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Multiple state transport deduced by weak antilocalization and electron–electron interaction effects in Sbx Te1−x layers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 095802 (http://iopscience.iop.org/0953-8984/26/9/095802) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.119.168.112 This content was downloaded on 14/06/2017 at 00:54 Please note that terms and conditions apply.
You may also be interested in: Transport properties in a Sb-Te binary topological-insulator system Y Takagaki, A Giussani, J Tominaga et al. Transport properties of topological insulators films and nanowires Liu Yi, Ma Zheng, Zhao Yan-Fei et al. Robust topological surface transport with weak localization bulk channels in polycrystalline Bi2Te3 films H B Zhang, J D Yao, J M Shao et al. Semicoherent growth of Bi2Te3 layers on InP substrates by hot wall epitaxy Y Takagaki, B Jenichen, V Kopp et al. Hot wall epitaxy of Sb2Te3 layers: coherent hetero-epitaxy on InAs(1 1 1) and Sb substitution in Cu-mediated growth Y Takagaki, B Jenichen, U Jahn et al. Spin–orbit coupling and weak antilocalization in the thermoelectric material beta-K2Bi8Se13 J Hu, J Y Liu and Z Q Mao Hot wall epitaxy of topological insulator films Y Takagaki, B Jenichen, U Jahn et al. Synthesis of SmB6 films by Mg-assisted solid state reaction Y Takagaki, A Wirsig, M Ramsteiner et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 095802 (5pp)
doi:10.1088/0953-8984/26/9/095802
Multiple state transport deduced by weak antilocalization and electron–electron interaction effects in SbxTe1−x layers Y Takagaki, U Jahn, A Giussani and R Calarco Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5–7, D-10117 Berlin, Germany E-mail: [email protected] Received 15 October 2013, revised 23 December 2013 Accepted for publication 2 January 2014 Published 13 February 2014
Abstract
Quantum corrections to the conductivity due to the weak antilocalization (WAL) and electron–electron interaction (EEI) effects are investigated in Sb–Te layers to evaluate the number of independent conduction channels in the topological insulator system. We separate the two contributions in the logarithmic temperature dependence of conductivity relying on their distinct response to a magnetic field. For the WAL effect, the amplitude parameter α being −1 observed in magnetoconductivity is confirmed. The magnitude of the EEI contribution is too large to be produced by one transport channel. The mixing between the surface and bulk states is thus indicated to be weak in the Sb–Te system. In addition, the disorder scattering appears to be less influential for the EEI effect than for the WAL effect. Keywords: topological insulator, antimony telluride, weak antilocalization, electron–electron interaction (Some figures may appear in colour only in the online journal)
antilocalization (WAL) effect in a disordered system having strong spin–orbit interaction [7]. The magnitude of the Berry phase effect, which we henceforth refer to as the WAL effect as it has become customary, should be doubled for ideal TI layers as the surface states exist at both of the surfaces of the layers. However, the values observed experimentally in Bi2 Se3 and Bi2 Te3 layers have been merely 1/3 ∼ 1/2 of the expectation [8]. The discrepancy is attributed to the fact that the layers were unintentionally doped as crystalline defects generated free carriers. The bulk states consequently contributed to the transport and the surface states at the two sides of the layers were mixed through frequent scattering with the bulk states [9–11]. That is, the whole system acted as one transport channel. In ordinary conductors, electron–electron interaction (EEI) provides another quantum correction in the presence of disorder [12, 13]. The EEI effect has been observed also in TI materials [14]. It was noticed in a Bi2 Se3 layer that the magnitude of the EEI effect was too large to be accounted
Topological insulators (TIs) [1, 2] host gapless conductive surface states including the energy range of the bulk band gap. The dispersion of the surface states is linear, similar to the Dirac cone in graphene. In contrast to graphene, in which the spin–orbit coupling is negligible, the momentum and the spin of the TI surface states are locked to each other due to strong spin–orbit coupling. One consequence of the spin–momentum locking is that the spin orientation is opposite between the states associated with a wavevector with positive and negative signs. As nonmagnetic impurities do not flip spin, the surface states are protected from backscattering, which is attractive for device applications. One may anticipate that the weak localization phenomenon is absent in TIs as forward and backward propagating waves cannot interfere with each other due to their opposite spin orientations [3]. The π Berry phase carried by the surface states, however, gives rise to a quantum correction to the conductivity at low temperatures [4, 5]. The temperature and magnetic field dependencies of the conductivity change [6] have turned out to be identical in form to those of the weak 0953-8984/14/095802+05$33.00
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c 2014 IOP Publishing Ltd
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J. Phys.: Condens. Matter 26 (2014) 095802
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for by a single transport channel implied by the WAL effect [15]. The inconsistency was confirmed by [16] and by [17] using Bi2 Te3 layers. A possible explanation emerged when the WAL effect in layers of Sb–Te binary compounds was revealed to possess the amplitude corresponding to ideal TI layers [8, 18]. The magnitude of the EEI effect in the Sb2 Te3 layer was reasonable if we interpreted the WAL effect as indicating the presence of two independent transport channels [8]. The observations may imply that the timescale relevant for the EEI effect is shorter than the scattering time between the surface and bulk states. The surface states at the two surfaces of the layers may, in general, be uncoupled so long as the EEI effect is concerned. This is not unreasonable considering that the Shubnikov–de Haas oscillations arising from the surface states [19, 20] can be observed in spite of the carrier occupation in the bulk states [21]. The Sb–Te system, including Sb2 Te3 , has received rather little attention so