We report the electric transport study of the three-dimensional topological
insulator TlBiSe$_2$. We applied a newly developed analysis procedure and
precisely determined two-carrier transport properties. Magnetotransport
properties revealed a multicarrier conduction of high- and low-mobility
electrons in the bulk, which was in qualitative agreement with angle-resolved
photoemission results~[K. Kuroda $et~al.$, Phys. Rev. Lett. $\bm{105}$, 146801
(2010)]. The temperature dependence of the Hall mobility was explained well
with the conventional Bloch-Gr{\"u}neisen formula and yielded the Debye
temperature $\varTheta_{\rm{D}}=113 \pm 14$~K. The results indicate that the
scattering of bulk electrons is dominated by acoustic phonons.Comment: 6 pages, 5 figures, to be published in Physical Review 

Ando, Y.

Eguchi, G.

Kimura, A.

Kuroda, K.

Shinjo, T.

Shirai, K.

Shiraishi, M.

Physical Review B

English

arXiv

We report the electric transport study of the three-dimensional topological insulator TlBiSe2. We propose an analysis procedure which determines the two-carrier transport properties in precision. Magnetotransport properties reveal a multicarrier conduction of high- and low-mobility electrons in the bulk, which is in qualitative agreement with angle-resolved photoemission results [K. Kuroda et al., Phys. Rev. Lett. 105, 146801 (2010)]. The temperature dependence of the Hall mobility is explained well with the conventional Bloch-Grüneisen formula and yields the Debye temperature Θ(D)=113±14 K. The results indicate that the scattering of bulk electrons is dominated by acoustic phonons

