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\title{Spectroscopic Evidence for a Three-Dimensional Charge Density Wave in Kagome Superconductor CsV$_3$Sb$_5$}


\author{Cong Li$^{1,2,\sharp,*}$, Xianxin Wu$^{3,4,\sharp}$, Hongxiong Liu$^{5,\sharp}$, Craig Polley$^{6}$, Qinda Guo$^{1}$, Yang Wang$^{5}$, Xinloong Han$^{7}$, Maciej Dendzik$^{1}$, Magnus H. Berntsen$^{1}$, Balasubramanian Thiagarajan$^{6}$, Youguo Shi$^{5}$, Andreas P. Schnyder$^{3}$, Oscar Tjernberg$^{1,*}$
}

\affiliation{
\\$^{1}$Department of Applied Physics, KTH Royal Institute of Technology, Stockholm 11419, Sweden
\\$^{2}$Department of Applied Physics, Stanford University, Stanford, CA 94305, USA
\\$^{3}$Max-Planck-Institut f$\ddot{u}$rFestk$\ddot{o}$rperforschung, Heisenbergstrasse 1, D-70569 Stuttgart, Germany
\\$^{4}$CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
\\$^{5}$Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
\\$^{6}$MAX IV Laboratory, Lund University, 221 00 Lund, Sweden
\\$^{7}$Kavli Institute of Theoretical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, China
\\$^{\sharp}$These people contributed equally to the present work.
\\$^{*}$Corresponding authors: conli@kth.se, oscar@kth.se
}

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{\bf 1. Calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions and their orbital nature}

Figure S1 shows the calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions ($\Gamma$-K-M-$\Gamma$-A-H-L-A) and their orbital nature which corresponding the calculated band structure in Fig. 1g. It can be clearly observed that the electronic-like band around the Brillouin zone (BZ) center is dominated by the $p_z$ orbital of Sb atoms.\\

{\bf 2. Calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions when considering the 2$\times$2$\times$2 lattice reconstruction}

Figure S2a-S2d show the crystal structure of CsV$_3$Sb$_5$ when considering the 2$\times$2$\times$2 lattice reconstruction due to the CDW caused by the alternate stacking of star of David (SoD)- and tri-hexagonal (TrH)-like distortions along $c$ axis (Fig. S2a), the alternate stacking of SoD- and TrH-like distortions with in-plane $\pi$ phase shift along $c$ axis (Fig. S2b), the alternate stacking of the TrH-like distortions and its in-plane $\pi$ phase shift along $c$ axis (Fig. S2c) and the alternate stacking of the SoD-like distortions and its in-plane $\pi$ phase shift along $c$ axis (Fig. S2d). In this case, due to the lattice constant expands to 2$\times$2$\times$2 times as pristine, the corresponding Brillouin zone (BZ) should shrinks to 0.5$\times$0.5$\times$0.5 times as before which will result in the reconstruction of the electronic structure in the new BZ. Fig. S2e-S2h show the unfolded calculated bands of CsV$_3$Sb$_5$ along high-symmetry directions ($\Gamma$'-K'-M'-$\Gamma$'-A'-H'-L'-A') when considering the 2$\times$2$\times$2 lattice reconstruction that corresponding to Fig. S2a-S2d. \\


{\bf 3. Detailed Fermi surface mapping of CsV$_3$Sb$_5$}

Figure S3 shows the Fermi surface map of CsV$_3$Sb$_5$ measured at 150 K (20 K) in the sample cleaved at high temperature (HTC, 150 K) with the photon energy of 57 eV (Fig. S3a for 150 K, Fig. S3e for 20 K), 62 eV (Fig. S3b for 150 K, Fig. S3f for 20 K) and 120 eV (Fig. S3c for 150 K, Fig. S3g for 20 K) and measured at 120 K (20 K) in the sample cleaved at low temperature (LTC, 20 K) with the photon energy of 95 eV (Fig. S3d for 120 K, Fig. S3h for 20 K). Comparing the data above (150 K and 120 K) and below (20 K) the CDW transition temperature (94 K), no obvious in-plane Fermi surface folding caused by a CDW was observed.\\ 

