Phonon linewidth due to electron-phonon interactions with strong forward scattering in FeSe thin films on oxide substrates
PPhonon linewidth due to electron-phonon interactions with strong forward scatteringin FeSe thin films on oxide substrates
Yan Wang, Louk Rademaker, Elbio Dagotto,
1, 3 and Steven Johnston Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA Kavli Institute for Theoretical Physics, University of California Santa Barbara, California 93106, USA Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA (Dated: March 7, 2017)The discovery of an enhanced superconducting transition temperature T c in monolayers of FeSegrown on several oxide substrates has opened a new route to high- T c superconductivity throughinterface engineering. One proposal for the origin of the observed enhancement is an electron-phonon ( e -ph) interaction across the interface that peaked at small momentum transfers. In thispaper, we examine the implications of such a coupling on the phononic properties of the system.We show that a strong forward scattering leads to a sizable broadening of phonon lineshape, whichmay result in charge instabilities at long-wavelengths. However, we further find that the inclusion ofCoulombic screening significantly reduces the phonon broadening. Our results show that one mightnot expect anomalously broad phonon linewidths in the FeSe interface systems, despite the fact thatthe e -ph interaction has a strong peak in the forward scattering (small \bfq ) direction. PACS numbers: 74.70.Xa, 74.20.Pq, 74.25.Kc, 74.78.-w \bfI . \bfI \bfN \bfT \bfR \bfO \bfD \bfU \bfC \bfT \bfI \bfO \bfN
Due to its structural simplicity, FeSe has played a lead-ing role in many experimental and theoretical studies onFe-based superconductors since its discovery in 2008. The enduring interest in this compound is partially owedto the high T c (ranging between 55--100 K) achieved whenmonolayer FeSe films are grown on SrTiO substrates (FeSe/STO), a ten-fold enhancement from the T c \sim Intriguingly,the high T c in the interfacial system proves to be robustfor various oxide substrates, including SrTiO (001), BaTiO (001), SrTiO (110), anatase TiO (001), and rutile TiO (100). These oxide substrates, termi-nated at TiO surface when interfaced with FeSe, havelattice parameters significantly larger than that of bulkFeSe and thus apply strong tensile strain on FeSe thinfilms. The anatase and rutile TiO substrates even in-duce rather different strains along a and b axes of themonolayer FeSe. The T c 's, however, are consistentlyabove 55 K, as measured by angle-resolved photoemis-sion spectroscopy (ARPES). This observation appears torule out a direct correlation between the enhanced super-conductivity and the tensile strain. The electronic structure of the interfaces displaying en-hanced T c 's are also remarkably similar across the vari-ous substrates. For instance, the Fermi surface measuredby ARPES consists of only electron pockets at the cor-ners of the two-Fe Brillouin zone, indicating substantialelectron doping from the parent compound. This obser-vation poses a challenge to theories for the high T c basedon the pairing mediated by spin fluctuations that arestrongly enhanced by Fermi surface nesting. One poten-tial solution to this problem is the involvement of bandsbelow the Fermi level in pairing (so-called incipient bandpairing). Another possibility is the involvement of adifferent type of pairing mediator such as nematic fluc- tuations or phonons particular to the interface. Evidence for the latter has been provided by the com-mon observation of replica bands in the electronic struc-ture of superconducting FeSe monolayers on SrTiO , BaTiO , and rutile TiO . The replica bands observed by ARPES are exact copiesof the original bands crossing the Fermi level in momen-tum space but with a weaker spectral weight. They areinterpreted as being generated by an electron-phonon ( e -ph) interaction between the FeSe electrons and oxygenphonons in the substrate. This view is supportedby the fact that the \sim
100 meV energy offset betweenthe primary and the replica band coincides with thephonon energy of oxygen modes in SrTiO , BaTiO , and TiO . Due to the particular properties of the inter-face, this interaction is strongly peaked for forwardscattering (i.e., peaked at small momentum | \bfq | trans-fer), as found by analyzing the electrostatic potentialfrom the dipole induced by the oxygen modes andby first-principles calculations. This unique mo-mentum structure accounts for the fact that the replicassharply trace the dispersion of the primary band, whichrequires the e -ph interactions be forward-focused. Sucha coupling can also significantly enhance T c , due to thelinear dependence of T c on the dimensionless couplingconstant \lambda m , as opposed to the exponential depen-dence obtained for the usual BCS case. For example,assuming a narrow width q for the forward scatteringpeak, some of the current authors found \lambda m \sim . . T c \sim Ref. 23 has obtained similar re-sults after extending this approach to a more realisticband structure.Many aspects of the influence of the e -ph interactionswith strong forward scattering on electronic propertiesand superconductivity are summarized in Refs. 18, 21, a r X i v : . [ c ond - m a t . s up r- c on ] M a r and 24. In comparison, there are no qualitative or quanti-tative studies of the phononic properties for the problemat hand. Here, we have carried out such a study to ad-dress two issues. First, Zhang et al. recently measuredthe phonon linewidth of a \sim
90 meV phonon mode pen-etrating from the SrTiO substrate into thin FeSe Filmsusing high resolution electron energy loss spectroscopy(HREELS) and concluded a mode-specific e -ph couplingconstant \lambda \sim .