far as TIs. Hsieh [22] investigated the band structure in Sb2 Te3 using angle-resolved photo emission spectroscopy. Due to intrinsic p-type doping, which is typical for the Sb–Te system, the Fermi level was located in the bulk valence band continuum. The existence of a Dirac point for Sb2 Te3 was, therefore, able to be evidenced merely indirectly by the agreement of the measured valence band structure below the Fermi level with the prediction by ab initio density-functional calculations. The Sb–Te compounds are well-known, on the contrary, as a phase-change material system [23] as well as thermoelectric materials. The transport properties have been investigated, as a consequence, mainly from the viewpoint of applications for phase-change and thermoelectric [24, 25] devices. We note that there are reports on the WAL effect in Sb2 Te3 nanowires [26, 27]. Interestingly, GeTe–Sb2 Te3 superlattices have been predicted to become a TI for certain lattice periods [28]. Giant magnetic responses reminiscent of ferromagnetic junctions were observed in the resistance of the TI superlattice [29]. The spin–momentum locking for the TI states is anticipated to be responsible for the unusual magnetic properties. In this paper, we examine the magnitude of the temperature dependence in the conductivity originating from the EEI and WAL effects in a number of Sbx Te1−x layers. In consistent with the weak coupling scenario, we find the effective number of transport channels contributing to the EEI correction remain larger than one when the composition x is varied. The Sb–Te system is an infinite series of TIs. The Sb–Te compounds form a layered structure (Sb2 Te3 )m (Sb2 )n , where the quintuple layer Sb2 Te3 and the bilayer Sb2 are stacked sequentially in the hexagonal c-axis direction. The Sb–Te natural superlattices thus resemble the GeTe–Sb2 Te3 superlattices [28, 29], where the Sb2 layer plays the role of the GeTe layer. In the Sb–Te system, there are three stable phases Sb2 Te3 , SbTe and Sb2 Te in equilibrium [30]. Ab initio density-functional calculations predict, at least, Sb2 Te3 and SbTe to be TIs [18]. The Dirac point in Sb2 Te3 is located in the middle of the bulk band gap, similar to Bi2 Se3 . In SbTe, the location of the Dirac point is predicted to be at the bulk valence band edge, resembling Bi2 Te3 . We grew Sbx Te1−x layers on the Si(111) substrates by molecular beam epitaxy (MBE). The substrate temperature
Figure 1. (a) Distribution of structural domains in Sb0.52 Te0.48 layer (#7) revealed using the EBSD technique. The inset shows the scanning electron micrograph of the same area. The blue and green areas show the twin domains associated with the favored in-plane ¯ and [1100] ¯ alignment (the [1120] directions of the layers being ¯ and [112] ¯ directions of the Si(111) substrates, parallel to the [110] respectively). The in-plane tolerance around the preferred direction is ±15◦ . The in-plane alignment is beyond the tolerance for the red area. (b) Probability P(φ) of the in-plane rotation angle being smaller than φ. The layers are Sb0.52 Te0.48 (≈SbTe, #7) and Sb0.64 Te0.36 (≈Sb2 Te, #8) for the solid and dotted curves, respectively. The steeper the slope, the larger the existence probability for a given angle φ. The line colors correspond to the area colors in (a). The statistics were obtained from 2.5 × 2.5 µm2 areas.
was 250 ◦ C. The growth was carried out with various intensities of the Sb and Te fluxes. We determined the fraction x in the layers using energy dispersive x-ray spectroscopy. In some of the layers, compositions intermediate between Sb2 Te3 and SbTe were realized taking advantage of the nonequilibrium growth in MBE. Our MBE-grown layers were all (0001)-oriented. While there was a favored in-plane epitaxial orientation relationship, the layers were not completely single-crystalline as the lattice mismatch between the layers and the substrate is large. In figure 1(a), we show the crystal domain structure in an Sb0.52 Te0.48 (≈SbTe) layer revealed using the electron backscatter diffraction (EBSD) technique. The scanning electron micrograph in the same area is shown in the inset. The ¯ and [1100] ¯ [1120] directions of the layer were preferentially ¯ and [112] ¯ directions of the aligned to be along the [110] substrate, respectively. The blue and green areas in figure 1(a) display the distribution of twin domains for the component having the favored alignment [31]. In the red area, the range of the in-plane rotation is shifted by 30◦ in comparison to that for the dominant orientation. The domains are seen to 2
J. Phys.: Condens. Matter 26 (2014) 095802
Y Takagaki et al
form submicrometer-size islands. The island sizes are nearly the same regardless of the degree of the lattice mismatch with the substrate. Even the islands having unfavorable in-plane alignments are indicated to grow larger at the same rate once they were generated. A similar domain structure was observed also for an Sb0.64 Te0.36 (≈Sb2 Te) layer (not shown). The in-plane alignment was less accurate in the Sb2 Te layer in comparison to that in the SbTe layer, which might have reflected the larger lattice mismatch. (The mismatch is 11.2% for the Sb2 Te layer and 11.0% for the SbTe layer [32].) As a quantitative demonstration, we plot in figure 1(b) the probability P(φ) of the in-plane rotation angle being smaller than φ for the SbTe and Sb2 Te layers. The pronounced rises of P at φ ≈ 30◦ and 60◦ for the solid curve demonstrate the well-established alignment in the SbTe layer. The increases are gradual for the dotted curve, evidencing large fluctuations in the alignment in the Sb2 Te layer. The conductivity change δσ originating from the WAL effect depends on temperature T as [6, 7] e 2 τϕ e2 T δσ L (T ) = −α ln = αp ln , (1) πh τ πh TL where τ and TL are characteristic constants. The phase breaking time τϕ was assumed to vary with T as ∝ T − p . For each side of TI layers, α = −1/2 is predicted for the surface state. Assuming that the EEI effect in TI layers is identical to that in ordinary conductors [12, 13], we expect it to behave as T 3 e2 n 1 − F ln , (2) δσ I (T ) = πh 4 TI
Figure 2. Temperature dependence of sheet conductivity in (a)
Sb0.43 Te0.57 layer (#3) and (b) Sb0.64 Te0.36 layer (#8). A magnetic field of B = 1 T was applied perpendicular to the layer for the lower curve in (a) and for (b). The lines indicate logarithmic behavior.