{\bf 4.  Distinguish the surface states of CsV$_3$Sb$_5$}

Figure S4 shows the calculated band structure of CsV$_3$Sb$_5$ along the $\Gamma$-K-M-$\Gamma$ directions for a ten-unit-cell-thick slab, which can help us distinguish the surface states from bulk states in CsV$_3$Sb$_5$. The bands marked by arrows in Fig. S4a are the surface states. Based on the analysis of Fig. S1, the orbital nature of the surface states can be determinate as the $p_z$ orbital of Sb atoms. Further analysis showed that the surface states marked in Fig. S4a are contributed by Sb1 atoms (the Sb atom in the V-Sb plane, blue lines in Fig. S4b). When the sample surface is terminated by Cs atoms, the surface states can be clearly observed (Fig. S4b). However, the surface states vanish in the sample where the surface is terminated by Sb atoms as shown in Fig. S4c.\\


{\bf 5. Detailed photon energy dependent ARPES spectral intensity map of CsV$_3$Sb$_5$ at 20 K.}

Figure S5 shows the detailed photon energy dependent ARPES spectral intensity map of CsV$_3$Sb$_5$ measured at 20 K in the sample cleaved at low temperature (20 K) at the binding energy of 0 eV (Fig. S5a), 0.1 eV (Fig. S5b), 0.2 eV (Fig. S5c), 0.3 eV (Fig. S5d), 0.4 eV (Fig. S5e) and 0.5 eV (Fig. S5f) with the photon energy range from 25 eV to 110 eV. It can be clearly observed that there are some periodic structures along the $k_z$ direction.\\

{\bf 6. Photon energy dependent spectra for photoemission intensity at the BZ center of CsV$_3$Sb$_5$.}

Figure S6 shows the photon energy dependent spectra for photoemission intensity at the BZ center of CsV$_3$Sb$_5$ measured at 20 K which correspond to the measurments in Fig. 4e-4g in the main manuscript. \\



{\bf 7. Distinguish the CDW configurations that result in a 2$\times$2$\times$2 lattice reconstruction in CsV$_3$Sb$_5$.}

Fig. S7a-S7d shows the calculated band structure of CsV$_3$Sb$_5$ along the M-K-$\Gamma$-K-M direction when considering the 2$\times$2$\times$2 lattice reconstruction due to the CDW caused by the alternate stacking of SoD- and TrH-like distortions (Fig. S7a) (TrH-like distortions with in-plane $\pi$ phase shift, Fig. S7b) along the $c$ axis, the alternate stacking of the TrH-like (Fig. S7c) (SoD-like, Fig. S7d) distortions and its in-plane $\pi$ phase shift. Fig. S7e and Fig. S7g show the band structure along M-K-$\Gamma$-K-M direction measure at 20 K (Fig. S7e) and 120 K (Fig. S7g) with the photon energy of 70 eV. In order to observe band features more clearly, we do the EDC second derivative image for Fig. S7e and Fig. S7g which are shown in Fig. S7f and Fig. S7h. It can be clearly observed that there are two bands ($\delta_{1}$ band and $\delta_{2}$ band) around the M point at 20 K but only one band ($\delta$ band) around the M point at 120 K. It can be inferred that the emergent $\delta_{1}$ band at 20 K is the band folding due to the CDW. In addition, it can be found that only the CDW configurations with the alternate stacking of SoD- and TrH-like distortions (or TrH-like distortions with in-plane $\pi$ phase shift) along the $c$ axis can induce two bands ($\delta_{1}$ band and $\delta_{2}$ band) around the M point which is in agreement with our experimental data. Therefore, it can be inferred that the 3-dimensional CDW configuration of CsV$_3$Sb$_5$ along the $c$ direction is likely the alternate stacking of SoD- and TrH-like distortions (or TrH-like distortions with in-plane $\pi$ phase shift).\\


\vspace{3mm}

\newpage

\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS1_BandNature.png}
\end{center}
\caption{\textbf{Calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions and their orbital nature.} Calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions across the Brillouin zone without considering the CDW. The orbital nature of bands are marked by different colors that pink is the $p_{x/y}$ orbital of Sb atoms, cyan is the $p_z$ orbital of Sb atoms, red is the $d_{xy/x^{2}-y^{2}}$ orbital of V atoms, green is the $d_{xz/yz}$ orbital of V atoms and blue is the $d_{z^{2}}$ orbital of V atoms.
}
\end{figure*}


