landau damping [28]) − 0.5 − 2 400 are taken into account. The total linewidth of the satel- − 1.0 1.0 lite may thus be approximated as the sum of the widths − 4 300 of the coupling-strength weighted hole-plasmon joint den- − 1.5 − 6 sity of states, the quasiparticle spectral function and the 0.5 plasmon lineshape. In spectroscopic experiments on real − 2.0 200 Energy (eV) samples, additional phonon and disorder broadening as Energy (meV)− 8 − 2.5 well as broadening due to extrinsic losses occur [29, 30]. 0.0 100 − 10 Also at nonzero wave vectors, the peak of Jk(ω) is − 3.0 located at an energy ω0 below the quasiparticle energy 0 − 3.5 − 12 Ek (see appendix). In other words: the satellite band is a rigidly shifted copy of the quasiparticle band. Sur- 0.0 0.5 1.0 0.0 0.5 1.0 prisingly, this means that the effective mass of the satel- k/kF k/kF lite is the same as the quasiparticle effective mass irre- spective of the plasmon effective mass. We confirm our FIG. 2. a) Spectral functions (in 1/eV) of doped graphene on conclusions by carrying out GW+C calculations of the a silicon carbide substrate from GW plus cumulant theory. 3DEG at rs = 4.0 (corresponding to metallic sodium) b) Spectral functions (in 1/eV) of a one-dimensional electron using the frequency-dependent RPA dielectric function, gas from GW plus cumulant theory. see Fig. 1(b). Generalizing our findings for the plasmon satellite properties of the 3DEG to real materials is straightfor- spectral function at√ the Dirac point [34], we find that 2 ward. Fig. 1(c) shows the spectral functions of crystalline Jk=0(ω) ∝ Θ(ω +ω ˜)/ ω +ω ˜, whereω ˜ = β /(4vF ) is the silicon obtained from ab initio GW+C calculations [31]. separation between the quasiparticle and satellite peaks. Because of the parabolic dispersion of the valence quasi- Again, the plasmon satellite lineshape is highly asym- particle bands near the band extrema and the parabolic metric.√ The dependence of β on the charge density n, dispersion of the plasmon, the plasmon satellite band β ∝ n [16], gives rise to small changes in the lineshape structure appears as a rigidly shifted copy of the quasi- as function of the carrier density. Comparing the ex- particle band structure, but significantly broadened. pression for Jk=0 of doped graphene to the result for the Previous models of plasmon satellites in three- 3DEG [see Eq. (10)], we observe that no drastic changes dimensional systems [6, 32, 33] assumed that plasmon in the asymmetry of the lineshape occur as function of dispersion is a minor effect and approximated the satel- the carrier density. lite simply as a shifted, broadened copy of the quasiparti- Figure 2(a) shows the spectral functions of doped cle peak. Such approaches fail to describe the asymmet- graphene on a silicon carbide substrate from GW+C cal- ric lineshape of the satellite and also cannot be applied culations [35]. We observe that the plasmon satellite straightforwardly to lower-dimensional systems, which band is not a shifted copy of the quasiparticle band, but we discuss below. that the two bands merge at the Fermi wave vector kF . Plasmon satellites in two and one dimensions.—In Finally, we analyze the plasmon satellite properties in three-dimensional systems, the satellite feature is sep- a one-dimensional metallic system, the 1DEG. In this arated from the quasiparticle peak by the lowest plas- system, the plasmon dispersion relation at long wave- p mon frequency. In metallic two-dimensional systems, the lengths is given by ωq = βq log(1/ql) with l denoting a plasmon energy is proportional to the square root of the cutoff distance [36–38]. Assuming a parabolic quasiparti- √ 2 plasmon wave vector, i.e. ωq = β q [20], and it is not a cle dispersion, i.e. Ek = αk , we find again that the first priori clear where the satellite peak is located. plasmon satellite exhibits a highly asymmetric lineshape. We now apply our GW+C-based analysis of plas- Fig. 2(b) shows GW+C spectral functions for the 1DEG mon satellite properties to two-dimensional systems and at rs = 1.4 [39]. In contrast to graphene and the 3DEG, choose electron-doped graphene as a test case. Within the plasmon satellite is relatively weak and appears as a the Dirac model approach, the two bands in the vicinity shoulder-like feature near a strong quasiparticle band. of the Fermi energy are described by a linear dispersion Summary.