Lifshitz-Kosevich Formula Research Articles

Rashba-splitting-induced topological flat band detected by anomalous resistance oscillations beyond the quantum limit in ZrTe5

Topological flat bands — where the kinetic energy of electrons is quenched — provide a platform for investigating the topological properties of correlated systems. Here, we report the observation of a topological flat band formed by polar-distortion-assisted Rashba splitting in the three-dimensional Dirac material ZrTe5. The polar distortion and resulting Rashba splitting on the band are directly detected by torque magnetometry and the anomalous Hall effect, respectively. The local symmetry breaking further flattens the band, on which we observe resistance oscillations beyond the quantum limit. These oscillations follow the temperature dependence of the Lifshitz–Kosevich formula but are evenly distributed in B instead of 1/B at high magnetic fields. Furthermore, the cyclotron mass gets anomalously enhanced about 102 times at fields ~ 20 T. Our results provide an intrinsic platform without invoking moiré or order-stacking engineering, which opens the door for studying topologically correlated phenomena beyond two dimensions.

Optical Shubnikov–de Haas oscillations in two-dimensional electron systems

We report on dynamic Shubnikov–de Haas (SdH) oscillations that are measured in the optical response, subterahertz transmittance of two-dimensional systems, and reveal two distinct types of oscillation nodes: “universal” nodes at integer ratios of radiation and cyclotron frequencies and “tunable” nodes at positions sensitive to all parameters of the structure. The nodes in both real and imaginary parts of the measured complex transmittance are analyzed using a dynamic version of the static Lifshitz-Kosevich formula. These results demonstrate that the node structure of the dynamic SdH oscillations provides an all-optical access to quantization- and interaction-induced renormalization effects, in addition to parameters one can obtain from the static SdH oscillations. Published by the American Physical Society 2024

Inhomogeneity identification by measuring magnetic quantum oscillations

This study explores the identification of sample inhomogeneity via magnetic quantum oscillations analysis in semimetal NbSb2. By doping Bi and Cr, we obtained a homogenous Bi-doped sample and an inhomogeneous Cr-doped sample, whose homogeneity was confirmed by comparing the magnetic quantum oscillation before and after grinding the samples. The magnetic quantum oscillations in the inhomogeneous sample exhibited a distinct phase shift and unusual field-dependent amplitude, believed to result from a non-uniform Fermi energy. The analysis of the magnetic quantum oscillations demonstrated that the homogeneous Bi-doped sample can be interpreted by the symmetric and Lorentzian effective Fermi energy distribution, while the inhomogeneous Cr-doped sample exhibited an asymmetric distribution, illustrating an unconventional violation of the Lifshitz-Kosevich formula. This research provides a novel method for identifying material inhomogeneity and mitigating potential misinterpretations of magnetic quantum oscillations' unusual phase, commonly seen as a nontrivial Berry phase indicator in topological materials studies.

High-sensitivity specific heat study of the low-temperature–high-field corner of the H−T phase diagram of FeSe

We report on an extensive experimental study of the low-temperature--high-field corner of the $H\text{\ensuremath{-}}T$ phase diagram in FeSe using high-sensitivity specific heat $C$ measurements. Indeed, the superconducting gap and Fermi and Zeeman energies are surprisingly close in this compound, possibly leading to the existence of different superconducting phases as well as Lifshitz transitions in the electronic structure. The nature of this part of the phase diagram hence remains debated, and we show here that two distinct anomalies are visible in $C(H)$ (below $\ensuremath{\sim}3$ K): a (smeared) jump at the superconducting transition and a kink in $C(H)$. This second structure lies either below (for $H$ close to the $ab$ plane) or above (for field orientations close to the $c$ direction) the superconducting transition, indicating that it is most probably related to a field-induced change in the electronic structure. Moreover, quantum oscillations are clearly observed below $\ensuremath{\sim}2$ K, and $C(H)$ can be well described by the Lifshitz-Kosevich formula for fields both above and below the kink (with an effective mass ${m}^{*}\ensuremath{\sim}4{m}_{e}$ and a frequency $F\ensuremath{\sim}200$ T, in good agreement with the values previously inferred from magnetotransport measurements for the hole sheet).