The WAL correction is quenched when the time-reversal symmetry is broken by applying a magnetic field perpendicular to the layer. The temperature dependence was hence measured in the absence and presence of a magnetic field. The temperature dependence at a magnetic field of B = 1 T in figure 2(a) is almost entirely attributed to the EEI effect. The magnetic field dependence of the EEI effect is negligible at B = 1 T, and so the difference in the slopes of the ln T behaviors provides the magnitude of the WAL effect. We plot the variation with B of the temperature coefficient f defined as δσ (T ) = f ln (T ) in figure 3 by the filled circles. For the Sb0.43 Te0.57 layer, therefore, we obtain α = −1.2 from the change in f between B = 0 and 1 T and n(1 − 34 F) = 2.9 from the value of f at B = 1 T. The coefficient α can be evaluated independently from the magnetoconductivity. Theory predicts the dependence on B to be given as [6]
where F is the screening parameter and should satisfy the condition 0 ≤ F ≤ 1. The EEI effect vanishes at the characteristic temperature TI . We have explicitly specified the number of independent transport channels by n. We point out that the layers investigated in the present work are thinner than the phase coherence length L φ = (Dτϕ )1/2 and the thermal diffusion length L T = (h¯ D/k B T )1/22 . Here, D = 13 v F le is the diffusion coefficient with v F and le being the Fermi velocity and the elastic mean free path, respectively. The system is thus two-dimensional (2D) concerning both the WAL and EEI effects. That the dephasing is governed by the 2D electron–electron scattering means p = 1, as verified experimentally in an Sb2 Te3 layer in [8]. The Sb–Te layers exhibit p-type conduction. The details of the layers, which are numbered #i, can be found in [18]. The thicknesses are in a range of 16–20 nm. The hole concentration increases and the mobility decreases with increasing x. As we show in figure 2, a logarithmic dependence is indeed observed at low temperatures. Below, we compare its magnitude with the theoretical predictions. Notice that the magnitudes of the quantum corrections are on the order of the conductance quantum e2 / h. The coefficient of the ln T term was thus most accurately estimated for the sample having the smallest conductivity, which is the Sb0.43 Te0.57 layer shown in figure 2(a).
1σ L (B) = δσ L (B) − δσ L (B = 0) " ! !# e2 1 h¯ h¯ =α ψ + − ln , πh 2 4eB L 2φ 4eB L 2φ
(3)
where ψ(x) is the digamma function. We show a fitting result of equation (3) to experimental data in figure 4. As we have previously reported in [18], α = −1 is found in the Sb–Te layers. We emphasize that α = −1, which is expected for ideal TI layers, has never been observed in Bi2 Se3 and Bi2 Te3 layers except when the transport properties were tuned by gating [8]. The Sb–Te layers are hence remarkable as the value of α implies the absence of mixing between the surface and
As we reported in [18] L φ in the layers is longer than 0.3 µm at T = 0.3 K and L T at T = 1 K is estimated to be ∼ 0.2 µm. We used in the estimation the hole effective mass of Sb2 Te3 reported in [34].
