DOI: 10.5061/dryad.jwstqjqmq
title: Direct measurement of the quantum metric tensor in solids

description: The quantum metric tensor is a central geometric quantity in modern
 physics that determines the distance between nearby quantum states.
 Despite numerous studies highlighting its relevance to fundamental
 physical phenomena in solids, the method for measuring the complete
 quantum metric tensors in real solid-state materials remains unknown.
 Here, we report the first direct measurement of the full quantum metric
 tensors of Bloch electrons in solids using black phosphorus as a
 representative material. The key idea is to extract the momentum space
 distribution of the pseudospin texture of the valence band from the
 polarization dependence of angle-resolved photoemission spectroscopy
 measurement. The spectroscopic probe of all possible quantum metric tensor
 components is poised to significantly advance our understanding of quantum
 geometric responses in a wide class of crystalline systems.
description: Theoretical data is collected from the density functional theory
 calculations and electronic band calculations based on the tight-binding
 model. Also, the experimental data is collected from the angle-resolved
 photoemission spectroscopy.
description: # Direct measurement of the quantum metric tensor in solids
 [https://doi.org/10.5061/dryad.jwstqjqmq](https://doi.org/10.5061/dryad.jwstqjqmq) We use three methods to collect the data: 1. We calculate the eigenstate and the eigenvalues of the model Hamiltonian based on the tight-binding model. For this calculation, we use Mathematica. This method is used to calculate the data in the folders Fig1, Fig3, Fig4, FigS1, FigS2, FigS11, and FigS12. 2. We directly calculate the electronic properties of the black phosphorus through the density functional theory calculations. This method is used to calculate the data in the folders Fig3, Fig4, FigS7, FigS8, FigS9, and FigS10. 3. We perform the angle-resolved photoemission spectroscopy (ARPES) for bulk black phosphorus and measure the photoemitted electron intensity. This data is in the folders Fig2, FigS4, FigS5, and FigS6. ## Description of the data and file structure Here, we explain how the data in each folder is calculated or measured. * Detailed description for the ARPES data We carried out ARPES measurements at Beamline 7.0.2 (MAESTRO), Advanced Light Source. This end-station is equipped with the cryogenic 6-axis manipulator and the hemispherical analyzer. The orientation of this electron analyzer is in situ rotatable, which makes it possible to measure the dispersion along kx and ky with the scattering plane unchanged. The photoemission scattering plane was aligned to the y (zigzag) axis in parallel with the glide-mirror plane of the sublattice pair AB, and the azimuthal angle of samples with respect to the y axis. Data were taken at the sample temperature of 20 ~ 80 K to reduce the thermal broadening with the photon energy of 86−104 eV, which covers half the Brillouin zone from Gamma to Z in the kz direction. With these settings, energy and k resolutions were better than 0.02 eV and 0.01 Å^(-1), respectively. The single-crystal bulk black phosphorus samples were glued on conductive sample holders with silver epoxy and transferred into the ultrahigh vacuum chamber with the base pressure of 5.0*10^(–11) torr. We cleaved samples in this condition to prepare a clean surface, which is scanned by the high-flux photon beam of about 50 μm in diameter to find optimal spots. * Fig1A We calculated the energy band structure of the monolayer black phosphorus. Although it has four band structure due to the four atoms in one unit cell, the two energy bands near the Fermi level can be descibed by two-by-two Hamiltonian. We listed the two energy bands near the Fermi level at red_line.txt file and other bands at black_line.txt file. We plot the energy band along the high-symmetry line and its unit is eV. Also, we attached the Mathematica code to reproduce the band structure.  