Kyoto University Research Information Repository

Title Precise determination of two-carrier transport properties in thetopological insulator   TlBiSe 2Author(s)Eguchi, G.; Kuroda, K.; Shirai, K.; Ando, Y.; Shinjo, T.;Kimura, A.; Shiraishi, M.CitationPhysical Review B (2015), 91(23)Issue Date2015-06-11URL http://hdl.handle.net/2433/201544Right©2015 American Physical SocietyType Journal ArticleTextversionpublisherKyoto UniversityPHYSICAL REVIEW B 91, 235117 (2015)Precise determination of two-carrier transport properties in the topological insulator TlBiSe2G. Eguchi,1,* K. Kuroda,2,† K. Shirai,2 Y. Ando,1 T. Shinjo,1 A. Kimura,2 and M. Shiraishi11Department of Electronic Science and Engineering, Graduate School of Engineering, Kyoto University, Kyoto 615-8510, Japan2Graduate School of Science, Hiroshima University, Kagamiyama, Higashi-Hiroshima 739-8526, Japan(Received 6 April 2015; revised manuscript received 11 May 2015; published 11 June 2015)We report the electric transport study of the three-dimensional topological insulator TlBiSe2. We proposean analysis procedure which determines the two-carrier transport properties in precision. Magnetotransportproperties reveal a multicarrier conduction of high- and low-mobility electrons in the bulk, which is in qualitativeagreement with angle-resolved photoemission results [K. Kuroda et al., Phys. Rev. Lett. 105, 146801 (2010)]. Thetemperature dependence of the Hall mobility is explained well with the conventional Bloch-Gru¨neisen formulaand yields the Debye temperature D = 113 ± 14 K. The results indicate that the scattering of bulk electrons isdominated by acoustic phonons.DOI: 10.1103/PhysRevB.91.235117 PACS number(s): 72.10.Bg, 72.10.Di, 72.80.−rI. INTRODUCTIONElectric and spin transport in three-dimensional (3D)topological insulators has received a great deal of attentionbecause of the insulators’ surface Dirac fermions with theunique spin texture. The presence of the spin texture hasbeen confirmed by several experiments employing angle-resolved photoemission spectroscopy (ARPES) and a varietyof spin-dependent transport have been proposed [1–8]. In realmaterials, however, unavoidable bulk conduction often hinderssurface conduction.Electric transport in 3D topological insulators has beenanalyzed with the so-called two-carrier model, i.e., the parallelconduction of carriers with different mobility [9–11]. Withinthe model, the resistivity ρxx in a magnetic field B = μ0H isexpressed byρxx(B) = n1q1μ1 + n2q2μ2 + (n1q1μ2 + n2q2μ1)μ1μ2B2(n1q1μ1 + n2q2μ2)2 + (n1q1 + n2q2)2μ21μ22B2,(1)where q1 and q2, n1 and n2, and μ1 and μ2 are, respectively,the charges, densities, and mobilities of the carriers. Despiteits usability, the model contains four active parameters, whichlead to large errors in most cases. The errors have not beendiscussed and their statistical reliability remains unclear. Thisis a fundamental issue requiring a more elaborate analysisprocedure.In this article, we report the electric transport of the 3Dtopological insulator TlBiSe2. The material is known as one ofthe simplest and most practical materials for transport studies[12–17]. Two-carrier transport properties were determined inprecision by applying a procedure proposed in this study. Themagnetotransport properties were well explained, accountingfor high- and low-mobility electrons in the whole temperaturerange. The carrier densities were (0.61 ± 0.01) × 1019 cm−3and (4.45 ± 0.01) × 1019 cm−3, which were 105 times thevalues expected from the surface. These results indicate that*geguchi@kuee.kyoto-u.ac.jp†Current address: Institute for Solid State Physics, The Universityof Tokyo, Kashiwanoha 5-1-5, Kashiwa, Chiba 277-8581, Japan.multicarrier conduction originated from the bulk electronsand the scattering was insensitive to the magnetic field. Thetemperature dependence of the Hall mobility