⇦Chinese Physics Letters, 2023, Vol. 40, No. 7, Article code 077302 Gate-Dependent Nonlinear Hall Effect at Room Temperature in Topological Semimetal GeTe N. N. Orlova, A. V. Timonina, N. N. Kolesnikov, and E. V. Deviatov* Affiliations Institute of Solid State Physics of the Russian Academy of Sciences, Chernogolovka, Moscow District, 2 Academician Ossipyan str., 142432, Russia Received 1 April 2023; accepted manuscript online 7 June 2023; published online 27 June 2023 *Corresponding author. Email: dev@issp.ac.ru Citation Text: Orlova N N, Timonina A V, Kolesnikov N N et al. 2023 Chin. Phys. Lett. 40 077302    Abstract We experimentally investigate nonlinear Hall effect as zero-frequency and second-harmonic transverse voltage responses to ac electric current for topological semimetal GeTe. A thick single-crystal GeTe flake is placed on the Si/SiO$_2$ substrate, where the p-doped Si layer serves as a gate electrode. We confirm that electron concentration is not gate-sensitive in thick GeTe flakes due to the gate field screening by bulk carriers. In contrast, by transverse voltage measurements, we demonstrate that the nonlinear Hall effect shows pronounced dependence on the gate electric field at room temperature. Since the nonlinear Hall effect is a direct consequence of a Berry curvature dipole in topological media, our observations indicate that Berry curvature can be controlled by the gate electric field. This experimental observation can be understood as a result of the known dependence of giant Rashba splitting on the external electric field in GeTe. For possible applications, the zero-frequency gate-controlled nonlinear Hall effect can be used for the efficient broad-band rectification.
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 DOI:10.1088/0256-307X/40/7/077302 © 2023 Chinese Physics Society Article Text Recent interest to the nonlinear Hall (NLH) effect[1] is connected with its significance for both the fundamental physics and possible applications. In the linear response, there is no Hall voltage in the presence of time-reversal symmetry. NLH effect is predicted[1] as a transverse voltage response in zero magnetic field due to the Berry curvature dipole in momentum space.[2-11] Thus, for the fundamental physics, NLH effect is the direct manifestation of finite Berry curvature in topological media. Since Berry curvature often concentrates in regions where two or more bands cross, three classes of candidate materials have been proposed,[1] i.e., topological crystalline insulators, two-dimensional transition metal dichalcogenides, and three-dimensional Weyl and Dirac semimetals. Being defined by the bulk energy spectrum, NLH effect is a rare example of macroscopic quantum phenomenon, which does not obligatory require low temperatures to be observed. A search for the room-temperature effect points to three-dimensional systems, primary topological semimetals.[12] In Weyl semimetals, every band touching point splits into two Weyl nodes with opposite chiralities due to the time reversal or inversion symmetries breaking. The materials with broken time-reversal symmetry are bulk ferromagnets or antiferromagnets, while Weyl semimetals with broken inversion symmetry have to obtain bulk ferroelectric polarization.[12,13] Due to the gapless bulk spectrum,[12] ferroelectric polarization makes the non-magnetic Weyl semimetal the natural representation of the novel concept of the intrinsic polar metal, or the ferroelectric conductor.[14-17] As a transverse response to longitudinal ac excitation, NLH effect can be observed at both zero and twice the excitation frequency.[3] The second-harmonic response is much easier to be measured by standard lock-in technique.[18-23] On the other hand, zero-frequency NLH response can be used for high-frequency (even terahertz or infrared) detection,[24,25] which is important for wide-band communications,[26] wireless charging, energy harvesting, etc. An advantage of the NLH rectification is the absence of thermal losses, since it originates from the Berry curvature dipole. The latter can be in principle controlled by electric field, which has been demonstrated[27] for two-dimensional WTe$_2$. Thus, both physics and applications require new materials for the room-temperature NLH effect, which allow electric field control of Berry curvature dipole. Among these materials, GeTe is of special interest[28,29] due to the reported giant Rashba splitting.[30-32] GeTe is predicted to be topological semimetal in low-temperature ferroelectric $\alpha$-phase.[33,34] The bulk Dirac points evolve either into pairs of Weyl nodes or into mirror-symmetry protected nodal loops upon breaking inversion symmetry.[33] Moreover, ferroelectric $\alpha$-GeTe is unveiled to exhibit an intriguing multiple nontrivial topology of the electronic band structure due to the existence of triple-point and type-II Weyl fermions.[34] As an additional advantage, the Rashba parameter is known to depend on the external electric field in GeTe.[30,35,36] In this Letter, we experimentally investigate nonlinear Hall effect as zero-frequency and second-harmonic transverse voltage responses to ac electric current for topological semimetal GeTe. We confirm that electron concentration is not gate-sensitive in thick conductive flakes due to the gate field screening by bulk carriers in GeTe. In contrast, by transverse voltage measurements, we demonstrate that the nonlinear Hall effect shows pronounced dependence on the gate electric field at room temperature. Samples and Technique. GeTe single crystals were grown by physical vapor transport in the evacuated silica ampule. The initial GeTe load was synthesized by direct reaction of the high-purity (99.9999%) elements in vacuum. For the crystals growth, the initial GeTe load serves as a source of vapors: it was melted and kept at 770–780 ℃ for 24 h. Afterward, the source was cooled down to 350 ℃ at the 7.5 ℃/h rate. The GeTe crystals grew during this process on the cold ampule walls somewhat above the source.
