Abstract
The geometric phase of an electronic wave function, also known as Berry phase, is the fundamental basis of the topological properties in solids. This phase can be tuned by modulating the band structure of a material, providing a way to drive a topological phase transition. However, despite significant efforts in designing and understanding topological materials, it remains still challenging to tune a given material across different topological phases while tracing the impact of the Berry phase on its quantum transport properties. Here, we report these two effects in a magnetotransport study of ZrTe_{5}. By tuning the band structure with uniaxial strain, we use quantum oscillations to directly map a weaktostrong topological insulator phase transition through a gapless Dirac semimetal phase. Moreover, we demonstrate the impact of the straintunable spindependent Berry phase on the Zeeman effect through the amplitude of the quantum oscillations. We show that such a spindependent Berry phase, largely neglected in solidstate systems, is critical in modeling quantum oscillations in Dirac bands of topological materials.
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Introduction
Topological phase transitions in many materials can be described by a massive Dirac equation at a time reversal invariant momenta. When the energy gap closes and reopens, the mass term in the corresponding Dirac equation changes sign, which can be reflected by the evolution of Berry phase along the Fermi surface if the system is slightly doped^{1,2,3,4,5,6}. Both the evolution of the band gap and the Berry phase can be probed in quantizing magnetic fields, where Shubnikov de Haas (SdH) oscillations from Landau level quantization provides a tool to probe “Fermiology”^{7,8,9,10,11}. In addition to Landau level quantization, the Zeeman effect splits the spindegenerate Landau levels. The difference in the Berry phases between the spinsplit Landau levels plays a crucial rule in the SdH oscillations. Interestingly, while the Zeeman effect has been extensively studied over the past decades^{12,13}, its connection to band topology and nontrivial Berry phase has emerged only recently. A Dirac band with a finite mass gap hosts spindependent Berry phase which can modify the SdH oscillations through the Zeeman effect, as recently demonstrated in the spinzero effect^{14,15}. Such a spindependent Berry phase is expected to be fundamentally generic for Diraclike bands, and is band parameter dependent. It is therefore desirable to conduct a comprehensive study of such effect in a system with tunable band parameters.
Despite the significant developments in topological materials study in recent years, it remains a challenge to tune between different topological phases, and correspondingly the Berry phase, in the same material. Strain is a widely proposed mechanism for driving a topological phase transition^{16,17,18,19,20,21,22,23,24,25,26}. And transition metal pentatelluride ZrTe_{5} is a promising material for such study. ZrTe_{5} is a van der Waals layered material with orthorhombic lattice structure, with layer planes extending along the a and c lattice directions and stacking along the b direction. Its electronic properties of is largely dominated by the Fermi surface centered around the Γ point with a Dirac dispersion, but can also have contribution from the parabolic side bands with Fermi surfaces centered between the R and E points in the Brillouin zone (see Supplementary Note 2). ZrTe_{5} hosts intriguing properties such as a resistance peak (Lifshitz transition)^{27,28,29}, chiral magnetic effect^{30}, and 3D quantum Hall effect^{31}. In its 3D bulk form, ZrTe_{5} has a Diraclike low energy band structure, with sampledependent mass gap rendering the material from Dirac semimetal to topological insulator^{30,32,33,34,35,36}, suggesting extreme sensitivity to lattice deformations. Recently a straininduced weak topological insulator (WTI) to strong topolgical insulator (STI) transition has been proposed^{37,38}, followed by experimental evidence in angleresolved photoemission spectroscopy (ARPES) study^{39}, and indirect charge transport evidences via the chiral anomaly effect^{40}. A charge transport study of the quantum oscillations which directly map the topological phase transition is still lacking. Such bulk transport measurements would be complementary to the surfacesensitive ARPES study, and also reveal the spindependent Berry phase over tunable band parameters.