2
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J. Phys.: Condens. Matter 26 (2014) 095802
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experimental result in the Sb0.43 Te0.57 layer. This could mean that the surface states at the two sides of the layer as well as the bulk states are independently contributing to the EEI effect. In figure 3, the results obtained from layers having other compositions (x = 0.50 and 0.64) are also shown using open symbols. We point out that the magnitudes of the logarithmic temperature dependence at B = 0.5 and 1.0 T are nearly the same in the layer with x = 0.50. This confirms that the WAL effect is almost completely quenched at these magnetic fields, whereas the EEI effect is practically unaffected in the range of the magnetic field that we utilized, similar to our detailed examinations in [8, 15]. These samples lead us to the same conclusions, i.e., α ≈ −1 for the WAL effect and several independent transport channels contribute to the EEI effect. Our analysis of the temperature dependence thus provides further evidence that the mixing between the surface and bulk states is weak in the Sb–Te system. It is noteworthy that there are reports of giant electron–electron interaction effects in topologically trivial conductors. As discussed in detail by [16, 33], the values of F were deduced to be negative. The discrepancy with the theoretical expectation of F ≥ 0 may be interpreted as evidence for independent participation of multiple bulk states in the transport. One notices in figure 2(a) that the conductivity exhibits deviation from the logarithmic behavior at very low temperatures. As we show in figure 2(b), the temperature dependence became weak for temperatures below about 1 K in the Sb0.64 Te0.36 layer. The tendency of saturation is not due to the heating effect as the conductivity values were unchanged when the measurement current was varied. As the deviation is a positive conductivity correction, it may be the WAL contribution from an additional transport channel. The deviation survives in the presence of a magnetic field of 1 T. The WAL effect is quenched when the magnetic flux penetrating through the phase coherent area L 2φ becomes on the order of the flux quantum h/e. The weak dependence on magnetic field may suggest that L φ associated with this new channel is small, thereby its contribution appears only at very low temperatures. We speculate that the additional contribution stems from the bulk states. Its small magnitude, in fact, explains why the bulk states appear to be irrelevant for the WAL magnetoconductivity and α is fixed at −1. In conclusion, we have investigated the logarithmic temperature dependence in the conductivity in a series of Sbx Te1−x layers. Multiple transport channels have been indicated to contribute to the EEI effect. In addition, our analysis supports the observation of α = −1 in the magnetoconductivity for the WAL contribution.
Figure 3. Coefficient f of logarithmic temperature dependence in
sheet conductivity δσ (T ) = f ln (T ). The magnetic field applied perpendicular to the layers was varied between 0 and 1 T. The composition x in the Sbx Te1−x layers is 0.43 (#3), 0.50 (#6), and 0.64 (#8) for the filled circles, open circles, and open triangles, respectively. The horizontal lines indicate the magnitudes of e2 /(π h) multiplied by 1, 2 and 3, i.e., n(1 − 34 F) = 1, 2 and 3. The separation between adjacent lines corresponds to α = −1.
Figure 4. Dependence of sheet conductivity on magnetic field in Sb0.43 Te0.57 layer (#3) at temperature T = 0.33 K. The curve shows a fit to equation (3) with parameters of α = −1 and L φ = 0.38 µm. Magnetoresistance 1R/R in the layer is shown in the inset.
bulk states. We also note that the layer is intriguing as the magnetoresistance 1R/R is linear [18], as we show in the inset of figure 4. Similar to our previous comparisons [8, 15], the values of α derived from the magnetic field and temperature dependencies agree reasonably well with each other. As for the EEI effect, the magnitude is again much larger than the amount that one transport channel can be responsible for. In fact, at least three transport channels are required to explain the
References Hasan M Z and Kane C L 2010 Rev. Mod. Phys. 82 3045 Qi X-L and Zhang S-C 2011 Rev. Mod. Phys. 83 1057 Bergmann G 1984 Phys. Rep. 107 1 Suzuura H and Ando T 2002 Phys. Rev. Lett. 89 266603 McCann E, Kechedzhi K, Falko V I, Suzuura H, Ando T and Altshuler B L 2006 Phys. Rev. Lett. 97 146805 [6] Tkachov G and Hankiewicz E M 2011 Phys. Rev. B 84 035444
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[7] Hikami S, Larkin A I and Nagaoka Y 1980 Prog. Theor. Phys. 63 707 [8] Takagaki Y, Giussani A, Perumal K, Calarco R and Friedland K-J 2012 Phys. Rev. 86 125137 [9] He H-T, Wang G, Zhang T, Sou I-K, Wong G K L, Wang J-N, Lu H-Z, Shen S-Q and Zhang F-C 2011 Phys. Rev. Lett. 106 166805 [10] Wang H et al 2013 Crossover between weak antilocalization and weak localization of bulk states in ultrathin Bi2 Se3 films arXiv: 1310.5194 [11] Lang M et al 2013 Nano Lett. 13 48–53 [12] Lee P A and Ramakrishnan T V 1985 Rev. Mod. Phys. 57 287 [13] Altshuler B L and Aronov A G 1985 Electron–Electron Interactions in Disordered Systems ed A L Efros and M Pollak (Amsterdam: North-Holland) p 1 [14] Wang J, DaSilva A M, Chang C-Z, He K, Jain J K, Samarth N, Ma X-C, Xue Q-K and Chan M H W 2011 Phys. Rev. B 83 245438 [15] Takagaki Y, Jenichen B, Jahn U, Ramsteiner M and Friedland K-J 2012 Phys. Rev. B 85 115314 [16] Chiu S-P and Lin J-J 2013 Phys. Rev. B 87 035122 [17] Roy A, Guchhait S, Sonde S, Dey R, Pramanik T, Rai A, Movva H C P, Colombo L and Banerjee S K 2013 Appl. Phys. Lett. 102 163118 [18] Takagaki Y, Giussani A, Tominaga J, Jahn U and Calarco R 2013 J. Phys.: Condens. Matter 25 345801 [19] Qu D-X, Hor Y S, Xiong J, Cava R J and Ong N P 2010 Science 329 821
