Transient Hot Electron Dynamics in Single-Layer TaS 2
Federico Andreatta, Habib Rostami, Antonija Grubišić Čabo, Marco Bianchi, Charlotte E. Sanders, Deepnarayan Biswas, Cephise Cacho, Alfred J. H. Jones, Richard T. Chapman, Emma Springate, Phil D. C. King, Jill A. Miwa, Alexander Balatsky, Søren Ulstrup, Philip Hofmann
TTransient Hot Electron Dynamics in Single-Layer TaS Federico Andreatta, ∗ Habib Rostami, ∗ Antonija Grubiˇsi´c ˇCabo, Marco Bianchi, Charlotte E.Sanders, Deepnarayan Biswas, Cephise Cacho, Alfred J. H. Jones, Richard T. Chapman, EmmaSpringate, Phil D. C. King, Jill A. Miwa, Alexander Balatsky,
2, 5
Søren Ulstrup, and Philip Hofmann † Department of Physics and Astronomy, Interdisciplinary Nanoscience Center, Aarhus University, 8000 Aarhus C, Denmark Nordita, 106 91 Stockholm, Sweden Central Laser Facility, STFC Rutherford Appleton Laboratory, Harwell, United Kingdom SUPA, School of Physics and Astronomy, University of St. Andrews, St. Andrews, United Kingdom Institute for Materials Science, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA (Dated: January 24, 2019)Using time- and angle-resolved photoemission spectroscopy, we study the response of metallic sin-gle layer TaS in the 1H structural modification to the generation of excited carriers by a femtosecondlaser pulse. A complex interplay of band structure modifications and electronic temperature increaseis observed and analyzed by direct fits of model spectral functions to the two-dimensional (energyand k -dependent) photoemission data. Upon excitation, the partially occupied valence band is foundto shift to higher binding energies by up to 150 meV, accompanied by electronic temperatures ex-ceeding 3000 K. These observations are explained by a combination of temperature-induced shiftsof the chemical potential, as well as temperature-induced changes in static screening. Both contri-butions are evaluated in a semi-empirical tight-binding model. The shift resulting from a change inthe chemical potential is found to be dominant. INTRODUCTION
Two-dimensional (2D) electron systems can be dra-matically altered and driven into a number of distinctphases by the application of external fields. A prime ex-ample of this is the ability to change the electron-electroninteraction by electric field control of the charge carrierdensity in 2D electron gasses (2DEGs) confined in highelectron mobility transistors [1, 2]. Tuning such systemsinto extreme conditions can create an electronic instabil-ity such as a metal-insulator transition [3] or lead to theemergence of superconductivity [4]. Even when merelydoped by using different substrates or adsorbates, Fermilevel shifts on the order of electron volts can be induced insemimetallic 2D materials such as graphene [5–8], provid-ing access to the dependence of the electronic self-energyon the electron/hole density over a wide energy range[9]. In principle, the electronic self-energy also dependson the temperature but such effects are usually negli-gible because k B T at reachable temperatures tends tobe much smaller than the electronic bandwidth [10, 11].The emergence of interesting physics therefore requiresnarrow electronic bandwidths or high electronic tempera-tures. These conditions are ideally reached in 2D systemswhere non-trivial temperature effects are expected to in-fluence density-dependent many-body effects [12, 13].By employing an intense optical excitation in a pump-probe scheme such as time- and angle-resolved photoe-mission (TR-ARPES) to drive (semi) metallic 2D ma-terials out of equilibrium, an extremely wide range oftransient electronic temperatures can be accessed [14].This approach is effective for studying instabilities in theelectronic system under extreme conditions and on ultra-fast time scales where transient charge order effects and metal-insulator transitions may completely alter the elec-tronic spectrum around the Fermi energy as observed inmetallic transition metal dichalcogenides (TMDCs) [15–19].Here we probe the electronic response of a metallic 2DTMDC, single layer (SL) TaS of the 1H polymorph, tothe excitation of electrons by a femtosecond laser pulsewith a photon energy of 2.05 eV using TR-ARPES. Weobserve that this leads to a strongly excited thermal dis-tribution of the electrons within the time resolution ofour experiment, with electronic temperatures exceeding3000 K. Interestingly, the elevated electronic tempera-ture is accompanied by a binding energy shift of theband structure of up to ≈
100 meV. Such a change ofthe electronic structure is not necessarily expected fora 2D metal. While dynamic band structure renormal-ization is not uncommon in 2D semiconductors due tothe strongly enhanced screening by degenerate transientdoping [20–22], and complex band shifts have even beenobserved for bulk insulators [23], such effects are absentin semimetallic systems such as graphene [14] or bilayergraphene [24].This paper is structured as follows: this Introduc-tion section is followed by an Experimental section thatprovides the details of the sample preparation, staticARPES, and TR-ARPES. The Results and Discussionsection is divided into five subsections that give (i) apresentation of the experimental results, (ii) a descrip-tion of an approach to fit the full (
