Exciton-plasmon states in nanoscale materials: breakdown of the Tamm-Dancoff approximation
aa r X i v : . [ c ond - m a t . m t r l - s c i ] S e p Exciton–plasmon states in nanoscale materials: breakdown of the Tamm–Dancoffapproximation
Myrta Gr¨uning, Andrea Marini, and Xavier Gonze European Theoretical Spectroscopy Facility (ETSF), Universit´e Catholique de Louvain,Unit´e de Physico-Chimie et de Physique des Mat´eriaux, B-1348 Louvain-la-Neuve, Belgium European Theoretical Spectroscopy Facility (ETSF),CNR-INFM Institute for Statistical Mechanics and Complexity,CNISM and Dipartimento di Fisica, Universit´a di Roma “Tor Vergata”,via della Ricerca Scientifica 1, 00133 Roma, Italy
Within the Tamm–Dancoff approximation ab initio approaches describe excitons as packets ofelectron-hole pairs propagating only forward in time. However, we show that in nanoscale materialsexcitons and plasmons hybridize, creating exciton–plasmon states where the electron-hole pairsoscillate back and forth in time. Then, as exemplified by the trans -azobenzene molecule and carbonnanotubes, the Tamm–Dancoff approximation yields errors as large as the accuracy claimed in abinitio calculations. Instead, we propose a general and efficient approach that avoids the Tamm–Dancoff approximation, and correctly describes excitons, plasmons and exciton–plasmon states.
PACS numbers: 73.20.Mf,78.67.-n,71.15.Qe,31.15.-p
The Bethe–Salpeter (BS) [1] and the Time-DependentDensity Functional Theory (TDDFT) [2] equations al-low the accurate calculation of the polarization func-tion of many physical systems without relying on ex-ternal parameters. Within these frameworks neutralexcitations are described as combination of electron-hole (e-h) pairs of a noninteracting system. However fornanoscale materials, the huge number of e-h pairs in-volved makes the solution of the BS/TDDFT equationextremely cumbersome. Consequently, the increasing in-terest in the excitation properties of such materials hasjustified the use of ad-hoc approximations. The most im-portant and widely-used is the Tamm-Dancoff approx-imation (TDA) [3] where only positive energy e-h pairsare considered. Within the TDA the interaction betweene-h pairs at positive and negative (antipairs) energies isneglected, and only one e-h pair is assumed to propagatein any time interval. The main advantage of the TDA isthat the non-Hermitian
BS/TDDFT problem reduces toa
Hermitian problem, that can be solved with efficientand stable iterative methods [4].In Solid State Physics—a major field of application ofthe BS/TDDFT equation—the success of TDA is basedon the sharp distinction between excitonic and plasmonicexcitations. Excitons are localized packets of e-h pairsbound together by the Coulomb attraction and are ob-served in optical absorption experiments. Plasmons are,instead, delocalized collective oscillations of the electronicdensity that induce a macroscopic polarization effect andare observed in electron energy loss (EEL) experiments.In contrast to the case of excitons, the TDA is knownto misdescribe plasmons in solids [5] as the density os-cillations involve the excitation of e-h antipairs. Nev-ertheless, the success in describing optical absorption ofsolids and the remarkable numerical advantages have mo-tivated the application of the TDA to very different sys- tems. Nowadays the BS/TDDFT equation within theTDA is becoming a standard tool to study excitations innanostructures [6, 7], and in molecular systems [8].In this Letter we argue that for confined systems—such as nanostructures or π -conjugated molecules—theexcitations appearing in the response function show amixed excitonic–plasmonic behavior. As a consequencethe e-h pair-antipair interaction becomes crucial and theTDA does not hold anymore. A paradigmatic exampleis the trans -azobenzene molecule, where the TDA overes-timates the static polarizability by ∼ ab initio calculations.Even more intriguing is the case of carbon nanotubesthat, because