Large magnetoelectric resistance in the topological Dirac semimetal α-Sn.

The spin-momentum locking of surface states in topological materials can produce a resistance that scales linearly with magnetic and electric fields. Such a bilinear magnetoelectric resistance (BMER) effect offers a new approach for information reading and field sensing applications, but the effects demonstrated so far are too weak or for low temperatures. This article reports the first observation of BMER effects in topological Dirac semimetals; the BMER responses were measured at room temperature and were substantially stronger than those reported previously. The experiments used topological Dirac semimetal α-Sn thin films grown on silicon substrates. The films showed BMER responses that are 106 times larger than previously measured at room temperature and are also larger than those previously obtained at low temperatures. These results represent a major advance toward realistic BMER applications. Significantly, the data also yield the first characterization of three-dimensional Fermi-level spin texture of topological surface states in α-Sn.

INTRODUCTION

Surface electronic states in topological materials exhibit spin-momentum locking. That is, the spin direction of the relevant conduction electrons is locked to the momentum at right angles. In applicable two-dimensional (2D) momentum space, this locking manifests itself as a chiral Fermi contour upon which the spins point along the tangential direction everywhere, as sketched in Fig. 1A. Such a Fermi contour usually takes a circular shape. It, however, can morph into a hexagonally warped contour, as in Fig. 1B, if the material structure has threefold rotational symmetry (–). As one walks along such a hexagonal contour, the spins still rotate in a chiral manner as on the circular contour but now have an out-of-plane component. Such hexagonal warping exists in thin films of topological materials with (111)-oriented cubic structures or c axis–oriented rhombohedral structures.

Fig. 1.

BMER due to spin-momentum locking ( ).

(A) Spin texture on a circular Fermi contour of topological surface states (TSS). (B) Spin texture on a hexagonally warped Fermi contour. (C) Electric field E-induced generation of a pure spin current Js. (D) Magnetic field H-induced partial conversion of Js to a charge current Jc, giving rise to a lower resistance. (E) In comparison with (D), a flip in H leads to a flip in Jc and a higher resistance. The short red arrows denote spin directions.

The spin-momentum locking gives rise to two important effects: prohibition of electron backscattering and efficient conversion between charge and spin currents. The first effect opens the possibility of dissipation-free information transport. The second allows one to use charge currents to control magnetic properties (–).

There is another newly reported spin-momentum locking effect: a bilinear magnetoelectric resistance (BMER) response (–). This effect occurs in topological materials with hexagonal warping contours as introduced above. The warping results in a resistance that scales linearly with magnetic and electric fields and varies with the direction of the electric current relative to the crystallographic axes of the material. Such responses were first observed in the topological insulator Bi2Se3, with a BMER coefficient of χ = 0.6 nm2A−1 Oe−1 (). Here, χ is the normalized BMER per unit magnetic field and per unit current density. The effect was then also observed for 2D electron gas on surfaces of SrTiO3 crystals but with a substantially larger coefficient, χ = 500 nm2A−1 Oe−1 (). These observations were made at relatively low temperatures (60 K for Bi2Se3; 7 K for SrTiO3). More recent work shows that room-temperature BMER is also possible (). This later response was seen in the Weyl semimetal WTe2 but was significantly weaker, with χ=0.001 nm2A−1 Oe−1 only.

Figure 1 (C to E) provides a schematic guide to the BMER effect (). First, when an electric field E (black arrow) is applied, as in Fig. 1C, in addition to the first-order correction to the electron distribution that yields an ordinary charge current, there is a second-order correction to the distribution that results in equal numbers of electrons populated in the surface states with opposite spins and velocities. The net effect is a pure spin current Js (red arrow) that scales with the square of the electric field, namely, E2. Second, when a magnetic field H (blue arrow) is applied, as in Fig. 1D, the flux of electrons with opposite spins is no longer balanced. As a result, Js is partially converted to a charge current Jc (green arrow) that adds to the ordinary charge current and thereby gives rise to a lower resistance. Last, if H is reversed, as in Fig. 1E, Jc is also reversed and gives rise to a higher resistance.