E, k )-dependence ofthe measured photoemission intensity, (iii) a presentationof the band shifts and electronic temperatures resultingfrom our analysis, (iv) a theoretical description of theexpected band shifts in the single particle picture, and(v) a section accounting for the effect of temperature- a r X i v : . [ c ond - m a t . s t r- e l ] J a n dependent screening. Finally, the main results and theirimplications are summarized in a Conclusions section. EXPERIMENTAL
SL TaS was grown on bilayer graphene by ultra-highvacuum (UHV) van der Waals epitaxy, in a manner sim-ilar to that used previously for the growth of SL MoS [25]. Tantalum from an e-beam evaporator was depositedon a graphene bilayer on top of a buffer layer on theSi-face of a 6H-SiC(0001) substrate (TanKeBlue Semi-conductor Co.) [26]. The Ta deposition took place in aH S atmosphere of ∼ − mbar for two minutes. Subse-quently, the sample was annealed for 20 minutes at 590 Kin the same background pressure of H S. Repetition ofthis procedure allowed for an increase in coverage. Forthe sample used in this experiment, two growth cycleswere performed, resulting in a coverage on the order ofone monolayer, as judged by the reduced photoemissionintensity from the graphene π -band. Due to the weakinteraction between the SL TaS and the underlying bi-layer graphene, TaS domains are found to be randomlyoriented with respect to the substrate; this is clearly ev-idenced by both low energy electron diffraction (LEED)and scanning tunnelling microscopy (STM). Indeed, theinteraction is so weak that individual domains can bemoved on the surface using the tip of the STM.The growth of SL TaS on bilayer graphene and theresulting electronic structure were monitored using corelevel spectroscopy [27] and ARPES at the SGM3 beam-line at the ASTRID2 synchrotron light source using aphoton energy of 25 eV at a sample temperature of ≈
44 K[28]. The energy and angular resolution were set to25 meV and 0.2 ◦ , respectively. Since the typical size ofthe SL TaS islands (less than 10 nm) is much smallerthan the spot size of the synchrotron beam (100 µ m),the random orientation of the islands results in the ob-servation of an azimuthally averaged band structure, incontrast to the well-defined Dirac cone of the underlyingbilayer graphene [27].TR-ARPES experiments were performed at theArtemis facility, Rutherford Appleton Laboratory [29].The sample was transferred from the growth chamber atASTRID2 to the TR-ARPES facility under UHV condi-tions, which was necessary because the chemical sensitiv-ity of TaS causes degradation of the samples if they areexposed to air. A 1 kHz Ti:sapphire amplified laser sys-tem with a fundamental wavelength of 785 nm was usedto generate p -polarized high harmonic probe pulses withan energy of 25 eV for photoemission and s -polarizedpump pulses with a photon energy of 2.05 eV for opticalexcitation of our sample. The angular and time resolu-tion were 0.3 ◦ and 40 fs, respectively, while the energyresolution varied between 300 and 800 meV depending onthe beamline and detector settings and the fluence of the optical pulses. Different fluences and sample tempera-tures were investigated. The spot size of the laser beamsare also of the order of 100 µ m, leading to the same av-eraging of the electronic structure from azimuthally dis-ordered SL TaS areas as for the experiments using thespot coming from the synchrotron light source. RESULTS AND DISCUSSION(i) Time-resolved ARPES on TaS The electronic structure of undoped SL TaS is char-acterized by a half-filled band with a Fermi surface con-sisting of hole pockets around the ¯Γ and ¯ K points of the2D Brillouin zone (BZ) [30]. A TR-ARPES spectrumtaken near ¯Γ without optical pumping is shown in Fig.1(a). It shows a dispersive feature with a highest bindingenergy of ≈
400 meV at ≈ − . It crosses the Fermilevel at ≈ − and is unoccupied at ¯Γ. In view ofthe azimuthal disorder between different domains of SLTaS on the sample, this observed dispersion does notcorrespond to any particular high-symmetry direction inthe calculated band structure of Fig. 1(b), but has to beinterpreted as an average over all possible orientations,roughly corresponding to an average over the ¯Γ − ¯ M and¯Γ − ¯ K directions marked in Fig. 1(b) [30]. The maindifferences between these directions are the higher maxi-mum binding energy reached along ¯Γ − ¯ M and the strongspin-orbit splitting along ¯Γ − ¯ K . However, for small | k | ,the dispersion in the two directions is sufficiently simi-lar for the average to still show the hole pocket charac-ter. The main effect of integrating over all directions isa broadening of the features at higher binding energiesin the data shown. The