The tight-binding hopping parameters to get the energy band are given in the paper "PHYSICAL REVIEW B 96, 155427 (2017)". Note that the Mathematica code is copied in Fig1A.pdf. * Fig2 Here, we listed the ARPES signal data for each polarization. In Fig2D.csv, and Fig2E.csv, the ARPES signals for the linearly polarized light are listed. In Fig2F.csv, and Fig2G.csv, the ARPES signals for the circularly polarized light are listed. In each file, the first column is ky (Å^(-1)), and the first row is kx(Å^(-1)). * Fig3C We calculated the energy band structure of the bulk black phophorus from the ab initio calculation. The first column is the crystal momentum along the high-symmetry line of the bulk black phosphorus with an unit of Å^(-1). The second column is the energy of the band with an unit of eV. The third column is the weight of the pz orbital for each band, which has no unit. The detailed description for the ab initio calculation is explained in Code/Software. * Fig3D We calculated the pseudospin of the black phosphorus using the tight-binding model. In thetaTB.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the pseudospin angle. Also, we uploaded the Mathematica file to reproduce the pseudospin angle data. Note that the Mathematica code is copied in Fig3D.pdf. * Fig3E From the ARPES signal, we calculated the pseudospin angle based on the plane wave approximation. In Epeak_pseudospin.csv, the first column is ky (Å^(-1)), the first row is kx (Å^(-1)), and the others are pseudospin angle at each kx, and ky. Also, we uploaded the Mathematica file to reproduce the Figure 3E. Note that the Mathematica code is copied in Fig3E.pdf. * Fig3F From the ARPES signal, we calculated the pseudospin angle without supposing the plane wave final state. In polarization_LV.csv, the first row is the photon energy with (-1.2450725 + j 0.001611992)eV, the first column is the (kx,ky) with {-0.71889 + (0.0092165)(j - 1)Å^(-1), -0.72965 + (0.011401)*(i - 1)Å^(-1)} for 143i+j th row, and the others are corresponding ARPES signal for the x-directional polarized light. Note that the value NAN represents 'Not a number' which means that the ARPES signal is too small to be captured in the detector. In pseudospin_ARPES.csv, the first column is ky (Å^(-1)), the first row is kx (Å^(-1)), and the others are the energy band of the bulk black phosphorus. In Projection_pol.csv, the first column is ky (Å^(-1)), the first row is kx (Å^(-1)), and the others are the corresponding APRES signal for the x-directional polarized light at the peak of the energy distribution curve. In thetaGFS.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the pseudospin angle calculated from ARPES signal without supposing the plane wave final state. Also, we uploaded the Mathematica file to reproduce thetaGFS.txt. Note that the Mathematica code is copied in Fig3F.pdf. * Fig4A-D From the pseudospin angle extracted from the ARPES signal, we calcaulated the quantum metric tensor. polarization_LV.csv, Projection_pol.csv, and pseudospin_ARPES.csv files are the same as Fig. 3F. In Fig4A.txt, Fig4B.txt, Fig4C.txt, and Fig4D.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the components of the quantum metric tensor, Gxx, Gxy, Gyy, and Tr[G], at each (kx,ky). Also, we uploaded the Mathematica file to reproduce Fig4A.txt, Fig4B.txt, Fig4C.txt, and Fig4D.txt. Note that the Mathematica code is copied in Fig4A-D.pdf. * Fig4E-H In Quantum_metric.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), the third column is kz (Å^(-1)), the fourth, fifth, and sixth column are ab initio calculated quantum metric tensor, Gxx, Gxy, Gyy, at each (kx,ky). The detailed description of the ab initio calculation is explained in Code/software. * FigS1 By changing the hopping parameter of the black phosphorus, we fit the energy band to the ARPES experimental data. In VB_dispersion.csv, the second column is ky (Å^(-1)), the second row is kx (Å^(-1)), and the others are the energy band extracted from the ARPES signal with a unit of eV. In FigS1A.