Supplementary Figures a Sample A Sample Sample B mm Sample A a Sample B Supplementary Figure : Laue patterns and piture of the single rystals. (a,) Laue patterns of sample A (a) and sample B (). () Piture of the single rystals (sample A and sample B), whose rystallographi orientations are shown with white arrows.
ρ (Ωm)... ρ (Ωm).. 2 3 4 5 T (K) (K) a-axis (sample B) -axis (sample A) 2 3 T (K) Supplementary Figure 2: Temperature dependene of the eletrial resistivity along the -axis (sample A) and a-axis (sample B). 2
a - mm - - 2- - S - S d - 2-2 2 Supplementary Figure 3: Sample piture and Laue patterns. (a-d) Sample piture and Laue patterns of samples S (sample ross setion F = 25 245 µm 2 ), (F = 2 25 µm 2 ), and (F = 8 6 µm 2 ). From these Laue patterns, we identify the rystallographi orientation of the rystal ars. Transport properties were measured in (a,)-plane for S and and along -axis for. 3
a 3 µm S5 27 µm S4 24-23- 33-22- 32-2- - Supplementary Figure 4: Sample piture and Laue patterns. (a,) Sample piture and Laue pattern of samples S4 (F = 27 25 µm 2 ) and S5 (F = 3 6 µm 2 ), whih were prepared y utting a single rystal into two piees. The sample piture and Laue pattern are taken from one side of the single rystal. 4
ρ (Ωm) - -3 ~ 3 mev S S4 and S5 S : 2 ~ 4 mev : 2 ~ 4 mev : 2 ~ 3 mev S4 and S5 : 2 ~ 4.2 mev..2 /T (K - ) Supplementary Figure 5: Eletrial resistivity as a funtion of /T. Two energy gaps aove 3 K ( 3 mev) and elow K ( 2 4 mev for S and, 3 mev for, and 4.2 mev for S4 and S5) were evaluated using the thermally ativated funtion ρ = ρ exp( 2k B T ) as plotted y yellow lines. The previous studies of impurity effets and magnetotransport properties suggest that is the gap etween the ondution and valene ands formed y Fe 3d oritals, and 2 is the gap etween the ondution and impurity ands. 3 5
a ρ (Ωm) xx ρ xx (Ωm) m) ρ xy (Ωm) d ρ (Ωm) xx xx(ω ρ xx(ω ρ xx(ω 2.5.5 2 3 4 5 6 7 µ.8.6.4.2 2 3 4 5 6 7 µ.5 ρ.5 xx(ω ρ S 2 3 4 5 6 7 µ 2.5.5 S4 and S5 2 K 3 K 25 K 8 K K 5 K 2 3 4 5 6 7 µ e ρ xy (Ωm) f ρ xy (Ωm) g ρ xy (Ωm) -. -.2 -.3 -.4 ρ -.5 ρ yx (Ω yx (Ω -.6 K -.7 S 8 K -.8 2 3 4 5 6 7 µ -. -.2 -.3 -.4 -.5 -.6 -.7 2 3 4 5 6 7 µ -.5 -. -.5 -.2 -.25 ρ -.3 yx (Ω 2 K 5 K 3 K 25 K -.35 -.4 2 3 4 5 6 7 µ g ρ xy (Ωm) -.2 -.4 -.6 ρ yx (Ω -.8 - S4 and S5 2 3 4 5 6 7 µ Supplementary Figure 6: Magneti field dependene of the resistivity tensors ρ xx and ρ yx. (a-d) Magneti field dependene of the resistivity tensor ρ xx of S,,, and S4(and S5). (e-f) Magneti field dependene of the resistivity tensor ρ xy of S,,, and S4(and S5). 6
a - σ (/Ωm) - ) xy - σ xy (/Ωm) - ) σ xy (Ω -.2 -.4 -.6 -.8 - -.2 2 K 25 K -.4 2 3 4 5 6 7 µ -.5 - -.5-2 S 8 K K 3 K 5 K σ -2.5-3 -3.5 2 3 4 5 6 7 µ xy (Ω σ xy (/Ωm) - - ) d σ (/Ωm) - xy - ) -.2 -.4 -.6 -.8 - σ -.2 xy (Ω -.4 -.6 2 3 4 5 6 7 µ S4 and S5 -.5 - σ -.5 xy (Ω -2 2 3 4 5 6 7 µ Supplementary Figure 7: Magneti field dependene of the ondutivity tensor σ xy. (a-d) Magneti field dependene of the ondutivity tensor σ xy of S,,, and S4(and S5). Solid urves are the alulation using Supplementary Eq. (see Supplementary Note ). 7
a µ µ (m (m 2 2 V 2 /Vs) - s - ) n (/m (m -3 3 ) 6 5 4 4 S ρ~exp( /2k B T) S ~ 3.5 mev ~ 4.5 mev ~ 4. mev S4 and S5 ~ 3.8 mev S4 and S5.5..5 /T (/K) 3 T (K) T -3/2 - C -2-3.5..5 T (K) Supplementary Figure 8: Temperature dependene of the physial parameters. (a-) Temperature dependene of the arrier onentration n, moility µ, and low moility omponent C evaluated from the fitting results shown in Supplementary Fig. 7. The energy gap is evaluated from n (T) using the ativation funtion as shown y dotted lines in Supplementary Fig. 8a. The slight size dependene of the energy gaps (3.5meV < 2 < 4.5 mev) represents that the eletroni states of all samples are almost idential: the