spin-orbit splitting along the¯Γ − ¯ K direction has no direct consequence for the ob-served band, since the upper spin-split branch is abovethe Fermi energy.Figure 1(c) shows the result of optically pumping thesystem with a photon energy of 2.05 eV at a fluence of F =7.8 mJ/cm , displaying TR-ARPES data collectedat a time delay of 40 fs between the pump and the probepulse, corresponding to the peak excitation of the system.Pumping leads to drastic changes in the spectrum: Theobserved dispersion now extends well above the Fermienergy, indicating the presence of hot electrons. This isclearly seen when considering the difference between theexcited spectrum and the equilibrium spectrum in Fig.1(d). Excited carriers are expected to be generated fromdirect optical transitions involving occupied valence bandand unoccupied conduction band states, as indicated bythe arrows in Fig. 1(b). The presence of a continuousdistribution of hot electrons is then merely indicative ofa very fast thermalization of these excited carriers thattakes place at a timescale lower than our temporal reso-lution. t < 0 E F t = 40 fs lowhigh neg.pos. t = 40 fs (a)(b) (c)(d) 0.80.40.0 F = 7.8 mJ/cm /(arb. u.) -2 M Γ K E b i n ( e V ) k (Å -1 ) k (Å -1 ) /(arb. u.) E b i n ( e V ) FIG. 1. (Color online) TR-ARPES measurement of the SLTaS dispersion around E F : (a) Left: Measured spectrumbefore optical excitation ( t < k value ofthe band’s highest binding energy as given in the text. Thedashed red line is an estimate of the peak position. (b) Calcu-lated dispersion from Ref. 30 with examples of possible directelectron (filled circles) and hole (open circles) excitation pro-cesses (arrows). The region enclosed by a green square marksthe ( E, k )-space probed in the TR-ARPES experiment. (c)TR-ARPES data as in (a) but at the peak of optical excita-tion ( t = 40 fs). Right: Energy distribution curve taken as in(a). (d) Difference spectrum: intensity difference obtained bysubtracting the intensity for t < t = 40 fsin (c). A much more surprising result of the optical excitationis that the entire dispersion is shifted to higher bind-ing energies by > ∼
100 meV, as seen clearly by the shiftbetween the maxima of the energy distribution curves(EDCs) in Fig. 1(a) and (c). Such shifts are not neces-sarily expected for metallic systems. Indeed, for a simplefree-electron like 2D system, one might not expect anyshift at all because of the energy-independent density ofstates. (ii) Fit to simulated model spectral function
A quantitative comparison of the observed effects tocalculations requires an accurate determination of theband structure changes and the electronic temperatureas a function of pump fluence and time delay. The com-plexity of the situation and the many unknown param- eters render the conventional approach of fitting energyor momentum distribution curves by simple models im-practical. Indeed, extracting the electronic temperaturefrom such fits is already problematic even for very simplesituations [31–33]. Instead, we introduce an approach inwhich the energy and k − dependent photoemission inten-sity, such as in Fig. 1(a), is fitted to a model based on aresolution-broadened spectral function that can, in prin-ciple, include the single particle dispersion, many-bodyeffects and the electronic temperature.The photoemission intensity measured in an idealARPES experiment is given by I ( E, k ) ∝ |M k if | A ( E, k ) f ( E, T ) , (1)where A ( E, k ) is the hole spectral function, f ( E, T ) isthe Fermi-Dirac distribution and M k if is the energy- and k − dependent matrix element for the transition from theinitial state i to the final state f . If, as in the present case,the data are only collected for a small range of energy and k and for a fixed photon energy and polarization, M k if isexpected to vary only weakly. In our experiment, the fi-nite energy and k − resolution of the setup must be takeninto account, such that the actual measured intensity ismodelled by a convolution of I ( E, k ) with the appropri-ate resolution functions G (∆ E ) and G (∆ k ), assumed tobe Gaussian. The measured intensity thus needs to befitted to I conv ( E, k ) = I ( E, k ) ∗ G (∆ E ) ∗ G (∆ k ).The azimuthally averaged photoemission intensityfrom SL TaS is phenomenologically modelled as I TaS ( E, k ) = ( O + P E + Q k ) × π − (cid:0) α + βE + γE (cid:1) ( E − ( ak + bk + c )) + ( α + βE + γE ) × ( e ( E − E (cid:48) F ) /k B T e + 1) − , (2)where E (cid:48) F is the experimentally determined Fermi en-ergy in the spectrum and the three parameters O , P , Q in the first term are used to match the calculated pho-toemission intensity to the experimental results, allow-ing for the possibility of a linear dependence on E and k that could arise from, e.g. , small matrix element varia-tions. The second term represents the spectral functionin which the single particle dispersion is approximatedby a parabola as ak + bk + c , with its minimum po-sition constrained to k min = b/ a = 0 .