of the quasi-one-dimensional (1D) struc-ture, behave either as extended or isolated system de-pending on the direction of the perturbing field. Thus,for transverse perturbations the excitons acquire a plas-monic nature and the TDA overestimates the position ofthe π plasmon peak appearing in both absorption andEEL spectra by almost 1 eV. By exploiting the symme-try properties of the BS and TDDFT kernels we devise arobust and efficient iterative approach to calculate thefrequency-dependent response, beyond the TDA. Thisapproach benefits from the same numerical advantagesof Hermitian techniques, and correctly describes excitons,plasmons and exciton–plasmon states.To introduce and understand the reasons beyond thebreakdown of the TDA we need to study in detail thestructure of the TDDFT and BS equations. These arecommonly rewritten as a Hamiltonian problem [9], byexpanding the single particle states in the Kohn–Shambasis. Then, the BS/TDDFT Hamiltonian H is a matrixin the Fock space of the e-h pairs | eh i and antipairs | f he i ,and it has the block-form H = (cid:18) R C − C ∗ − R ∗ (cid:19) . (1)The resonant block R is Hermitian and the couplingblock C is symmetric (see Appendix B of Ref. [9]).The dielectric function ε ( ω ) is written in terms ofthe resolvent of H , ( ω − H ) − , as ε ( ω ) = 1 − (8 π ) / Ω h P | ( ω − H + i + ) − | P i , where Ω is the simula-tion volume. In the limit of large Ω the polarizabilityis given by α ( ω ) ∝ ε ( ω ). | P i is a ket whose compo-nents along the | eh i space are the optical oscillators: h P | eh i ∼ h e | ~d · ~ξ | h i , with ~d the electronic dipole, and ~ξ the light polarization factor. Energy (eV) P o l a r i z ab ili t y ( a . u . x ) -4 -2 0 2 4 6e-h pairs energy (eV) E xc i t a t i on a m p li t ude Full: 275 (a.u.) TDA: 194 (a.u.) { Static polarizability
TDA breakdown
FIG. 1:
Dynamical polarizability of the trans -azobenzenemolecule calculated within the TDLDA either by using the fullHamiltonian (solid line) or the TDA (dashed line). For comparisonthe results obtained by diagonalization (circles) are also reported.The TDA largely underestimates the static polarizability, by al-most 40 %. More importantly the amplitude function (inset) of themost intense peak of the polarizability (see text) differs dramati-cally in the TDA and in the full Hamiltonian. The TDA misses ofan important contribution from the spatially extended e-h antipairwith energy ∼ − . The matrix elements of R and C have different def-initions in the BS and TDDFT Hamiltonians. Withinthe BS equation R ee ′ hh ′ = E eh δ ee ′ δ hh ′ + h eh | W − V | e ′ h ′ i ,and C ee ′ hh ′ = h eh | W − V | g h ′ e ′ i . E eh is the energy of theindependent e-h pair, W is the statically screened e-h at-traction and ¯ V is the bare Coulomb interaction withoutthe long-range tail [9]. Within TDDFT W is replaced by − f xc , with f xc the exchange–correlation kernel.The TDA assumes that the effect induced by the e-hantipairs is negligible. Consequently, the C block, thatcouples pairs and antipairs, is neglected, and the Hamil-tonian H is approximated by R . The coupling describedby C is dominated by the contribution arising from thebare Coulomb interaction ¯ V . This term, called Hartreecontribution, measures the degree of inhomogeneity ofthe electronic density. The larger this inhomogeneity, thestronger the corresponding polarization and the inducedlocal fields that counteract the external perturbing field.Therefore, the inhomogeneity of the electronic densityand the strength of local fields discriminate whether anexcitation is well-described by the TDA. Furthermore the TDA is known to fail for plasmonic excitations [5] that,causing an oscillation of the density, involve the creationof e-h pairs and antipairs.The electronic density of confined systems, likemolecules and nanostructures, is typically strongly in-homogeneous. Moreover the excitons can be spread allover the molecule, involving the excitations of most of theelectrons. Thus, in contrast to solids, it is not possible todistinguish between excitonic and plasmonic excitations,and the arguments commonly used to sustain the TDAfail.Indeed, the striking failure of the TDA is clearlydemonstrated by the dynamical polarizability ℑ [ α ( ω )] of trans -azobenzene, calculated within the time-dependentlocal density approximation (TDLDA) [10, 11]. InFig. 