25. Not only does this echo the discoveryof replica bands by the ARPES experiments in the samesystem, but it also calls for a theoretical consideration onthe HREELS measurements. Doing so would corrobo-rate both the total coupling strength and momentum de-pendence of the e -ph coupling in FeSe/STO system withthose inferred from the ARPES measurements. Second,when a strong e -ph coupling is distributed over a subsetof wave vectors, one expects tendencies towards charge-density-wave formation that can compete with super-conductivity. Such tendencies will manifest themselvesas Kohn anomalies in the phonon dispersion and broadphonon linewidths. One can, therefore, address this issuedirectly by examining the phononic self-energy.Here, we examine the phonon linewidth due to e -ph in-teractions with strong forward scattering using the samemodel adopted in Ref. 18 to study the electronic spec-tral function. We first describe the details of the modeland method in Sec. II. Next, in Sec. III we give someanalytical results for the normal state phonon proper-ties in the perfect forward scattering limit, where theinteraction is treated as a delta function at \bfq = 0. Ournumerical results for both normal and superconductingstate with finite q are given in Sec. IV. Here, our resultsshow that the forward focused peak in the e -ph couplingresults in very broad phonon lineshapes. However, inSec. V we reintroduce Coulomb screening, which subse-quently undresses the phonon propagator and suppressesthese effects. Finally, in Sec. VI we summarize our re-sults and make some concluding remarks in relation tothe HREELS experiment of Zhang et al. \bfI \bfI . \bfM \bfO \bfD \bfE \bfL \bfA \bfN \bfD \bfM \bfE \bfT \bfH \bfO \bfD Our model Hamiltonian describes a single band modelof FeSe electrons coupled to an optical phonon branchvia a momentum-dependent coupling, which reads H = \sum \bfk ,\sigma \xi \bfk c \dagger \bfk ,\sigma c \bfk ,\sigma + \sum \bfq \omega \bfq \biggl( b \dagger \bfq b \bfq + 12 \biggr) + 1 \surd N \sum \bfk , \bfq ,\sigma g ( \bfk , \bfq ) c \dagger \bfk + \bfq ,\sigma c \bfk ,\sigma ( b \dagger - \bfq + b \bfq ) . (1)Here, c \dagger \bfk ,\sigma ( c \bfk ,\sigma ) creates (annihilates) an electron withwavevector \bfk and spin \sigma , b \dagger \bfq ( b \bfq ) creates (annihilates)a phonon with wavevector \bfq ; \xi \bfk is the electronic banddispersion measured relative to the chemical potential \mu ; \omega \bfq is the phonon dispersion ( \hbar = 1); and g ( \bfk , \bfq ) is themomentum dependent e -ph coupling.We take a simple electronic band dispersion \xi \bfk = - t [cos( k x a ) + cos( k y a )] - \mu , where a is the in-plane lat-tice constant. We set t = 0 .
075 eV and \mu = - .
235 eV,which produces around \Gamma point an electronlike Fermipocket with k \mathrm{F} = 0 . /a , a Fermi velocity v \mathrm{F} = 0 .
12 eV \cdot a/ \hbar along the k y = 0 line, and an effective electron bandmass m \ast x,y = \Bigl( \partial \xi \bfk \hbar \partial k x,y \Bigr) - \bfk =0 = \hbar ta = 3 . m e , which is sim-ilar to the electron pocket at the M point in FeSe/STOseen in ARPES experiments. Since we have a singleband model, it only takes a trivial \bfQ = ( \pi /a, \pi /a ) shiftto map our \Gamma -point pocket onto the electron pocket inthe real system centered at the M point and any physi-cal quantities depending only on the momentum transfer \bfq = \bfk - \bfk \prime , such as phonon linewidth, do not depend onthe position of the pocket. Since we are not consideringthe effects of an unconventional pairing mechanism here,we do not need to consider the possibility of d -wave in-stabilities due to scattering between the electron pockets.As such, a single band model is sufficient for our purpose.Throughout we approximate the experimental phonondispersion with a dispersionless Einstein mode \omega \bfq \approx \omega \mathrm{p}\mathrm{h} = 100 meV according to the observed energy separa-tion between the replica band and the primary band, as well as the phonon dispersion of the interface, asmeasured by HREELS. We neglect the fermion mo-mentum dependence in the coupling g ( \bfk , \bfq ) = g ( \bfq ),where \bfq is the momentum transfer and adopt g ( \bfq ) = g \sqrt{} \pi / ( aq ) exp( - | \bfq | /q ) as derived from simple mi-croscopic model. Here, g is adjusted to fix thetotal dimensionless coupling strength of the interactionand q sets the range of the interaction in momentumspace. The normalization factor \sqrt{} \pi / ( aq ) is chosensuch that \langle g ( \bfq ) \rangle \bfq \approx g for q \ll \pi , where \langle F \bfq \rangle \bfq = a \int \int \mathrm{B}\mathrm{Z} F \bfq dq x dq y / (2 \pi ) denotes an momentum integralover the first Brillouin zone. We will typically set thein-plane lattice constant a = 1 below; however, we willoccasionally write it out for clarity.The values of \omega \bfq and g ( \bfq ) we use in the calculation in-clude all the screening effect within the oxide substrate,but none from the FeSe film. Thus, we refer to themas the ``bare"" or ``unscreened"" quantities. In Sec. V, weshow that such a treatment is justified in calculating theelectron self-energy using the ``unscreened"" phonon prop-agator and the ``unscreened"" coupling g ( \bfq ), but it overes-timates the phonon self-energy, especially the imaginarypart (phonon linewidth) at \bfq = 0, by overlooking thestrong screening effect at \bfq = 0 from the FeSe film. Thedifference between the fully screened phonon frequency \omega \mathrm{p}\mathrm{h} (by both the substrate and the FeSe film) and par-tially screened \omega \bfq (only by the substrate itself) is small,however, so we do not distinguish them ( \omega \bfq \approx \omega \mathrm{p}\mathrm{h} ) insections II, III, and IV. Our calculation in Sec. V showsthat the difference is within 10\% for most parameters.The experimental measurements in Ref. 25 on phononfrequency in SrTiO with and without FeSe deposited Γ(a) (b)= + + + · · · (c)
FIG. 1. The Feynman diagram for the electron self-energy(a) and the phonon self-energy (b). The extra external legs(gray lines) are not part of self-energy but are attached forclarity. The lines (double-lines) with an arrow in the middlerepresent bare (dressed) electron propagators; the wiggly-lines(double-wiggly-lines) represent bare (dressed) phonon propa-gators. The gray triangle represents the vertex part. (c) Thescreened electron-phonon vertex, approximated by a seriesinvolving Coulomb interactions (dashed lines) and neglectingvertex corrections from the crossing diagrams. also support this conclusion.The electron and phonon self-energies due to e -ph in-teraction are calculated using Migdal-Eliashberg theory,where the vertex part \Gamma (i \omega n , \bfk ; i \omega \nu , \bfq ) is approximatedwith the zeroth order vertex function g ( \bfq ). Here, \omega n ( \omega \nu )is the fermionic (bosonic) Matsubara frequency. This isshown in Fig. 1. As discussed in Ref. 24, in the forwardscattering limit the vertex corrections are of order \lambda m ,and can thus be neglected in the weak coupling regime \lambda m \sim . .