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the energy band extracted from the ARPES signal with a unit of eV. In Blueline_FigS1B.txt, the first column is the momentum point from (kx,ky)=(0Å^(-1), -0.15Å^(-1)) to (0Å^(-1),0Å^(-1)) to (0.15Å^(-1),0Å^(-1)) with a unit of Å^(-1), the second column is the energy band calculated from the tight-binding model in the paper "PHYSICAL REVIEW B 96, 155427 (2017)". In Blueline_FigS1B.txt, the first column is the same as that of Blueline_FigS1B.txt, and the second column is the energy band calculated from the tight-binding model fitted to the ARPES signal. In Reddot_FigS1B.txt and Reddot_FigS1C.txt, the first column is the same as that of Blueline_FigS1B.txt, and the second column is the energy band extracted from the ARPES signal. Also, we uploaded the Mathematica file to reproduce text files. Note that the Mathematica code is copied in FigS1.pdf. * FigS2 We calculated the energy band and the quantum metric tensor of the bulk black phosphorus while changing z-directional momentum. In blackline_FigS2A.txt, and blackline_FigS2E.txt, the energy band (unit: eV) along the high-symmetry line with kz=0, 0.3pi/lz (lz: distance between the phosphorus layers). In FigS2B.txt, FigS2C.txt, and FigS2D.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the components of the quantum metric tensor, Gxx, Gxy, and Gyy, at kz=0. In FigS2F.txt, FigS2G.txt, and FigS2H.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the components of the quantum metric tensor, Gxx, Gxy, and Gyy, at kz=0.3pi/lz. Also, we uploaded the Mathematica file to reproduce text files. Note that the Mathematica code is copied in FigS2.pdf. * FigS4 (FigS4A.csv, FigS4B.csv, FigS4C.csv, and FigS4D.csv) Energy dispersion of black phosphorus taken along kx with the photon polarization of LH (FigS4A.csv) and LV (FigS4B.csv). By subtracting those in A and B, we obtained the LD-ARPES intensity pattern in FigS4C.csv. Likewise, we took the CD-ARPES intensity pattern in FigS4D.csv by subtracting those taken with LC-polarized light and RC-polarized light. The first column is the energy (eV), and the first row is kx (Å^(-1)) (FigS4E.csv, FigS4F.csv, FigS4G.csv, and FigS4H.csv) Constant energy maps taken at –0.54 eV with light polarization as marked at the upper left of each panel. The first column is the ky (Å^(-1)), and the first row is kx (Å^(-1)). * FigS5 LD-ARPES (FigS5A.csv) and CD-ARPES (FigS5B.csv) intensity pattern of black phosphorus taken by varying the photon energy in the range of 85–104 eV (from Gamma to Z in kz). The binding energy is set at 0.37 eV. The first column is the photon energy (eV), and the first row is kx (Å^(-1)). * FigS6 LD-ARPES (FigS6A.csv, FigS6B.csv, and FigS6C.csv) and CD-ARPES (FigS6D.csv, FigS6E.csv, and FigS6F.csv) intensity pattern of black phosphorus taken with the scattering plane of 0 deg in FigS6A.csv and FigS6D.csv, 15 deg in FigS6B.csv and FigS6E.csv, and 30 deg in FigS6C.csv and FigS6F.csv with respect to the ky axis. The binding energy is set at 0.54 eV. The first column is the ky (Å^(-1)), and the first row is kx (Å^(-1)). * FigS7 From the ab initio calculation, we calculated the quantum metric tensor with two-band approximation and the general equation. In Quantum_metric.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), the third column is kz (Å^(-1)), the fourth, fifth, and sixth column are ab initio calculated quantum metric tensor, Gxx, Gxy, Gyy, at each (kx,ky). In Quantum_metric_10-11.txt, the meaning of each column is the same as that of the corresponding column in Quantum_metric.txt, but calculated based on the two-band approximation. * FigS8 From the ab initio calculation, we directly calculated the band structure of the bulk black phosphorus using the DFT and Wannier tight-binding model, respectively. In DFTband.txt, the first column is the momentum point along the high-symmetry line of the bulk black phosphorus with an unit of Å^(-1), and the second column is the