Abstract
Noncentrosymmetric conductors are an interesting material platform, with rich spintronic functionalities^{1,2} and exotic superconducting properties^{3,4} typically produced in polar systems with Rashbatype spin–orbit interactions^{5}. Polar conductors should also exhibit inherent nonreciprocal transport^{6,7,8}, in which the rightward and leftward currents differ from each other. But such a rectification is difficult to achieve in bulk materials because, unlike the translationally asymmetric p–n junctions, bulk materials are translationally symmetric, making this phenomenon highly nontrivial. Here we report a bulk rectification effect in a threedimensional Rashbatype polar semiconductor BiTeBr. Experimentally observed nonreciprocal electric signals are quantitatively explained by theoretical calculations based on the Boltzmann equation considering the giant Rashba spin–orbit coupling. The present result offers a microscopic understanding of the bulk rectification effect intrinsic to polar conductors as well as a simple electrical means to estimate the spin–orbit parameter in a variety of noncentrosymmetric systems.
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Main
The effect of the lattice symmetry on the electronic states is a fundamental issue in condensed matter physics. In particular, broken inversion symmetry in the crystal structure generally causes spin band splitting which modifies the electronic ground state, affecting transport properties represented by the superconductivity in noncentrosymmetric systems^{3,4} or spinrelated transport in nonmagnetic materials^{1,2}. Among them, Rashba^{5} and Dresselhaus^{9}type spin–orbit interactions are two wellknown textbook models which have succeeded in explaining a variety of exotic phenomena in systems without inversion symmetry.
Although the Rashba effect has been conventionally studied at surfaces or interfaces^{10,11,12,13,14}, the recent discovery of threedimensional (3D) materials which host a large Rashbatype band splitting^{15,16,17,18} pave the way towards exploring novel transport originating from the 3D chiral spintexture of the electronic band. BiTeX (X = I, Br) is one such bulk polar semiconductor, in which Bi, Te and X layers are stacked alternately so that the mirror symmetry along the c axis is broken (Fig. 1a). The resultant Rashbatype spin splitting of the electronic bands has been confirmed by angleresolved photoemission spectroscopy (ARPES)^{15,16} and the transport signatures of the split Fermi surface have been reported by quantum oscillations in resistance^{19} or thermoelectric coefficients^{20}. However, characteristic magnetotransport reflecting the spin polarization in the electronic band or polarity of the crystal has been elusive, except for photocurrent experiments on BiTeBr^{21}. One of the manifestations of lattice symmetry breaking in electric transport is the rectification effect. In the presence of an inplane magnetic field, the Rashbatype spin band splitting is modulated to become asymmetric along a direction perpendicular to both the polar axis and the magnetic field (Fig. 1b). Reflecting this asymmetric nature of the electronic band structure, the electrical resistivity is expected to be different, depending on the current direction, as illustrated in Fig. 1c. Although there have been several reports of such an asymmetric electric transport at interfaces^{6,7,8} or in chiral nanostructures^{22,23}, only one bulk material with chirality is known to show such a nonreciprocity^{24}. In addition, the microscopic mechanism underlying the observed nonreciprocal electric transport is unclear, so that a strategy to enhance this rectification effect is still lacking.
In this letter, we report the observation of nonreciprocal electric charge transport in polar bulk materials. The rectification effect in BiTeBr satisfies the characteristic selection rule for polar systems, and the magnitude of the signal is greatly enhanced compared with those observed at nonmagnetic interfaces or in chiral materials, reflecting the giant spin splitting. Moreover, the experimental results are quantitatively explained on the basis of a simple Rashba model making use of the Boltzmann equation, indicating that nonreciprocal transport measurements can become a new probe with which to estimate Rashba spin splitting. Our result offers not only a new functionality, in terms of the 3D Rashba system, but also the first quantitative explanation for nonreciprocal electric transport in terms of a microscopic mechanism.