[20] Analytis J G, McDonald R D, Riggs S C, Chu J-H, Boebinger G S and Fisher I R 2010 Nature Phys. 6 960 [21] Taskin A A, Sasaki S, Segawa K and Ando Y 2012 Phys. Rev. Lett. 109 066803 [22] Hsieh D et al 2009 Phys. Rev. Lett. 103 146401 [23] Lankhorst M H R, Ketelaars B W S M M and Wolters R A M 2005 Nature Mater. 4 347 [24] Peranio N, Winkler M, Aabdin Z, K¨onig J, B¨ottner H and Eibl O 2012 Phys. Status Solidi 209 289 [25] Zastrow S, Gooth J, Boehnert T, Heiderich S, Toellner W, Heimann S, Schulz S and Nielsch K 2013 Semicond. Sci. Technol. 28 035010 [26] Zuev Y M, Lee J S, Galloy C, Park H and Kim P 2010 Nano Lett. 10 3037 [27] Hamdou B, Gooth J, Dorn A, Pippel E and Nielsch K 2013 Appl. Phys. Lett. 102 223110 [28] Sa B, Zhou J, Sun Z, Tominaga J and Ahuja R 2012 Phys. Rev. Lett. 109 096802 [29] Tominaga J, Simpson R E, Fons P and Kolobov A V 2011 Appl. Phys. Lett. 99 152105 [30] Ghosh G 1994 J. Phase Equilib. 15 349 [31] Takagaki Y, Jenichen B and Tominaga J 2013 Phys. Rev. B 87 245302 [32] Poudeu P F P and Kanatzidis M G 2005 Chem. Commun. 2672–4 [33] Wu C Y, Jian W B and Lin J J 1995 Phys. Rev. B 52 15479 [34] Seibel C, Maa H, Ohtaka M, Fiedler S, J¨unger C, Min C-H, Bentmann H, Sakamoto K and Reinert F 2012 Phys. Rev. B 86 161105
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Multiple state transport deduced by weak antilocalization and electron–electron interaction effects in Sbx Te1−x layers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 095802 (http://iopscience.iop.org/0953-8984/26/9/095802) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 128.119.168.112 This content was downloaded on 14/06/2017 at 00:54 Please note that terms and conditions apply.
You may also be interested in: Transport properties in a Sb-Te binary topological-insulator system Y Takagaki, A Giussani, J Tominaga et al. Transport properties of topological insulators films and nanowires Liu Yi, Ma Zheng, Zhao Yan-Fei et al. Robust topological surface transport with weak localization bulk channels in polycrystalline Bi2Te3 films H B Zhang, J D Yao, J M Shao et al. Semicoherent growth of Bi2Te3 layers on InP substrates by hot wall epitaxy Y Takagaki, B Jenichen, V Kopp et al. Hot wall epitaxy of Sb2Te3 layers: coherent hetero-epitaxy on InAs(1 1 1) and Sb substitution in Cu-mediated growth Y Takagaki, B Jenichen, U Jahn et al. Spin–orbit coupling and weak antilocalization in the thermoelectric material beta-K2Bi8Se13 J Hu, J Y Liu and Z Q Mao Hot wall epitaxy of topological insulator films Y Takagaki, B Jenichen, U Jahn et al. Synthesis of SmB6 films by Mg-assisted solid state reaction Y Takagaki, A Wirsig, M Ramsteiner et al.
Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 095802 (5pp)
doi:10.1088/0953-8984/26/9/095802
Multiple state transport deduced by weak antilocalization and electron–electron interaction effects in SbxTe1−x layers Y Takagaki, U Jahn, A Giussani and R Calarco Paul-Drude-Institut für Festkörperelektronik, Hausvogteiplatz 5–7, D-10117 Berlin, Germany E-mail: [email protected] Received 15 October 2013, revised 23 December 2013 Accepted for publication 2 January 2014 Published 13 February 2014
Abstract
Quantum corrections to the conductivity due to the weak antilocalization (WAL) and electron–electron interaction (EEI) effects are investigated in Sb–Te layers to evaluate the number of independent conduction channels in the topological insulator system. We separate the two contributions in the logarithmic temperature dependence of conductivity relying on their distinct response to a magnetic field. For the WAL effect, the amplitude parameter α being −1 observed in magnetoconductivity is confirmed. The magnitude of the EEI contribution is too large to be produced by one transport channel. The mixing between the surface and bulk states is thus indicated to be weak in the Sb–Te system. In addition, the disorder scattering appears to be less influential for the EEI effect than for the WAL effect. Keywords: topological insulator, antimony telluride, weak antilocalization, electron–electron interaction (Some figures may appear in colour only in the online journal)
antilocalization (WAL) effect in a disordered system having strong spin–orbit interaction [7]. The magnitude of the Berry phase effect, which we henceforth refer to as the WAL effect as it has become customary, should be doubled for ideal TI layers as the surface states exist at both of the surfaces of the layers. However, the values observed experimentally in Bi2 Se3 and Bi2 Te3 layers have been merely 1/3 ∼ 1/2 of the expectation [8]. The discrepancy is attributed to the fact that the layers were unintentionally doped as crystalline defects generated free carriers. The bulk states consequently contributed to the transport and the surface states at the two sides of the layers were mixed through frequent scattering with the bulk states [9–11]. That is, the whole system acted as one transport channel. In ordinary conductors, electron–electron interaction (EEI) provides another quantum correction in the presence of disorder [12, 13]. The EEI effect has been observed also in TI materials [14]. It was noticed in a Bi2 Se3 layer that the magnitude of the EEI effect was too large to be accounted