This article reports a large room-temperature BMER response with a coefficient that is orders of magnitude larger than that in previous work. The experiments used a topological Dirac semimetal (TDS) α-Sn, in contrast to the topological insulators, 2D electron gas, and Weyl semimetals in previous work. (111)-oriented TDS α-Sn thin films in the 4- to 6-nm thickness range were grown on silicon substrates by sputtering at room temperature. Room-temperature measurements showed a resistance that scaled linearly with both the magnetic field and the charge current and strongly depended on the current direction relative to the crystalline axes of the films. The data showed χ values in the range of 2900 nm2A−1 Oe−1. This coefficient is 106 times larger than that obtained at room temperature for Weyl semimetals (). It is also substantially larger than previous values obtained at low temperatures (, ). Such a giant effect is most likely due to the low carrier density and spatial asymmetry of the α-Sn films.

Furthermore, the data also provide evidence for Fermi contour warping in TDS α-Sn films. The spins on the hexagonal contour (Fig. 1B) lie in-plane at the hexagon vertexes but tilt ±30° out-of-plane at the middle points of the hexagon sides.

Before bringing in the key data, it is useful to establish three important points. First, the BMER effect offers a totally new approach for information reading and field sensing technologies. While previous results were too weak or manifested only at low temperatures, the very large, room-temperature BMER reported here points to the possibility of real applications. This technological importance is further highlighted by the following: (i) The effects have been obtained in a single-material, single-element thin film; this can significantly simplify device fabrication. (ii) The α-Sn films were grown on silicon, a common industrial substrate, by sputtering, an industry-friendly technique. Second, this work shows the technological potential for surface states in TDS α-Sn. There have been major efforts to develop spintronic applications of surface states in topological insulators (–), but corresponding efforts for TDS systems up to now have been limited. Third, this work also represents the first-ever measurement of 3D Fermi contours in α-Sn. There have been extensive angle-resolved photoemission spectroscopy (ARPES) measurements on surface and bulk states in α-Sn (–), but none of them have even touched on the 3D aspect of the Fermi-level spin texture in α-Sn.

α-Sn has a diamond cubic structure. Unstrained α-Sn is a gapless semiconductor in which the quadratic conduction and valence bands touch each other at the Γ point near the Fermi level. In the presence of a tensile strain along the [001] or [111] direction, however, the two bands cross each other near the Fermi level. This band crossing forms two Dirac points and gives rise to a TDS phase (, –). One of these Dirac points is shown in Fig. 2. The topological nature of the TDS α-Sn originates from the band inversion: The bands () near the Fermi level are derived from p electrons, while the s electron–derived band () with opposite parity is below the Fermi level, as illustrated in Fig. 2. The topological surface states (TSS) bridge the conduction band and the valence band (, ). The Fermi level cut across the TSS cone corresponds to the circular contour in Fig. 1A.

Fig. 2.

Band structure in TDS α-Sn thin films (, ).

In this work, (111)-oriented α-Sn thin films were grown on Si substrates by sputtering (, ). The film growth details and structural properties are given in the Supplemental Materials. The lattice constant of α-Sn (6.489 Å) is larger than that of Si (5.431 Å). This mismatch yields a perpendicular tensile strain in the α-Sn films. This strain pushes the films into a TDS phase (, –). The (111) orientation leads to the hexagonal warping depicted in Fig. 1B.

RESULTS

Figures 3 and 4 present the main results, namely, the BMER data obtained on 4-nm-thick α-Sn films. As in previous work (, , ), the experiments were based on Hall bar structures and measurements of the second-harmonic resistance in response to an AC current. The Hall bar drive current takes the form I(t) = I0 cos (ωt), where I0 is the current amplitude, and ω is the frequency. The total longitudinal resistance is taken to be composed of an ordinary resistance R0 and a BMER component CI(t). Then, the longitudinal voltage in the α-Sn film can be evaluated asDivision of Eq. 1 by I0 gives the longitudinal resistance as

Fig. 3.

Second-harmonic resistance (R2ω) as a function of the magnetic field angle for a 4-nm-thick α-Sn film.