Fermi surface mapping of the kagome superconductor RbV3Sb5 using de Haas-van Alphen oscillations

We present the results from torque magnetometry studies of the kagome superconductor ${\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$ under applied fields up to 45 T and temperatures down to liquid $^{3}\mathrm{He}$ temperature (0.32 K). The torque signal shows clear de Haas--van Alphen (dHvA) oscillations with eight distinct frequencies ranging from $\ensuremath{\approx}150$ to 3000 T. Among these, five are above 500 T. Angle-dependent measurement of dHvA oscillations shows that all frequencies follow 1/$\mathrm{cos}\ensuremath{\theta}$ dependence, where $\ensuremath{\theta}$ is the tilt angle with respect to the applied field direction, and the oscillations disappear above $\ensuremath{\theta}$ = ${60}^{\ensuremath{\circ}}$, which confirms that the Fermi surfaces corresponding to these frequencies are two dimensional. The Berry phase ($\ensuremath{\phi}$), calculated by constructing a Landau level fan diagram, is found to be $\ensuremath{\approx}\ensuremath{\pi}$, which strongly supports the nontrivial topology of ${\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$. Using the Lifshitz-Kosevich formula, we estimate the effective mass (${m}^{*}$) of charge carriers in ${\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$, and it is found to be heavier ($\ensuremath{\approx}0.7{m}_{o}$, where ${m}_{o}$ is the free electron mass) than that for other topological insulators. The findings of high frequencies up to 3000 T in ${\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$ have not been reported previously, and the results regarding the Fermi surface of ${\mathrm{RbV}}_{3}{\mathrm{Sb}}_{5}$ are crucial for understanding the charge density wave order, superconductivity, and nontrivial topology in $A{\mathrm{V}}_{3}{\mathrm{Sb}}_{5}$ ($A$ = K, Rb, and Cs), as well as the interplay among them.

Orbit topology analyzed from π phase shift of magnetic quantum oscillations in three-dimensional Dirac semimetal

With the emergence of Dirac fermion physics in the field of condensed matter, magnetic quantum oscillations (MQOs) have been used to discern the topology of orbits in Dirac materials. However, many previous researchers have relied on the single-orbit Lifshitz-Kosevich (LK) formula, which overlooks the significant effect of degenerate orbits on MQOs. Since the single-orbit LK formula is valid for massless Dirac semimetals with small cyclotron masses, it is imperative to generalize the method applicable to a wide range of Dirac semimetals, whether massless or massive. This report demonstrates how spin-degenerate orbits affect the phases in MQOs of three-dimensional massive Dirac semimetal, NbSb2 With varying the direction of the magnetic field, an abrupt π phase shift is observed due to the interference between the spin-degenerate orbits. We investigate the effect of cyclotron mass on the π phase shift and verify its close relation to the phase from the Zeeman coupling. We find that the π phase shift occurs when the cyclotron mass is half of the electron mass, indicating the effective spin gyromagnetic ratio as g s = 2. Our approach is not only useful for analyzing MQOs of massless Dirac semimetals with a small cyclotron mass but also can be used for MQOs in massive Dirac materials with degenerate orbits, especially in topological materials with a sufficiently large cyclotron mass. Furthermore, this method provides a useful way to estimate the precise g s value of the material.

Shubnikov-de-Haas oscillation and possible modification of effective mass in CeTe3 thin films

Magnetoresistance measurements have been performed in CeTe3 thin film devices in a temperature range from 2.1 to 20 K up to 8 T. A clear Shubnikov-de-Haas oscillation was observed in the whole temperature range. The temperature dependence of the oscillation amplitude was found to deviate from the Lifshitz–Kosevich formula below the magnetic transition temperature at TN1 ≈ 3 K. This indicates a significant interplay between the magnetic ordering and the conduction electrons, which could lead to a modification of the effective cyclotron mass. By analyzing the temperature dependence of the oscillation amplitude, we have estimated the effective mass, quantum lifetime and quantum mobility of the material both in the paramagnetic and antiferromagnetic states.

Description of the Magnetization Oscillations of a Silicon Nanostructure in Weak Fields at Room Temperature. The Lifshitz–Kosevich Formula with Variable Effective Carrier Mass

A formalism of the statistical approach to describing de Haas–van Alphen oscillations known as the Lifshitz–Kosevich formula is developed as applied to a low-dimensional system with an effective carrier mass depending on an external magnetic field. The statistical approach makes it possible to perform more detailed interpretation of the experimental results and analyze the interrelation of the dependence found by us of the effective carrier mass with the individual features of the structure of a silicon nanosandwich caused by the formation of negative-U δ barriers in its composition.

Intrinsic Bulk Quantum Oscillations in a Bulk Unconventional Insulator SmB6.