In this work, we study charge transport and SdH oscillations in ZrTe_{5} under tunable uniaxial strain. In a magnetic field perpendicular to the a–c plane and the applied current, SdH oscillations and their evolution over strain allow direct mapping of the WTISTI transition through the closing and reopening of the Dirac band mass gap. The dependence of the SdH oscillation amplitude on the Fermi energy and Dirac mass gap, tunable through the strain, is analyzed and compared with the quantum oscillation theory. Our results reveal that the spindependent Berry phase intrinsic to the Dirac bands, which has been largely neglected in previous studies, is critical in modeling the SdH oscillations in such topological materials.
Results and discussion
Magnetotransport under tunable strain
The samples studied in this work are ZrTe_{5} microcrystals mechanically exfoliated onto flexible Polyimide substrates (Fig. 1a–c), which allow application of external tensile and compressive strains along the aaxis through substrate bending, over a wide temperature range from room temperature down to below 4K (see Methods: Sample fabrication and characterization, and Supplementary Note 3). Our ZrTe_{5} exhibits a straintunable resistance peak at T_{p} ≈ 140K(Fig. 1e). At T = 20K a nonmonotonic resistance versus strain dependence is observed (Fig. 1f), consistent with the report in macroscopic ZrTe_{5} crystals^{40}. A large strain gauge factor ranging 10^{2}−10^{3} generally presents throughout the temperature range up to room temperature. The gauge factor observed here is very large compared to that of the metal thin films (~2), and is comparable to typical single crystal semiconductors such as Si and Ge (~100). The large gauge factor manifests a strainsensitive band structure and a small Fermi surface.
Next, we characterize the electrical resistance in magnetic field perpendicular to the a–c crystal plane. At the base temperature of ≈4K, SdH oscillations are clearly visible on the magnetoresistance background and evolve with changing strain (Fig. 2a). Plotting the oscillatory part ΔR versus inverse magnetic field 1/B, equally spaced resistance oscillations can be resolved (Fig. 2b). In the limits of large compressive and tensile strains applied here, the oscillation amplitude monotonically decreases with increasing 1/B. Under mild compressive strains, however, the SdH oscillation amplitude appears nonmonotonic, suggesting contributions from more than a single Fermi surface.
Analysis of SdH oscillations
Our analysis of the SdH oscillations focuses on mapping out the straindependent mass gap, Fermi level, and the spindependent Berry phase. We model ZrTe_{5} by an anisotropic Dirac Hamiltonian (see Methods: Computing the gfactor). Starting from a full Hamiltonian including all the Bloch bands, we divide the Hilbert space into a low energy subspace consisting of the Dirac bands, and a high energy subspace containing all the other bands. When a magnetic field is applied, the vector potential couples the two subspaces and must be downfolded into the low energy bands at the Fermi level, which gives rise to the nonzero gfactor terms in the twoband Dirac model for the low energy subspace, g_{p} and g_{s}. We note that such a down folding process includes the contribution to the orbital magnetic moments (the importance of which in the analysis of quantum oscillations has been recently pointed out^{11}) from the high energy bands and the contribution from the low energy bands can be captured by the Landau quantization with the Berry curvature being considered^{15,41}.
In quantizing magnetic fields, the extremal crosssectional area of the Fermi surface is S_{F} = π(μ^{2} − Δ^{2})/(ℏ^{2}v^{2}). The corresponding cyclotron mass is \({m}_{c}=\frac{{\hslash }^{2}\partial {S}_{k}}{2\pi \partial \mu }=\frac{\mu }{{v}^{2}}\). Here μ is the Fermi energy, Δ is the mass gap, and v ≈ 5 × 10^{5} m s^{−1} is the Dirac band velocity in the a–c plane^{32}. The SdH oscillations can be modeled with the Lifshitz–Kosevich (LK) formula: up to the second order in the B field, each single band contributes to the SdH oscillations:
We note that the conventional (parabolic band) LK formula and its Dirac version^{42} have the same mathematical form with their corresponding band parameters (see Supplementary Note 5). In Eq. (1) R_{0} is the zero magnetic field resistance. Defined for Dirac band, \({R}_{D}=\exp (2{\pi }^{2}{k}_{B}{T}_{d} \mu  /\hslash eB{v}^{2})\) is the amplitude reduction factor from disorder scattering, where T_{d} is the Dingle temperature characterizing the level of disorder,and k_{B} is the Boltzmann constant. \({R}_{T}=\xi /\sinh (\xi )\) is the amplitude reduction factor from temperature, where ξ = 2π^{2}k_{B}Tμ/ℏeBv^{2}. ϕ_{B} is the Berry phase, and δ = ± π/4 in 3D materials.