The circles show the data, the curves show sinusoidal fits, and the lines show linear fits. The field strength H and the current amplitude I0 are indicated. In (C), the red curve shows the fit in (A), and the black curve shows the fit in (B); the blue curve is the sum of the red and black curves. In (D), ΔR2ω is the amplitude of the sinusoidal R2ω versus angle response.

Fig. 4.

Second-harmonic resistance (R2ω) as a function of the magnetic field angle for a 4-nm α-Sn film.

The data in (A) were measured with a Hall bar whose length is along of the α-Sn film, while those in (B) were measured with a Hall bar with its length along .

One can see that the coefficient of cos(ωt) in Eq. 2 is the ordinary resistance, whereas that of cos(2ωt) is one-half the BMER. Second-harmonic signal measurements, therefore, allow for a direct measurement of the BMER response. In the discussions below, a new parameter, R2ω, is used to denote the second-harmonic resistance .

Figure 3 gives the basic R2ω data. Figure 3 (A to C) shows the R2ω data obtained by rotating the field H in different planes. For clarity, the measurement configurations are shown in the upper diagrams in the panels. The red and black curves in Fig. 3 (A and B) are fits to sine and cosine functions, respectively. The red and black curves in Fig. 3C are the same as those in Fig. 3 (A and B). The blue curve corresponds to the sum of the red and black curves. Figure 3D shows ∆R2ω, the amplitude of the sinusoidal R2ω, versus angle response, as a function of the current amplitude I0 and the field strength H. The current or the Hall bar length is along the axis of the α-Sn film. The effects of the current direction are discussed shortly.

Four important results are evident from the data in Fig. 3. (i) The data in Fig. 3A indicate that R2ω takes the maximum and the minimum for a magnetic field along and , respectively. This is consistent with the picture depicted in Fig. 1 (D and E). It suggests that the projection of the spin vector on the xy plane points along the y axis, as in the bottom diagram in Fig. 3A. The response in Fig. 3A also represents a unidirectional magnetoresistance (UMR) along the y axis; the resistance is the largest for and is the smallest for . (ii) The data in Fig. 3B show a UMR along the z axis. This confirms that the spin has a nonzero out-of-plane component, as depicted in the bottom diagram in Fig. 3B. (iii) Considered together, the data in Fig. 3 (A and B) suggest that the spin vector lies in the yz plane. This is confirmed by the data in Fig. 3C. Furthermore, the data show a R2ω maximum at a field angle of ϕ ≈ 120° and a minimum at ϕ ≈ 300°. This indicates that the spin vector is tilted ~30° out of the xy plane, as in the bottom diagram in Fig. 3C. These results are strongly supported by the nice agreement between the experimental response and the blue curve in Fig. 3C. (iv) The data in Fig. 3D show that ∆R2ω scales linearly with both I0 and H.

Figure 4 shows the R2ω versus field angle responses for the currents applied along two different crystalline axes: and , which correspond to the and momentum space lines in Fig. 1B, respectively. The data in Fig. 4A are the same as in Fig. 3 (A to C). The relevant crystalline axes are defined in the Supplementary Materials. The data in Fig. 4A show that when the current is applied along , the spins are in the yz plane and tilt at an angle of about 30° away from the y axis, as discussed above. In contrast, if the current is along , then R2ω exhibits a notable angle dependence for field rotation in the xy plane but shows almost no angle dependence for field rotation in the xz plane, as shown in Fig. 4B. These responses indicate that the spins point along the y axis and have a zero out-of-plane component. Such spin orientation is consistent with the spin texture depicted in Fig. 1B.