exhibited ametallic behavior over the whole temperature range and wasexplained well by the Bloch-Gru¨neisen formula. This indicatesthat the scattering of the bulk electrons was dominated byacoustic phonons. It should be noted that the analyses proposedin this article allow a precise identification of the two-carriertransport properties and are therefore useful for all othermulticarrier systems.II. EXPERIMENTThe TlBiSe2 sample used in this study was taken fromthe same batch in Ref. [16]. As reported in Ref. [16], asignificant variation of the chemical composition was revealedbetween the starting substances and synthesized crystals; thevariation resulted in the spontaneous electron doping. Thesample, which was synthesized from the starting composi-tion Tl:Bi:Se = 1:1:2 in stoichiometric ratio, correspondedto Tl1−xBi1+xSe2−δ (x = 0.064,δ = 0.29) [16,18]. Electrictransport experiments were performed using the conventionalsix-probe technique down to 2 K with a commercial appa-ratus (Quantum Design PPMS). Electric contacts were madeonto the cleaved surface with room-temperature-cured silverpaste under atmospheric conditions. Geometries including thedistances of electrodes and the thickness of the sample weredetermined from optical microscope images. The schematicsof the crystal and parallel conduction of the surface and bulkare presented in Figs. 1(a) and 1(b). The optical microscopeimages are also shown in Fig. 1(c). The sample had a thicknessof 0.380 mm. Magnetic fields (maximum strength of ±9 T)were applied parallel to the c axis.III. RESULTS AND ANALYSISThe temperature dependence of the longitudinal resistivityρxx is presented in Fig. 2(a). The inset shows the Hallresistivity ρyx at 300 and 2 K. The positive temperaturecoefficient of ρxx indicates metallic conduction. The negativemagnetic coefficients of ρyx indicate electron conduction.These results are consistent with the ARPES results; theFermi level was located in the bulk conduction bands [14–16].1098-0121/2015/91(23)/235117(5) 235117-1 ©2015 American Physical SocietyG. EGUCHI et al. PHYSICAL REVIEW B 91, 235117 (2015)0.380 mmTlSeBica(a) (b) Electric currentBulkSurfacecc1 mmc(c)FIG. 1. (Color online) (a) Schematic of the TlBiSe2 crystal withhexagonal coordinates (space group R ¯3m [13]). (b) Schematic ofparallel conduction of the spin-polarized surface and bulk. Thecleaved surface forms thallium islands on the selenium layer [19].(c) Optical images of the top surface and side surface. The samplethickness was d = 0.380 mm.The ρyx exhibited a linear magnetic field dependence at eachtemperature. The temperature dependence of the Hall coeffi-cient |RH| = |ρyx/B| and the Hall mobility μH = |RH|/ρxxare presented in Fig. 2(b). As shown in the figure, the|RH| exhibited a small temperature dependence; μH alsoexhibited a temperature variation above 15 K. The |RH| at2 K was (1.775 ± 0.006) × 10−1 cm3/C and the μH at thesame temperature was 342 ± 6 cm2/V s. Because μH is inproportion to the scattering time τ , the temperature variationof μH is attributed to the temperature dependence of τ . Wediscuss the dependence later.The longitudinal resistivity ρxx exhibited a magneticfield dependence as can be clearly seen in Fig. 3(a). Hereρxx/ρxx = [ρxx(B) − ρxx(0)]/ρxx(B) is the magnetoresis-tance. For further insight, we transformed the two-carriermodel [20]. Within the model, ρxx is expressed by Eq. (1).Equation (1) was transformed to Eq. (2) using n+ = (n1 +n2), n− = (n1 − n2), μ+ = (μ1 + μ2)/2, μ− = (μ1 − μ2)/2,0 100 200 3000.10.20.301234T (K)| RH | (cm3/C)H (102  cm2 /Vs)(a)(b)0 100 200 300051015T (K)xx (10-4cm)–6 0 6–100100H (T)yx (10-5cm) 300 K2 KFIG. 2. (Color online) (a) Temperature dependence of the longi-tudinal resistivity ρxx in the range 2–300 K. The inset is the Hallresistivity ρyx at 2 and 300 K. Negative magnetic field dependencesindicate electron conduction. A linear fit of ρyx at 2 K is alsopresented. (b) Temperature dependence of the Hall coefficient |RH| =|ρyx/μ0H | and the Hall mobility μH = |RH|/ρxx ; |RH| at 2 K was(1.775 ± 0.006) × 10−1 cm3/C and μH at the same temperature was342 ± 6 cm2/V s.N = n−/n+, and M = μ−/μ+. In the present case, thecharges were q1 = q2 = −e and thus −μ+ > 0 and −1 M  1. We also defined n1  n2, i.e., 0  N  1. Finally,ρxx(B), which was applied in further analyses, was obtainedas Eq. (3) and the magnetoresistance was expressed as Eq. (4).