25 considered here. (Here, \lambda m measuresthe Fermi surface average of the mass enhancement dueto the e -ph interaction, see Ref. 18.) Note that the ver-tex correction is independent of the adiabatic parameter \omega \mathrm{p}\mathrm{h} /E \mathrm{F} , in contrast to the standard Migdal's approx-imation for | \omega \nu | / | \bfq | \ll v \mathrm{F} . (The vertex correction isalways proportional to \lambda m for either | \omega \nu | / | \bfq | \ll v \mathrm{F} or | \omega \nu | / | \bfq | \gtrsim v \mathrm{F} , so our argument also applies for theforward-focused e -ph interaction.) There are alterna-tive treatments that do not make use of Migdal's ap-proximation in the nonadiabatic regime for momen-tum independent interaction g ( \bfk , \bfq ) = g . These ap-proaches are beyond the scope of this work, which in-stead focuses on a momentum dependent interaction.Furthermore, we calculate the dressed electron Green'sfunction (electron propagator) from the self-energy us-ing the bare phonon Green's function (phonon propaga-tor) [see Fig. 1(a)] and then insert this into the bub-ble diagram for the phonon self-energy [see Fig. 1(b)].This approach is the so-called ``unrenormalized Migdal-Eliashberg"" scheme, where the phonon self-energy isnot fed back into the electron self-energy self-consistently.As we will show in section V, this treatment is justifiedwhen one includes the Coulomb screening of the e -ph in-teraction in the problem.Adopting Nambu's 2-spinor scheme, the electron self-energy \^\Sigma ( \bfk , i \omega n ) = i \omega n [1 - Z ( \bfk , i \omega n )]\^ \tau + \chi ( \bfk , i \omega n )\^ \tau + \phi ( \bfk , i \omega n )\^ \tau and the dressed electron Green's function \^ G - ( \bfk , i \omega n ) = i \omega n \^ \tau - \xi \bfk \^ \tau - \^\Sigma ( \bfk , i \omega n ) are matrices inNambu space with \^ \tau i being the Pauli matrices; \omega n =(2 n + 1) \pi /\beta are fermionic Matsubara frequencies with \beta = 1 /T the inverse temperature ( k \mathrm{B} = 1); Z ( \bfk , i \omega n ) and \chi ( \bfk , i \omega n ) renormalize the single-particle mass and banddispersion, respectively; and \phi ( \bfk , i \omega n ) is the anomalousself-energy. The electron self-energy is self-consistentlycalculated from the one-loop diagram in Fig. 1(a) as fol-lows \^\Sigma ( \bfk , i \omega n ) = - N \beta \sum \bfq ,\nu \Bigl[ | g ( \bfq ) | D ( \bfq , i \omega \nu )\^ \tau \^ G ( \bfk - \bfq , i \omega n - i \omega \nu )\^ \tau \Bigr] , (2)where D ( \bfq , i \omega \nu ) = - \omega \bfq \omega \bfq + \omega \nu is the ``bare"" phonon prop-agator.Once we obtain the electron Green's function self-consistently, the polarization bubble in Fig. 1(b) is givenby P ( \bfq , i \omega \nu ) = 1 N \beta \sum \bfk ,n Tr \Bigl[ \^ \tau \^ G ( \bfk , i \omega n )\^ \tau \^ G ( \bfk - \bfq , i \omega n - i \omega \nu ) \Bigr] , (3)and \Pi ( \bfq , i \omega \nu ) = | g ( \bfq ) | P ( \bfq , i \omega \nu ) is the phonon self-energy and \gamma ( \bfq , \omega ) = - Im \Pi ( \bfq , i \omega \nu \rightarrow \omega + i \eta ) is thephonon linewidth, which has been analytically contin-ued to the real frequency axis. To perform the analyticcontinuation we use the spectral representation of thedressed Green's functionIm \Pi ( \bfq , \omega ) = - | g ( \bfq ) | \pi \int \infty - \infty d\omega \prime \Biggl\{ [ n \mathrm{F} ( \omega \prime - \omega ) - n \mathrm{F} ( \omega \prime )]1 N \sum \bfk Tr \Bigl[ \^ \tau \^ A ( \bfk , \omega \prime - \omega )\^ \tau \^ A ( \bfk + \bfq , \omega \prime ) \Bigr] \Biggr\} , (4)where n \mathrm{F} ( x ) = 1 / ( e \beta x +1) is the Fermi-Dirac distributionfunction and \^ A ( \bfk , \omega ) = - \pi Im \^ G ( \bfk , \omega + i\eta ) . (5)\^ G ( \bfk , \omega + i\eta ) is obtained by the same iterative analyticcontinuation method we used in Ref. 18.Finally, we find the dressed phonon propagator using D ( \bfq , \omega ) = 2 \omega \mathrm{p}\mathrm{h} \omega - \omega + 2i \gamma ( \bfq , \omega ) \omega \mathrm{p}\mathrm{h} , (6)and phonon spectral function