Topological insulators (TIs) [1, 2] host gapless conductive surface states including the energy range of the bulk band gap. The dispersion of the surface states is linear, similar to the Dirac cone in graphene. In contrast to graphene, in which the spin–orbit coupling is negligible, the momentum and the spin of the TI surface states are locked to each other due to strong spin–orbit coupling. One consequence of the spin–momentum locking is that the spin orientation is opposite between the states associated with a wavevector with positive and negative signs. As nonmagnetic impurities do not flip spin, the surface states are protected from backscattering, which is attractive for device applications. One may anticipate that the weak localization phenomenon is absent in TIs as forward and backward propagating waves cannot interfere with each other due to their opposite spin orientations [3]. The π Berry phase carried by the surface states, however, gives rise to a quantum correction to the conductivity at low temperatures [4, 5]. The temperature and magnetic field dependencies of the conductivity change [6] have turned out to be identical in form to those of the weak 0953-8984/14/095802+05$33.00
1
c 2014 IOP Publishing Ltd
Printed in the UK
J. Phys.: Condens. Matter 26 (2014) 095802
Y Takagaki et al
for by a single transport channel implied by the WAL effect [15]. The inconsistency was confirmed by [16] and by [17] using Bi2 Te3 layers. A possible explanation emerged when the WAL effect in layers of Sb–Te binary compounds was revealed to possess the amplitude corresponding to ideal TI layers [8, 18]. The magnitude of the EEI effect in the Sb2 Te3 layer was reasonable if we interpreted the WAL effect as indicating the presence of two independent transport channels [8]. The observations may imply that the timescale relevant for the EEI effect is shorter than the scattering time between the surface and bulk states. The surface states at the two surfaces of the layers may, in general, be uncoupled so long as the EEI effect is concerned. This is not unreasonable considering that the Shubnikov–de Haas oscillations arising from the surface states [19, 20] can be observed in spite of the carrier occupation in the bulk states [21]. The Sb–Te system, including Sb2 Te3 , has received rather little attention so far as TIs. Hsieh [22] investigated the band structure in Sb2 Te3 using angle-resolved photo emission spectroscopy. Due to intrinsic p-type doping, which is typical for the Sb–Te system, the Fermi level was located in the bulk valence band continuum. The existence of a Dirac point for Sb2 Te3 was, therefore, able to be evidenced merely indirectly by the agreement of the measured valence band structure below the Fermi level with the prediction by ab initio density-functional calculations. The Sb–Te compounds are well-known, on the contrary, as a phase-change material system [23] as well as thermoelectric materials. The transport properties have been investigated, as a consequence, mainly from the viewpoint of applications for phase-change and thermoelectric [24, 25] devices. We note that there are reports on the WAL effect in Sb2 Te3 nanowires [26, 27]. Interestingly, GeTe–Sb2 Te3 superlattices have been predicted to become a TI for certain lattice periods [28]. Giant magnetic responses reminiscent of ferromagnetic junctions were observed in the resistance of the TI superlattice [29]. The spin–momentum locking for the TI states is anticipated to be responsible for the unusual magnetic properties. In this paper, we examine the magnitude of the temperature dependence in the conductivity originating from the EEI and WAL effects in a number of Sbx Te1−x layers. In consistent with the weak coupling scenario, we find the effective number of transport channels contributing to the EEI correction remain larger than one when the composition x is varied. The Sb–Te system is an infinite series of TIs. The Sb–Te compounds form a layered structure (Sb2 Te3 )m (Sb2 )n , where the quintuple layer Sb2 Te3 and the bilayer Sb2 are stacked sequentially in the hexagonal c-axis direction. The Sb–Te natural superlattices thus resemble the GeTe–Sb2 Te3 superlattices [28, 29], where the Sb2 layer plays the role of the GeTe layer. In the Sb–Te system, there are three stable phases Sb2 Te3 , SbTe and Sb2 Te in equilibrium [30]. Ab initio density-functional calculations predict, at least, Sb2 Te3 and SbTe to be TIs [18]. The Dirac point in Sb2 Te3 is located in the middle of the bulk band gap, similar to Bi2 Se3 . In SbTe, the location of the Dirac point is predicted to be at the bulk valence band edge, resembling Bi2 Te3 . We grew Sbx Te1−x layers on the Si(111) substrates by molecular beam epitaxy (MBE). The substrate temperature
Figure 1. (a) Distribution of structural domains in Sb0.52 Te0.48 layer (#7) revealed using the EBSD technique. The inset shows the scanning electron micrograph of the same area. The blue and green areas show the twin domains associated with the favored in-plane ¯ and [1100] ¯ alignment (the [1120] directions of the layers being ¯ and [112] ¯ directions of the Si(111) substrates, parallel to the [110] respectively). The in-plane tolerance around the preferred direction is ±15◦ . The in-plane alignment is beyond the tolerance for the red area. (b) Probability P(φ) of the in-plane rotation angle being smaller than φ. The layers are Sb0.52 Te0.48 (≈SbTe, #7) and Sb0.64 Te0.36 (≈Sb2 Te, #8) for the solid and dotted curves, respectively. The steeper the slope, the larger the existence probability for a given angle φ. The line colors correspond to the area colors in (a). The statistics were obtained from 2.5 × 2.5 µm2 areas.