Together, the results above clearly demonstrate the presence of the BMER in the 4-nm α-Sn films. The same responses have been confirmed in multiple samples. As an example, Fig. 5 gives the data measured on an α-Sn film that is thicker. Figure 5A shows the R2ω versus field angle responses for the Hall bar axis and current applied along , which correspond to in momentum space. Figure 5B gives companion data measured for a current 60° away from , along in momentum space. The data show two key results. First, the field angle dependences in Fig. 5A are about the same as those presented above for the 4-nm film. This consistency supports the results discussed above. Second, when the current direction is rotated from to , the φ dependence remains the same, but the θ response flips in sign. This indicates that a rotation in the current direction does not affect the in-plane component of the spin vector but reverses the sign of the out-of-plane component. This is consistent with the chiral spin texture depicted in Fig. 1B. In addition, the data show that ∆R2ω is smaller than that for the 4-nm film. This result, together with the data in fig. S8, indicates that the BMER decreases with an increase in the film thickness. Possible reasons include that as the thickness increases, the phase of a Sn film transforms from pure α to a mixture of α and β and then to β dominant (). This is discussed further in the Supplementary Materials. It is worthy to note that although the first-harmonic resistance is irrelevant to the BMER and is therefore not presented in Figs. 3 to 5, it yields important information about the carrier density and mobility of the bulk and surface states in the α-Sn films, as reported previously ().

Fig. 5.

BMER responses in a 5.8-nm α-Sn film.

(A) R2ω versus field angle responses measured with currents applied along . (B) R2ω versus field angle responses measured with currents applied along . (C) Spin texture on the Fermi contour.

DISCUSSION

BMER coefficient

Previous studies have characterized the BMER effect in terms of a coefficient ()where (w, Hall bar width; t, film thickness) is the current density. With the data in Fig. 3, one obtains χ ≈ 2900 nm2A−1 Oe−1. This coefficient is 106 times larger than the value obtained previously at room temperature for the Weyl semimetal WTe2 (). It is also markedly larger than the values measured previously at low temperatures (, ).

The large BMER in TDS α-Sn films possibly originates from the following facts. First, the 2D carrier density (n) of the TSS in α-Sn films is relatively low. One can show theoretically that, in the presence of sizable warping, the BMER coefficient χ exhibits a strong inverse dependence on the carrier density: for an in-plane magnetic field and for an out-of-plane magnetic field (see section S5). A response has been reported for SrTiO3 (). Previous experiments have shown n ≈ 1.8 × 1012 cm−2 for TSS in TDS α-Sn films (), 3.0 × 1013 cm−2 for TSS in Bi2Se3 films (), and 5.3 × 1013 to 7.7 × 1014 cm−2 for 2D electron gas in SrTiO3 (). One can see that n in TDS α-Sn films is more than one order of magnitude smaller than in either Bi2Se3 or SrTiO3.

Second, the α-Sn film in this work does not have spatial inversion symmetry. Specifically, the film bottom surface at the interface with the Si substrate is smooth, while the top surface is relatively rough. For this reason, the electrons on the top surface may undergo more frequent scattering from nonmagnetic disorders than those on the bottom surface. The net effect is that the measured BMER can be attributed mainly to the spin-momentum locking at the bottom surface. If the film has inversion symmetry, the BMER responses from the two surfaces would cancel each other and thereby give rise to very weak overall responses, if not zero. This partially explains why χ in Bi2Se3 is only 0.6 nm2A−1 Oe−1 (). Note that among these two facts, the first one may play a more dominant role than the second, as discussed in section S5.

Spin texture of TSS

Together, the data in Figs. 4 and 5 show the 3D aspect of the spins of the Fermi-level TSS in TDS α-Sn. Specifically, the spin in the middle point of a side of the hexagonal Fermi contour is not exactly along the hexagon side but tilts 30° out-of-plane. As one goes along the hexagonal contour to a vertex, the spin rotates into the plane. As one goes further to the middle point of the next hexagon side, the spin tilts −30° out-of-plane. This spin rotation is illustrated in Fig. 5D. There have been considerable ARPES works on the band structure of α-Sn (–), including spin-resolved ARPES measurements (, ). However, none of those studies have touched on the 3D features of the spins on the Fermi contours of the TSS reported here.

