Observation of ultrahigh mobility surface states in a topological crystalline insulator by infrared spectroscopy

Topological crystalline insulators possess metallic surface states protected by crystalline symmetry, which are a versatile platform for exploring topological phenomena and potential applications. However, progress in this field has been hindered by the challenge to probe optical and transport properties of the surface states owing to the presence of bulk carriers. Here, we report infrared reflectance measurements of a topological crystalline insulator, (001)-oriented Pb1−xSnxSe in zero and high magnetic fields. We demonstrate that the far-infrared conductivity is unexpectedly dominated by the surface states as a result of their unique band structure and the consequent small infrared penetration depth. Moreover, our experiments yield a surface mobility of 40,000 cm2 V−1 s−1, which is one of the highest reported values in topological materials, suggesting the viability of surface-dominated conduction in thin topological crystalline insulator crystals. These findings pave the way for exploring many exotic transport and optical phenomena and applications predicted for topological crystalline insulators.

with several months between one another. All the data in the main text are from sample 2.
Supplementary Figure 1a shows the reflectance ratio data R sample /R Al at zero field obtained in Exp #1, which is the reflectance of sample divided by that of a reference aluminum mirror. Because of the lack of in situ gold coating in this experiment, the R sample /R Al spectra are not absolute R(ω).
Nevertheless, the plasma minimum (dip in R sample /R Al ) at the screened plasma frequency ! = ! / ! shown in Supplementary Figure 1a should be fairly accurate owing to the frequency independent R(ω) of aluminum in this range. Here, the bare plasma frequency ! is given by ! ! = 4 ! / , ! represents all high-energy contributions to the dielectric constant other than the free carrier contribution. The data in Supplementary Figures 1a and 1b show that the peak in the , / ( , = 0T) spectrum is exactly at the plasma minimum frequency ! at zero field measured in the same experiment.
In order to obtain the absolute R(ω) spectrum at zero field, we performed Exp #2 with in situ gold coating technique, after which , / ( , = 0T) were measured again in Exp #3. We find that ! changes slightly from ~34.35 meV to ~32.49 meV (Supplementary Figure 1c), and then to ~31 meV in Exp #1-3. This observation probably arises from self-doping due to Se vacancies caused by the so-called "Se loss phenomenon" in selenides, namely, thermal energy assisted surface Se atom escape, which is very common in selenides such as TiSe 2 , SnSe and Bi 2 Se 3 as discussed in ref. [1] and references therein. The sample is cooled down to ~4.5K or ~10K then warmed up to 300K for multiple times during different measurements, between which it is briefly exposed to air. Every thermal cycling could introduce Se loss, especially for the Se atoms at the surface with the weakest 12 chemical bonding. The Se loss effect could lead to changes in the carrier density n and therefore the change of ! with time, since ! ∝ . Importantly, we find that the frequencies of the features in , / ( , =0T) below 60 meV change proportionally with ! . As shown in It has been a longstanding challenge in the research of selenides to control of the sample quality.
Even for samples with the same nominal Pb/Sn ratio, the actual carrier density is sample dependent, which depends strongly on the self-doping effect of Se vacancy defects in each crystal. In the Bridgman crystal growth, the details can be slightly different for every batch, such as vacuum control and the specific solidification temperature at the solid-liquid interface. Moreover, the vacancy defect has higher density closer to the surface. All of these factors fluctuate in each growth, which leads to sample-dependent defect density and therefore carrier density. The samples used in our IR study are from a batch with low carrier density, but the carrier density is batch dependent even for the same nominal Pb/Sn ratio. The lineshape of , is determined by real and imaginary parts of !! and !" based on Eq.(4) in the main text. Although there is a sharp peak due to SS in Re !! , the contributions from all real and imaginary parts of !! and !" in Supplementary Figure 3 and 4 lead to a relatively smooth lineshape for , in this spectral range, which is a consequence of the spectral properties (lineshapes) of optical response functions typically described by magneto-Drude-Lorentz oscillators.

Supplementary Note 3. Comparing R(ω, B) data with the conventional magnetoplasma effect in semiconductors
Conventional doped semiconductors exhibit the so-called magnetoplasma effect in magnetic field [5,6]: the plasma minimum (edge) in the R(ω) spectra in zero field splits into two minimums (edges) in R(ω, B) in magnetic field, with the high (low) energy edge moving to higher (lower) energies with increasing field. This effect is observed in the regime of ! << ! , where ! is the frequency of the plasma minimum in the R(ω). Such a behavior in R(ω, B) spectrum arises from a single CR mode at ! in the optical conductivity !! , that is well separated in energy from other resonances such as interband LL transitions [5,6]. In our measurements of Pb 1−x Sn x Se, the observed features in R(ω, B) below 50 meV are entirely different from the magnetoplasma effect described above. Our observation results from the overall contributions of two resonances in this energy range: the CR mode of the SS and the LL +0 →LL +1 transition from the bulk states (Fig. 4c of the main text).
We stress that the LL +0 →LL +1 transition around 30-40 meV is the LL transition with the lowest energy from the bulk states, which can not produce the dramatic changes below ~25 meV in R(ω, B) with increasing magnetic field. The latter observation suggests the existence of a low energy resonance below the LL +0 →LL +1 transition of the bulk, which we demonstrate to be the CR mode of the SS. 16