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Fig. 1. (a) The x-ray powder diffraction pattern (Cu $K_\alpha$ radiation), which is obtained for the crushed GeTe single crystal. The single-phase $\alpha$-GeTe is confirmed with the space group $R3m$ (No. 160). The upper inset shows optical image of the sample with a 0.5-µm-thick GeTe flake on the insulating SiO$_2$ substrate. The Au leads geometry is shown below; and 100-nm-thick, 5 $µ\rm{m}$ separated leads form Hall-bar geometry with several pairs of current and potential contacts. The ac current is applied between C1 and C3 leads (80 µm distance), while the transverse (Hall) voltage $U_{xy}$ is measured between the C2 and C6 potential probes (60 µm). Also, the longitudinal $U_{xx}$ component can be measured between the C6 and C5 (5 µm separation). The p-doped bulk Si layer under the SiO$_2$ surface serves as a gate electrode to apply the gate electric field through the 200-nm-thick SiO$_2$ layer.
The GeTe composition is verified by energy-dispersive x-ray spectroscopy. The powder x-ray diffraction analysis confirms single-phase GeTe, see Fig. 1(a), the known structure model[30] is also refined with single crystal x-ray diffraction measurements. Ferroelectric polarization was previously reported for the epitaxial films, microwires and bulk GeTe crystals,[28] it is defined by the non-centrosymmetric distorted rhombohedral structure ($\alpha$-GeTe) with space group $R3m$ (No. 160).[30] The gate-dependent ferroelectric polarization was also demonstrated in capacitance measurements,[36] the results confirmed giant Rashba splitting[30] in our GeTe single crystals. The upper inset to Fig. 1 shows a top-view image of the sample. The topological semimetals are essentially three-dimensional objects,[12] so we have to select relatively thick (above 0.5 µm) flakes. A thick flake also ensures sample homogeneity for correct determination of $xx$- and $xy$-voltage responses, however, the desired experimental geometry cannot be defined by usual mesa etching for thick flakes. Thick flakes require special sample preparation procedure: the mechanically exfoliated GeTe flake is transferred on the Au leads pattern, which is defined by lift-off technique on the SiO$_2$ surface after thermal evaporation of 100 nm Au, as depicted in the inset to Fig. 1. We choose $\sim$ $100$ µm wide flakes with defect-free surface by an optical microscope. After initial single-shot pressing by another oxidized silicon substrate, the flake is firmly connected to the Au leads. This procedure provides high-quality contacts to the flake, while the Au leads pattern defines the desired experimental geometry. The obtained samples are electrically and mechanically stable even in different cooling cycles to liquid helium temperatures. This sample preparation technique has been verified before for a wide range of materials.[37-44] We investigate transverse ($xy$-) first- and second-harmonic voltage responses by standard four-point lock-in technique.[20,22,23] The ac current is applied between C1 and C3 contacts in Fig. 1(b), while the transverse (Hall) voltage $U_{xy}$ is measured between the C2 and C6 potential probes. Also, the longitudinal $U_{xx}$ component can be measured between the C6 and C5. The Au leads are of 10 µm width, the current ones C1 and C3 are separated by 80 µm distance. The voltage probe separation is 60 µm for the $xy$-configuration (C2 and C6) and 5 µm for the $xx$-one (C6 and C5), respectively. The signal is confirmed to be independent of the ac current frequency within 100 Hz–10 kHz range, which is defined by the applied filters. The p-doped bulk Si layer under the SiO$_2$ surface serves as a gate electrode to apply the gate electric field through the 200-nm-thick SiO$_2$ layer. Even for relatively thick flakes, ferroelectric polarization is sensitive[36,45] to the gate electric field, since the relevant (bottom) flake surface is directly adjoined to the SiO$_2$ layer. We check it by an electrometer in which there is no measurable leakage current in the gate voltage range $\pm 50$ V. Since the ferroelectric $\alpha$-GeTe phase exists[46] below $700$ K, all the measurements are performed at room temperature under ambient conditions. This may also be important in the case of possible applications of the observed effects. For thick flakes, it is only important to protect the GeTe surface with Au contacts (the bottom one in Fig. 1). In the present technique, this surface is protected from any contamination (also oxygen and moisture) by the SiO$_2$ substrate, as it has been demonstrated for sensitive materials like black phosphorus.