Zeeman effect and spindependent Berry phase
Now we consider the Zeeman effect, which splits a spindegenerate band into two, each with a different Fermi surface extremal crosssectional area: S_{F↑/↓} = S_{F} ± αB, and Berry phase: ϕ_{B↑/↓} = ϕ_{B} ± ϕ_{s}^{15}. Here α describes the splitting of extremal crosssectional area of the Fermi surface and ϕ_{s} is the spindependent part of the Berry phase. Due to the strong spinorbit coupling in ZrTe_{5} spin is not a good quantum number at general momenta: here spin up and spin down refer to “pseudo spins” defined as the two eigenstates of the mirror reflection operator through the k_{z} = 0 plane, which contains the extremum projection of the Fermi surface. The overall quantum oscillation is a summation of the two oscillation terms from the two spin bands: \(\cos \left(\frac{\hslash {S}_{F\uparrow }}{eB}+{\phi }_{B\uparrow }+\pi +\delta \right)+\cos \left(\frac{\hslash {S}_{F\downarrow }}{eB}+{\phi }_{B\downarrow }+\pi +\delta \right)=2\cos \left(\frac{\hslash {S}_{F}}{eB}+{\phi }_{B}+\pi +\delta \right)\cos \left(\frac{\hslash \alpha }{e}+{\phi }_{s}\right)\) Interestingly S_{F} and ϕ_{B}, which are the common (spinindependent) part of the two spin bands before the Zeeman splitting, determines the period and phase offset of the quantum oscillations; while the difference between the two spin bands determines the amplitude.
For systems with nontrivial Berry curvature, the Berry phase splitting ϕ_{s} along the Fermi surface will be nonzero. In particular, for a Dirac electron system, we find (see Methods: Quantum oscillations)
Defining the Zeeman effectinduced SdH amplitude reduction factor: \({R}_{s}=\cos \left(\frac{\hslash \alpha }{e}+{\phi }_{s}\right)\), we derive for the Dirac band:
Further, the coefficient α is computed from the gfactor tensor obtained from firstprinciple calculations at given values of Δ and μ_{F}.
Under large compressive and large tensile strains, we observed approximately single SdH oscillation frequencies and a monotonic decay of SdH oscillation amplitude over increasing 1/B, suggesting that SdH oscillations are dominated by single bands there. The linear dependence of the Landau level indices on 1/B (Fig. 2c) extrapolates to the index values of close to 0 and 1/2 at 1/B = 0, corresponding to a SdH oscillation phase change from approximately zero to π. Fitting the 1/B and T dependence of the SdH oscillation amplitude to the LK formula (examples shown in Fig. 2d–f) provides estimations to the cyclotron mass and the Dingle temperature at these particular strains.
More generally, the SdH oscillations show a complex 1/B dependence which can be modeled considering contributions from two bands: ΔR = ΔR_{1} + ΔR_{2}. Here band 1 is a Dirac band (with corresponding Fermi surface centered around the Γ point) with a common Berry phase of π for its two spins bands and hence a overall SdH phase which is much smaller than π (Fig. 2c). Band 2, a “trivial” parabolic band (with corresponding Fermi surface between the R and E points) possess a overall SdH phase which is close to π. We note that evidence of charge transport contribution from a secondary band has been reported previously^{28}. The change of SdH phase shown in Fig. 2c is due to the strainevolution of the relative contribution from the two bands, instead of band topology transition within the same band.