B ( \bfq , \omega ) = - \pi Im D ( \bfq , \omega ) . (7)In the numerical calculations, we solve the electronself-energy self-consistently on a 256 \times k -grid. Theconvergence for the self-energy is reached if the differenceof the self-energies from two consecutive iterations is lessthan 10 - meV. The small imaginary part included inthe iterative analytic continuation is \eta = 3 meV. \bfI \bfI \bfI . \bfA \bfN \bfA \bfL \bfY \bfT \bfI \bfC \bfA \bfL \bfR \bfE \bfS \bfU \bfL \bfT \bfS \bfF \bfO \bfR \bfT \bfH \bfE \bfP \bfE \bfR \bfF \bfE \bfC \bfT \bfF \bfO \bfR \bfW \bfA \bfR \bfD \bfS \bfC \bfA \bfT \bfT \bfE \bfR \bfI \bfN \bfG \bfC \bfA \bfS \bfE We begin by examining the perfect forward scatter-ing limit, where several analytical results can be ob-tained. Here, we consider only the normal state in thelow-temperature limit ( T c < T \ll | \xi \bfk | ), because manyqualitative features of the phonon linewidth are alreadymanifested there.For a normal metal with a parabolic band \xi \bfk = k m - E \mathrm{F} , i.e., electron gas in three-dimensions (3D),the analytical result of Eq. (3) is the Lindhard func-tion. The corresponding result for electron gas in two-dimension (2D) is given in Refs. 34 and 35. With-out the e -ph interaction, we can apply the 2D elec-tron gas result to our single band model, due to thesmall size of the Fermi pocket from the band disper-sion \xi \bfk = - t [cos( k x a ) + cos( k y a )] - \mu \approx k m \ast - E \mathrm{F} ,where k = | \bfk | = \sqrt{} k x + k y , m \ast = t , E \mathrm{F} = k m \ast , and k \mathrm{F} = \sqrt{} \mu t . This approximate band dispersion is ex-act at the band bottom and suitable for small k . Withthis approximation, the imaginary part of the electronpolarization without e -ph interaction isIm P ( \bfq , \omega ) = - N \mathrm{F} \~ q \Bigl[ \Theta (1 - \nu - ) \sqrt{} - \nu - - \Theta (1 - \nu ) \sqrt{} - \nu \Bigr] , (8)where \~ q = | \bfq | /k \mathrm{F} , \nu \pm = \omega / (2 E \mathrm{F} \~ q ) \pm \~ q/ N \mathrm{F} = m \ast /\pi is density of states of two spins, and the step-function\Theta ( x ) = 1 for x > x ) = 0 for x < e -ph interaction, the self-energy in Eq. (2) is nonzero but diagonal in the nor-mal state. In the perfect forward scattering limit | g ( \bfq ) | = g ( N \delta \bfq , ) = \lambda m \omega ( N \delta \bfq , ), where \lambda m \equiv \langle | g ( \bfq ) | \rangle \bfq /\omega = g /\omega . The (1 , \Sigma ( \bfk , i \omega n ) = a\omega i \omega n - \xi \bfk - b\omega \mathrm{p}\mathrm{h} - \omega (1 - b )i \omega n - \xi \bfk + b\omega \mathrm{p}\mathrm{h} , (9)where a = \lambda m / tanh \beta \omega \mathrm{p}\mathrm{h} and b = tanh \beta \omega \mathrm{p}\mathrm{h} tanh \beta \xi \bfk .Using this self-energy and Dyson's equation, we find thatat low temperatures ( T \ll | \xi \bfk | and T \ll \omega \mathrm{p}\mathrm{h} ), the dressedGreen's function acquires a two-pole form G ( \bfk , i \omega n ) = A \mathrm{M} i \omega n - \xi \mathrm{M} \bfk + A \mathrm{R} i \omega n - \xi \mathrm{R} \bfk , (10)where A \mathrm{M},\mathrm{R} = ( \surd \lambda m \pm / (2 \surd \lambda m ) and \xi \mathrm{M},\mathrm{R} \bfk = \xi \bfk + sgn( \xi \bfk ) \omega \mathrm{p}\mathrm{h} (1 \mp \surd \lambda m ). Here, ``M"" and ``R""denote the main and replica band, respectively. To sim-plify the calculation, we shift the two bands by the sameenergy - sgn( \xi \bfk ) \omega \mathrm{p}\mathrm{h} (1 - \surd \lambda m ) (which is small if q [1/ a ] ω [ e V ] − I m P ( q , ω ) / N F (a) q [1/ a ] ω [ e V ] − I m P ( q , ω ) / N F (b) FIG. 2. Normalized imaginary part of the electron polariza-tion - Im P ( q, \omega ) /N \mathrm{F} without e -ph interaction (a) and withforward scattering e -ph interaction (b). \lambda m = 0 .
16 is used inpanel (b). The parabolic band approximation for FeSe/STOmodel \xi \bfk \approx k m \ast - E \mathrm{F} is assumed, so P ( q = | \bfq | , \omega ) is isotropicin momentum space. k \mathrm{F} \approx /a , v \mathrm{F} \approx . \cdot a \hbar , E \mathrm{F} \approx .