[42] Experimental Results. We confirm the correctness of experimental conditions by demonstrating standard Ohmic behavior for the first-harmonic $U^{1\omega}_{xx}$ longitudinal voltage component, see Fig. 2. $U^{1\omega}_{xx}(I_{\rm ac})$ shows strictly linear dependence on the applied ac current $Iac$, which is also confirmed by negligibly small second-harmonic $U^{2\omega}_{xx}$ response. The measured $U^{1\omega}_{xx}(I_{\rm ac})$ slope corresponds to the $\sim$ 10 m$\Omega$ sample resistance between the C6 and C5 probes in Fig. 1. In contrast to the standard two-dimensional materials such as graphene or field-effect transistors, carrier concentration in thick three-dimensional GeTe flakes is not sensitive to the gate electric field due to the perfect field screening by bulk carriers. This is experimentally confirmed in the inset to Fig. 2: The longitudinal $U^{2\omega}_{xx}$ response is nearly independent of the gate voltage with 0.1% accuracy in a wide gate voltage range $\pm50$ V. Maximum variation of the carrier concentration can be estimated in the capacitor approximation as $\delta n / n \sim \delta R /R \approx 10^{-3}$.
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Fig. 2. Standard Ohmic behavior for the first-harmonic $U^{1\omega}_{xx}$ longitudinal voltage component. $U^{1\omega}_{xx}(I_{\rm ac})$ is strictly linear, so the second-harmonic $U^{2\omega}_{xx}$ component is negligibly small. The measured $U^{1\omega}_{xx}(I_{\rm ac})$ slope corresponds to the $\sim$ $10$ m$\Omega$ GeTe resistance between the C6 and C5 probes in Fig. 1. Inset shows the first-harmonic $U^{1\omega}_{xx}$ response independence of the gate voltage in a wide $\pm 50$ V range, as it should be expected for the gate field screening in thick conductive GeTe flakes.
Figure 3 shows typical behavior of the nonlinear Hall effect[18-21] as a quadratic transverse Hall-like response $U^{2\omega}_{xy}$ to ac excitation current $I$. The $\sim$ $I^2$ dependence is directly demonstrated in the inset to Fig. 3(a) for zero gate voltage. The longitudinal second-harmonic voltage $U^{2\omega}_{xx}$ is one order of magnitude smaller, which confirms well-defined Au leads geometry and high sample homogeneity. The nonlinear Hall $U^{2\omega}_{xy}$ curves show clearly visible dependence on the gate voltage in Figs. 3(a) and 3(b). For the whole current range, $U^{2\omega}_{xy}$ values are diminishing both for positive gate voltages in Fig. 3(a), and for the negative ones in Fig. 3(b). This behavior is directly demonstrated in the inset to Fig. 3(b) as the gate voltage scan $U^{2\omega}_{xy}(V_{\rm g})$ at the fixed current value $I_{\rm ac}=4$ mV. The scan indeed shows symmetrical $U^{2\omega}_{xy}$ diminishing within $\sim$ $10$% for both gate voltage signs, while the $U^{2\omega}_{xy}(V_{\rm g})$ curve is centered at $V_{\rm g}=-10$ V. It is important that this symmetric $U^{2\omega}_{xy}(V_{\rm g})$ dependence cannot be ascribed to the gate-field effect on the carrier concentration in GeTe: (i) There is no noticeable dependence in the inset to Fig. 2. (ii) The concentration should depend asymmetrically on the gate voltage sign. If one demonstrates NLH effect at twice the frequency of the excitation current, it is natural to expect NLH rectification as the zero-frequency dc voltage $U^{\rm dc}_{xy}$. Indeed, we observe finite $U^{\rm dc}_{xy}$ values for the applied ac current $Iac$ in Fig. 3(c). $U^{\rm dc}_{xy}(I_{\rm ac})$ is clearly nonlinear, while it does not show clear $\sim$ $I^2$ dependence in a whole ac current range. The dc signal is noisy in comparison with the second-harmonic one due to the direct measurements by digital dc voltmeter after a broad-band preamplifier. Direct measurements of low signals by a voltmeter may also be a reason to have distorted nonlinear $U^{\rm dc}_{xy}(I_{\rm ac})$ dependence in comparison with the clear $\sim$ $I^2$ signal from lockin. In both the cases, we check the signal to be antisymmetric if one exchanges the Hall voltage probes. Also, $U^{\rm dc}_{xy}(I_{\rm ac})$ shows symmetrical gate voltage dependence in Fig. 3(c), despite the dc voltage is increasing for both signs of $V_{\rm g}$. As a result, we observe finite gate-dependent nonlinear transverse dc voltage $U^{\rm dc}_{xy}(I_{\rm ac})$, which confirms NLH rectification in GeTe at room temperature.
