Supplementary material for “Low-energy excitations in type-II Weyl semimetal
T
d-MoTe
2evidenced through optical conductivity”
D. Santos-Cottin,1,∗ E. Martino,1, 2 F. Le Mardelé,1 C. Witteveen,3, 4 F. O. von Rohr,3, 4 C. C. Homes,5 Z. Rukelj,1, 6 and Ana Akrap1,†
1Department of Physics, University of Fribourg, 1700 Fribourg, Switzerland 2
IPHYS, EPFL, CH-1015 Lausanne, Switzerland
3
Department of Chemistry, University of Zürich, CH-8057 Zürich, Switzerland
4Physik-Institut der Universitat Zürich, CH-8057 Zürich, Switzerland 5
Condensed Matter Physics and Materials Science Division, Brookhaven National Laboratory, Upton, New York 11973, USA
6Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, HR-10000 Zagreb, Croatia (Dated: December 27, 2019)
In the Supplementary Materials we show the Drude Lorentz fits of the optical conductivity, and the spectral weight.
DRUDE LORENTZ FITS OF OPTICAL CONDUCTIVITY
The optical conductivity has been fit using a non-linear least-squares technique. Typically, only the real part of the conductivity would be fit, using the residual rj = |σ1,j − f1,j|, where j = 1 .. len(σ1) indexes the
experimentally-determined data, and ˜f (ω) is the (complex) function being fitted. Starting with the dielectric function, ˜(ω) = 1(ω) + i2(ω), the complex conductivity is defined ˜σ(ω) = σ1(ω) + iσ2(ω) = 2πi [∞− ˜(ω)]/Z0, where ∞ is the real
part of the dielectric function at high frequency, and Z0' 377 Ω is the permittivity of free space.
0 500 1000 1500 2000 2500 3000 Wave number (cm−1) 0 1 2 3 4 σ 1 ( ω ) (1 0 3 Ω − 1 cm − 1 )
MoTe2(Ekab) Data (10 K) Fit
0 100 200 300
Photon energy (meV)
0 1000 2000 −1 0 1 2 3 σ 2 ( ω ) (1 0 3Ω − 1cm − 1)
Fig. S1. The Drude-Lorentz model fits (dashed line) to the real and imaginary (inset) parts of the in-plane optical conductivity of MoTe2 (solid line) decomposed into the narrow and broad Drude components, as well as several bound excitations at 10 K.
Using a simple Drude model for the free carriers, ˜(ω) = ∞− ωp2/(ω2+ iω/τ ), where ωp is a plasma frequency, and
1/τ is the scattering rate, the real and imaginary parts of the conductivity are (neglecting the contribution from ∞), σ1(ω) = σ0 1 + ω2τ2, and σ2(ω) ' σ0ωτ 1 + ω2τ2, ∗david.santos@unifr.ch †ana.akrap@unifr.ch Typeset by REVTEX
2 0 500 1000 1500 2000 2500 3000 Wave number (cm−1) 0 1 2 3 4 σ 1 ( ω ) (1 0 3 Ω − 1 cm − 1 )
MoTe2(Ekab) Data (300 K) Fit
0 100 200 300
Photon energy (meV)
0 1000 2000 −1 0 1 σ 2 ( ω ) (1 0 3Ω − 1cm − 1)
Fig. S2. The Drude-Lorentz model fits (dashed line) to the real and imaginary (inset) parts of the in-plane optical conductivity of MoTe2(solid line) decomposed into the narrow and broad Drude components, as well as several bound excitations at 300 K.
where σ0= 2π ωp2τ /Z0. If the scattering rate is small, say 1/τ ' 1 meV, then it would not be unusual for most of the
real part of the Drude optical conductivity to fall below the lowest measured frequency, making it difficult to fit the Drude parameters. However, the presence of a frequency term in the numerator of the imaginary part of the Drude conductivity results in σ2 being considerably broader than σ1, allowing even very small (sub-meV) values of 1/τ to
be reliably determined. We have therefore fit the real and imaginary parts of the optical conductivity simultaneously using the residual
rj = |σ1,j− f1,j| + |σ2,j− f2,j|.
The results of the fit using the two-Drude model to the real and imaginary parts of the optical conductivity of MoTe2
for light polarized in the a-b planes at 10 and 300 K, shown in Figs. S1and S2, respectively, are decomposed into the individual Drude and Lorentz contributions; the agreement between data and fit is excellent.
SPECTRAL WEIGHT
The spectral weight is defined here as the integral of the real part of the optical conductivity from the lowest measured frequency (∼ 0) up to a chosen cutoff frequency ω:
SW (ω) = Z ω
0
σ1(ω0)dω0.
The temperature-dependence of the low-energy spectral weight is shown in Fig. S3. The dramatic reduction of the Drude contribution from 10 to 300 K results in a significant transfer of spectral weight from the far-infrared to the mid-infrared region, evident in Fig.S3above and below ' 100 meV.
3 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 0 1 2 3 4 S p e c tr a l w e ig h t ( 10 6 Ω -1 c m -2 ) P h o t o n e n e r g y ( m e V ) 3 0 0 K 2 7 5 K 2 5 0 K 2 2 5 K 2 0 0 K 1 7 5 K 1 2 5 K 7 5 K 5 0 K 2 5 K 1 5 K 1 0 K
Fig. S3. The temperature dependence of the spectral weight as a function of photon energy, showing the transfer of spectral weight from high to low energy with decreasing temperature.