SummaryThe finding of bulk quantum oscillations in the Kondo insulator SmB6 proved a considerable surprise. Subsequent measurements of bulk quantum oscillations in other correlated insulators including YbB12 lent support to our discovery of a class of bulk unconventional insulators that host bulk quantum oscillations. Here we perform a series of experiments to examine evidence for the intrinsic character of bulk quantum oscillations in floating zone-grown single crystals of SmB6 that have been the subject of our quantum oscillation studies. We present results of thermodynamic, transport, and composition analysis experiments on pristine floating zone-grown single crystals of SmB6 and compare quantum oscillations with metallic LaB6 and elemental aluminum. These results establish the intrinsic origin of quantum oscillations from the insulating bulk of floating zone-grown SmB6. The similarity of the Fermi surface in insulating SmB6 with the conduction-electron Fermi surface in metallic hexaborides is at the heart of a theoretical mystery.

Quantum Oscillations and Electronic Structure in the Large-Chern-Number Topological Chiral Semimetal PtGa*Supported by the National Key Research and Development Program of China (Grant Nos. 2019YFA0308602, 2018YFA0305700 and 2016YFA0300600), the National Natural Science Foundation of China (Grant Nos. 11874422 and 11574391), the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant Nos. 19XNLG18

We report the magnetoresistance (MR), de Haas-van Alphen (dHvA) oscillations and the electronic structures of single-crystal PtGa. The large unsaturated MR is observed with the magnetic field B ∥ [111]. Evident dHvA oscillations with the B ∥ [001] configuration are observed, from which twelve fundamental frequencies are extracted and the spin-orbit coupling (SOC) induced band splitting is revealed. The light cyclotron effective masses are extracted from the fitting by the thermal damping term of the Lifshitz–Kosevich formula. Combining with the calculated frequencies from the first-principles calculations, the dHvA frequencies F1/F3 and F11/F12 are confirmed to originate from the electron pockets at Γ and R, respectively. The first-principles calculations also reveal the existence of spin-3/2 Rarita–Schwinger–Weyl fermions and time-reversal doubling of the spin-1 excitation at Γ and R with large Chern numbers of ± 4 when SOC is included.

Quantum oscillation beyond the quantum limit in pseudospin Dirac materials

Recently, many unexpected fine structures in electric, magnetic, and thermoelectric responses at extremely magnetic fields in topological materials have attracted tremendous interest. We propose a new mechanism of quantum oscillation beyond the strong-field quantum limit for Dirac fermions. The amplitude of the oscillation is far larger than the usual Shubnikov--de Haas oscillation. The oscillation tends to be periodic in the magnetic field B, instead of 1/B. The period of the oscillation does not depend on the Fermi energy. These behaviors cannot be described by the famous Lifshitz-Kosevich formula. The oscillation arises from a mechanism that we refer to as the inversion of the lowest Landau level, resulted from the competition between the pseudospin Dirac-type Landau levels and real-spin Zeeman spitting beyond the quantum limit. This inversion gives rise to the oscillation of the Fermi energy and conductivity at extremely large magnetic fields. This mechanism will be useful for understanding the unexpected fine structures observed in the strong-field quantum limit in Dirac materials.

Thickness-dependent magneto-transport properties of topologically nontrivial DyPdBi thin films

DyPdBi (DPB) is a topological semimetal which belongs to the rare-earth-based half-Heusler alloy family. In this work, we studied the thickness-dependent structural and magneto-transport properties of DPB thin films (20 to 60 nm) grown using pulsed laser deposition. The DPB thin films show (110) oriented growth on MgO(100) single crystal substrates. Longitudinal resistance data indicate metallic surface states dominated carrier transport and the suppression of semiconducting bulk state carriers for films ≤40 nm. We observe the weak antilocalization (WAL) effect and Shubnikov–de Hass (SdH) oscillations in the magneto-transport data. The presence of a single coherent transport channel (α∼ −0.50) is observed in the Hikami–Larkin–Nagaoka (HLN) fitting of WAL data. The power law temperature dependence of phase coherence length (LØ) ∼ T−0.50 indicates the observation of the 2D WAL effect and the presence of topological nontrivial surface states for films ≤40 nm. The 60 nm sample shows semiconducting resistivity behavior at higher temperature (>180 K) and HLN fitting results (α∼ −0.72, LØ∼ T−0.68) indicate the presence of partial decoupled top and bottom surface states. The Berry phase ∼π is extracted for thin films ≤40 nm, which further demonstrates the presence of Dirac fermions and nontrivial surface states. Band structure parameters are extracted by fitting SdH data to the standard Lifshitz–Kosevich formula. The sheet carrier concentration and cyclotron effective mass of carriers decrease with increasing thickness (20 nm to 60 nm) from ∼1.35 × 1012 cm−2 to 0.68 × 1012 cm−2 and from ∼0.26 me to 0.12 me, respectively. Our observations suggest that samples with a thickness ≤40 nm have transport properties dominated by surface states and samples with a thickness ≥60 nm have contributions from both bulk and surface states.