ρxx(B) = 2n+μ+[q1(1 + N )(1 + M) + q2(1 − N )(1 − M) + [q1(1 + N )(1 − M) + q2(1 − N )(1 + M)](1 − M2)μ2+B2[q1(1 + N )(1 + M) + q2(1 − N )(1 − M)]2 + [q1(1 + N ) + q2(1 − N )]2(1 − M2)2μ2+B2],(2)ρxx(B) = 1n+e(−μ+)[1 + NM + (1 − M2)(1 − NM)μ2+B2(1 + NM)2 + (1 − M2)2μ2+B2], (3)ρxxρxx= (1 − N2)M2(1 − M2)μ2+B2(1 + NM)2 + (1 − N2M2)(1 − M2)μ2+B2. (4)Within the model, RH and μH at H = 0, which are presented in Fig. 2(b), are expressed as Eqs. (5) and (6). Using μH,Eq. (4) was rewritten as Eq. (7). Because μH was an experimental value, ρxx/ρxx is determined by employing two additionalparameters N and M . Furthermore, ρyx(B) was rewritten as Eq. (8) using RH, μH and was thus determined by N and M .RH = − 1n+e[1 + 2NM + M2(1 + NM)2], (5)μH = |μ+|[1 + 2NM + M21 + NM], (6)235117-2PRECISE DETERMINATION OF TWO-CARRIER TRANSPORT . . . PHYSICAL REVIEW B 91, 235117 (2015)ρxxρxx= (1 − N2)M2(1 − M2)μ2HB2(1 + 2NM + M2)2 + (1 − N2M2)(1 − M2)μ2HB2, (7)ρyx(B) = RH[ (1 + 2NM + M2)3B + (1 + NM)2(1 − M2)2μ2HB3(1 + 2NM + M2)3 + (1 + 2NM + M2)(1 − M2)2μ2HB2]. (8)The magnetoresistance ρxx/ρxx at 5 K and the fit with Eq. (7)are presented in Fig. 3(a); the fit yielded N = 0.76 ± 0.01 andM = −0.586 ± 0.003. Also presented are ρxx/ρxx at 300 Kand the curve expected from the N and M values with RHand μH at 300 K. The figure shows that the curve at 300 Kis consistent with the experiments. Presented in Fig. 3(b)are ρyx(B) and the curves expected from the same N andM values. The curves also agree well with the experimentalresults. We thus conclude that the N and M values obtainedare invariant within the studied temperature range and thetransport properties of the sample are explained well with thetwo-carrier model.Once the four parameters of the two-carrier model RH,μH, N , and M were obtained, the densities and mobilitiesof high- and low-mobility electrons were determined. Thedensities and the mobilities at 5 K are summarized in Table I.Here n1 = n+(1 + N )/2, n2 = n+(1 − N )/2, μ1 = μ+(1 +M), and μ2 = μ+(1 − M).IV. DISCUSSIONAs demonstrated above, the magnetotransport propertieswere well explained by the two-carrier model. We next discussthe origin of these conduction channels with the electronicyx (10-4cm)–10 –5 0 5 10050H (T)xx /xx (%) 300 K5 K–10 –5 0 5 10–2–10120H (T)300 K5 K(a)(b)FIG. 3. (Color online) (a) Magnetoresistance ρxx/ρxx =[ρxx(B) − ρxx(0)]/ρxx(B) at 300 and 5 K. The fit with Eq. (7) at 5 Kis shown by the solid line. The curve expected at 300 K is shown bythe dashed line. (b) Hall resistivity ρyx at 300 and 5 K. The curvesexpected at each temperature are shown by dashed lines.band structure. A schematic of the structure revealed byARPES is presented in Fig. 4 [15,16,21]. The figure showssurface Dirac fermions and two Fermi surfaces of the bulk;one of the Fermi surfaces was around the  point and theother around the F point. For the surface Dirac fermions, thetwo-dimensional carrier density ns = k2F/4π was estimatedas ∼8 × 1012 cm−2, where kF was the Fermi wave numberobtained from the ARPES spectrum (Fig. 2 of Ref. [15]).The value of ns yielded an effective 3D carrier densityns/d = 2.1 × 1014 cm−3, where d = 0.380 mm was thesample thickness. The estimated value of ns/d was smallerthan the experimental values by five orders of magnitude. Thisindicates that the contribution of the Dirac