energy band calculated from DFT with an unit of eV. In WANband.txt, each column has the same meaning of that of DFTband.txt, but calculated from Wannier tight-binding model. In S8-Fatband-px.txt, S8-Fatband-py.txt, and S8-Fatband-pz.txt, the weights of px, py, and pz orbitals for each band are listed. * FigS9 From the Wannier tight-binding model calculation, we calculated the quantum metric tensor with two-band approximation and the general equation. In Quantum_metric_tot.txt, the first column is kx (Å^(-1)), the second column is ky (Å^(-1)), the third column is kz (Å^(-1)), the fourth, fifth, and sixth column are Wannier model calculated quantum metric tensor, Gxx, Gxy, Gyy, at each (kx,ky). In Quantum_metric_10-11.txt, the meaning of each column is the same as that of the corresponding column in Quantum_metric.txt, but calculated based on the two-band approximation. * FigS10 We simulated the ARPES signal based on the plane wave approximation. In ARPES_data_011.txt, and ARPES_data_100.txt, the first column is  kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the simulated ARPES intensity for the light polarization (0,1,1), and (1,0,0), respectively. They are plotted in arpes_011_plot.png and arpes_100_plot.png, respectively. And they are plotted in ARPES-sim_-0.54Ef_new.png with an red box [-0.4Å^(-1), 0.4Å^(-1)]*[-0.6Å^(-1), 0.6Å^(-1)]. * FigS11 We fit the energy distribution curve of ARPES signal by the Lorentian function. In Bluedot_FigS11.txt, the first column is the energy (eV), and the second column is ARPES signal at (kx,ky,kz)=(-0.141Å^(-1),-0.695Å^(-1), 0.588Å^(-1)). In Blackline_FigS11.txt, the first column is the energy (eV), and the second column is values of the Lorentizian function fitted to the ARPES signal. We uploaded the Mathematica file to reproduce text files. Note that the Mathematica code is copied in FigS11.pdf. Also,  polarization_LV.csv, and pseudospin_ARPES.csv files are the same as Fig. 3F.  * FigS12 We calculated the quantum metric tensor from the pseudospin angle estimated based on the plane wave approximation. In FigS12A.txt, FigS12B.txt, and FigS12C.txt, the first column is  kx (Å^(-1)), the second column is ky (Å^(-1)), and the third column is the components of the quantum metric tensor, Gxx, Gxy, and Gyy, at (kx,ky) based on plane wave approximation. In Epeak_pseudospin.csv, the first column is ky (Å^(-1)), the first row is kx (Å^(-1)), and the others are pseudospin angle estimated based on the plane wave approximation. Also, we uploaded the Mathematica file to reproduce text files. Note that the Mathematica code is copied in FigS12.pdf. ### Files and variables #### File: FigureData.zip **Description:** In this file, we list folders and Excel files whose names are the figures in the paper. In each folder and file, we include the raw data and code to reproduce that figure. ## Code/software We use Mathematica ver. 10.4 to calculate the energy band and the quantum metric tensor based on the tight-binding model of the black phosphorus. DFT details: Vienna ab-initio simulation package (VASP) using plane-wave basis set are used to perform density-functional theory (DFT) calculations. HSE06 hybrid functional are used for electronic band structure calculations. Wannier90 (*Phys. Rev. B* **65**, 035109 (2001)) was used to obtain maximally localized Wannier functions with ![](file:///C:/Users/김선제/AppData/Local/Temp/msohtmlclip1/01/clip_image002.gif) orbitals as projector. In DFT, quantum metric tensor is computed by reading the Bloch wave functions from output WAVECAR similar as what VASPBERRY code (*Phys. Rev. Lett*. **128**, 046401 (2022)) does. In tight-binding model level, quantum metric tensor can be calculated using Wannier Hamiltonian. The ARPES simulation is done by using *chinook* code considering plane wave final states approximation (Day, R.P., Zwartsenberg, B., Elfimov, I.S. & Damaselli, A. Computational framework *chinook* for angle-resolved photoemission spectroscopy. *npj Quantum Materials* **4**, 54 (2019)).