05 eV,and \omega \mathrm{p}\mathrm{h} = 0 . \lambda m \ll G ( \bfk , i \omega n ) = A \mathrm{M} i \omega n - \xi \bfk + A \mathrm{R} i \omega n - \xi \mathrm{R} \bfk , (11)where the shifted \xi \mathrm{R} \bfk = \xi \bfk + sgn( \xi \bfk )\Delta \omega and \Delta \omega = \omega \mathrm{p}\mathrm{h} \surd \lambda m . Here, A \mathrm{M} + A \mathrm{R} = 1. Physically, Eq. (11)clearly indicates that the replica band exactly follows thedispersion of the main band, and its energy offset fromthe main band is +\Delta \omega ( - \Delta \omega ) for the part of the mainband above (below) the Fermi level.Using Eq. (11), the imaginary part of the electron po-larization with the e -ph interaction in perfect forwardscattering limit can be expressed in terms of the nonin-teracting electron polarization as followsIm P ( \bfq , \omega ) = A Im P ( \bfq , \omega )+ 2 A \mathrm{M} A \mathrm{R} Im P ( \bfq , \omega - sgn( \omega )\Delta \omega ) \Theta ( | \omega | - \Delta \omega )+ A Im P ( \bfq , \omega - sgn( \omega )2\Delta \omega ) \Theta ( | \omega | - \omega ) . (12)Here, sgn( \omega ) is the sign of \omega . Equation (12) isalso a good approximation when the coupling function g ( \bfq ) \propto exp ( - | \bfq | /q ) has a sharp peak ( q \ll \pi /a ).Then, the phonon linewidth is given by \gamma ( \bfq , \omega ) = - | g ( \bfq ) | Im P ( \bfq , \omega ).In Fig. 2 we show - Im P ( \bfq , \omega ) calculated from Eq. (8)and from Eq. (12) in panel (a) and (b), respectively.Fig. 2(a) manifests the electron-hole continuum for 2Delectron gas at low temperature, while Fig. 2(b) showsmultiple scattering processes at low temperature corre-sponding to the three terms in Eq. (12): one withinthe main band for | \omega | > | \omega | > \Delta \omega = \omega \mathrm{p}\mathrm{h} \surd \lambda m , and onewithin the replica band for | \omega | > \omega , in a descendingorder of weights ( A , 2 A \mathrm{M} A \mathrm{R} , and A ). As shown inFig. 2(b), at the fixed frequency \omega = \omega \mathrm{p}\mathrm{h} , the magnitudeof the imaginary part of the electron polarization has asharp upturn at a finite momentum, leading to a peakthat slowly decreases at larger momentum. This quali-tative feature persists in the full numerical result in thenext section. Note that since the coupling constant is adelta function, the phonon linewidth \gamma ( \bfq , \omega ) is zero at all \bfq values despite the fact that the polarization P ( \bfq , \omega ) isnonzero. \bfI \bfV . \bfN \bfU \bfM \bfE \bfR \bfI \bfC \bfA \bfL \bfR \bfE \bfS \bfU \bfL \bfT \bfS We now turn to the polarization and phonon linewidthfor the case of an e -ph interaction with a small butnonzero width in momentum space. Figure 3 shows theimaginary part of the electron polarization - Im P ( \bfq , \omega )and the phonon linewidth \gamma \bfq = - Im \Pi ( \bfq , \omega \mathrm{p}\mathrm{h} ) for vari-ous temperatures. Here, we have parameterized the total e -ph coupling using the double Fermi-surface averageddefinition \lambda = 2 \omega \mathrm{p}\mathrm{h} \= N \mathrm{F} N \sum \bfk , \bfk \prime | g ( \bfk - \bfk \prime ) | \delta ( \xi \bfk ) \delta ( \xi \bfk \prime ) , (13)where \= N \mathrm{F} is the density of states per spinand N - \sum \bfk , \bfk \prime \delta ( \xi \bfk ) \delta ( \xi \bfk \prime ) = \= N . We haveused this definition because the \bfq -averaged \lambda m = \langle | g ( \bfq ) | \rangle \bfq /\omega = g /\omega equals the mass en-hancement factor - Re \partial \Sigma ( \omega ) \partial \omega \bigm| \bigm| \bigm| \omega =0 only in the limit ofperfect forward scattering, while \lambda approximates themass enhancement factor when the e -ph interaction ismore uniform. The latter case occurs for the largervalues of q used in Fig. 3. In addition, \lambda as definedin Eq. (13) does not depend on temperature where as \lambda m does. Empirically, we find \lambda m \propto ( q a ) \lambda (see Ref. 24for the proportionality constant), which can be usedto approximately convert between the two definitions.In Fig. 3 we have set \lambda = 0 .
8, which is equivalent to \lambda m = 0 .
16 for q = 0 . /a and within the suitable rangeof values that simultaneously fit both high T c value andthe measured spectral weight of the replica bands. At low temperature and \omega = \omega \mathrm{p}\mathrm{h} , the imaginary partof the polarization in Fig. 3(a) has a peak appearing at | \bfq | = \sqrt{} m \ast \omega \mathrm{p}\mathrm{h} , which is a feature of the electron-holecontinuum; with increasing temperature, the - Im P ( \bfq =0 , \omega \mathrm{p}\mathrm{h} ) increases, and the rate of increase is faster for q = 0.1 [1/ a ] q = 0.5 [1/ a ] q = 1 [1/ a ] q = ( q x , 0) [1/ a ] − I m P ( q , ω ph ) [ / e V ] T [K]10 50 100 150 200 250 300 q = 2 [1/ a ] (a) q = 0.1 [1/ a ] q = 0.5 [1/ a ] q = 1 [1/ a ] q = ( q x , 0) [1/ a ] − I m Π ( q , ω ph ) [ e V ] T [K]10 50 100 150 200 250 300 q = 2 [1/ a ] (b) FIG. 3. (a) Momentum and temperature dependence of theimaginary part of the electron polarization - Im P ( q, \omega ) fora fixed frequency \omega = \omega \mathrm{p}\mathrm{h} and various momentum width pa-rameters q = 0 .
1, 0 .
5, 1, and 2 in the e -ph coupling function g ( \bfq ) \propto exp( - | \bfq | /q ). The double Fermi-surface averaged cou-pling constant (defined in the text) is fixed at \lambda = 0 .