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Fig. 3. [(a), (b)] Typical NLH behavior[18-21] of the transverse second-harmonic voltage component at different gate voltages. $U^{2\omega}_{xy}\sim I^2$ is confirmed in the inset to (a) as strictly linear $U^{2\omega}_{xy}(I^2)$ dependence for zero gate voltage. The longitudinal second-harmonic voltage $U^{2\omega}_{xx}$ is one order of magnitude smaller. The curves are shown for positive (a) and negative (b) gate voltages, they are symmetrically affected by the gate electric field of both signs. Inset to (b) demonstrates the gate voltage scan $U^{2\omega}_{xy}(V_{\rm g})$ at fixed ac current $I_{\rm ac}=4$ mA, $U^{2\omega}_{xy}$ is symmetrically diminishing within $\sim$ $10$%. (c) Zero-frequency transverse voltage component $U^{\rm dc}_{xy}$ as a function of the applied ac current $Iac$ at three fixed gate voltages $V_{\rm g}=0, \pm50$ V. The curves are strongly nonlinear, $U^{\rm dc}_{xy}$ also symmetrically depends on the gate voltage of both signs. The data are obtained at room temperature.
Qualitatively similar $\sim$ $I^2$ NLH dependence can be demonstrated for different samples with different distances between the voltage probes, see Fig. 4, as depicted by solid and dashed lines, respectively. The curves are quite similar for the samples of the same dimensions, see the dashed curve in Fig. 4 and the curves in Fig. 3. For the 20 µm spaced voltage leads (solid line in Fig. 4), the $U^{2\omega}_{xy}$ values are twice smaller in comparison with Fig. 3(a). The zero-frequency $U^{\rm dc}_{xy}(I_{\rm ac})$ curves are presented in the inset to Fig. 4 for the smallest (20 µm) sample. The curves are qualitatively similar to ones in Fig. 3(c): the obtained values are twice smaller, the curves are strongly nonlinear, $U^{\rm dc}_{xy}$ also symmetrically depends on the gate voltage for both signs of $V_{\rm g}$. Thus, sample asymmetry can be excluded as a source of NLH signal, since similar results are obtained for different samples and different current directions.
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Fig. 4. Room-temperature NLH behavior of the transverse second-harmonic voltage component $U^{2\omega}_{xy}\sim I^2$ for two different samples, as depicted by solid (60 µm spacing between the voltage probes) and dashed (20 µm spacing) lines, respectively. The curves are obtained at zero gate voltage. Inset shows the gate voltage dependence of the zero-frequency transverse voltage component $U^{\rm dc}_{xy}$ for one of the samples (solid curve in the main field). The curves are strongly nonlinear, $U^{\rm dc}_{xy}$ also symmetrically depends on the gate voltage of both signs, similarly to Fig. 3(c).