To perform transport experiments in single polar domains, we have fabricated microsize devices of BiTeBr (as shown in Fig. 1c). Figure 2a shows the temperature dependence of the resistivity of a measured sample. The observed magnitude of the resistivity (ρ ∼ 1 mΩ cm) and carrier density estimated by the Hall effect (n = 1.1 × 10^{19} cm^{−3}) are similar to those in a previous report^{21}. We have studied the rectification effect in this sample due to the symmetry breaking of the lattice—that is, the difference in the electric resistance depending on the current direction. Electrical resistance is phenomenologically described as
up to secondorder terms^{6,22}, where R_{0}, I, β and B represent the resistance at zero magnetic field, electric current, coefficient of the normal magnetoresistance and external magnetic field, respectively. The second term on the righthand side denotes the normal magnetoresistance, whereas the third term which depends on both the electric current and magnetic field corresponds to the nonreciprocal resistance. The coefficient tensor γ, which describes the ratio of nonreciprocal resistance to the normal resistance, satisfies the characteristic selection rule reflecting the lattice symmetry. For example, γ = 0 in centrosymmetric systems, which means the rectification effect is absent if the crystal has inversion symmetry. On the other hand, γ is finite when the magnetic field, electric current and polar axis are perpendicular to each other in the case of polar systems (Fig. 1c). When the electric current flows along the x axis and the magnetic field is applied along the inplane direction, the nonreciprocal resistance can be written as
with the polarization P (∥ z) and angle θ between the y axis and the magnetic field (Fig. 2c inset). This currentdependent resistance leads to a nonlinear voltage drop that can be detected as the secondharmonic signal in lockin measurements. If we apply an a.c. input current (I = I_{0} sinωt), considering the above selection rule, the nonlinear voltage due to the nonreciprocal components can be expressed as follows:
During the measurement, we have measured a ycomponent of the secondharmonic signal with a π/2 phase shift, and confirmed that the xcomponent of the second harmonic is almost zero. The nonreciprocal magnetoresistance is defined as
Figure 2b shows the secondharmonic signals in the cases of θ = 0° and 90°. Here, we plot ΔR^{2ω}, the antisymmetric components with respect to B, to remove any background signals arising from nonuniformity of the samples. When the magnetic field is parallel to the electric current (θ = 90°), there is no discernible signal in the second harmonic. On the other hand, if the magnetic field and electric current are perpendicular (θ = 0°), a finite nonreciprocal resistance signal proportional to the magnetic field appears. This characteristic selection rule (presence or absence of the nonreciprocal signals depending on the relative angle of the magnetic field and the electric current) is also confirmed by the angle dependence measurement under a magnetic field of 9 T in Fig. 2c. Around θ = 90° or 270°, ΔR^{2ω} = 0, whereas ΔR^{2ω} shows finite negative (or positive) values at θ = 0° (or 180°). The angle dependence of the ΔR^{2ω} signal is fitted well by a cosθ curve, which is consistent with the phenomenological discussion. In addition, the observed ΔR^{2ω} signal increases linearly with the electric current (Fig. 2d). Similar behaviour of the ΔR^{2ω} signals has been observed for all samples measured in the experiments (see Supplementary Information). The estimated value of γ, which characterizes the magnitude of the rectification effect, is γ ∼ 1 A^{−1} T^{−1} at the lowest temperature, which is much larger than those reported so far (γ ∼ 10^{−3} A^{−1} T^{−1} for a Bi helix^{22}, γ ∼ 10^{−2} A^{−1} T^{−1} for chiral organic materials^{24}, and γ ∼ 10^{−1} A^{−1} T^{−1} for Si FET interfaces^{6}), possibly reflecting the giant spin splitting in BiTeBr.
In Fig. 3, we show the nonreciprocal magnetoresistance for several samples with different carrier densities and their temperature dependences. Since γ is the coefficient of electric current (I_{0}) rather than the current density, and thus depends on the sample size, we have calculated the value of γ′ ≡ γA = γL_{y}L_{z} (where L_{i} (i = x, y, z) is the length of the sample along the i direction and A = L_{y}L_{z} is the cross sectional area of the sample) to compare several samples with different dimensions and theoretical calculations. The nonreciprocal electric signal is significantly large in samples with low carrier densities, and is enhanced at low temperature in all samples. (Note that the carrier density n is estimated experimentally from the Hall coefficient, whereas n determines the position of the chemical potential in the theoretical calculation.) Both the steep increase of γ′ in samples with low carrier densities and its enhancement at low temperature, together with its magnitude, are described well by the theoretical calculation (solid lines) using the simple 3D Rashba model as discussed below.