was 250 ◦ C. The growth was carried out with various intensities of the Sb and Te fluxes. We determined the fraction x in the layers using energy dispersive x-ray spectroscopy. In some of the layers, compositions intermediate between Sb2 Te3 and SbTe were realized taking advantage of the nonequilibrium growth in MBE. Our MBE-grown layers were all (0001)-oriented. While there was a favored in-plane epitaxial orientation relationship, the layers were not completely single-crystalline as the lattice mismatch between the layers and the substrate is large. In figure 1(a), we show the crystal domain structure in an Sb0.52 Te0.48 (≈SbTe) layer revealed using the electron backscatter diffraction (EBSD) technique. The scanning electron micrograph in the same area is shown in the inset. The ¯ and [1100] ¯ [1120] directions of the layer were preferentially ¯ and [112] ¯ directions of the aligned to be along the [110] substrate, respectively. The blue and green areas in figure 1(a) display the distribution of twin domains for the component having the favored alignment [31]. In the red area, the range of the in-plane rotation is shifted by 30◦ in comparison to that for the dominant orientation. The domains are seen to 2
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form submicrometer-size islands. The island sizes are nearly the same regardless of the degree of the lattice mismatch with the substrate. Even the islands having unfavorable in-plane alignments are indicated to grow larger at the same rate once they were generated. A similar domain structure was observed also for an Sb0.64 Te0.36 (≈Sb2 Te) layer (not shown). The in-plane alignment was less accurate in the Sb2 Te layer in comparison to that in the SbTe layer, which might have reflected the larger lattice mismatch. (The mismatch is 11.2% for the Sb2 Te layer and 11.0% for the SbTe layer [32].) As a quantitative demonstration, we plot in figure 1(b) the probability P(φ) of the in-plane rotation angle being smaller than φ for the SbTe and Sb2 Te layers. The pronounced rises of P at φ ≈ 30◦ and 60◦ for the solid curve demonstrate the well-established alignment in the SbTe layer. The increases are gradual for the dotted curve, evidencing large fluctuations in the alignment in the Sb2 Te layer. The conductivity change δσ originating from the WAL effect depends on temperature T as [6, 7] e 2 τϕ e2 T δσ L (T ) = −α ln = αp ln , (1) πh τ πh TL where τ and TL are characteristic constants. The phase breaking time τϕ was assumed to vary with T as ∝ T − p . For each side of TI layers, α = −1/2 is predicted for the surface state. Assuming that the EEI effect in TI layers is identical to that in ordinary conductors [12, 13], we expect it to behave as T 3 e2 n 1 − F ln , (2) δσ I (T ) = πh 4 TI
Figure 2. Temperature dependence of sheet conductivity in (a)
Sb0.43 Te0.57 layer (#3) and (b) Sb0.64 Te0.36 layer (#8). A magnetic field of B = 1 T was applied perpendicular to the layer for the lower curve in (a) and for (b). The lines indicate logarithmic behavior.
The WAL correction is quenched when the time-reversal symmetry is broken by applying a magnetic field perpendicular to the layer. The temperature dependence was hence measured in the absence and presence of a magnetic field. The temperature dependence at a magnetic field of B = 1 T in figure 2(a) is almost entirely attributed to the EEI effect. The magnetic field dependence of the EEI effect is negligible at B = 1 T, and so the difference in the slopes of the ln T behaviors provides the magnitude of the WAL effect. We plot the variation with B of the temperature coefficient f defined as δσ (T ) = f ln (T ) in figure 3 by the filled circles. For the Sb0.43 Te0.57 layer, therefore, we obtain α = −1.2 from the change in f between B = 0 and 1 T and n(1 − 34 F) = 2.9 from the value of f at B = 1 T. The coefficient α can be evaluated independently from the magnetoconductivity. Theory predicts the dependence on B to be given as [6]
where F is the screening parameter and should satisfy the condition 0 ≤ F ≤ 1. The EEI effect vanishes at the characteristic temperature TI . We have explicitly specified the number of independent transport channels by n. We point out that the layers investigated in the present work are thinner than the phase coherence length L φ = (Dτϕ )1/2 and the thermal diffusion length L T = (h¯ D/k B T )1/22 . Here, D = 13 v F le is the diffusion coefficient with v F and le being the Fermi velocity and the elastic mean free path, respectively. The system is thus two-dimensional (2D) concerning both the WAL and EEI effects. That the dephasing is governed by the 2D electron–electron scattering means p = 1, as verified experimentally in an Sb2 Te3 layer in [8]. The Sb–Te layers exhibit p-type conduction. The details of the layers, which are numbered #i, can be found in [18]. The thicknesses are in a range of 16–20 nm. The hole concentration increases and the mobility decreases with increasing x. As we show in figure 2, a logarithmic dependence is indeed observed at low temperatures. Below, we compare its magnitude with the theoretical predictions. Notice that the magnitudes of the quantum corrections are on the order of the conductance quantum e2 / h. The coefficient of the ln T term was thus most accurately estimated for the sample having the smallest conductivity, which is the Sb0.43 Te0.57 layer shown in figure 2(a).