1 we compare ℑ [ α ( ω )] calculated either using theTDA, or by solving the full H eigenproblem. TDAyields a blueshift of 0.2 eV and a large overestimationof the intensity of the main peak. More importantlythe TDA causes a 40% underestimation of the staticpolarizability, ℜ [ α ( ω = 0)]. The reason for this fail-ure can be understood looking at the amplitude function A λ ( ω ) of the eigenstates | λ i of H , defined as A λ ( ω ) = P η = { eh } , { f eh } |h η | λ i| δ ( ω − E η ). The A λ max function forthe state λ max corresponding to the most intense peakof the α spectrum is shown in the inset of Fig. 1. In theTDA, the λ max state is decomposed only in positive e-hpairs. However, the solution the full Hamiltonian revealsan important contribution from the e-h antipair with en-ergy ∼ − π ∗ → π transition—changes the characterof the excitation. Whereas in the TDA the most intenseexcitation in the spectrum can be identified with a sin-gle e-h pair, in the full H solution it acquires a morecollective, plasmon-like character.The example of the trans -azobenzene makes clear thata proper description of the electron-electron correlationsin confined systems requires the solution of the full BS/TDDFT Hamiltonians, beyond the TDA. However,for larger nanostructures with many degrees of freedom,the size of the H matrix can be as large as 10 × and consequently the problem is impossible to treat ifnot using iterative methods. The TDA reduces H to aHermitian Hamiltonian, and makes possible to use theefficient and stable Hermitian iterative approaches [4].Therefore we need an iterative approach for calculatingthe resolvent of the full non-Hermitian Hamiltonian H asefficient and stable as for the Hermitian case.In what follows, we show that it is indeed possible todesign such an iterative approach by observing that H belongs to a class of non-Hermitian Hamiltonians with areal spectrum —that is real eigenvalues. As establishedby Mostafazadeh [12], the reality of the spectrum is re-lated to the existence of a positive-definite inner prod-uct with respect to which the Hamiltonian is Hermitian.We show that for the BS (TDDFT) Hamiltonian—and ingeneral for all the Hamiltonians of this form—this innerproduct does exist (thus the spectrum is real) and, moreimportantly, it is explicitly known . The knowledge of thisproduct allows one to conveniently transform the itera-tive approach designed for the Hermitian TDA Hamilto-nian to treat the full non-Hermitian Hamiltonian.Following Zimmermann [13] we can write H as theproduct of two noncommuting Hermitian matrices, H = F ¯ H = (cid:18) − (cid:19) (cid:18) R CC ∗ R ∗ (cid:19) . (2)One can check that ¯ HH = H † ¯ H . This key propertyof H is called ¯ H -pseudo-Hermicity [14]. Through the¯ H operator, we can define a positive-definite ¯ H -innerproduct [15] h·|·i ¯ H := h·| ¯ H |·i , and a corresponding ¯ H -expectation value h·| O |·i ¯ H := h·| ¯ HO |·i . With respect tothis ¯ H -expectation value, H is Hermitian as can be ver-ified by using the ¯ H -pseudo-Hermicity, h v | H | v ′ i ¯ H = h v ′ | H † ¯ H | v i ∗ = h v ′ | ¯ HH | v i ∗ =: h v ′ | H | v i ∗ ¯ H . (3)It follows that the ¯ H -expectation value of the resolventof H , ( ω − H ) − , is Hermitian as well. Then, to eval-uate h P | ( ω − H ) − | P i , and thus ǫ ( ω ), we rewrite it interms of Hermitian ¯ H -expectation values by using thecompleteness relationship I = P k | q k ih q k | ¯ H h P | ( ω − H ) − | P i = X k h P | q k ih q k | ( ω − H ) − | P i ¯ H , (4)where {| q k i} is a complete basis, orthonormal with re-spect