Figure 2b compares the measured SdH oscillations with the twocomponent LK formula simulations at base temperatures. Here the Dirac band SdH oscillations are modeled with Eq. (4), while the SdH oscillations from the trivial band are modeled with the conventional LK formula, with parameters including T_{d}, Fermi energy and cyclotron mass m_{c}. The band parameters used in the simulations are chosen so that they evolve smoothly with changing strain, and yield SdH oscillations which agrees with the experimental observations (with small deviations which may be attributed to the choices of background curves when extracting the oscillatory part of the data). The more general comparisons including temperature dependence are shown in the Supplementary Note 4. Figure 3 plots the key simulation parameters. Here we also include the estimated uncertainties of the parameters beyond which the simulated SdH oscillations deviate noticeably from the measurements. Focusing on the the Dirac band, a closing and reopening of mass gap with tuning external strain happens at a compressive strain of ≈ − 0.2%. This, remarkably, is a direct transport evidence of the theoretically predicted WTI to STI transition. Associated with such transition, the Fermi energy (measured from the center of the mass gap) shows a nonmonotonic strain dependence, reaching a minimum close to the transition strain where mass gap vanishes. From the mass gap and the Fermi energy, we calculate the electron doping: μ − Δ, which characterizes the energy of the Fermi level in relation to the bottom of the conduction band. The result indicates a maximum electron doping at the WTISTI transition, which decreases when straintuned away from the transition. Generally the analysis of the band parameters suggests a band evolution with strain which is depicted in Fig. 4a.
Compared to the previous (ARPES and chiral magnetic effect) reports on WTISTI transition, a similar order of magnitude of strain was applied in our studied here ( ≈ few − 0.1%). The absolute value of the strain at the transition point, however, is randomly shifted by a random builtin stress (typically tensile), resulting from the hottransfer fabrication process. We note that the observed STIWTI transition coincides with the minimum resistance point (Fig. 1e), consistent with the previous experimental study on chiral magnetic effect in ZrTe_{5} under strain^{40}. The straintunable bandgap observed through our magnetotransport, within the range of strain of a ≈ few − 0.1%, varies by a few 10 meV. This is in good agreement with the previous report on ARPES study of ZrTe_{5} under strain^{39}.
Next we focus on the impact of Zeeman effect on SdH oscillation amplitude (R_{s}). Experimentally we obtain R_{s} at every strain by quantitative simulation of the SdH oscillations. We then compare the results with the theoretical expectation: \({R}_{s}=\cos \left(\frac{\hslash \alpha }{e}+{\phi }_{s}\right)\). Here \(\frac{\hslash \alpha }{e}\), which is associated with the splitting of extremal crosssectional area of the Fermi surface, follows (see Methods: Quantum oscillations):
We adopt the gfactors for the s and p orbitals \({g}_{z}^{p}=9.66,{g}_{z}^{s}=6.45\), as computed from firstprinciple calculations (DFTmBJ)^{41}. We note that Eq (5) represents a generalization of the calculation in Ref. ^{11} by including the effect of higher energy nonDirac bands, which cause g_{p} and g_{s} to deviate from one. ϕ_{s}, which is associated with the Berry phase splitting along the Fermi surface (Fig. 4b), is calculated from the mass gap and Fermi energy obtained from simulating the SdH oscillations: \({\phi }_{s}=\pi \frac{\Delta }{\mu }\). The theory shows good qualitative agreement with the data on the strain dependence of R_{s}, as illustrated in Fig. 3f with the band parameters tuned along the trajectory in the (Δ, μ) parameter space shown in Fig. 3e. By contrast, the conventional modeling of Zeeman effect on SdH oscillations which only considers the Fermi surface splitting (Eq. (5)) completely fails to match with the experimental observations. This comparison definitively highlights the importance of spindependent Berry phase in the Zeeman effect. We also note that in our theoretical model, the value of R_{S} is dependent on both the amplitude and the sign of the energy gap Δ. The quantitative comparison between the theoretical model and our data on R_{S}, as shown in Fig. 3e and f, therefore reveals a signchange in band gap near where its amplitude vanishes. This is consistent with the occurrence of a topological phase transition.