Modification of Shubnikov–de Haas oscillation and quantum Hall effect for Bi2Se3 crystals by Fe

Shubnikov–de Haas (SdH) oscillations and the quantum Hall effect (QHE) can give insights into quantum oscillations of charge carriers. We report the SdH oscillations and QHE of magnetically doped topological insulator FexBi2−xSe3 crystals, in which the lattice constant c decreases with the increasing dopant concentration but increases when x is greater than 0.08. From the fitting of low-temperature magnetoresistance data to the Lifshitz–Kosevich formula, the effective mass of the carriers and the Dingle temperature of the samples were extracted and show a similar trend of change as the lattice constant. A constant step size of 1/Rxy observed during Hall measurements at low temperatures implies two-dimensional-like transport behavior associated with the layered structure of the doped Bi2Se3 samples. Two-dimensional and three-dimensional charge carrier densities were calculated, and the latter had a similar trend of change as the lattice constant. A systematic study of the changes in all these properties with increasing dopant concentration suggests that they are highly correlated.

Multi-frequency Shubnikov-de Haas oscillations in topological semimetal Pt2HgSe3

Monolayer jacutingaite (Pt2HgSe3) has been recently identified as a candidate quantum spin Hall system with a 0.5 eV band gap, but no transport measurements have been performed so far on this material, neither in monolayer nor in the bulk. By using a dedicated high-pressure technique, we grow crystals enabling the exfoliation of 50-100 nm thick layers and the realization of devices for controlled transport experiments. Magnetoresistance measurements indicate that jacutingaite is a semimetal, exhibiting Shubnikov-de Haas (SdH) resistance oscillations with a multi-frequency spectrum. We adapt the Lifshitz-Kosevich formula to analyze quantitatively the SdH resistance oscillations in the presence of multiple frequencies, and find that the experimental observations are overall reproduced well by band structure ab-initio calculations for bulk jacutingaite. Together with the relatively high electron mobility extracted from the experiments (≈ 2000 cm2/V−1s−1, comparable to what is observed in WTe2 crystals of the same thickness), our results indicate that monolayer jacutingaite should provide an excellent platform to investigate transport in 2D quantum spin Hall systems.

Quantum oscillations in Stoner susceptibility: A theory

Oscillatory effects in magnetic susceptibility of free electrons in a strong magnetic field is well known phenomenon and is well captured by Lifshitz-Kosevich formula. In this paper we point out similar oscillatory effects in Stoner susceptibility which makes the system to oscillate between paramagnetic phase and ferromagnetic phase alternatively as a function of external magnetic field strength. This effect can happen in a material which is tuned near to its magnetic instability. We suggest an experimental set-up to observe this effect. We also suggest that our result can be exploited to control a quantum critical system around its quantum critical point to study its thermodynamical or transport properties.

Electronic Transport Evidence for Topological Nodal-Line Semimetals of ZrGeSe Single Crystals

Although the band topology of ZrGeSe has been studied via magnetic torque technique, the electronic transport behaviors related to the relativistic Fermions in ZrGeSe are still unknown. Here, we first report systematic electronic transport properties of high-quality ZrGeSe single crystals under magnetic fields up to 14 T. Resistivity plateaus of temperature dependent resistivity curves both in the presence and absence of magnetic fields as well as large, non-saturating magnetoresistance in low-temperature region were observed. By analyzing the temperature- and angular-dependent Shubnikov-de Haas oscillations and fitting it via the Lifshitz-Kosevich (LK) formula with the Berry phase being taken into account, we proved that Dirac fermions dominate the electronic transport behaviors of ZrGeSe and the presence of non-trivial Berry phase. First principles calculations demonstrate that ZrGeSe possesses Dirac bands and normal bands near Fermi surface, resulting in the observed magnetotransport phenomena. These results demonstrate that ZrGeSe is a topological nodal-line semimetal, which provides a fundamentally important platform to study the quantum physics of topological semimetals.