to the ¯ H -inner product. The h q k | ( ω − H ) − | P i ¯ H are conveniently calculated within the standard Lanczos–Haydock (LH) iterative method [16]—the same used inthe TDA Hermitian case [17]— provided that the ¯ H -innerproduct replaces the standard one . The LH method recur-sively builds the {| q k i} basis in which H is represented bya one-dimensional semi-infinite chain of sites with onlynearest-neighbors interactions. For such a system theevaluation of the matrix elements of ( ω − H ) − reducesto the calculation of a continued fraction.This approach allows us to treat systems as large ascommonly done using the TDA, but using the full Hamil-tonian. Computationally, it requires a single matrix–vector multiplication at each iteration —as in the Hermi-tian case [18]. Since h P | ( ω − H ) − | P i in Eq. (4) is con-verged after a number of iterations much smaller thanthe dimension of H , this method is far more efficientthan performing the diagonalization of the Hamiltonian.Specifically, for the trans -azobenzene, the number of op-erations performed in the diagonalization is about twoorders of magnitude larger than in our approach [19],while the results are indistinguishable (Fig. 1). FIG. 2: Polarized (angle φ with respect to the CNT axis) ab-sorption (left stack) and φ -dependent EEL (right stack) spectra of zig-zag (8,0) CNT calculated within the BSE either by using the full(solid line) or the TDA (dashed line) Hamiltonian. The dotted lineindicates the position of the quasiparticle band gap [7]. The grayarea shows the experimental energy blueshift (0.5 eV) between theparallel and transverse plasmon mode [22], correctly reproduced byfull Hamiltonian calculations. Instead, the TDA predicts the trans-verse plasmon mode to red-shift, in striking disagreement with theexperiment. As an application we consider the frequency-dependentresponse of carbon nanotubes (CNTs) within the BSequation. Recently, experimental studies have shownCNTs to be characterized by strongly anisotropic elec-tronic and optical properties [21, 22]. Moreover, the mea-sured optical and EEL spectra have revealed that excita-tions appearing at energies higher than 4 eV have a collec-tive character [21, 22]. Using the present method we showthat the external field polarization and the excitonic–plasmonic character of the optical excitations of CNTsare intimately related.So far ab initio studies have been limited to the ex-citonic effects appearing below ∼ . × e-h pairs in the solutionof the BS equation, without using the TDA .Figure 2 shows the absorption (left stack) and q → EEL (right stack) spectrum of CNTs calculated withinthe BS [10, 20] equation as a function of the angle φ that the perturbing field forms with the tube axis. Wecompare the full solution of the BS equation (full line)with the result of the TDA (dashed line).For a parallel perturbing field ( φ = 0) the CNT be-haves as an extended solid. As expected, in this case theTDA describes well the excitonic peaks in the absorptionspectrum [7]. On the contrary the TDA completely failsin reproducing the longitudinal plasmonic peak, overesti-mating its frequency by 1.5 eV. By increasing the φ anglethe perturbing field acquires a perpendicular component,and a peak at ∼ ǫ ( ω ) function. Thevery same peak occurs in the EEL spectrum, confirm-ing that it corresponds to a mixed excitonic–plasmonicexcitation. This mixed behavior is misdescribed by theTDA both in the absorption and in the EEL spectra.The TDA performs even worse for a transverse perturb-ing field ( φ = 90 ◦ ), where the excitations are confinedwithin the tube radius. The ǫ ( ω ) and ǫ − ( ω ) functionsbecome very similar— that is, the system behaves as anisolated molecule. Thus, like in the case of the trans -azobenzene, the contribution from e-h antipairs cannotbe neglected, and the TDA yields an error on the posi-tion of the main peak in the absorption/EEL spectra of ∼ ℑ ( ǫ ) arenot all purely excitonic. When