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Non-Drude magneto-transport behavior in a topological crystalline insulator/band insulator heterostructure Chieh-Wen Liu, Feng Wei, Kasun Premasiri, Shuhao Liu, Song Ma, Zhidong Zhang, and Xuan P.A. Gao Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.8b03113 • Publication Date (Web): 05 Sep 2018 Downloaded from http://pubs.acs.org on September 10, 2018
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Non-Drude magneto-transport behavior in a topological crystalline insulator/band insulator heterostructure Chieh-Wen Liu1,†, Feng Wei2,3,† , Kasun Premasiri1, Shuhao Liu1, Song Ma2,* , Zhidong Zhang2, and Xuan P. A. Gao1,* 1
Department of Physics, Case Western Reserve University, 2076 Adelbert Road, Cleveland, Ohio, 44106, United States
2
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, 72 Wenhua Road, Shenyang 110016, China 3
University of Chinese Academy of Sciences, Beijing 100049, China
†
These authors contributed equally to this work.
*
Email:
[email protected] (S.M.),
[email protected] (X.P.A.G.)
Abstract The Drude model is one of the most fundamental models used to understand the electronic carrier transport in materials, including recently discovered topological materials. Here, we present a magneto-transport study revealing the non-Drude transport behavior in a heterostructure of topological crystalline insulator (TCI) SnTe and band insulator PbTe which exhibits a non-saturating linear magneto-resistance (MR) effect, a novel phenomenon widely observed in topological materials with gapless dispersion. It is shown that in the van der Pauw geometry in which the longitudinal and transverse magneto-resistances are measured to extract the magneto-conductivity, the two-band Drude model is not sufficient to self-consistently describe both the longitudinal and transverse magneto-conductivities. Furthermore, in the Corbino geometry which directly measures the longitudinal magneto-conductivity σxx(B) for a straightforward comparison with the Drude model, the MR, 1/σxx(B), still reveals a large linear MR effect, in direct discrepancy with the Drude model. While shining further light on the multiple carrier transport in TCI, this study highlights an unusual magneto-transport character of topological materials that challenges the standard Drude picture of electron transport.
Keywords: magneto-transport, Drude model, topological insulator, linear magneto-resistance Transport study is an essential first step in characterizing and understanding electronic materials and paves the way to further device design. Conventional theories and experiments have demonstrated that for most metals, the resistance has a quadratic dependence with magnetic field (B) and then saturates at high field.1 A nonsaturating and linear magnetoresistance (MR) is one particularly intriguing phenomenon and has continued to attract attention for decades.1-15 Linear MR has been reported in a variety of materials, such as semimetals,13, 14 small bandgap semiconductors4, 5 and multilayer graphene.6 A growing
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number of studies on MR have also been made in topological insulator (TI) materials,16-19 a class of materials in which gapless metallic surface states exist within the bulk band gap. It is apparent that in TIs with linearly dispersive gapless surface states (Bi2Se3, Bi2Te3, etc),7-11 large linear MR appears to be ubiquitous in experiments. More recently, Fu proposed a new class of TI called topological crystalline insulator (TCI)18, arises from crystal symmetry protection.
20, 21
whose topological nature
18
The metallic surface states in TCIs are protected by mirror symmetry of crystal instead of time-reversal symmetry as in the TIs. Such new topological phases of matter have been predicted and observed in the SnTe material class. A linear MR has also been reported in In-doped SnTe nanoplates,12 Pb0.6Sn0.4Te15 and SnTe/PbTe heterojunction.22 While the mechanism of linear MR in TI materials is still unresolved, it has been suggested that it may arise from quantum MR effect in gapless Dirac-like surface states5, sample.
23
or charge inhomogeneity and mobility fluctuations in the
2, 24, 25
But the understanding of the nature of MR transport of topological surface states is hindered by the existence of bulk states which also contribute to the overall electronic transport.