The significant straintunability of the spindependent Berry phase ϕ_{s} is shown in Fig. 4c. Accompanied by the topological phase transition where the mass gap vanishes, ϕ_{s} passes through zero when the system is in the Dirac semimetal phase. We also compute the effective gfactor by comparing R_{s} with the conventional Zeeman effectinduced SdH amplitude reduction factor \(\cos (\pi g{m}_{c}/(2{m}_{e}))\)^{7,32}. This leads to g = 2m_{e}(ℏα/e + ϕ_{s})/(πm_{c}), whose strong strain tunability is shown Fig. 4d.
Conclusion
In conclusion, we have carried out a magnetotransport study on the evolution of band topology and nontrvial Berry phase over straintunable band parameters in ZrTe_{5}. The straindependent SdH oscillations allow direct mapping of the closing and reopening of the Dirac band mass gap, which is consistent with a WTISTI transition in ZrTe_{5}. Moreover we observed the nontrivial Berry phase and its dependence on band parameters over the transition. Such a spindependent Berry phase is generic and intrinsic to Dirac band structure, and is a critical factor in modeling Zeeman effect in SdH oscillations in topological materials.
Methods
Sample fabrication and characterization
The samples studied in this work are ZrTe_{5} microcrystals mechanically exfoliated onto flexible Polyimide substrates (Fig. 1a, b), which allow application of tensile and compressive strains over a wide temperature range from room temperature down to 4K. ZrTe_{5} single crystals were synthesized by chemical vapor transport method, with iodine as transport agency. Stoichiometry amounts of Zr(4N) and Te(5N) powder, together with 5mg/mL I_{2}, were loaded into a quartz tube under argon atmosphere. The quartz tube was flame sealed and then placed in a twozone furnace, a temperature gradient from 480 °C to 400 °C was applied. After 4 weeks reaction, golden, ribbonshaped single crystals were obtained, of typical size about 0.6 × 0.6 × 5 mm. The Xray diffraction characterization of the crystals is shown in the Supplementary Note 1.
To avoid degradation of the material from ambient exposure, the crystals are exfoliated and presscontacted on predefined gold electrodes on 120 μmthick Polyimide substrates in Ar environment, and are encapsulated with poly(methyl methacrylate) (PMMA) which both protects the crystals from degradation and facilitates the application of strain. A detailed description on the sample fabrication and on the contact/crystal interface can be found in reference^{43}. Uniaxial strain is applied through substrate bending. The strain homogeneity is facilitated by the total emersion of the crystal in PMMA, and more importantly the extremely small ratio of crystal thickness (~190 nm) to the total crystal length (~100 μm, we note that the length of the crystal outside the electrodes also helps creating an uniform strain throughout the thickness of the crystal). All measurements were done with electric current applied along the aaxis. Typical contact resistance over a 10 μm^{2} contact area is between few hundred ohms to a few kiloohms, and does not change significantly over the application of strain through substrate bending. Uniaxial strain is applied to the samples using a fourpoint bending setup, with motorized precision control to the substrate curvature^{44,45}. Because the substrate is much thicker than the microcrystals, significant uniaxial strain can be achieved at relatively mild substrate curvatures under which the strain on the microcrystals is predominantly tensile/compressive due to the elongation/compression of the top substrate surface where the microcrystals are PMMApinned. Besides external strain applied through substrate bending, both the crystals, the substrates, and the encapsulating PMMA layer go through thermal expansion/contraction upon temperature change. Because of such complication it is difficult to precisely characterize the absolute strain on the crystals. Here we focus on the dependence of the transport characteristics on the change of strain induced by substrate bending, which is described in the discussion below as “external strain”. In magnetotransport measurements, we limit the temperature range between 3 and 20K, where the change of thermal expansion is small in comparison to the range of the external strain. All charge transport measurements were performed in a Oxford Variable Temperature Insert with a superconducting magnet, using standard lockin technique.