The bulk conduction band of BiTeX under an inplane magnetic field is described well by the Rashba Hamiltonian
where we set ℏ = 1, σ_{x}, σ_{y}, σ_{z} are the Pauli matrices, and λ is the magnitude of the spin–orbit interaction (with λ = 3.85 eV Å for BiTeI and λ = 2.00 eV Å for BiTeBr^{16}). The magnetic field B_{y} in equation (5) has units of energy, and should be regarded as an abbreviation for gμ_{B}B_{y}/2, with g and μ_{B} being the gfactor and Bohr magneton, respectively. We set m_{⊥} = 0.15m_{e} and m_{∥} = 5m_{⊥} based on the previous density functional theory result^{25}. (m_{e} is the electron mass in the vacuum.) The energy dispersion reads as
Here, we first focus on the simple twodimensional (2D) case by setting k_{z} = 0. The schematic energy dispersion as a function of k_{x} at k_{y} = 0 for B_{y} < m_{⊥}λ^{2} and B_{y} > m_{⊥}λ^{2} is shown in the lower part of Fig. 4a. The position of the chemical potential μ determines the topology of the Fermi surfaces and defines the regions I, I′, II, III of the phase diagram (Fig. 4a). In the case of B_{y} < m_{⊥}λ^{2}, which is relevant for BiTeBr, there exist three regions, depending on the chemical potential μ: region I, −B_{y} − ((m_{⊥}λ^{2})/2) < μ < B_{y} − ((m_{⊥}λ^{2})/2); region II, B_{y} − ((m_{⊥}λ^{2})/2) < μ < ((B_{y}^{2})/2m_{⊥}λ^{2}); region III, μ > ((B_{y}^{2})/2m_{⊥}λ^{2}). In region I, there is one Fermi surface which does not enclose the origin. In region II (III), on the other hand, there are two Fermi surfaces with the same (opposite) spin helicity enclosing the origin (see also Supplementary Fig. 3). The calculated nonreciprocal current behaves differently depending on the region.
In region II, the firstorder current in the electric field E_{x} is given by solving the Boltzmann equation with the singlerelaxationtime (τ) approximation as
This expression contains only even order terms in the magnetic field B_{y} (J_{x, 2D}^{(1)}(−B_{y}) = J_{x, 2D}^{(1)}(B_{y})) and describes the normal resistance, which is independent of the current direction. On the other hand, the secondorder current in E_{x} of region II is given by
which contains a linear term in B_{y} (J_{x, 2D}^{(2)}(−B_{y}) = −J_{x, 2D}^{(2)}(B_{y})). Differently from the firstorder term in equation (7), this secondorder current leads to an electrical conductivity which depends on the direction of the electric field, and thus describes the nonreciprocal electric transport. Although it is difficult to obtain analytic solutions in region I, we can neglect it for the present analysis of the experimental results. Note that m_{⊥}λ^{2} = 75 meV for BiTeBr, and hence only the leftmost regime of Fig. 4a is relevant for analysis of the present experiment because we are interested in the region of low magnetic fields of the order of ∼1 T. However, the nonreciprocal magnetoresistance in Fig. 2b seems to deviate slightly from a straight line. In BiTeBr, the typical energy scale of the Rashba spin–orbit coupling (B_{y} = m_{⊥}λ^{2}) is B_{y} ∼ 43 T, assuming the same large gfactor (g = 60) as BiTeI^{26}. Thus, the effect of the higherorder term (∝ B^{3}) might appear around 9 T. In addition, the larger B_{y}/(m_{⊥}λ^{2}) in Fig. 4a also becomes relevant for systems with weaker Rashba splitting. For example, m_{⊥}λ^{2} for the (In, Ga)As/(In, Al)As heterostructure is of the order of 5 meV (ref. 11). The nonreciprocal current in region III is zero (small) for the 2D (3D) case in the singlerelaxationtime approximation since the inner and outer Fermi surfaces have opposite spin helicities above the bandcrossing point, and respond to the magnetic field B_{y} so that the contributions to J_{x, 2D}^{(2)} cancel.
In the theoretical analysis above, we have calculated the current density as a function of the applied electric field; J_{x} = J_{x}^{(1)} + J_{x}^{(2)} = σ_{1}E_{x} + σ_{2}E_{x}^{2} (see Supplementary Information). On the other hand, the experimental observable is the voltage drop as a function of the electric current: V_{x} = R_{0}I_{x}(1 + γB_{y}I_{x}), and γ is expressed as
where I_{x} = L_{y}L_{z}J_{x} and V_{x} = L_{x}E_{x}. Therefore, the τdependence of σ_{2} ∝ τ^{2} and σ_{1} ∝ τ indicates that γ is independent of τ. In Fig. 4b, c, we plot the calculated γ′ value for the 2D and 3D cases as a function of both the chemical potential and temperature. (For the 3D calculation, we take into account dispersion along the caxis, as explained in Supplementary Information.) It is noted that, in both the 2D and 3D cases, γ′ diverges as the electron density approaches zero at the lowest temperature, associated with strong suppression with increasing temperature.
These features as well as the magnitude of γ′ are in good agreement with the experimental results, as shown in Fig. 3a, b. This indicates that the singleτ approximation holds in the parameter region for the present measurements. More importantly, the magnitude of the nonreciprocal electric transport γ′ can be regarded as an intrinsic quantity of the band structure and determined by λ, m_{⊥} and m_{∥}, irrespective of τ. The situation is similar to the Hall effect, in which the Hall coefficient is determined simply by the carrier density.