81 ˚A − , as de-termined from a high-resolution ARPES spectrum. Theelectronic self-energy appearing in the spectral functionis a complex quantity with the real part re-normalizingthe dispersion and the imaginary part Γ broadening thefeatures. Here we assume that the real part is zero andthat Γ = α + βE + γE . This will always result in anincreased broadening at higher energies that accounts forthe azimuthal averaging over the somewhat anisotropicband structure. Care should thus be taken, to assignphysical significance to Γ. In the third term, the pop- lowhigh (a) (b) k F k min i n t en s i t y ( a r b . u . ) EDC at k F EDC at k min BG(c) (d)0.90.60.30.0 E b i n ( e V ) E bin (eV) E bin (eV) k (Å -1 ) k (Å -1 ) k F k min Data Fit
FIG. 2. (Color online) (a) Static ARPES data of the SL TaS parabolic state with the band minimum located at k min andthe Fermi level crossing at k F . (b) Modelled intensity overthe measured region of ( E, k )-space shown in (a). (c)-(d)Example EDCs of the measured data and intensity fit takenalong the dashed vertical lines shown in (a)-(b) at (d) k F and(e) k min , respectively. The background intensity in the fit isshown as a light gray line marked ”BG”. ulation of the states is dictated by the Fermi-Dirac dis-tribution with T e referring to the electronic temperaturein the SL TaS . In addition to the description of the SLTaS spectral function, it is necessary to account for thebackground intensity which is described in further detailin the supporting information [27].In order to determine the equilibrium dispersion pa-rameters, this model for the photoemission intensityis fitted to high-resolution experimental data taken atASTRID2 at a photon energy of 25 eV, i.e. the samephoton energy as used for the probe pulse in the TR-ARPES experiment. Fig. 2 shows the resulting excel-lent agreement between (a) measured and (b) modelledspectral function. Fig. 2(c) and (d) further show the de-gree of agreement in the form of EDC-cuts at the Fermilevel crossing ( k F ) and at the band minimum ( k min ). Inthis fit, the highest binding energy reaches a value of350 meV. The light gray lines in these cuts represent thebackground (BG) function that is not included in equ. (2). (iii) Extracting electronic temperature and bandshifts The approach of fitting I conv ( E, k ) to a model spectralfunction now permits the precise determination of param-eters such as the electronic temperature T e and changesof the dispersion. Fig. 3 shows representative results forthe application of this fitting method to time-resolveddata sets. Fig. 3(a)-(c) show the measured dispersionin equilibrium, at maximum excitation ( t =40 fs) andat t =350 fs, while Fig. 3(f)-(h) show modelled spectralfunctions, obtained as described in section (ii) above. Inorder to obtain these fits, we have varied the electronictemperatures T e , the dispersion offset c in equ. (2), andthe constant and linear coefficient of the linewidth α and β . A redistribution of background intensity followingphotoexcitation is also taken into account [27]. The ap-plication of this fitting procedure results in an excellentdescription of the data in Fig. 3(a)-(c) for all experimen-tal parameters (delay time, fluence). This is illustratedin the comparison between measured (Fig. 3(d) and (e))and fitted (Fig. 3(i) and (j)) intensity differences for timedelays of 40 and 350 fs and in the direct comparison be-tween measured and fitted EDC-cuts at k F and k min , asshown in Fig. 3(k) and (l). Fits of similar quality areobtained for data sets taken at different sample tempera-tures and pump laser fluence [27]. Given the high qualityof the fits obtained, we conclude that the hot electron gasis always observed to be in thermal equilibrium with awell-defined temperature, as might be expected given thetime resolution of the experiment.The extracted dynamic changes of the dispersion andelectronic temperature T e are given in Fig. 4. In order toquantify the band shift, we introduce the quantity ∆ W ,defined as the difference between the band minimum en-ergy in the excited and equilibrium state, that is thechange of the parameter c in equ. (2). Fig. 4(a) showsthat this shift is very substantial - more than 100 meV atpeak excitation ( i. e. roughly a third of the total occu-pied bandwidth). Fig. 4(b) shows the shift takes place atthe same time delays as the peak temperature of the elec-tron gas exceeding 3000 K. The time dependence of both∆ W and T e can be described by a double-exponential de-cay with a relaxation time τ well below 1 ps and a slower τ . We tentatively assign τ to a decay process involv-ing the