Computing the gfactor
To compute the effect of Zeeman coupling on quantum oscillations, we study the k ⋅ p Hamiltonian near Γ, which is an anisotropic gapped Dirac Hamiltonian^{41}:
in the basis \(\left\vert p,\frac{1}{2}\right\rangle\), \(\left\vert p,\frac{1}{2}\right\rangle\), \(\left\vert s,\frac{1}{2}\right\rangle\), \(\left\vert s,\frac{1}{2}\right\rangle\), where \(\pm \frac{1}{2}\) refers to the zcomponent of spin. In Eq. (6), Δ is half the mass gap; v_{x,y,z} are the anisotropic Dirac cone velocities; σ_{x,y,z} are the Pauli matrices; and σ_{0} is the 2 × 2 identity matrix. The eigenvalues of H(k) are
where all bands are doublydegenerate due to the combination of timereversal and inversion symmetry. The eigenvectors are given by
where u_{1} = (1, 0)^{T}, u_{2} = (0, 1)^{T} and we have defined the rescaled coordinates p_{i} = v_{i}k_{i}/v.
The Zeeman coupling for the fourband model is:
where \({g}_{x,y,z}^{s,p}\) are the matrixvalued gfactors for the s and p orbitals, which are anisotropic due to the crystal symmetry and which differ from their bare value due to coupling with all the other bands not in the fourband model.
To compute the quantum oscillations, we must downfold this fourband model into the two bands at the Fermi level. We seek an effectivegfactor for the two degenerate bands that make up the Fermi surface, such that when projected onto those bands, the Zeeman Hamiltonian takes the form:
where \({\mu }_{B}=\frac{e\hslash }{2{m}_{e}}\) is the Bohr magneton and \(m,{m}^{{\prime} }\) index the two bands at the Fermi surface.
The effective gfactor has two contributions:
where g_{0} is the projection of g^{s} and g^{p} (defined by \({H}_{0}^{Z}\)) onto the two conduction bands at the Fermi level and g_{D} is an extra orbital contribution that we will describe below. For a magnetic field in the z direction, using the expressions for the conduction band eigenstates from Eq. (8),
where the Pauli matrix σ_{z} acts in the basis of the two bands at the Fermi level. Ultimately, we will only need the gfactor at the extremum of the Fermi surface, where p_{z} = 0 and \({\hslash }^{2}{v}^{2}({p}_{x}^{2}+{p}_{y}^{2})={\mu }^{2}{\Delta }^{2}\). In this case,
where \(\gamma =\frac{\mu }{\Delta }\) is a dimensionless constant.
Returning to the second term in Eq. (11), g_{D} is the orbital contribution to the gfactor from the Dirac cone^{46}:
where ψ_{m,k} are the eigenvectors of Eq. (6).
The first step to evaluate g_{D} is to find the derivatives of the eigenstates:
where we have dropped the subscript/superscript ± on E and ψ to reduce clutter. After some algebra, it follows that:
which yields the matrix element
The terms symmetric under the exchange of the i and j indices (i.e., the last two terms) will cancel in the sum in Eq. (14). Thus, applying Eq. (14), the extra orbital contribution to the gfactor is given by:
where \({g}^{L}\equiv \frac{{m}_{e}{v}^{2}}{\Delta }\) and \(\tilde{\gamma }\equiv \frac{E}{\Delta }\) are dimensionless constants.
We now simplify this result by taking B in the zdirection and restricting to the boundary of the extremal crosssection, where p_{z} = 0 and \({\hslash }^{2}{v}^{2}({p}_{x}^{2}+{p}_{y}^{2})={\mu }^{2}{\Delta }^{2}\). Plugging this into Eq. (18), we obtain the zcomponent of g:
The total effective gfactor at the extremal crosssection of the Fermi surface is found by combining Eqs. (13) and (19):
This expression shows that the gfactor in the \(\hat{z}\)direction is independent of momentum on the boundary of the extremal crosssection and is diagonal, i.e., \(\vert {\psi }_{m,{{{{{{{\boldsymbol{k}}}}}}}}}\rangle\) are eigenstates of the Zeeman coupling.