fermions to ρxxand ρyx was negligibly small, i.e., the transport properties areattributed to the bulk electrons. Figure 4 shows that the Fermisurface around the  point has a larger Fermi cross section anda larger band mass than that around the F point. These bandcharacters are consistent with the two-carrier model, wherethe Fermi surface around the  point yields higher carrierdensity and lower mobility, i.e., n1 and μ1. It is also concludedthat the minor electrons with carrier density n2 and mobilityμ2 can be attributed to the Fermi surface around the F point.As discussed above, surface transport is hidden in bulkmetallic conduction with carrier density ∼1019 cm−3. In con-trast, a bulk insulating state with carrier density < 1017 cm−3was demonstrated for Tl1−xBi1+xSe2−δ (x  0.028,δ = 0.28)and the surface metallic conduction was distinguished [16,17].These results indicate the presence of a metal-insulator transi-tion in bulk between x = 0.064 and 0.028 and surface transportcan be distinguished only in the bulk insulating samples.Finally, we discuss the temperature dependence of scatter-ing realized in the system. The scattering time is in proportionto the mobility and the temperature dependence of μ+ isthus attributed to the temperature dependence of the averagescattering time τ . Note that the agreement with the two-carrier model indicates that the scattering was insensitive tothe magnetic field. As discussed above, the temperaturedependences of N and M were negligibly small. These resultsindicate that μH can be expressed as μH = 0.816|μ+| usingthe N and M values obtained [see Eq. (6)] and that thetemperature dependence of μH can be attributed to τ . Thetemperature dependence of μ−1H is presented in Fig. 5. The fitTABLE I. Electric transport properties at 5 K obtained from thetwo-carrier model. Here n1 and n2, and μ1 and μ2 are, respectively,the densities and mobilities of the electrons.n1 (4.45 ± 0.01) × 1019 cm−3n2 (0.61 ± 0.01) × 1019 cm−3μ1 176 ± 2 cm2/V sμ2 676 ± 2 cm2/V s235117-3G. EGUCHI et al. PHYSICAL REVIEW B 91, 235117 (2015)EF0.20.40.60.8Binding energy (eV)Bulk band gap:~0.3 eVF kSurfaceSmallerband massLarger band massBulkFIG. 4. (Color online) Schematic of the energy band dispersionof TlBiSe2 revealed by ARPES [15,16,21]. The smaller bandcurvature around the  point results in a larger band mass of electrons.The larger band curvature around the F point results in smaller bandmass. The amplitude of the minimum bulk band gap is also presented.with the Bloch-Gru¨neisen formula1τ∝(T 5Θ6D)∫ ΘD/T0x5(ex − 1)(1 − e−x)dx, (9)which describes the temperature dependence of acousticphonon scattering, is also presented. Here ΘD is the Debyetemperature determined in the present transport experiment.As shown in Fig. 5, the temperature dependence of 1/μH ∝1/τ agrees well with the formula, which yields ΘD = 113 ±14 K. Note that ΘD agrees with the Debye temperaturedetermined from the specific heat [22]. The result indicatesthat the scattering was dominated by acoustic phonons. Similarresults are often seen for metals of a weakly correlated electronsystem, indicating that the scattering of bulk electrons isexplained by the conventional theory of metals.V. CONCLUSIONWe investigated the transport properties of the 3D topo-logical insulator TlBiSe2. Two-carrier transport properties0 100 200 30000.511.2T (K)H-1  (10-2  Vs/cm2 )D = 114 ± 13 KBloch-GrüneisenFIG. 