Discussion. As a result, we observe second-harmonic NLH effect and NLH rectification at room temperatures in GeTe topological semimetal, both effects are sensitive to the gate electric field. The effect of the gate voltage is about 10% in Fig. 3. However, it is not only important to control the observed $U^{2\omega}_{xy}$ and $U^{\rm dc}_{xy}$, but also allows to rule out possible influence of thermopower on the transverse voltage response. In principle, topological materials are characterized by strong thermoelectricity,[47,48] which also appears as a second-harmonic signal.[49-51] For an inhomogeneous sample, it is possible to expect that Hall voltage probes are not perfectly symmetric in respect to the current path. In this case, Joule heating $\sim$ $R I^2$ can produce temperature gradient between the Hall voltage probes, so $U_{xy} \sim R I^2$ could be expected from thermoelectricity.[20,52] However, the sample resistance $R=U^{1\omega}_{xx}/I_{\rm ac}$ does not depend on the gate voltage in the inset to Fig. 2. In contrast, $U^{2\omega}_{xy}(V_{\rm g})$ shows the symmetric 10% variation in Fig. 3, so the observed $U^{2\omega}_{xy}(V_{\rm g})$ behavior is not connected with thermoelectricity.[20] The nonlinear Hall effect[18-21] arises from the Berry curvature dipole in momentum space.[1] In the simplified picture, the longitudinal current generates the effective sample magnetization, which leads to the Hall effect even in zero external magnetic field. Hall voltage is therefore proportional to the square of the excitation current, so it can be detected as the second-harmonic $U^{2\omega}_{xy}$ or the zero-frequency $U^{\rm dc}_{xy}$ transverse voltage components, as we observe in Figs. 3 and 4. It is obvious that the Berry curvature dipole can be controlled by dc electric field, as it has been demonstrated[27] for two-dimensional WTe$_2$: in-plane dc electric field $E_{\rm in}$ affects the measured second-harmonic NLH signal $U^{2\omega}_{xy}$ if $E_{\rm in}$ is parallel to the ac excitation current $Iac$. This argumentation cannot be applied to our experimental conditions, where the NLH voltage is affected by out-of-plane gate electric field $E_{\rm g}$ in Figs. 3 and 4. Since the carrier concentration is independent of the gate electric field in our samples, see the inset to Fig. 2, it is the Rashba parameter $\alpha_{\scriptscriptstyle{\rm R}}$ which defines the measured NLH signal $U^{2\omega}_{xy}$. In non-magnetic topological semimetals, the Berry curvature is defined by Weyl nodes separation due to the breaking of inversion symmetry,[12,13] so the Berry curvature is directly connected with the Rashba parameter $\alpha_{\scriptscriptstyle{\rm R}}$ in GeTe. On the other hand, the dependence of the Rashba parameter on the ferroelectric polarization is known for giant Rashba splitting in GeTe from theoretical[30] and experimental[35,36] investigations. Even for relatively thick flakes, ferroelectric polarization is sensitive[36,45] to the gate electric field, since the relevant (bottom) flake surface is directly adjoined to the SiO$_2$ layer. Variation of $\alpha_{\scriptscriptstyle{\rm R}}$ can be as large[30,36] as 20% in the gate voltage range of Fig. 3. The smaller value of the effect (about 10% in Fig. 3) seems to be due to $\alpha_{\scriptscriptstyle{\rm R}}$ that is changed only at the bottom (adjoined to the SiO$_2$ layer) flake surface. Thus, the measured NLH signal $U^{2\omega}_{xy}$ is affected by gate electric field through the field dependence of the giant Rashba spin-orbit coupling in GeTe.[30,35,36] Another possible contribution to the nonlinear Hall effect is skew scattering with nonmagnetic impurities in time-reversal-invariant non-centrosymmetric materials.[53] It has been demonstrated from symmetry analysis that NLH effect should be of pure disorder origin for some point groups.[54,55] Similar analysis seems to be quite sophisticated for GeTe, which exhibits an intriguing multiple nontrivial topology of the electronic band structure due to the existence of triple-point and type-II Weyl fermions.[34] However, it is most important that any disorder effects should depend on the carrier concentration. In contrast, the carrier concentration is independent of the gate electric field in our samples, while the measured NLH signal is clearly demonstrated to be gate-voltage-dependent. This seems to be a serious argument to connect the measured NLH signal with the Berry curvature in our samples. In conclusion, we observe second-harmonic NLH effect and NLH rectification at room temperatures in GeTe topological semimetal, both effects are sensitive to the gate electric field. The observed behavior is important for both the fundamental physics and possible applications. For physics, NLH effect is a direct consequence of a Berry curvature dipole, which is here demonstrated to be controlled by the gate electric field in GeTe. For possible applications, the gate-controlled NLH rectification can be used for the efficient broad-band detection. Acknowledgments. We wish to thank S. S. Khasanov for x-ray sample characterization and A. A. Avakyants and D. Yu. Kazmin for the help in sample preparation. We gratefully acknowledge financial support by the Russian Science Foundation (Grant No. RSF-23-22-00142), https://rscf.ru/project/23-22-00142/. 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