To account for both the topologically nontrivial surface (or edge) conduction and trivial bulk conduction, carrier transport in topological materials is usually interpreted within the two-band Drude model.26-29 The Drude model has been an essential model used to distinguish the surface and bulk conductions in TIs. For the simplest materials with only one carrier type, the Dude model predicts a magnetic field (B) independent longitudinal MR ρxx(B) and a linear B-dependent transverse MR, ρxy(B). In the presence of two different carrier types in a system, the MR is predicted to show a low field quadratic B-dependence followed by a saturation behavior at high fields and ρxy(B) has a non-linear field-dependent characteristic. While no real material can be regarded as a perfect single band Drude system with all the carriers having the same mass and mobility, transport in many metals and doped semiconductors is well described by the Drude model with a dominant carrier band, exhibiting weak quadratic MR and linear ρxy(B).1 For topological materials, since the conduction generally has both surface and bulk contributions, the two-band model is commonly used to analyze the electronic transport data for the extraction of key transport parameters such as surface and bulk carrier densities and mobilities. However, the MR observed in topological materials usually shows a strong linear field dependence, in contrast to the standard Drude MR picture. A recent progress in the magneto-resistance study of graphene heterointerface devices30 attributed the linear MR to the two-band Drude MR in the intermediate field range before the saturation occurs. This gives a simple yet physically motivated interpretation of the large linear MR in systems involving multiple types of carriers. Here, we report a magneto-transport study on TCI SnTe/PbTe heterostructure and elucidate the large linear MR effect by measurements in both the van der Pauw geometry and the
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Corbino geometry where the longitudinal magneto-conductivity is directly extracted and examined within the Drude model. Surprisingly, the longitudinal magneto-conductivity,
σxx(B), directly measured in the Corbino geometry shows a reciprocal linear field dependence distinct from the Drude formula. Our results suggest that the large linear MR observed in TCI involves physics beyond the standard Drude model. The SnTe/PbTe heterostructures were grown on insulating SrTiO3(111) (STO) substrates with a Te capping layer using an ultrahigh vacuum molecular beam epitaxy (MBE) system with base pressure below 2×10-10 mbar. Detailed MBE growth procedure and structural and composition analysis of the samples can be found elsewhere.22 Briefly, PbTe thin film was first deposited on the STO substrate at 200 ºC followed by an annealing at 200 ºC for 90 min. Next, SnTe thin film was deposited on the PbTe thin film at 200 ºC and annealed at the same temperature for 90 min. It is noted that the low growth and annealing temperature is essential for the formation of a sharp SnTe/PbTe interface and the maintenance of the lattice mirror symmetry down to low temperatures.22 Unless noted, the thicknesses for the (001) oriented PbTe and SnTe layers are 15 and 24 nm for the samples studied in this work. To avoid sample degradation or oxidation, a 7 nm thick Te layer was deposited on the surface of SnTe as the final step of the MBE growth. The transport measurements were performed in a Quantum Design Physical Property Measurement System with a superconducting magnet providing magnetic field up to 9 T using standard lockin technique with indium contacts placed on the sample either in the van der Pauw or Corbino geometry. A Hall bar sample was also studied to compare with the van der Pauw results. A small sinusoidal voltage (typically a few hundred microvolts) at low frequency (7 Hz) was applied on the sample by the lockin to drive a current. Both the current and four probe voltage (Vxx or Vxy) were measured by the lockin amplifiers to allow four-probe measurements of Rxx or Rxy and Ohmic behavior was confirmed. A schematic of the overall structure is shown in the inset of Fig.1a. Figure 1a shows the temperature (T) dependence of the longitudinal resistivity (sheet resistance) ρxx in the absence of magnetic field (B=0). The ρxx data shows a metallic behavior with resistivity ratio