Anomalous quantum oscillations and evidence for a non-trivial Berry phase in SmSb

Topologically non-trivial electronic structures can give rise to a range of unusual physical phenomena, and the interplay of band topology with other effects such as electronic correlations and magnetism requires further exploration. The rare earth monopnictides X(Sb,Bi) (X = lanthanide) are a large family of semimetals where these different effects may be tuned by the substitution of rare-earth elements. Here we observe anomalous behavior in the quantum oscillations of one member of this family, antiferromagnetic SmSb. The analysis of Shubnikov-de Haas (SdH) oscillations provides evidence for a non-zero Berry phase, indicating a non-trivial topology of the α-band. Furthermore, striking differences are found between the temperature dependence of the amplitudes of de Haas-van Alphen effect oscillations, which are well fitted by the Lifshitz-Kosevich (LK) formula across the measured temperature range, and those from SdH measurements which show a significant disagreement with LK behavior at low temperatures. Our findings of unusual quantum oscillations in an antiferromagnetic, mixed valence semimetal with a possible non-trivial band topology can provide an opportunity for studying the interplay between topology, electronic correlations and magnetism.

Shubnikov–de Haas and de Haas–van Alphen oscillations in the topological semimetal CaAl4

We report the magneto-transport properties of CaAl$_4$ single crystals with $C2/m$ structure at low temperature. CaAl$_4$ exhibits large unsaturated magnetoresistance $\sim$3000$\%$ at 2.5 K and 14 T. The nonlinear Hall resistivity is observed, which indicates the multi-band feature. The first-principles calculations show the electron-hole compensation and the complex Fermi surface in CaAl$_4$, to which the two-band model with over-simplified carrier mobility can't completely apply. Evident quantum oscillations have been observed with B//c and B//ab configurations, from which the nontrivial Berry phase is extracted by the multi-band Lifshitz-Kosevich formula fitting. An electron-type quasi-2D Fermi surface is found by the angle-dependent Shubnikov-de Haas oscillations, de Haas-van Alphen oscillations and the first-principles calculations. The calculations also elucidate that CaAl$_4$ owns a Dirac nodal line type band structure around the $\Gamma$ point in the $Z$-$\Gamma$-$L$ plane, which is protected by the mirror symmetry as well as the space inversion and time reversal symmetries. Once the spin-orbit coupling is included, the crossed nodal line opens a negligible gap (less than 3 meV). The open-orbit topology is also found in the electron-type Fermi surfaces, which is believed to help enhance the magnetoresistance observed.

Thermopower Quantum Oscillations in the Charge Density Wave State of the Organic Conductor $$\upalpha $$ α -(BEDT-TTF) $$_{2}\text {KHg}(\text {SCN})_{4}$$ 2 KHg ( SCN ) 4

We report on experimental studies of magnetic quantum oscillations of the interlayer thermopower in the charge density wave state of the multiband organic conductor $$\upalpha $$ -(BEDT-TTF) $$_{2}\text {KHg}(\text {SCN})_4$$ . The magnetic field is along the direction perpendicular to the conducting layers. The interlayer thermopower or Seebeck effect has been measured in magnetic fields of up to 32 T and temperatures down to 0.5 K. The thermopower magnetic field dependence was measured at two temperatures 0.5 K and 4 K. Quantum oscillations observed in thermopower, at fields above 8 T and temperature 4 K, originate only from the $$\alpha $$ orbit, whereas at 0.5 K the energy spectrum consists of Landau levels of not only the $$\alpha $$ orbit but also the second harmonic as obtained from the fast Fourier transform analysis. In addition to the $$\alpha $$ and 2 $$\alpha $$ frequencies, the oscillation spectrum reveals existence of a low-frequency peak at $$F_{\lambda } = 180$$ T which was previously observed in both magnetoresistance and magnetization. The behavior of thermopower magnetic quantum oscillations amplitude is analyzed using the Lifshitz–Kosevich formula, and the influence of different damping factors such as the temperature, spin splitting, Dingle and magnetic breakdown factors is studied. In addition, we analyze how the second harmonic of fundamental $$\alpha $$ frequency observed at low temperatures in the $$\text {CDW}_0$$ state at field perpendicular to the conducting layers affects the thermopower quantum oscillations amplitude. At low temperature, we find that on entering the low-field state, the scattering rate is observed to increase dramatically. We also observe an apparent increase in the effective mass of first harmonic from $$m_{\alpha }^*=1.7 m_{\text{e}}$$ , in the low-field state to $$m_{\alpha }^*=3.3 m_{\text{e}}$$ , in the high-field state. For the second harmonic, a constant effective mass of $$m_{2\alpha }^*=3.2 m_{\text{e}}$$ is estimated above and below the kink field.

De Haas–van Alphen Effect and Oscillation of the Metal Magnetization

In the focus of our research is the existing theory of the de Haas–van Alphen effect. We show that the Lifshitz–Kosevich formula represents, up to a factor, the amplitude of Fraunhofer diffraction by a circular aperture with radius dependent on the extremal value of the cross-sectional area of Fermi surface.