Quantum oscillations
We now discuss the quantum oscillations of the anisotropic Dirac semimetal. As discussed in the main text, the quantum oscillations for a doubly degenerate electronlike Fermi surface are proportional to
where ϕ_{B,↑/↓} = ϕ_{B} ± ϕ_{s} are the Berry phases around the extremal Fermi surface for each of the two spins and S_{F,↑/↓} = S_{F} ± αB are the extremal areas of the Fermi surface for each spin.
The Berry connection is defined by^{46}: \({{{{{{{{\boldsymbol{A}}}}}}}}}_{m{m}^{{\prime} }}({{{{{{{\boldsymbol{k}}}}}}}})={\sum }_{k}i\langle {\psi }_{m,{{{{{{{\boldsymbol{k}}}}}}}}} \frac{\partial }{\partial {k}_{k}} {\psi }_{{m}^{{\prime} },{{{{{{{\boldsymbol{k}}}}}}}}}\rangle {{{{{{{{\boldsymbol{e}}}}}}}}}_{k}\). By plugging in the expressions for the eigenstates and derivatives of eigenstates in Eqs. (8) and (15), we find the explicit expression:
To find the Berry phase, we ultimately need A(k) ⋅ dk on the extremal crosssection, where p_{z} = 0 and dk is in the x–y plane, which yields:
where σ_{z} is acting in the space of the two conduction bands at the Fermi level. We find the Berry phase around each extremal ring of the Fermi surface by integrating over the the boundary of the extremal crosssection:
The loop integral can be turned into an area integral using Stokes theorem:
which yields:
where we have added 2π in order to ensure the Berry phase in the range of 0 to 2π.
The last term that we need to evaluate the LifschitzKosevich formula is the area of the extremal Fermi surfaces. Since the effective gfactor at the extremal Fermi surfaces (Eq. (20)) is diagonal in the basis of the two conduction bands, the two extremal Fermi surfaces satisfy the equation:
accounting for the fact that the magnetic field is in the z direction and the extremal Fermi surfaces have p_{z} = 0. Thus, each extremal crosssection forms an ellipse whose area is given by:
Thus, the areasplitting term α in Eq. (21) is given by:
where we have used μ_{B} = eℏ/2m_{e}. Plugging in the result for \({{{{{{{{\boldsymbol{g}}}}}}}}}_{z}^{{{{{{{{\rm{ext}}}}}}}}}\) from Eq. (20),
Plugging in the calculation of the Berry phase (Eq. (27)) and the area difference (Eq. (31)) to the LifschitzKosevich formula (Eq. (21)), we derive the expression for quantum oscillations:
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
X.D. acknowledges support from the National Science Foundation (NSF) under award DMR1808491. X.D. and J.C. thank Aris Alexandradinata for insightful discussions. J.C. acknowledges the support of the Flatiron Institute, a division of Simons Foundation, and support from the National Science Foundation under Grant No. DMR1942447.
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A.G. and X.D. fabricated the samples and performed the measurements. S.S., J.C., and X.D. formulated the theory. P.W. and L.Z. synthesized the ZrTe_{5} single crystals. X.D. performed the data analysis and designed the experiment. S.S., J.C., X.D., and Xu.D. wrote the paper.
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Communications Materials thanks Ryo Noguchi, Jian Zhou and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Primary Handling Editors: Toru Hirahara and Aldo Isidori. Peer reviewer reports are available.
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Gaikwad, A., Sun, S., Wang, P. et al. Straintuned topological phase transition and unconventional Zeeman effect in ZrTe_{5} microcrystals. Commun Mater 3, 94 (2022). https://doi.org/10.1038/s43246022003165
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DOI: https://doi.org/10.1038/s43246022003165
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