ρxx(300K)/ρxx(10K) ~ 588. The lower inset shows the van der Pauw geometry for four-terminal measurement of a roughly square sample. To minimize the sample shape asymmetry and contact misalignment effect, two configurations were measured where we rotated the current/voltage probes by 90 degrees. The data shown here are the average values of these two configurations. The normalized MR, ∆ / 0 at different temperatures are shown in Fig. 1b. At low temperatures, the large and non-saturating MR (~1000% at 9 T) at the lowest temperature resembles the peculiar large MR observed in other materials with gapless dispersion.5,11-15,19-22 As the temperature increases, the magnitude of MR decreases
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and turns into a weak quadratic dependence, which is predicted by the two-band Drude model. The data at 77 K and 300 K are multiplied by factors of 50 and 500 respectively for clarity. Figure 1c represents the Hall resistivity ρxy as a function of B at various temperatures. The nonlinearity of the data is observed at T = 2.5-77 K, suggesting the relevance of multiple carrier transport in our sample. The sign of ρxy demonstrates the dominance of n-type conduction between 2.5 and 77 K. At higher temperatures, Hall resistivity data show that the dominant carrier type changes to p-type with increasing temperature. (a)
ρxx (Ω/Ǵ Ǵ)
80 Te SnTe(001) PbTe(001) STO(111)
60 40 20 0 0
50 100 150 200 250 300
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0.6 0.4 0.2 0.0 0
Figure 1.
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8
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Temperature and magnetic-field dependent transport in SnTe/PbTe in the van der
Pauw geometry. (a) Longitudinal sheet resistance ρxx as a function of temperature. The top inset shows a schematic of the SnTe/PbTe heterostructure. The lower inset illustrates a schematic of the van der Pauw configuration for four-probe resistance measurement. (b) Magnetoresistance ∆ / 0 (defined as [ − 0]/ 0) as a function
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of magnetic field at different temperatures. (c) Hall resistivity as a function of magnetic field at T = 2.5-77 K. The gray dashed line serves as a guide to show the change of the Hall slope as increasing magnetic field.
(a)
(b) T=300 K
Data Fit
69.6
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Data Two-band Drude fit Expectation from fitting σxy
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Figure 2. Drude analysis of magneto-transport at 300 K and 77 K. (a) Hall resistivity data at 300 K. (b) Longitudinal resistivity ρxx as a function of magnetic field at 300 K. The red line is the fit to the data according to = [1 + ]. (c) Hall conductivity σxy and (d) longitudinal conductivity σxx data at 77 K. Red (blue) solid line shows the fit to the data according to a two-band Drude model. Blue (red) dashed line is plotted using the fitting parameters of σxx (σxy) data. The perfectly linear B dependence of ρxy at 300 K (Fig. 2a) suggests the dominance of a single carrier band in the transport and the hole carrier density p can be determined by the Hall slope RH to be 1.43×1015/cm2 according to = 1⁄ where e is the elementary charge. This leads to a Hall mobility value of µ = 64 cm2/Vs using the single band Drude formula ρ = 1/peµ and zero-field sheet resistance value of 68.9 Ω/sq. Together with the linear
ρxy(B), a weak quadratic longitudinal magneto-resistance is observed (Fig. 2b), consistent
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with the single band dominated conduction in the Drude model as seen in typical metals. Interestingly, in the Drude theory, the mobility µ can also be estimated according to ≈ [1 + ] which yields a good fit (red line) to the data with mobility 118 cm2/Vs. The comparable values of µ from analyzing the and further affirm the consistency of the single band Drude model for the system at 300K. The high value of hole density at 300K (equivalent to ~1021/cm3) is also consistent with the fact that SnTe tends to grow with rich p-type defects. Thus, the sample's transport at room temperature follows the Drude model for a single band of hole carriers in the SnTe bulk. Since the Hall resistivity starts to show non-linear behavior at 77 K (Fig. 1c), it is thus necessary to adapt the two-band Drude formula to analyze the data. Since each carrier type follows the Drude equation with its own characteristic parameters and adds to the total conductivity of the system, it is natural and more convenient to analyze the magneto-conductivity. The Hall conductivity σxy(B) and longitudinal conductivity σxx(B) data at 77 K are shown in Figs. 2c and 2d respectively. Practically, fitting either σxy(B) or σxx(B) curve alone is able to extract the density and mobility of both bands, as commonly done in literature.28 Red solid line in Fig. 2c shows the fit to the converted data to the two-band Drude model =
! "
+
! "
#,
(1)
