Classification of collective modes in a charge density wave by momentum-dependent modulation of the electronic band structure

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Classification of collective modes in a charge density wave by momentum-dependent modulation of the electronic band structure

D. Leuenberger, J. A. Sobota, S.-L. Yang, A. F. Kemper, P. Giraldo-Gallo, R. G. Moore, I. R. Fisher, P. S. Kirchmann, T. P. Devereaux, and Z.-X. Shen

Phys. Rev. B 91, 201106(R) – Published 21 May 2015

Abstract

We present time- and angle-resolved photoemission spectroscopy (trARPES) measurements on the charge density wave system ${CeTe}_{3}$ . Optical excitation transiently populates the unoccupied band structure and reveals a gap size of $2 Δ = 0.59 eV$ . The occupied Te- $5 p$ band dispersion is coherently modified by three modes at $Ω_{1} = 2.2, Ω_{2} = 2.7$ , and $Ω_{3} = 3 THz$ . All three modes lead to small rigid energy shifts whereas $Δ$ is only affected by $Ω_{1}$ and $Ω_{2}$ . Their spatial polarization is analyzed by fits of a transient model dispersion and density-functional theory frozen phonon calculations. We conclude that modes $Ω_{1}$ and $Ω_{2}$ result from in-plane ionic lattice motions, which modulate the charge order and that $Ω_{3}$ originates from a generic out-of-plane $A_{1} g$ phonon. We thereby demonstrate how the rich information from trARPES allows identification of collective modes and their spatial polarization, which explains the mode-dependent coupling to charge order.

Received 5 December 2014
Revised 1 May 2015

DOI:https://doi.org/10.1103/PhysRevB.91.201106

Authors & Affiliations

D. Leuenberger^1,2, J. A. Sobota^1,2,3, S.-L. Yang^1,2,4, A. F. Kemper³, P. Giraldo-Gallo^1,2, R. G. Moore^1,2, I. R. Fisher^1,2, P. S. Kirchmann^1,*, T. P. Devereaux^1,2, and Z.-X. Shen^1,2,4,†

¹Stanford Institute for Materials and Energy Sciences, SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA
²Department of Applied Physics, Geballe Laboratory for Advanced Materials, Stanford University, Stanford, California 94305, USA
³Lawrence Berkeley National Laboratory, 1 Cyclotron Road, Berkeley, California 94720, USA
⁴Department of Physics, Stanford University, Stanford, California 94305, USA

^*kirchman@stanford.edu
^†zxshen@stanford.edu

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Vol. 91, Iss. 20 — 15 May 2015

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Images

Figure 1
(a) Structure of the Te planes where the CDW forms along the $c$ axis with lattice displacement $Δ z$ indicated. The FS is described in a TB model with the overlap integrals $t_{p σ}$ along and $t_{p π}$ perpendicular to the Te- $5 p$ orbitals. (b) Cartoon of the trARPES experiment on ${CeTe}_{3}$ in the CDW region of the FS (dotted lines). The blue plane illustrates an energy versus momentum cut with Te- $5 p$ bands and a pump pulse exciting electrons into unoccupied states. The blue line indicates the momentum cut in the experiment across the gapped region (dotted lines) of the calculated 2D FS, and the green arrow denotes the CDW wave vector $\vec{q}$ . trARPES spectra for selected pump-probe delays, (c) 330 fs before the arrival of the pump pulse and (d) during the presence of the pump pulse. Intensities are rescaled exponentially as a function of energy for enhanced visibility of the transiently occupied features above $E_{F}$ . (e) TB fit (blue markers) according to Eqs. (1) and (2) to the extracted band dispersion (black markers) at $- 25 fs$ yields $2 Δ = 0.59 (2) eV$ . The size of the blue markers is proportional to the calculated spectral weight of the TB bands [24]. The brown lines indicate the calculated metallic bare band dispersion.
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Figure 2
(a) Measured dispersion of the occupied Te- $5 p$ band for selected delays (markers) and transient TB fits (lines). Sketched effect of (b) a transient rigid energy shift $E_{0} (t)$ and (c) a change in the CDW gap $Δ (t)$ on the band dispersion. (d) The $2 Δ (t)$ and $E_{0} (t)$ with smooth backgrounds (black lines) of a tenth-order polynomial indicated.
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Figure 3
(a) Residuals of the polynomial background fit shown in Fig. 2. (b) and (c) Fourier transformation (FT) of (a). The vertical lines and markers at $Ω_{1} = 2.19 (1), Ω_{2} = 2.69 (1)$ , and $Ω_{3} = 2.98 (1) THz$ indicate mode frequencies and amplitudes independently obtained from fitting damped cosine functions, see Supplemental Material [35].
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Figure 4
Mode-dependent modulations of the Te- $5 p$ bands (markers) by the three dominant modes is derived from momentum-dependent FT amplitudes for deposited energies of 28–103 meV per unit cell. Solid lines are obtained from the corresponding TB fits of the transient Te- $5 p$ bands, see Supplemental Material [35]. (a) The AM at $Ω_{1} = 2.2 THz$ modulates the in-plane lattice displacement $Δ z$ and leads to a strongly momentum- and fluence-dependent response. (b) The AM at $Ω_{2} = 2.7 (1) THz$ exhibits a weaker momentum and fluence dependence, which indicates a weaker influence on the in-plane CDW. (c) The out-of-plane $A_{1} g$ mode at $Ω_{3} = 3 THz$ only shifts the band rigidly as it does not couple to the in-plane charge order.
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