excitations of high energy optical phonons in theTaS layer [34] and the slower decay to a combination ofacoustic phonon excitations and anharmonic decay of theoptical phonons, similar to the situation seen in graphene[14].This presence of a transient hot electron distributionis thus accompanied by a substantial band shift that isnot a priori expected for a metallic system. Note that t < 0 t = 40 fs t = 350 fs t = 40 fs t = 350 fs 10-1 1 0 -1 i n t en s i t y ( a r b . u . ) i n t en s i t y ( a r b . u . ) (a) (b) (c) (d) (e)(f) (g) (h) (i) (j) (k)(l) t < 040 fs350 fs t < 040 fs350 fs k -1 )(Å k -1 )(Å k -1 )(Å k -1 )(Å k -1 )(Å E b i n ( e V ) E b i n ( e V ) neg.pos. lowhighE F k F k min T e = 300 K T e = 3080 K T e = 1050 K T e = 3080 K T e = 1050 K E bin (eV) k min k F FIG. 3. (Color online) Time dependence of the excited-state signal in SL TaS and spectral function simulations: (a)-(c)TR-ARPES data obtained at the given time delays for an optical excitation energy of 2.05 eV and a pump laser fluence of7.8 mJ/cm with the sample at a temperature of 300 K. The spectrum in (a) was taken before optical excitation. The fittedparabolic dispersions derived according to equ. (2) are shown on top of the spectra and coloured to distinguish the differenttime delays. (d)-(e) Difference spectra determined by subtracting the equilibrium spectrum in (a) from the excited state spectrain (b)-(c). (f)-(j) Simulated intensity (difference) corresponding to the measured data in (a)-(e). (k),(l) Comparison of EDCsfrom measurements (symbols) and simulations (lines) at k F and k min , respectively (see pink and purple lines in (f)). t (ps)Ƭ = (3±1) psƬ = (410±20) fs 300020001000 Ƭ = (200±10) fsƬ = (11±4) ps (a) (b)10 -1 W ( m e V ) ∆ -1 t (ps) T e ( K ) FIG. 4. (Color online) Extracted parameters from the dataset shown in Fig. 3. (a) Time dependence of the extractedband shift ∆ W . The fit to a double exponential function isshown (solid line) and the relaxation times τ and τ are given.(b) Corresponding data and fit for the electron temperature T e . the fit was constrained to a rigid band shift, withoutother changes and, in particular, without the boundarycondition of fixed k F values. This procedure was chosenbecause the position of the band minimum is an experi-mentally well accessible quantity, as this spectral region is least affected by the broad Fermi-Dirac distribution.Moreover, the change in k F resulting from a rigid shift ofthe dispersion is small ( < .
05 ˚A − ) and would be muchharder to resolve than the energy shift. Assuming merelya rigid shift of the band also appears justified because ofthe high fit quality at all time delays. However, minorchanges in the dispersion and of the Fermi wave vectorcannot be completely excluded.A strong correlation between ∆ W and T e emergeswhen we combine all data points obtained at different flu-ences and sample temperatures and plot ∆ W as a func-tion of T e in Fig. 5. The cause of this correlation isexplored in the next sub-sections. However, the relation∆ W ( T e ) is clearly not strictly linear, explaining the factthat the relaxation times τ and τ from the fit of thedata in Fig. 4(a) and (b) are not necessarily the same. (iv) Calculated chemical potential shifts While the observed correlation between T e and ∆ W does not imply causality, it is tempting to seek a sim-ple mechanism that can explain the band shift as causedby the high electronic temperature without invoking, forexample, substrate effects. In the present section we ex- ∆ W ( m e V ) F = 8 .4 mJ/cm F = 7 .8 mJ/cm F = 4 .4 mJ/cm E F = 430 meV E F = 460 meV E F = 490 meV T e (K) FIG. 5. (Color online) Temperature-induced band shift, usingdata for different choices of laser fluence and sample tempera-ture. The data points show the extracted experimental bandshift ∆ W as a function of electronic temperature T e . Thecurves are the calculated change in the occupied bandwidthas a function of T e for three different values of the system’sFermi energy (chemical potential at T e = 0.) plore how the band shift can be caused by a temperature-induced shift of the chemical potential, which is requiredto conserve the total charge in the system. In the fol-lowing sub-section we also address the possibility of ad-ditional shifts caused by the temperature-dependence ofthe electronic screening.Small changes of the chemical potential compared tothe zero temperature Fermi energy are always expectedin a metallic system, but since k B T in typical experi-mental conditions is much smaller than the Fermi energythe shifts are also small. This is clearly not the casehere. Indeed, the