The out-of-plane component (sz) of the spin depends on the properties of the TSS at the Fermi level. These properties include the Fermi velocity (vF), the Fermi wave number (kF), and the strength of the hexagonal warping (λ). If one denotes ∅ as the angle between the direction and the wave vector (the current direction), sz can be evaluated as (, )where ℏ is the reduced Planck constant. The cos(3∅) term dictates the dependence of the BMER on the current direction as presented in Figs. 4 and 5, as well as the threefold symmetry of the chiral spin texture illustrated in Fig. 5D. For the data in Figs. 4A and 5A, one has ∅ = 0 and s = sin 30°. Taken these parameters and eV · m and k = 3.39 × 108 m−1 from previous experiments (, ), one can use Eq. 4 to estimate the warping parameter as λ ≈ 2 eV · nm3. This value is about one order of magnitude larger than the values reported for Bi2Se3 (, , , ).

Final remarks

Five final remarks are as follows.

1) As discussed above, the BMER depends on the properties of the TSS at the Fermi level. As a result, further work is of great interest that explores the effects of the Fermi level on the BMER, with an aim to maximize the effect in the TDS α-Sn. Possible strategies to tune the Fermi level include doping (), capping (), and voltage gating.

2) Although this work reports the first measurements of the spin canting angle and warping parameter for the hexagonal Fermi contour of the TSS in TDS α-Sn, the measurements were somewhat indirect. It will be of great interest to directly map the 3D spin texture in the future. One note about the spin canting: In the TSS, the momentum is locked to real spin, and the spin canting arises from the cubic spin-orbit coupling at the surface. Because of the C3 rotation symmetry along the film normal and the time-reversal symmetry, the spin-orbit coupling takes the form , where k± = k ± ik, and σz denotes the z component of the spin. The term leads to a warped Fermi contour and spin tilting along certain k directions ().

3) The hexagonal warping contour in this work is a consequence of the threefold symmetry of the (111) α-Sn films, as mentioned above. In topological insulator α-Sn films that are (001) oriented and therefore do not have threefold symmetry, there are also magnetoresistances that scale linearly with magnetic and electric fields, as mentioned in (). However, these magnetoresistances have nothing to do with the hexagonal warping but are associated with electron scattering by spin-orbit structural defects ().

4) The BMER in this work is unidirectional in nature. However, it is different from the UMR found in bilayered structures consisting of a magnetic thin film and a heavy metal or topological insulator thin film (–). The latter has a different origin.

5) Recent work has demonstrated magnetoresistances in the semiconductor Ge and the topological insulator HgTe that scale linearly with both magnetic and electric fields but are independent of the current direction relative to the crystalline axes (, ).The effect in Ge results from the interaction of the external magnetic field and a pseudomagnetic field associated with spin-split subsurface states (), while that in HgTe originates from inhomogeneity-related scattering (, ).

MATERIALS AND METHODS

Film growth

The Sn thin films are grown on single-crystal (111)-oriented Si substrates by DC magnetron sputtering. The substrates are rinsed sequentially with acetone and isopropyl alcohol before being loaded into the sputtering chamber. Before sputtering, the chamber has a base pressure of 2.0 × 10−8 torr; substrate biasing is performed that includes several cycles of Ar+ ion sputtering of the substrate surface and the post-annealing of the substrate at 250°C. After the substrate biasing, the Sn deposition is then carried out at a substrate temperature of 7°C and at a growth rate of about 3 nm/min. The sputtering power is set to a moderate value of 10 W, to minimize the heating effect during the deposition. The major substrate biasing and sputtering control parameters are given in the Supplementary Materials.

Structural characterization

The surface morphology of the α-Sn thin films was characterized by atomic force microscopy. The crystalline properties of the α-Sn thin films were determined through x-ray diffraction (XRD) θ-2θ scan, XRD rocking curve, and XRD phi scan measurements. The detailed data are presented in the Supplementary Materials.

Device fabrication and measurements

To measure the electric transport properties of a Sn thin film, a Hall bar device is fabricated through photolithography and argon ion milling processes. To avoid heating the Sn film, a Hall bar pattern is first created on a Si substrate using a Microtech laser writer (LW405C). Then, the exposed sample surface is cleaned using ion milling. This process helps to remove any photoresist residue or SiO2 on the device surface. Subsequently, deposition of a Sn thin film is carried out at a substrate temperature of 7°C via sputtering. Details about the device fabrication process, the dimensions of the Hall bar structure, and the electric transport measurements are presented in the Supplementary Materials.