We note that spinrelated quantities in BiTeBr are also of great interest. Theoretically, the Boltzmann theory can also be applied to the spin current {j}_{{S}_{\alpha}}^{i}={v}_{i}{s}_{\alpha} in direction i with the spin polarization α in the present system. Here, v is the group velocity and s is the spin operator. The resultant longitudinal spin current flowing in the xdirection is spin polarized in the ydirection. For region II, that is calculated as
According to this equation, the longitudinal spin current linear in E_{x} vanishes in the absence of a magnetic field in the singleτ approximation. Note that spin current and spin polarization are different and should be distinguished. (There is actually a finite spin polarization due to the Edelstein effect even without a magnetic field^{27,28}.) Recent works reported the detection of spin polarization or spintocharge conversion due to spinmomentum locking at the surface of topological insulators^{29} or in 2D Rashba systems^{30,31}. In these phenomena, spin scattering at the interface between magnetic and nonmagnetic materials is essential and the observed spin signals are dependent on parameters such as the spinmixing conductance and the effective relaxation time (spin Hall angle), and thus not directly related to the spin–orbit parameter. On the other hand, the charge rectification effect we have reported in this work corresponds to the secondorder term in the charge current (equation (8)), which is different from the signals originating from the spin current. (Note that the spin current proportional to E_{x}^{2}B_{y} is absent.) In contrast to the interface spindependent phenomena discussed above, γ′, which characterizes the magnitude of the bulk rectification effect, includes only intrinsic band parameters such as the effective mass and Rashba parameters. Thus, the nonreciprocal magnetoresistance can serve as a new probe with which to estimate the magnitude of spin splitting. Experimental detection of the spin current in equation (10) originating from spinmomentum locking in this material is an important and interesting future problem.
In view of the rich families of noncentrosymmetric conductors, including interface / bulk Rashba systems^{10,11,12,13,14,15,16,17,18} and surface states of topological insulators^{29} with similar chiral spin textures, the present result offers a new strategy in the search for the giant rectification effect originating from exotic electronic band structures, as well as in constructing novel functionalities in exotic materials.
Methods
Device fabrication.
Single crystals of BiTeBr were prepared by chemical vapour transport (CVT) with the same conditions as in the previous work^{21}. Powders of Bi_{2}Te_{3}, and BiBr_{3} were mixed into quartz tubes (φ = 24 mm, L = 150) and the temperature was set to 380 °C in the charge zone and to 230 °C in the growth zone, respectively, for seven days. The BiTeBr crystals obtained were cleaved onto Si/SiO_{2} substrates using the Scotchtape method and isolated BiTeBr flakes with thicknesses of tens of nanometres were chosen with the aid of an optical microscope and an atomic force microscope (AFM). After an Ar ion milling treatment on the surfaces, Au (150nm)/Ti (5nm) electrodes were patterned into a Hall bar configuration by a standard ebeam lithography process.
Transport measurements.
All the transport properties were measured in a Quantum Design Physical Property Measurement System (PPMS) with a Horizontal Rotator Probe under a Hepurged environment. Both the first and secondharmonic signals of the a.c. resistance were measured by means of a lockin amplifier (Stanford Research Systems Model SR830 DSP) at a frequency of 13 Hz. During the a.c. resistance measurements, the phase of the first(second)harmonic signal was confirmed to be approximately 0 (π/2), which is consistent with the theoretical expectation. All the R^{ω} (or R^{2ω}) signals discussed in the main text are x (or y)components of the lockin measurement.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This research was supported by the following grants: JSPS GrantinAid for Specially Promoted Research (No. JP25000003), MEXT GrantinAid for Research Activity Startup (No. JP15H06133), Murata Science Foundation, MEXT KAKENHI (No. JP24224009, No. JP26103006, No. JP25400317 and No. JP15H05854), and ImPACT Program of Council for Science, Technology and Innovation (Cabinet office, Government of Japan), and GrantsinAid for JSPS Fellows (Grants No. JP16J08009).
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T.I. and Y.I. conceived and designed the experiments. Y.K. performed the growth and characterization of the bulk samples. T.I., S.K. and S.S. fabricated devices and measured transport properties. K.H. and M.E. performed the theoretical calculations. T.I., K.H., S.K., M.E., Y.T., N.N. and Y.I. led the physical discussions and wrote the manuscript. All authors commented on the manuscript.
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Ideue, T., Hamamoto, K., Koshikawa, S. et al. Bulk rectification effect in a polar semiconductor. Nature Phys 13, 578–583 (2017). https://doi.org/10.1038/nphys4056
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DOI: https://doi.org/10.1038/nphys4056
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