8. Thecolors (gray scales) of lines represent low (blue) to high (red)temperatures. (b) Similar to (a) but for phonon linewidth \gamma ( \bfq , \omega ) = - Im \Pi ( \bfq , \omega ) = - | g ( \bfq ) | Im P ( \bfq , \omega ). smaller values of q . The phonon linewidth, shown inFig. 3(b), strongly peaks at \bfq = 0 for q = 0 . g ( \bfq ) strongly suppressesthe peak in the polarizability appearing at the finite | \bfq | .As the value of q increases, however, the width of g ( \bfq )begins to overlap with the peak in the polarization, and acorresponding peak in the linewidth recovers at nonzero \bfq . In this case, both the temperature and the width ofthe coupling function g ( \bfq ) dictate the full \bfq dependenceof the phonon linewidth. Thus, due to its sensitivityto these parameters, the momentum dependence of thephonon linewidth can be used to determine not only theoverall strength of the e -ph interaction but also the widthof the coupling function.To reproduce the replica bands observed in the ARPESexperiments, the width of the e -ph coupling must be nar-row in momentum space with q \approx . /a --0 . /a . Basedon this observation, and the results shown in Fig. 3, onemight expect that the phonon linewidth in the vicinityof \bfq = 0 should be very large. In turn, the real part ofthe phonon self-energy will also develop significant Kohnanomaly, leading to an instability of the lattice. It turnsout that the Coulomb interaction will prevent this fromoccurring, as the divergence in the Coulomb interactionat \bfq = 0 effectively blocks the long-wavelength instabil-ity. We will discuss this issue in the next section. \bfV . \bfU \bfN \bfD \bfR \bfE \bfS \bfS \bfI \bfN \bfG \bfO \bfF \bfT \bfH \bfE \bfP \bfH \bfO \bfN \bfO \bfN \bfL \bfI \bfN \bfE \bfW \bfI \bfD \bfT \bfH \bfD \bfU \bfE \bfT \bfO \bfC \bfO \bfU \bfL \bfO \bfM \bfB \bfI \bfC \bfS \bfC \bfR \bfE \bfE \bfN \bfI \bfN \bfG In this section we examine the effects of Coulombscreening by the FeSe electrons on the e -ph vertex andthe phonon linewidth. Fig. 1(c) shows the diagramaticexpansion of the screened e -ph vertex evaluated at thelevel of the random phase approximation. The screenedvertex is\= g ( \bfq , i \omega \nu ) = g ( \bfq ) + g ( \bfq ) [ - V C ( \bfq ) \chi ( \bfq , i \omega \nu )]+ g ( \bfq ) [ - V C ( \bfq ) \chi ( \bfq , i \omega \nu )] + . . . = g ( \bfq )1 + V C ( \bfq ) \chi ( \bfq , i \omega \nu ) , (14)where \chi ( \bfq , i \omega \nu ) = - P ( \bfq , i \omega \nu ) is the charge susceptibil-ity and V C ( \bfq ) is the Fourier transform of the Coulombpotential. In the contiuum limit, V C ( \bfq ) = \pi e | \bfq | in threedimensions (3D) and V C ( \bfq ) = \pi e | \bfq | in two dimensions(2D). The corresponding phonon self-energy is obtainedby replacing the vertex function with the screened vertexwith \Pi ( \bfq , i \omega \nu ) = g ( \bfq ) [\= g ( \bfq , i \omega \nu )] \ast [ - \chi ( \bfq , i \omega \nu )]= - | g ( \bfq ) | \chi ( \bfq , i \omega \nu )1 + V C ( \bfq ) \chi ( \bfq , i \omega \nu ) , (15)where we have assumed V C ( \bfq ) \chi ( \bfq , i \omega \nu ) is real.Here, we are interested in the case of an FeSe mono-layer located a distance h above the oxide substrate. Weplace the FeSe electrons at z = 0 and the ions in thetermination layer of the substrate at z = - h . For thisgeometry, we introduce an anisotropic Coulomb poten-tial V C ( q, q z ) = 4 \pi e \epsilon a q + \epsilon c q z , (16)where q = \sqrt{} q x + q y is the momentum transfer a planeparallel to the FeSe monolayer, and \epsilon a and \epsilon c are the zero-frequency dielectric constants parallel and perpendicularto the plane. By inverse Fourier transform, the real spaceformula is V C ( x, y, z ) = e \surd \epsilon a \epsilon c \surd r + \= z , (17)where r = x + y and \= z = ( \epsilon a /\epsilon c ) z . After performingthe 2D fourier transforming for the in-plane coordinateswe arrive at V C ( q, z ) = 2 \pi e \surd \epsilon a \epsilon c e - q | \= z | q . (18) q = 0.1/ a λ q = 0.4/ a q = 1.2/ a M Γ X M M Γ X M γ ( q , ω ph ) [ m e V ] q = 2/ a (a) increasing λ q = 0.1/ a λ q = 0.4/ a q = 1.2/ a M Γ X M M Γ X M γ ( q , ω ph ) [ m e V ] q = 2/ a (b) increasing λ FIG. 4. The phonon linewidth \gamma ( \bfq , \omega = \omega \mathrm{p}\mathrm{h} ) along a highsymmetry path M -\Gamma - X - M at T = 30 K. Results are shownfor (a) \epsilon a = 1 = \epsilon c and (b) \epsilon a = 25, \epsilon c = 1. The line color(gray scale) encodes the different values of \lambda = 0 .
2, 0 .
4, 0 . .
8, as indicated by the color bar.