\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS2_222BandCalculation.png}
\end{center}
\caption{\textbf{Single crystal structure and calculated band structure of CsV$_3$Sb$_5$ along high-symmetry directions when considering the 2$\times$2$\times$2 lattice reconstruction.} (a-d) One unit cell of CsV$_3$Sb$_5$ when considering the 2$\times$2$\times$2 lattice reconstruction due to the CDW caused by the alternate stacking of SoD- and TrH-like distortions along $c$ axis (a), the alternate stacking of SoD- and TrH-like distortions with in-plane $\pi$ phase shift along $c$ axis (b), the alternate stacking of the TrH-like distortions and its in-plane $\pi$ phase shift along $c$ axis (c) and the alternate stacking of the SoD-like distortions and its in-plane $\pi$ phase shift along $c$ axis (d). (e-h) The unfolded calculated bands of CsV$_3$Sb$_5$ along high-symmetry directions ($\Gamma$'-K'-M'-$\Gamma$'-A'-H'-L'-A') when considering the 2$\times$2$\times$2 lattice reconstruction that corresponding to (a-d). $\Gamma$', K', M', A', H', L' are the high-symmetry points of the folded BZ due to the 2$\times$2$\times$2 lattice reconstruction which corresponding to $\Gamma$, K, M, A, H, L in the unfolded BZ.
}
\end{figure*}



\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS3_FS.png}
\end{center}
\caption{\textbf{Fermi surface mapping of CsV$_3$Sb$_5$.} (a-c) Fermi surface mapping of CsV$_3$Sb$_5$ measured at 150 K in the sample of high temperature cleavage (HTC) with the photon energy of 57 eV (a), 62 eV (b) and 120 eV (c). (d) Fermi surface mapping of CsV$_3$Sb$_5$ measured at 120 K in the sample of low temperature cleavage (LTC) with the photon energy of 95 eV. (e-g) Fermi surface mapping of CsV$_3$Sb$_5$ measured at 20 K in the sample of high temperature cleavage (HTC) with the photon energy of 57 eV (e), 62 eV (f) and 120 eV (g). (h) Fermi surface mapping of CsV$_3$Sb$_5$ measured at 20 K in the sample of low temperature cleavage (LTC) with the photon energy of 95 eV.
}
\end{figure*}


\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS4_SurfaceState.png}
\end{center}
\caption{\textbf{The calculated band structures of CsV$_3$Sb$_5$ along $\Gamma$-K-M-$\Gamma$ directions for a ten-unit-cell-thick slab.} (a) The original data. The bands marked by arrows are the surface state (SS). (b-c) The calculated band structures which show the orbitals contributed by Sb atoms in the sample with terminated surface covered by Cs atoms (b) and Sb atoms (c). The thickness of the lines represent the proportion of the orbital. The bule lines are the orbital from Sb1 atoms, and the red lines are the orbital from Sb2 atoms.
}
\end{figure*}


\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS5_HvFS.png}
\end{center}
\caption{\textbf{Photon energy dependent ARPES spectral intensity map of CsV$_3$Sb$_5$.} (a-f) Photon energy dependent ARPES spectral intensity map of CsV$_3$Sb$_5$ measured at 20 K in the sample of the low temperature cleavage (LTC) at the binding energy of 0 eV (a), 0.1 eV (b), 0.2 eV (c), 0.3 eV (d), 0.4 eV (e) and 0.5 eV (f).
}
\end{figure*}


\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS6_KzBand.png}
\end{center}
\caption{\textbf{Photon energy dependent spectra for photoemission intensity at the BZ center of CsV$_3$Sb$_5$.} (a) Photon energy dependent spectra for photoemission intensity at the BZ center. The intensity are normalized in the range of -0.8 eV to
-0.5 eV. (b) The zoom in of (a). The green open circles mark the peak position of main subbands which extracted from Fig. 4f. (c) The corresponding energy distribution curve (EDC) second derivative image of (b), but it is normalized around -0.8 eV.
}
\end{figure*}


\begin{figure*}[tbp]
\begin{center}
\includegraphics[width=1\columnwidth,angle=0]{FigS7_ComBandST.png}
\end{center}
\caption{\textbf{Distinguish the CDW configurations that result in a 2$\times$2$\times$2 lattice reconstruction in CsV$_3$Sb$_5$.} (a-d) Calculated band structure of CsV$_3$Sb$_5$ along M-K-$\Gamma$-K-M direction when considering the 2$\times$2$\times$2 lattice reconstruction due to the CDW with the alternate stacking of SoD- and TrH-like distortions along $c$ axis (a), the alternate stacking of SoD- and TrH-like distortions with in-plane $\pi$ phase shift along $c$ axis (b), the alternate stacking of the TrH-like distortions and its in-plane $\pi$ phase shift (c) and the alternate stacking of the SoD-like distortions and its in-plane $\pi$ phase shift (d). (e) and (g) Band structure along M-K-$\Gamma$-K-M direction measured at 20 K (e) and 120 K (g) with the photon energy of 70 eV. (f) and (h) The corresponding EDC second derivative image of (e) and (g).
}
\end{figure*}

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