where n1, n2, µ1, µ2 are the density and mobility of the two types of carriers. By applying constraints = 0 and lim"→
()* "
to equation (1), the number of fitting parameters
can be reduced to two.28 Such fitting of σxy(B) yields n1 = 2.65×1014/cm2, n2 = 9.32×1015/cm2, and µ1 = 1400 cm2/Vs, µ2 = 280 cm2/Vs for the surface and bulk carriers, which should also consistently describe σxx(B). Applying these parameters to the two-band Drude model for
σxx(B) =
! "
+
! "
# ,
(2)
we obtain an expected σxx(B) curve as shown by the red dashed line in Fig. 2d which only slightly deviates from the measured data at high fields. Alternatively, we can fit the measured σxx(B) data to Eq.(2) and obtain the blue solid line as the fit with n1 = 3.84×1014/cm2, n2 = 7.27×1015/cm2, and µ1 = 1100 cm2/Vs, µ2 = 350 cm2/Vs. These fitting parameters are not far from the ones obtained by fitting σxy(B). During the fitting, we used the measured σxx(B) at B=0 and replaced according to = 0 = + + + to reduce the number of fitting parameters to three. The curve expected from the fitting parameters of the fit is plotted in Fig. 2c as the blue dashed line. Again, it is seen that the σxy(B) curve expected from the parameter extracted in fitting agrees well
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with the measured data at low B and only shows some deviation at high fields. Therefore, two-band Drude model gives a good description of the system at 77 K. The two electron bands with carrier density n1 and n2 are likely the 2D Dirac electrons at the SnTe/PbTe interface and bulk electron carriers in n-type PbTe. The inset in Figure 2c shows a proposed schematic band diagram of SnTe/PbTe heterostructures.22 The SnTe/PbTe heterostructure is known to have a broken gap band alignment at low temperatures. Equilibration leads to electrons transfer from n-type PbTe to p-type SnTe and therefore a downward (upward) band-bending in for SnTe (PbTe). The Dirac point also shifts downward accordingly, giving rise to the Fermi level lying above the Dirac point at the interface and n-type Dirac electrons, despite bulk SnTe being p-type. In addition to the Dirac electrons at SnTe/PbTe interface (with parameters n1 and µ1 in the two-band Drude fitting in Fig.2c and d), the n-type bulk PbTe and p-type bulk SnTe can also contribute in the conduction, as indicated by the schematic band-diagram. The good fitting results in Fig.2c and d indicate that the p-type SnTe bulk does not have significant contribution to the total conduction at 77K and the two-band Drude model including n-type bulk PbTe and Dirac electrons at interface is sufficient to describe the data. Fitting the data to the three-band Drude model including the p-type SnTe bulk as the third band does not further improve the agreement between data and model (See Supporting Information, Fig. S1a and b). The self-consistency of multi-band Drude analysis becomes much worse at lower temperatures where a large MR is observed. The fitting procedure for the magneto-conductivity data at 5 K follows the two-band Drude analysis method described above, as shown in Figs. 3a and 3b. It can be seen that using the fitting parameters from the fit to σxy (σxx) data predicts a curve that is higher than the measured σxx (σxy) data and the discrepancy becomes larger at higher fields. Specifically, the difference between the experimental data and the expected σxy (σxx) curve from fitting σxx (σxy) reaches 140 - 600% at 9 Tesla, much greater than the 5 - 20% disagreement in Figs. 2c and 2d. Similar inconsistency is observed when analyzing additional MR data at low temperatures and there is little improvement by including a third band (Supporting Information, Fig.S1c and d and Fig.S2). This failure of multi-band Drude model appears to be a consequence of the large linear MR measured at low temperatures. As shown in Fig. 3c, the curve (blue dashed line) expected from the transport parameters from the two-band Drude fit to σxx data is not consistent with the measured data which exhibit a much stronger and linear MR effect. Using parameters from the two-band Drude model to the σxy(B) also yields similar that is much weaker than experimentally observed linear MR. Although in the two-band Drude model, there is an intermediate field range (between ~1/µ1 and 1/µ2) that the MR rises with the magnetic field in a quasi-linear fashion,30 we were not able to find a set of
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parameters that can consistently describe both σxx(B) and σxy(B) (thus ρxx(B)) within the two-band Drude analysis. Using alternative methods of fitting the data to the two-band Drude model (e.g. fitting ρxx(B) instead of σxx(B) or σxy(B) following Ref. 29, see Supporting Information, Fig. S3) also produces strong discrepancy between the fit and data. We also performed a Hall bar measurement on another sample which has similar layer sequence but with different film thickness and the results are similar (see Supporting Information Fig. S4).