width of the Fermi-Dirac function at3000 K exceeds the occupied bandwidth of SL TaS , andtemperature-induced shift of the chemical potential couldthus be considerable. Predicting the size or even the di-rection of the shift is not trivial because it involves thedetails of the entire occupied and unoccupied electronicband structure.In order to calculate the expected shift of the chemicalpotential, we start from a tight-binding (TB) model forSL TMDCs [35] and adapt it for the case of SL TaS .The model is based on the d -orbitals of Ta atoms on atriangular lattice where the first, second and third near-est neighbour hopping integrals are taken into account.Accurate TB parameters are obtained by fits to the re-sults of a density functional theory (DFT) calculation [27]. The resulting single particle dispersion at zero tem-perature is shown in Fig. 6(a) and the correspondingdensity of states in Fig. 6(b).Based on the known electronic structure, we can deter-mine the temperature-dependent chemical potential us-ing the following procedure. We start from the assump-tion of a fixed number of electrons N at any temperature.This number is given by N = (cid:90) ∞−∞ ρ ( E ) e β ( E − µ ) + 1 dE, (3)where β = 1 /k B T e and µ is the chemical potential. Thedensity of states, including lifetime broadening effects, isgiven by ρ ( E ) = L − (cid:80) k ∈ BZ A ( E, k ), where L is the sidelength of the sample. The spectral function A ( E, k ) isgiven by A ( E, k ) = 1 π (cid:88) σ = ± Γ k σ ( E − E k σ ) + Γ k σ , (4)in which the quasiparticle dispersion follows E k σ = E k σ + Σ k σ . (5)Note that E k σ stands for the bare dispersion as a functionof k and for a particular spin-split branch of σ = ± in thewhole BZ [27, 30]. Σ k σ is the real part of the on-shell self-energy correction, Σ k σ = Re[Σ( E, k , σ )] E → E k σ − µ , andΓ k σ = − Im[Σ( E, k , σ )] E → E k σ − µ is the correspondingimaginary part of the self energy resulting from electron-electron interactions at finite temperature. Note thatthe theoretical linewidth 2Γ k σ cannot be compared di-rectly to the experimental linewidth. The experimentallinewidth is expected to be larger, due to the contri-butions of electron-phonon and electron-defect scatter-ing not present in the theoretical model. Moreover, theexperimental linewidth at higher binding energies is af-fected by the presence of azimuthal disorder in the sam-ple.When defining the absolute band minimum in the BZ E k ◦ as the zero of the energy scale (see Fig. 6(a)), thetemperature-dependent occupied bandwidth measured inthe experiment is given by W ( T e ) = µ ( T e ) − Σ ◦ ( T e ) , (6)where Σ ◦ ( T e ) is the real part of the self-energy at theband minimum position k ◦ . W ( T e ) thus has two contri-butions, one from the temperature-dependent chemicalpotential and one from the electron-electron interactionthat affects the energy of the band minimum via the self-energy Σ ◦ ( T e ) (we tacitly assume that self-energy effectsdo not move the band minimum away from k ◦ . This isconfirmed by the calculations below). The temperature-induced change of the bandwidth when the system isheated from the equilibrium temperature T eq to T e thenreads∆ W ( T e ) = W ( T e ) − W ( T eq ) = ∆ µ ( T e ) − ∆Σ ◦ ( T e ) (7) o E k σ ( e V ) E ( e v ) µ ( e V ) T e = 300 K T e = 1000 K T e = 2000 K T e = 3000 K E F (eV) E F = 430 meV E F = 460 meV E F = 490 meV T e (K) Δ µ ( m e V ) FIG. 6. (Color online) (a) Calculated single-particle disper-sion of the topmost valence band of SL TaS . The solid lineis the result from a tight-binding calculation with parametersfitted to a density functional theory calculation (dashed line).Note that the bands are spin-split at K . The dashed blackline shows the position of the Fermi energy for a filling of theband with one electron per unit cell, as expected for the free-standing layer. (b) Resulting density of states. (c) Chemicalpotential versus the Fermi energy at different values of T e with Γ k σ = 10meV and Σ k σ = 0. (d)The chemical potentialshift versus T e for three different values of the Fermi energy.Solid (dashed) curves correspond to the absence (presence) ofself-energy effects on the chemical potential. where ∆ µ ( T e ) = µ ( T e ) − µ ( T eq ) ≈ µ ( T e ) − E F and∆Σ ◦ ( T e ) = Σ ◦ ( T e ) − Σ ◦ ( T eq ). Note that ∆ µ ( T e ) is alsoaffected by self-energy effects (if present), since these canlead to a rigid shift of the entire band which is then com-pensated by a change in the chemical potential. This willbe discussed in the next sub-section.We perform the energy integral in equ. (3) analyti-cally and then numerically