onentration and/or level of the impurity states are slightly different. Sine the analysis of n an exlude the ontriution of the moility, the energy gaps evaluated with this method are more preise ompared with those evaluated from the eletrial resistivity as shown in Supplementary Fig. 5. n and µ are almost independent of the rystal size and rystallographi orientation. µ inreases with dereasing the temperature with the slope of T 3/2, suggesting that the eletrons mainly sattered y phonons. The ontriutions of the low-moility omponents C to σ xy are muh smaller (less than %) than those of the high-moility arriers, and thus the high moility arriers are predominant to the transport properties. 8
S el (mv/k) K - ).8.6.4 S S4 and S5.2 2 3 4 T (K) Supplementary Figure 9: Seeek oeffiient of the eletron-diffusion part. Seeek oeffiient alulated from the Supplementary Eq. 2 (see Supplementary Note 2). 9
a S Fe a () (2) Intensity (ar. unit) (2) (2) () () (2) (22) (3) (2) (3) (22) (3) 3 4 5 6 2θ (deg.) Supplementary Figure : Crystal struture and powder X-ray diffration pattern. (a,) Crystal struture and powder X-ray diffration pattern of pulverized single rystals of FeS 2 after removing S-flux with nitri aid. Crystal struture of FeS 2 is the marasite-type FeS 2 struture with the spae group Pnnm (a = 5.83275, = 6.5334, = 3.963).
Supplementary Notes Supplementary Note : Sample dependene of the eletri transport properties The eletroni states elow 3 K an e preisely analyzed from the magnetotransport properties. As shown in Ref., the arrier onentration n and moility µ are evaluated from the magneti field H dependene of the ondutivity tensor σ xy [= ρ yx /(ρ 2 xx + ρ 2 yx)] with a two-arrier model desried as σ xy (H) = n xy eµ 2 xyh [ ] + (µ xy H) + C, () 2 in whih one arrier is of high moility and the other is of low moility (µh ). Here, C is the low-moility omponent. Supplementary Fig. 6 shows the H dependene of the resistivity tensors ρ xx and ρ yx. The est agreements of σ xy with Supplementary Eq. is shown y lak solid urves in Supplementary Fig. 7. Supplementary Note 2: Seeek oeffiient of the eletroni part Here, we disuss the Seeek oeffiient of the eletron-diffusion part. Sine this ompound shows the lear energy gap and small arrier onentration, the Seeek oeffiient an e alulated with a nondegenerate model expressed as 4 S el = ± k B [η (r + 5/2)]. (2) e Here, η is the redued Fermi energy written as n = 2( 2πm k B T ) 3/2 expη, and r is a sattering param- h 2 eter. In this alulation, we use n evaluated from σ xy, r = 3/2, and m = 5.4m (m is the are eletron mass), whih is estimated from the ylotron resonane experiment. Supplementary Fig. 9 plots the temperature dependene of the Seeek oeffiient evaluated from Supplementary Eq. 2. The Seeek oeffiient is almost mv/k from 8 to 3 K, whih is muh smaller than that of the experimental results. This result indiates that the eletron-diffusion part of the Seeek oeffiient plays a minor role in our ompounds. Supplementary Referenes. Takahashi, H. et al. Low-temperature magnetotransport of the narrow-gap semiondutor FeS 2. Phys. Rev. B 84, 2525 (2). 2. Takahashi, H. et al. Origin of the energy gap in the narrow-gap semiondutor FeS 2 revealed y high-pressure magnetotransport measurements. Phys. Rev. B 88, 6525 (23).
3. Takahashi, H. et al. Effets of ppm-level imperfetion on the transport properties of FeS 2 single rystals. J. Phys. S o. Jpn. 8, 5478 (2). 4. Sun, P. et al. FeS 2 : Prototype of huge eletron-diffusion thermoeletriity. Phys. Rev. B 79, 5338 (29). 5. Petrovi, C. et al. Anisotropy and large magnetoresistane in the narrow-gap semiondutor FeS 2 Phys. Rev. B 67, 5525 (23). 6. Jie, Q. et al. Eletroni thermoeletri power fator and metal-insulator transition in FeS 2. Phys. Rev. B 86, 52 (22). 2