To compute the screened e -ph interaction, we must usethe interaction at z = 0 for the Coulomb potential sincethe particle-hole pairs are created in the FeSe layer.Putting this all together, the phonon linewidth is givenby \gamma ( \bfq , \omega ) = \omega \bfq \omega \mathrm{p}\mathrm{h} Im | g ( \bfq ) | \chi ( \bfq , \omega )1 + V C ( q, z = 0) \chi ( \bfq , \omega ) /a (19)where we define the ``unscreened"" phonon energy as \omega \bfq = \sqrt{} \omega + [Re \Pi ( \bfq , \omega )] - Re \Pi ( \bfq , \omega ).We evaluated Eq. (19) for several values of q and \lambda ,and the results are shown in Fig. 4. Since the exactvalues of the dielectric constants are not known for theFeSe interface systems, we show results for \epsilon a = \epsilon c = 1 inFig. 4(a) and \epsilon a = 25, \epsilon c = 1 in Fig. 4(b). Note that thelatter values are close to the estimates obtained by Kuli\'cand Dolgov (Ref. 21) in the limit of perfect forward scat-tering. In both cases, we find that the phonon linewidthis dramatically suppressed once Coulomb screening isincluded; however, as the values of \epsilon a and \epsilon c are in-creased, the magnitude of the linewidth increases. Theseresults indicate that the long-range Coulomb interactionprevents the formation of a competing charge orderingat long-wavelengths, which is consistent with the notionthat extend Coulomb interactions can suppress insulatingbehavior. Our results also show that this effect will besomewhat senstive to the dielectric properties of the in-terface, which may offer a means to tune these properties.Finally, the undressing of the phonon linewidth observedhere also provides a rationale for adopting an unrenor-malized Migdal-Eliashberg scheme, where the phononself-energy is not fed back into the electron self-energyin a self-consistent manner. In this case, the calculatedphonon self-energy is small, justifying the use of the barephonon propagator in the electron self-energy diagrams.Comparing our results to the recent RHEELS measure-ments by Zhang et al. , we find that once the Coulombscreening is included, the computed linewidths are muchsmaller than those inferred experimentally. Moreover, inthe experimental data, the linewidth is finite at \Gamma pointand maximal around X point. Our calculated linewidthis exactly zero at \Gamma point because the screening from theCoulomb potential diverges at \bfq = 0. However, we havenot considered any impurity potential in our calculation,or other sources of broadening in the electron Green'sfunction, and, subsequently, the phonon linewidth oncethe charge susceptibility \chi is computed. Regardless, X point is not the maximal point for the linewidth in anyof our calculation results. This discrepancy could alsobe due to the limitation of our single band model. Thereal system is multiband in nature and also shows strongmagnetic fluctuations. \bfV \bfI . \bfS \bfU \bfM \bfM \bfA \bfR \bfY \bfA \bfN \bfD \bfC \bfO \bfN \bfC \bfL \bfU \bfS \bfI \bfO \bfN \bfS In this paper, we have calculated the phonon linewidth,i.e., the imaginary part of the phonon self-energy in anunrenormalized Migdal-Eliashberg scheme in the weak tointermediate coupling regime for strong forward scatter-ing e -ph interaction. Such an e -ph interaction dresses theelectron propagator by simply creating the replica bandsand shuffles the electron-hole continuum of 2D electrongas into three similar parts with descending weights be-ginning at | \omega | > | \omega | > \Delta \omega , and | \omega | > \omega . If we donot include Coulomb screening, the phonon linewidth isa simple product of coupling function | g ( \bfq ) | with a for-ward scattering peak around \bfq = 0 and the electron po-larization with a very similar momentum structure of theelectron-hole continuum of 2D electron gas. Dependingon the peak width q of the e -ph coupling constant g ( \bfq )and the peak of electron polarization at | \bfq | = \sqrt{} m \ast \omega \mathrm{p}\mathrm{h} , we find the linewidth \gamma ( \bfq , \omega \mathrm{p}\mathrm{h} ) has a maximum value at \bfq = 0 or | \bfq | \approx \sqrt{} m \ast \omega \mathrm{p}\mathrm{h} at low temperature, and thelinewidth is broad at \bfq = 0. Even if the latter happens,since the linewidth for small | \bfq | tends to increase withtemperature, the maximum may shift back to \bfq = 0at high temperature. The momentum resolved phononspectral function at \omega \approx \omega \mathrm{p}\mathrm{h} can be understood in thesame picture.The broad linewidths at \bfq = 0 would normally in-dicate an instability to a charge-ordered phase at longwavelengths. However, once the long-range Coulomb in-teraction screens the e -ph interaction we find that thephonons are undressed. Here, the anomalous broadeningat \bfq = 0 is suppressed by the divergence in the Coulombinteraction at \bfq = 0 while the total phonon linewidthis reduced throughout the Brillioun zone. In this case,a small peak remains at nonzero momentum transfers;however, the magnitude of this peak is much smaller thanthe linewidths measured by HREELS. Our results sug-gest that the broadening of the SrTiO phonons (with amaximum at X point) observed by Zhang et al. are notdue to the forward-focused e -ph coupling inferred fromthe ARPES measurements and are likely from some othersource. To resolve the forward-focused e -ph interaction,the HREELS experiments should focus on smaller valuesof \bfq , which will be challenging given the large backgroundsignal at \bfq = 0. \bfA \bfC \bfK \bfN \bfO \bfW \bfL \bfE \bfD \bfG \bfM \bfE \bfN \bfT \bfS We thank T. Berlijn, T. P. Devereaux, M. L. Kuli\'c,E. W. Plummer, and D. J. Scalapino for useful discus-sions. L. R. acknowledges funding from the Dutch Sci-ence Foundation (NWO) via a Rubicon Fellowship andby the National Science Foundation under Grant No.PHY11-25915 and Grant No. NSF-KITP-17-019. S. J.is funded by the University of Tennessee's Office of Re-search and Engagement's Organized Research Unit pro-gram. Y. W. and E. D. are supported by the U.S. De-partment of Energy, Office of Basic Energy Sciences, Ma-terials Sciences and Engineering Division. This researchused computational resources supported by the Univer-sity of Tennessee and Oak Ridge National Laboratory'sJoint Institute for Computational Sciences. F.-C. Hsu, J.-Y. Luo, K.-W. Yeh, T.-K. Chen, T.-W.Huang, P. M. Wu, Y.-C. Lee, Y.-L. Huang, Y.-Y. Chu,D.-C. Yan, and M.-K. Wu, Proc. Natl. Acad. Sci. \bfone \bfzero \bffive ,14262 (2008). Q.-Y. Wang, Z. Li, W.-H. Zhang, Z.-C. Zhang, J.-S. Zhang,W. Li, H. Ding, Y.-B. Ou, P. Deng, K. Chang, J. Wen,C.-L. Song, K. He, J.-F. Jia, S.-H. Ji, Y.-Y. Wang, L.-L.Wang, X. Chen, X.-C. Ma, and Q.-K. Xue, Chin. Phys.Lett. \bftwo \bfnine , 037402 (2012). J.-F. Ge, Z.-L. Liu, C. Liu, C.-L. Gao, D. Qian, Q.-K. Xue, Y. Liu, and J.-F. Jia, Nat. Mater. \bfone \bffour , 285 (2015). J. J. Lee, F. T. Schmitt, R. G. Moore, S. Johnston, Y.-T.Cui, W. Li, M. Yi, Z. K. Liu, M. Hashimoto, Y. Zhang,D. H. Lu, T. P. Devereaux, D.-H. Lee, and Z.-X. Shen,Nature \bffive \bfone \bffive , 245 (2014). R. Peng, H. C. Xu, S. Y. Tan, H. Y. Cao, M. Xia, X. P.Shen, Z. C. Huang, C. Wen, Q. Song, T. Zhang, B. Xie,X. Gong, and D. Feng, Nat. Commun. \bffive , 5044 (2014). P. Zhang, X.-L. Peng, T. Qian, P. Richard, X. Shi, J.-Z.Ma, B. B. Fu, Y.-L. Guo, Z. Q. Han, S. C. Wang, L. L.