Data Two-band Drude fit Expectation from fitting σxx
3
T=5 K
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(b)
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Data Two-band Drude fit Expectation from fitting σxy
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(c) ρxx (Ω/Ǵ Ǵ)
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Data Expectation from two-band Drude
1.0 0.8
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Figure 3. The failure of two-band Drude analysis at low temperatures where a large linear magnetoresistance exists. (a) Hall conductivity σxy and (b) longitudinal conductivity σxx data at 5 K. Red (blue) solid line shows the fit to the data according to the two-band Drude model. Blue (red) dashed line is plotted using the fitting parameters of σxx (σxy) data. (c) ρxx as a function of magnetic field at 5 K. The blue dashed line is expected from the two-band Drude model using the fitting parameters of σxx data.
To further study the applicability of the Drude description of SnTe/PbTe heterostructure independently, we performed a Corbino measurement where the current flows in the radial direction and the Hall voltage does not exist. The longitudinal conductivity σxx(B) can therefore be directly measured without the complication of measuring both ρxx(B) and ρxy(B) and magneto-resistivity tensor inversion as in the van der Pauw method. Here the Corbino measurement was performed on the same sample after finishing the van der Pauw measurement to allow a fair comparison of magneto-transport in both configurations. To make the Corbino measurement, indium solder was placed around the perimeter and at the center of the sample after finishing the van der Pauw measurement, as shown in the inset of Fig. 4a. These contacts were used as the current contacts. Two small indium spots were placed between the center and the outer contacts as the voltage probes to allow intrinsic four-probe measurement and remove the contact resistance effect. The normalized MR data ⁄ 0 and the normalized magneto-conductance data , /, 0 at 5 K for the Corbino geometry are shown in Figs. 4a and 4b. The most striking and surprising finding in the Corbino experiment is that in Fig. 4a, the MR shows a clear linear dependence above 2 T (blue line). Since there is no Hall signal in the Corbino geometry, the magneto-resistivity
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ρxx(B) is simply the inversion of magneto-conductivity: ρxx(B) = 1/σxx(B) and a linear field dependence is not expected in the Drude theory for magneto-conductivity. Although one can attempt to fit the magneto-conductance data in the Corbino measurement to Eq. (2) and obtain a seemingly reasonable fit, as shown by the red solid line in Fig. 4b, the difference between the two-band Drude fit (red dashed line) and the magneto-transport data in the Corbino geometry is clearly seen in the MR plot at B > 1 Tesla in Fig. 4a. The essential difference between the Drude description and data is that the Drude equation (Eq. (2)) contains parabolic field dependence that can only describe the low field data at B < 1 Tesla while the experimental data follow a simple linear function with B up to the highest measured magnetic field. It is obvious that the two-band Drude model cannot describe the linearly increasing MR data in the Corbino geometry adequately. Analyzing the data from the Corbino measurements at different temperatures shows that the Drude model works better at higher temperatures (Fig. S5, S6, Supporting Information), similar to the previous conclusion from the Drude analysis of van der Pauw data. 50
(b)
Corbino Data Expectation from two-band Drude Linear fit
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(a) Rxx(B)/Rxx(0)
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4
B (T)
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Vxx 8
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Figure 4. Corbino measurement of magneto-transport. (a) Magneto-resistance data ⁄ 0 at 5 K from Corbino measurement. Blue solid line is the linear fit to the data from 2T to 9T. The inset shows the four-terminal Corbino geometry. (b) Magneto-conductance , /, 0 at 5 K obtained from van der Pauw and Corbino measurement. Blue dashed line is expected from the linear fit in (a). Red solid line fits to the data according to the two-band Drude model and the magneto-resistance curve expected from the fitting parameters is shown in (a) by a red dashed line.