1σ L (B) = δσ L (B) − δσ L (B = 0) " ! !# e2 1 h¯ h¯ =α ψ + − ln , πh 2 4eB L 2φ 4eB L 2φ
(3)
where ψ(x) is the digamma function. We show a fitting result of equation (3) to experimental data in figure 4. As we have previously reported in [18], α = −1 is found in the Sb–Te layers. We emphasize that α = −1, which is expected for ideal TI layers, has never been observed in Bi2 Se3 and Bi2 Te3 layers except when the transport properties were tuned by gating [8]. The Sb–Te layers are hence remarkable as the value of α implies the absence of mixing between the surface and
As we reported in [18] L φ in the layers is longer than 0.3 µm at T = 0.3 K and L T at T = 1 K is estimated to be ∼ 0.2 µm. We used in the estimation the hole effective mass of Sb2 Te3 reported in [34].
2
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experimental result in the Sb0.43 Te0.57 layer. This could mean that the surface states at the two sides of the layer as well as the bulk states are independently contributing to the EEI effect. In figure 3, the results obtained from layers having other compositions (x = 0.50 and 0.64) are also shown using open symbols. We point out that the magnitudes of the logarithmic temperature dependence at B = 0.5 and 1.0 T are nearly the same in the layer with x = 0.50. This confirms that the WAL effect is almost completely quenched at these magnetic fields, whereas the EEI effect is practically unaffected in the range of the magnetic field that we utilized, similar to our detailed examinations in [8, 15]. These samples lead us to the same conclusions, i.e., α ≈ −1 for the WAL effect and several independent transport channels contribute to the EEI effect. Our analysis of the temperature dependence thus provides further evidence that the mixing between the surface and bulk states is weak in the Sb–Te system. It is noteworthy that there are reports of giant electron–electron interaction effects in topologically trivial conductors. As discussed in detail by [16, 33], the values of F were deduced to be negative. The discrepancy with the theoretical expectation of F ≥ 0 may be interpreted as evidence for independent participation of multiple bulk states in the transport. One notices in figure 2(a) that the conductivity exhibits deviation from the logarithmic behavior at very low temperatures. As we show in figure 2(b), the temperature dependence became weak for temperatures below about 1 K in the Sb0.64 Te0.36 layer. The tendency of saturation is not due to the heating effect as the conductivity values were unchanged when the measurement current was varied. As the deviation is a positive conductivity correction, it may be the WAL contribution from an additional transport channel. The deviation survives in the presence of a magnetic field of 1 T. The WAL effect is quenched when the magnetic flux penetrating through the phase coherent area L 2φ becomes on the order of the flux quantum h/e. The weak dependence on magnetic field may suggest that L φ associated with this new channel is small, thereby its contribution appears only at very low temperatures. We speculate that the additional contribution stems from the bulk states. Its small magnitude, in fact, explains why the bulk states appear to be irrelevant for the WAL magnetoconductivity and α is fixed at −1. In conclusion, we have investigated the logarithmic temperature dependence in the conductivity in a series of Sbx Te1−x layers. Multiple transport channels have been indicated to contribute to the EEI effect. In addition, our analysis supports the observation of α = −1 in the magnetoconductivity for the WAL contribution.
Figure 3. Coefficient f of logarithmic temperature dependence in
sheet conductivity δσ (T ) = f ln (T ). The magnetic field applied perpendicular to the layers was varied between 0 and 1 T. The composition x in the Sbx Te1−x layers is 0.43 (#3), 0.50 (#6), and 0.64 (#8) for the filled circles, open circles, and open triangles, respectively. The horizontal lines indicate the magnitudes of e2 /(π h) multiplied by 1, 2 and 3, i.e., n(1 − 34 F) = 1, 2 and 3. The separation between adjacent lines corresponds to α = −1.
Figure 4. Dependence of sheet conductivity on magnetic field in Sb0.43 Te0.57 layer (#3) at temperature T = 0.33 K. The curve shows a fit to equation (3) with parameters of α = −1 and L φ = 0.38 µm. Magnetoresistance 1R/R in the layer is shown in the inset.
bulk states. We also note that the layer is intriguing as the magnetoresistance 1R/R is linear [18], as we show in the inset of figure 4. Similar to our previous comparisons [8, 15], the values of α derived from the magnetic field and temperature dependencies agree reasonably well with each other. As for the EEI effect, the magnitude is again much larger than the amount that one transport channel can be responsible for. In fact, at least three transport channels are required to explain the
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