—By connecting the GW+C Green’s func- relation, i.e. Ek = ±vF k with vF denoting the graphene tion approach to a wavefunction-based perspective, Fermi velocity. Here, k is measured from the K or K0 we established the coupling-constant weighted electron- points of the graphene Brillouin zone. plasmon joint density of states Jk(ω) as a useful quan- Taking into account that electrons in the upper Dirac tity for analyzing plasmon satellites in electron spectral band give the dominant contribution to the satellite functions. We evaluated Jk(ω) for systems in one, two 5 and three dimensions and demonstrated how the prop- find that erties of plasmon satellites are related to the proper- ties of the underlying electrons and plasmons empha- Jk(ω) = (14) Z ∞ sizing the importance of plasmon dispersion. Our for- ω0 dq 2 2 malism for electron-plasmon interactions can be gener- Θ(2kq|α| − |ω + ω0 − αk + [β − α]q |). 4π|α|k 0 q alized straightforwardly to study the generation of hot electron-hole pairs in plasmonic devices for photovoltaics We now assume that β − α > 0 and distinguish the ∗ 2 ∗ and photocatalysis in the future. two cases: i) ωk ≡ ω + ω0 − αk > 0 and ii) ωk < 0. To Acknowledgments.—The authors would like to thank find the position of the peak of Jk(ω), it is sufficient to Prof. Feliciano Giustino for valuable discussions. J.L. consider case i) and we find that acknowledges support from EPRSC under Grant No. Z ∞ ω0 dq ∗ 2 EP/N005244/1 and also from the Thomas Young Centre Jk(ω) = Θ(2kq|α| − ωk − [β − α]q ) under Grant No. TYC-101. Via J.L.’s membership of 4π|α|k 0 q the UK’s HEC Materials Chemistry Consortium, which (15) √ is funded by EPSRC (EP/L000202), this work used the ω0 1 + 1 − fk = Θ(1 − fk) log √ , ARCHER UK National Supercomputing Service. S.G.L. 4π|α|k 1 − 1 − fk acknowledges support from the National Science Foun- ∗ 2 dation under grant DMR15-1508412. with fk = [β − α]ωk/(k|α|) . This function diverges as ∗ 2 ωk → 0 indicating that Jk(ω) is peaked at −[ω0 − αk ]. For case ii), we have to evaluate
APPENDIX " ∗ ω Z q dq J (ω) = 0 Θ(2kq|α| + ω∗ + [β − α]q2)+ k 4π|α|k q k Electron-Plasmon Joint Density of States in Three Di- 0 mensions.—We calculate the coupling-strength weighted (16) Z ∞ joint density of states Jk(ω) comprising only plasmon- dq ∗ 2 Θ(2kq|α| − ωk − [β − α]q ) hole pairs with total momentum k for a three- q∗ q √ dimensional homogeneous electron gas (3DEG). As ω 1 + 1 − f shown in the main text, this quantity is closely related = 0 log √ k , 4π|α|k −1 + 1 − f to the first satellite contribution to the electron spectral k function. Specifically, Jk(ω) is given by ∗ p ∗ with q = |ωk|/[β − α]. Note that the solutions above also describe the case of Z 3 d q 2 β − α < 0, but now Eq. (5) describes negative ω∗ and Jk(ω) = gqδ(ω − Ek−q + ωq), (11) k (2π)3 ∗ Eq. (6) describes positive ωk. where gq denotes the electron-plasmon coupling strength, 2 2 ωq = ω0 + βq is the plasmon dispersion and Ek = αk is the quasiparticle dispersion. Using a plasmon-pole model that conserves sum rules, ∗ [email protected] 2 2 2 [1] S. Mukherjee, F. Libisch, N. Large, O. Neumann, L. V. we find gq = vqω0/(2ωq) ≈ vqω0/2 with vq = 4π/q . For the special case of k = 0, we find Brown, J. Cheng, J. B. Lassiter, E. A. Carter, P. Nord- lander, and N. J. Halas, Nano Letters 13, 240 (2012). Z ∞ [2] C. Clavero, Nature Photonics 8, 95 (2014). ω0 2 [3] M. Moskovits, Nature Nanotechnology 10, 6 (2015). Jk=0(ω) = dqδ(ω + ω0 − [α − β]q ) (12) π 0 [4] B. Lundqvist, Physik der kondensierten Materie 9, 236 (1969). ω0 Θ(sgn(α − β)(ω + ω0)) = p , [5] J. Inglesfield, Journal of Physics C: Solid State Physics 2π |(α − β)(ω + ω0)| 16, 403 (1983). [6] M. Guzzo, G. Lani, F. Sottile, P. Romaniello, M. Gatti, which has a peak at −ω0. J. J. Kas, J. J. Rehr, M. G. Silly, F. Sirotti, and L. Rein- In the general case of nonzero k, we have to evaluate ing, Phys. Rev. Lett. 107, 166401 (2011). [7] J. Lischner, G. P´alsson, D. Vigil-Fowler, S. Nemsak, J (ω) = (13) J. Avila, M. Asensio, C. Fadley, and S. G. Louie, Phys- k ical Review B 91, 205113 (2015). ω Z ∞ Z 1 [8] M. Guzzo, J. J. Kas, L. Sponza, C. Giorgetti, F. Sot- 0 dq duδ(ω + ω − αk2 + 2αku + [β − α]q2). 2π 0 tile, D. Pierucci, M. G. Silly, F. Sirotti, J. J. Rehr, and 0 −1 L. Reining, Physical Review B 89, 085425 (2014). [9] L. Ley, S. Kowalczyk, R. Pollak, and D. Shirley, Physical R 1 Using that −1 duδ(A + Bu) = Θ(|B| − |A|)/|B|, we Review Letters 29, 1088 (1972). 6
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