Therefore, while the magneto-transport of TCI/band insulator SnTe/PbTe agrees well with the Drude theory at high temperatures (>~ 77 K), the two-band Drude theory becomes increasingly insufficient to describe the magneto-transport data in both the van der Pauw and Corbino measurements at low temperatures when the linear MR emerges. Note that although the Drude theory is a classical theory, it should remain applicable at low temperatures when
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there are quantum effects (e.g. quantum oscillations in MR only add to the Drude MR background in materials showing Shubnikov-de Haas MR oscillations at low temperatures). Linear MR in various gapless topological materials has been suggested to be related to the quantum MR in the extreme quantum limit (carriers occupy only one Landau level),23 but the high carrier density in our samples sets our experimental conditions to be far below the extreme quantum limit and the Drude theory should be valid. The mechanism for the observations reported here remains to be developed, but since the MR in the Corbino geometry shows a simple linear dependence (as shown by the blue line in Fig. 4a), this suggests that the magneto-conductance , /, 0 has an inversed linear B-dependence (the blue dashed line in Fig. 4b). Since in the Drude theory, =
-
! "
, one possible
implication of the linear MR in the Corbino geometry is that the mobility or scattering time has some peculiar field dependence that produces the apparent linear B-dependence. In conclusion, magneto-transport measurements were performed on topological crystalline insulator (TCI) SnTe and band insulator PbTe heterostructures in both the van der Pauw and Corbino geometries. It is found that although the standard Drude model can satisfactorily describe the data at high temperatures (>= ~ 77 K) where the MR is weak, the magneto-conductivity σxx(B) and σxy(B) significantly deviate from the two-band Drude theory in the field range where the MR is linear and large. Surprisingly, the MR measured in the Corbino geometry which is simply the inverse of the longitudinal magneto-conductivity, still reveals a large linear MR effect, in direct disagreement with the Drude model. Although the multi-band Drude theory may still serve as a starting point for analyzing the carrier transport in topological materials with both bulk and surface (or edge) transports, this study highlights an unconventional magneto-transport nature of topological materials and calls for further theoretical understanding. Supporting Information The Supporting Information is available free of charge on the ACS Publications website. More information on van der Pauw and Corbino measurements and comparison with two-band Drude model at additional temperatures and an alternative three-band Drude analysis and two-band Drude analysis of the magneto-resistance instead of magneto-conductivity. Additional data/analysis on a Hall bar sample. Notes The authors declare no competing financial interests. Acknowledgements X.P.A.G. thanks the NSF (grant number DMR-1607631) for supporting the work at CWRU. S.M. and Z.D.Z. thank the Natural Science Foundation of China (NSFC) with the grant Nos.
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51571195, 51331006, and 51590883.
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