Supplementary Note 4. Scattering rate and mobility of the surface states
As illustrated in Supplementary Figure 5, the width of the dip feature around 32 meV in R(ω, B) is directly related to the width of the CR mode of the SS in Re !! spectra. Therefore, the very narrow dip feature around 32 meV in the R(ω, B) data suggests that the scattering rate for the SS 1/ !! is very low. We find that Re !! spectra with 1/ !!~1 .2 ± 0.6 meV for the surface It is instructive to compare the surface mobility in TCIs to the mobility of graphene [7][8][9] since both materials feature massless Dirac fermions. In a simple Drude model, the mobility of carriers in graphene is given by [8]: where = ! / ! ! . In a previous study [9] of graphene samples with carrier density n~ 4.7×10 !" cm -2 (corresponding to m~0.04 m e ), the scattering rate is found to be 1/~30 meV based on IR study of Landau level transitions, which yields an IR mobility of !"~5 ,700 cm 2 V -1 s -1 based on Supplementary Eq. (1). The estimated !" is in reasonable agreement with the reported DC mobility !"~4 ,000 cm 2 V -1 s -1 [9]. The lower value of !" compared to !" might arise from disorder effects as discussed in [7]. The good agreement between !" and !" in graphene [9] supports our estimation of the surface mobility !! in TCIs using an equation similar to Supplementary Eq. (1). Moreover, a scattering rate of 1/~2 meV was observed in IR measurements of graphene/BN with mobility ~50,000 cm 2 V -1 s -1 [7]. Therefore, the scattering rate and mobility from previous studies of graphene [7,9] provide further support for our estimation of !!~4 0,000 cm 2 V -1 s -1 in TCIs.

Supplementary Note 5. Analysis of total Drude spectral weight in zero field
We use the Drude-Lorentz model to fit the data in zero field. One typical fit to and the corresponding ! are shown in Supplementary Figure 6 with the parameters for all oscillators in the model summarized in Supplementary Table 3. The optical conductivity = ! +i ! from Drude-Lorentz fit is consistent with that evaluated from Kramers-Kronig (KK) transformation of R(ω). The total Drude spectral weight (SW !"!#$ ) can be directly determined by the bare plasma frequency ! of the Drude oscillator from SW !"!#$ = The total (observed) plasma frequency ! for the Drude mode can be obtained from fitting the entire R(ω) spectrum shown in Supplementary Figure   6a, because it is related to the screened plasma frequency ! by: where !~2 60 cm !! corresponds to the plasma minimum in R(ω) and ! is determined by all Lorentzian oscillators obtained from fitting the overall R(ω) spectrum above ! . ! represents all electronic contributions to the dielectric constant other than the Drude conductivity, which is given by (see for example, [10]): where P denotes the Cauchy principal value, frequencies are in cm −1 , ! is in Ω !! cm !! , and ! is a frequency separating the Drude mode and the interband transitions ( !~2 00 cm !! ). The high-energy contribution of ! to ! is negligibly small because of the denominator of the integrand in Supplementary Eq. (3). Taking into account the uncertainties of ! from both Drude-Lorentz fit and KK transformation of R(ω) as well as those for ! , we estimate !~4 5 ± 9 using Supplementary Eq. (3).
In fact, the lineshape and absolute value of the R(ω) spectrum above ! directly reflects the value of ! . Because the R(ω) spectrum shows a typical metallic behavior near and below ! , we can use a simple Drude model to illustrate the dependence of R(ω) on ! . Supplementary Figure 7 displays model R(ω) spectra calculated from the Drude model different values of ! , which shows that the R(ω) spectrum above ! increases with increasing value of ! . Although R(ω) spectra from the Drude model in Supplementary Figure 7 only intends to qualitatively illustrate the effect of ! , it shows that the absolute value of the experimental R(ω) spectrum above ! can be reproduced by !~5 2 ± 10, which is in good agreement (within 15%) with the result from our full analysis ( !~4 5 ± 9) using Drude-Lorentz model. Therefore, our analysis demonstrates that ! = ! !~1 744 ± 174 cm −1 and SW !"!#$ ≈ (7.9 ± 1.6)× 10 ! Ω !! cm !! . This method of evaluating SW !"!#$ doesn't rely on the integral ! ! ! ! , so it is not limited by the lack of information on ! for the Drude mode below ~7meV.