Wang, Q.-K. Xue, J. P. Hu, Y.-J. Sun, and H. Ding, Phys.Rev. B \bfnine \bffour , 104510 (2016). G. Zhou, D. Zhang, C. Liu, C. Tang, X. Wang, Z. Li,C. Song, S. Ji, K. He, L. Wang, X. Ma, and Q.-K. Xue,Appl. Phys. Lett. \bfone \bfzero \bfeight (2016), 10.1063/1.4950964. C.-F. Zhang, Z.-K. Liu, Z.-Y. Chen, Y.-W. Xie, R.-H. He,S.-J. Tang, J.-F. He, W. Li, T. Jia, S. N. Rebec, E. Y. Ma,H. Yan, M. Hashimoto, D.-H. Lu, S.-K. Mo, Y. Hikita,R. G. Moore, H. Y. Hwang, D.-H. Lee, and Z.-X. Shen,arXiv:1608.01740 (2016). H. Ding, Y.-F. Lv, K. Zhao, W.-L. Wang, L. Wang, C.-L. Song, X. Chen, X.-C. Ma, and Q.-K. Xue, Phys. Rev.Lett. \bfone \bfone \bfseven , 067001 (2016). S. N. Rebec, T. Jia, C.-F. Zhang, M. Hashimoto, D.-H. Lu,R. G. Moore, and Z.-X. Shen, arXiv:1606.09358 (2016). Y. Bang, New J. Phys. \bfone \bfsix , 023029 (2014). X. Chen, S. Maiti, A. Linscheid, and P. J. Hirschfeld,Phys. Rev. B \bfnine \bftwo , 224514 (2015). A. Linscheid, S. Maiti, Y. Wang, S. Johnston, and P. J.Hirschfeld, Phys. Rev. Lett. \bfone \bfone \bfseven , 077003 (2016). V. Mishra, D. J. Scalapino, and T. A. Maier, Sci. Rep. \bfsix ,32078 (2016). J. Kang and R. M. Fernandes, Phys. Rev. Lett. \bfone \bfone \bfseven ,217003 (2016). S. Coh, M. L. Cohen, and S. G. Louie, New J. Phys. \bfone \bfseven ,073027 (2015). D.-H. Lee, Chin. Phys. B. \bftwo \bffour , 117405 (2015). L. Rademaker, Y. Wang, T. Berlijn, and S. Johnston, NewJ. Phys. \bfone \bfeight , 022001 (2016). B. Li, Z. W. Xing, G. Q. Huang, and D. Y. Xing, J. Appl.Phys. \bfone \bfone \bffive , 193907 (2014). Y. Wang, A. Linscheid, T. Berlijn, and S. Johnston, Phys.Rev. B \bfnine \bfthree , 134513 (2016). M. L. Kuli\'c and O. V. Dolgov, New J. Phys. \bfone \bfnine , 013020 (2017). Y. Zhou and A. J. Millis, Phys. Rev. B \bfnine \bfthree , 224506 (2016). A. Aperis and P. M. Oppeneer, arXiv:1701.08136 (2017). Y. Wang, K. Nakatsukasa, L. Rademaker, T. Berlijn, andS. Johnston, Supercond. Sci. Technol. \bftwo \bfnine , 054009 (2016). S. Zhang, J. Guan, X. Jia, B. Liu, W. Wang, F. Li,L. Wang, X. Ma, Q. Xue, J. Zhang, E. W. Plummer,X. Zhu, and J. Guo, Phys. Rev. B \bfnine \bffour , 081116 (2016). D. Liu, W. Zhang, D. Mou, J. He, Y.-B. Ou, Q.-Y. Wang,Z. Li, L. Wang, L. Zhao, S. He, Y. Peng, X. Liu, C. Chen,L. Yu, G. Liu, X. Dong, J. Zhang, C. Chen, Z. Xu, J. Hu,X. Chen, X. Ma, Q. Xue, and X. Zhou, Nat. Commun. \bfthree ,931 (2012). S. He, J. He, W. Zhang, L. Zhao, D. Liu, X. Liu, D. Mou,Y.-B. Ou, Q.-Y. Wang, Z. Li, et al. , Nat. Mater. \bfone \bftwo , 605(2013). L. P. Gor'kov, Phys. Rev. B \bfnine \bfthree , 054517 (2016). A. V. Chubukov, I. Eremin, and D. V. Efremov, Phys.Rev. B \bfnine \bfthree , 174516 (2016). B. Rosenstein, B. Y. Shapiro, I. Shapiro, and D. Li, Phys.Rev. B \bfnine \bffour , 024505 (2016). F. Marsiglio, Phys. Rev. B \bffour \bftwo , 2416 (1990). F. Marsiglio, M. Schossmann, and J. P. Carbotte, Phys.Rev. B \bfthree \bfseven , 4965 (1988). A. L. Fetter and J. D. Walecka,
Quantum theory of many-particle systems (Dover, 2003). F. Stern, Phys. Rev. Lett. \bfone \bfeight , 546 (1967). G. Giuliani and G. Vignale,
Quantum Theory of theElectron Liquid (Cambridge University Press, Cambridge,2005). D. N. Aristov and G. Khaliullin, Phys. Rev. B \bfseven \bffour , 045124(2006). D. Poilblanc, S. Yunoki, S. Maekawa, and E. Dagotto,Phys. Rev. B \bffive \bfsix\bffive \bfsix