integrate over the BZ. Fromthis, the temperature-dependent chemical potential canbe extracted, and thereby also the shift of the observeddispersion. In the initial iteration, we neglect the in-fluence of many-body effects on the band dispersion bysetting Σ k σ = 0 and assume a constant electronic lifetimebroadening of Γ k σ . Fig. 6(c) shows the resulting shift ofthe chemical potential as a function of electron filling inthe layer, expressed in terms of the Fermi energy ( i.e. ,the chemical potential at zero temperature). Evidently,the degree of change strongly depends on the position ofthe Fermi energy. For a very low filling of the band a temperature increase leads to a decrease of the chemicalpotential, whereas the opposite is the case for high filling.These opposite trends at different filling levels can beunderstood by considering the following thermodynami-cal identity in a fixed system volume [36]: ∂µ∂T e (cid:12)(cid:12)(cid:12) N = − ∂N∂T e (cid:12)(cid:12)(cid:12) µ (cid:18) ∂N∂µ (cid:12)(cid:12)(cid:12) T e (cid:19) − . (8)Then, utilizing the low- T e Sommerfeld expansion for thenumber of particles, N , we have [36] ∂µ∂T e (cid:12)(cid:12)(cid:12) N ≈ − π k T e ρ (cid:48) ( µ ) ρ ( µ ) , (9)where ρ (cid:48) ( µ ) = dρ ( µ ) /dµ stands for the derivative of thedensity of states. At low- T e we can approximate ρ (cid:48) ( µ ) ≈ ρ (cid:48) ( E F ). By looking at the density of states in Fig. 6(b),we can see that ρ (cid:48) ( E F ) has a positive (negative) sign inlow (high) doping. This can roughly explain ∂µ/∂T e < ∂µ/∂T e >
0) for low (high) filling as depicted in Fig.6(c).It is not entirely clear what choice of E F is most ap-propriate for a comparison with the experiment. Theband minimum at k min determined from the static high-resolution experiment is 350 meV. However, due to theazimuthal disorder this is not equal to the Fermi en-ergy but roughly to the average highest binding energyin the Γ − M and Γ − K directions. The actual Fermienergy corresponds to the highest binding energy alongΓ − M (marked k ◦ in Fig. 6(a)) which is ≈
80 meV higherthan the average highest binding energy. A choice of E F = 430 meV should thus be a good estimate of theFermi energy in the experimental data.We plot the temperature-induced shift in the chemicalpotential as a function of T e for three different values ofthe Fermi energy in Fig. 6(d). Due to the extreme tem-peratures reached and the small bandwidth, the effectof a temperature-induced chemical potential shift is es-sential in order to explain the observed band shift ∆ W .However, due to the potentially strongly temperature-dependent screening of the Coulomb interaction, a con-siderable contribution from the self-energy correction isalso expected. This is discussed in the following section. (v) Calculated effect of static screening It is conceivable that many-body effects also contributeto the observed band shift in addition to the effect stem-ming from the chemical potential. It is well-known, forinstance, that the electronic self-energy has a significanteffect on the observed dispersion of electronic states inARPES, even in the case of simple metals [37, 38]. In thepresent experiment the electronic temperature is changedover such a wide range that it is relevant to ask if thetemperature-induced change in electronic screening couldcontribute significantly to the observed changes in theelectronic structure.We investigate this by calculating the temperature-dependent static screened exchange self-energy using thesingle-band TB model of SL TaS . For this we first eval-uate the density-density susceptibility which is given by[10] χ ( q , T e ) = 1 L (cid:88) k ∈ BZ (cid:88) σ f ( E k σ , T e ) − f ( E k + q σ , T e ) E k σ − E k + q σ . (10)The effect of temperature-dependent screening is treatedthrough the static screened exchange which is given by[10]Σ k σ ( T e ) = − L (cid:88) q ∈ BZ v q − k − v q − k χ ( q − k , T e ) f ( E q σ , T e ) , (11)where the bare Coulomb interaction in 2D reads v q =2 πe / ( (cid:15) eff | q | ) with (cid:15) eff ∼ (1 + (cid:15) sub ) /
2. Note that (cid:15) sub ∼