5. (Color online) Temperature dependence of the inverseHall mobility μ−1H . The fit with the Bloch-Gru¨neisen formula yieldsthe Debye temperature ΘD = 114 ± 13 K. See the text for details.were determined in precision. The magnetotransport propertieswere well explained, accounting for high- and low-mobilityelectrons in the whole temperature range. This indicated thatthe scattering of electrons was insensitive to the magneticfield. The transport properties obtained were consistent withARPES results and contributions of the surface transport werenegligibly small over the entire temperature range studied.The scattering of bulk electrons was dominated by acousticphonons. The results indicate that the bulk transport propertiesof the TlBiSe2 sample were in the vicinity of those ofconventional metals. The simple nature of bulk conductionhints at the proper verification of surface transport properties. Itshould be emphasized that the analysis using the conventionaltwo-carrier model with four active parameters leads to largeerrors. Instead, the analysis proposed in this study exhibitedsufficiently small errors and therefore possessed statisticalreliability. We also note that the procedure should be usefulfor all other transport studies.ACKNOWLEDGMENTSThis work was partly supported by JSPS KAKENHI,a Grant-in-Aid for Scientific Research (B) (Grant No.23340105), and Innovative Areas “Nano Spin ConversionScience.” G.E. and K.K. acknowledge support from the JapanSociety for the Promotion of Science for Young Scientists.[1] M. Hasan and C. Kane, Rev. Mod. Phys. 82, 3045 (2010).[2] X.-L. Qi and S.-C. Zhang, Rev. Mod. Phys. 83, 1057 (2011).[3] Y. Ando, J. Phys. Soc. Jpn. 82, 102001 (2013).[4] Z.-H. Pan, E. Vescovo, A. V. Fedorov, D. Gardner, Y. S. Lee,S. Chu, G. D. Gu, and T. Valla, Phys. Rev. Lett. 106, 257004(2011).[5] C. H. Li, O. M. J. van ’t Erve, J. T. Robinson, Y. Liu, L. Li, andB. T. 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Sugawara, S. Souma,T. Takahashi, and Y. Ando, Phys. Rev. Lett. 105, 136802 (2010).235117-4PRECISE DETERMINATION OF TWO-CARRIER TRANSPORT . . . PHYSICAL REVIEW B 91, 235117 (2015)[15] K. Kuroda, M. Ye, A. Kimura, S. V. Eremeev, E. E. Krasovskii,E. V. Chulkov, Y. Ueda, K. Miyamoto, T. Okuda, K. Shimada,H. Namatame, and M. Taniguchi, Phys. Rev. Lett. 105, 146801(2010).[16] K. Kuroda, G. Eguchi, K. Shirai, M. Shiraishi, M. Ye,K. Miyamoto, T. Okuda, S. Ueda, M. Arita, H. Namatame,M. Taniguchi, Y. Ueda, and A. Kimura, Phys. Rev. B 91, 205306(2015).[17] G. Eguchi, K. Kuroda, K. Shirai, A. Kimura, and M. Shiraishi,Phys. Rev. B 90, 201307 (2014).[18] Samples of our preceding studies [15,19] were also synthesizedfrom the same starting composition Tl:Bi:Se = 1:1:2, thusexpected to be Tl1−xBi1+xSe2−δ (x = 0.064,δ = 0.29).[19] K. Kuroda, M. Ye, E. F. Schwier, M. Nurmamat, K. Shirai,M. Nakatake, S. Ueda, K. Miyamoto, T. Okuda, H. Namatame,M. Taniguchi, Y. Ueda, and A. Kimura, Phys. Rev. B 88, 245308(2013).[20] N. W. Ashcroft and N. D. Mermin, Solid State Physics, 1st ed.(Brooks/Cole, Pacific Grove, 1976).[21] The Fermi level of the Tl1−xBi1+xSe2−δ (x = 0.064, δ = 0.29),reported in Ref. [16], was almost identical to that of Ref. [15].Therefore, a wide-range mapping of the ARPES presentedin Fig. 3 in Ref. [15] represents the band structure of thesample.[22] V. Bazakutsa, M. Vasil’eva, V. Ustimenko, and L. Mokhir,J. Eng. Phys. 37, 1191 (1979).235117-5

Precise determination of two-carrier transport properties in the topological insulator TlBiSe 2

G. Eguchi

K. Kuroda

K. Shirai

Y. Ando

T. Shinjo

A. Kimura

M. Shiraishi

Crossref

Precise determination of two-carrier transport properties in the topological insulator<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>TlBiSe</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:math>

Precise determination of two-carrier transport properties in the
  topological insulator TlBiSe$_2$

Precise determination of two-carrier transport properties in the topological insulator TlBiSe $_2$

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Precise determination of two-carrier transport properties in the topological insulator TlBiSe2_22​

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Precise determination of two-carrier transport properties in the topological insulator TlBiSe $_2$