the excitation is forcedto be spatially confined, it acquires a mixed excitonic–plasmonic character, and the TDA breaks down. TheTDA performs well only for purely excitonic states thatare dominant only when the perturbing field is polarizedalong the tube axis. However, in practice it is not possi-ble to measure selectively only longitudinal polarizationas CNTs are generally randomly oriented in a sample.Even in the case of vertical aligned CNTs the tubes arefound to form angles of ∼ ◦ [22].Finally, an important confirmation of the accuracy ofthe present approach is given by the position of the plas-mon peaks in the EEL spectra. Using the full solutionof the BS equation, the plasmon energy is found to blue-shift of ∼ φ = 0 ◦ to φ = 90 ◦ . Thisis in striking agreement with the 0 . redshift of the plas-mon, in disagreement with the experimental results.In summary, we have shown that the TDA breaks downin nanoscale systems where dimensionality effects con-fine the optical excitation, inducing a mixed excitonic–plasmonic behavior. We propose a novel approach tosolve the BS/TDDFT equations beyond the TDA, keep-ing the numerical advantages of a Hermitian formulation.This approach successfully explains the experimental fea-tures in the optical and EEL spectra of CNTs, and opensthe way to a truly ab initio approach to linear responseproperties of nanoscale materials.This work was supported by the EU through theFP6 Nanoquanta NoE (NMP4-CT-2004-50019), the FP7ETSF I3 e-Infrastructure (Grant Agreement 211956), the Belgian Interuniversity Attraction Poles Program(P6/42), the Communaut´e Fran¸caise de Belgique (ARC07/12-003), the R´egion Wallone (WALL-ETSF). [1] E. E. Salpeter and H. A. Bethe, Phys. Rev. , 1232(1951).[2] E. Runge and E. K. U. Gross, Phys. Rev. Lett. , 997(1984).[3] A. L. Fetter and J. D. Walecka, Quantum Theory ofmany-particle systems (Dover, 2003), chap. 15, p. 565.[4]
Templates for the solution of algebraic problems: a prac-tical guide , edited by Z. Bai, J. Demmel, J. Dongarra, A.Ruhe, H. van der Vorst, (SIAM, Philadelphia, 2000).[5] V. Olevano and L. Reining, Phys. Rev. Lett. , 5962(2001).[6] See e.g. M. Del Puerto et al. Phys. Rev. Lett. , 096401(2006); B. Arnaud et al. ibid. , 026402 (2006); L. Wirtzet al., ibid. , 126104 (2006)[7] C.D. Spataru et al., Top. Appl. Phys. , 195 (2008).[8] S. Hirata and M. Head-Gordon, Chem. Phys. Lett. ,291 (1999).[9] G. Onida, et al., Rev. Modern Phys. , 601 (2002).[10] BS/TDDFT spectra are calculated with the yambo code, (A. Marini et al., the yambo abinit , X. Gonze et al. Comp. Mat.Science , 478 (2002); Z. Kristallogr. , 478 (2005).[11] We have included e-h pairs and antipairs in a window of30 eV and used a cut-off of 1.6 Ha for the ¯ V integrals.[12] A. Mostafazadeh, J. Math. Phys. , 3944 (2002).[13] R. Zimmermann, Phys. Stat. Sol. , 23 (1970).[14] A. Mostafazadeh, J. Math. Phys. , 205 (2002).[15] When the formation of excitons does not lead to a fun-damental change of ground state, ¯ H is positive definite,see Ref. [13].[16] R. Haydock, in Solid State Phys. , edited by H. Ehrenfest,F. Seitz,and D. Turnbull (Academic Press, 1980), vol. 35,p. 215–294.[17] L. X. Benedict and E. L. Shirley, Phys. Rev. B , 5441(1999).[18] Because of the interplay between H , F and ¯ H , evaluat-ing the ¯ H -inner product does not require extra operationwith respect to the standard inner-product.[19] The proposed approach requires mN operations, where m is the number of iterations and N the size of the sys-tem. For azobenzene N ∼ and m ∼ . Thus itis much faster and convenient compared to the standarddiagonalization, that requires O ( N ) operations.[20] We have included e-h pairs and antipairs from band 32to 96 in a window of 15 eV and used cut-offs of 4 Ha and1 Ha for the ¯ V and W integrals. The 1D Brillouin zoneis sampled by 64 k-points. Quasiparticle correction fromRef. [7].[21] Y. Murakami, et al., Phys. Rev. Lett. , 087402 (2005).[22] C. Kramberger, et al., Phys. Rev. Lett.100