22 is the dielectric constant of the graphene/SiC sub-strate [39].The q -dependent susceptibility of SL TaS is shownin Fig. 7(a) for four different temperatures along high-symmetry directions in the BZ and for E F = 430 meV, asobserved in the experiment. As seen in Fig. 7(a), the ab-solute magnitude of the static susceptibility (consideringthat the static susceptibility is always a negative value, i.e. | χ ( q , T e ) | = − χ ( q , T e )), and therefore the screening,decreases by increasing T e . This is a universal trend atlow temperature and low momentum. At the BZ centerand at low electronic temperature ( T e (cid:28) T F ), we canapproximate χ (0 , T e ) ≈ − ρ ( µ ) [10] and by using equ. (9)we have ∂ | χ (0 , T e ) | ∂T e (cid:12)(cid:12)(cid:12) N ≈ ∂µ∂T e (cid:12)(cid:12)(cid:12) N ρ (cid:48) ( µ ) < , (12)with T F being the Fermi temperature. In general thisnegative slope is not fulfilled at finite q or very high tem-perature. Under our experimental conditions, the elec-tronic temperature is very high, leading to a consider-able broadening of the Fermi-Dirac distribution functionwhich in fact becomes similar to the entire bandwidth( i.e. k B T e ∼ E F ). This strong broadening reduces theprobability of virtual transitions, E k σ → E k + q σ , for allvalues of q in the BZ. Because of this semi-Pauli-blockingeffect at very high temperature, the number of vir-tual electron-hole excitations is diminished, leading to aweaker screening effect. A similar reduction of screeningis predicted to emerge in the parabolic-band two dimen-sional electron gas (2DEG) [40–42], i.e. χ ( q , T e (cid:29) T F ) ≈ T F /T e [41]. On the other hand, in the masslessDirac fermion (MDF) model of graphene the screening ef-fect is predicted to have an increasing trend with temper-ature as χ MDF ( q , T e (cid:29) T F ) ≈ log(4) T e /T F [41]. This can M Γ K M6080100120140 −240−220−200−180−160−140(a) (b) T e = 300 K T e = 1000 K T e = 2000 K T e = 3000 K M Γ K M ∑ k σ ( m e V ) χ ( q , T e ) (( � a ) - ( e V ) - ) FIG. 7. (Color online) (a) Calculated temperature-dependentelectronic susceptibility along high-symmetry lines of the BZfor four different values of T e . (b) Real part of the electronicself-energy along high-symmetry lines of the BZ ( T e as in(a)). Note that the difference between up and down spins isnegligible for this set of parameters ( i.e. Σ k ↑ ∼ Σ k ↓ ). Notethat for this figure we set E F = 430meV. be due to the inter-band transitions in graphene whichdo not exist in our single-band metallic system.Having a weaker screening of the Coulomb interactionat high electronic temperature implies a stronger many-body effect. As depicted in Fig. 7(b), the explicit self-energy calculation indicates a very strong temperaturedependence, with changes of ≈
100 meV over the exper-imental temperature range. However, this self-energy ismostly comprised of a rigid shift of the whole band whichis compensated by an opposite shift in the chemical po-tential in order to conserve the particle number. Whenthese many-body effects are included in equ. (4) andthe chemical potential is calculated in a second iteration,this new estimate is thus strongly modified. This explainsthe large difference between the chemical potential shiftswith and without self-energy corrections in Fig. 6(d).Experimentally, neither the change in the chemical po-tential nor in the self-energy is directly observable butonly their combination in ∆ W , see equ. (7). Fig. 5 thuscompares the full theoretical result to the experimentaldata, showing an almost quantitative agreement betweenexperiment and calculation. A comparison of the theoret-ical results in Fig. 5 and 6(d) shows that including staticscreening to first order leads to a bandwidth change ofmerely ∼
15 meV at T e ∼ change in this binding energy is the same as the theoreticallycalculated ∆ W . CONCLUSIONS
We have demonstrated strong electronic heating andchanges in the occupied bandwidth upon optical pump-ing of a 2D metal, SL TaS . The data could be quan-titatively analyzed using a 2D fitting scheme of the en-tire resolution-broadened spectral function. The exper-imentally observed band shifts are explained by consid-ering the temperature-dependent many-body screeningeffect and the chemical potential shift required to con-serve charge neutrality in the presence of a hot electronpopulation.The possibility of very large band shifts in pumpedmetallic systems could potentially be used to create anumber of unconventional states of matter. We empha-size that neither the direction nor the magnitude of theshift is trivial but both result from the material’s bandstructure in a wide range around the Fermi energy. In-deed, much larger shifts still could be expected for SLTaS with a different band filling. Starting from an ap-propriate band structure, it could thus be possible to usetransient temperature-induced shifts in order, for exam-ple, to push a Van Hove singularity in the density ofstates close to the chemical potential, possibly creatingelectronic instabilities at high temperatures. ACKNOWLEDGEMENTS
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