Evidence for a strain-tuned topological phase transition in ZrTe5.

A phase transition between topologically distinct insulating phases involves closing and reopening the bandgap. Near the topological phase transition, the bulk energy spectrum is characterized by a massive Dirac dispersion, where the bandgap plays the role of mass. We report measurements of strain dependence of electrical transport properties of ZrTe5, which is known to host massive Dirac fermions in the bulk due to its proximity to a topological phase transition. We observe that the resistivity exhibits a pronounced minimum at a critical strain. We further find that the positive longitudinal magnetoconductance becomes maximal at the critical strain. This nonmonotonic strain dependence is consistent with the switching of sign of the Dirac mass and, hence, a strain-tuned topological phase transition in ZrTe5.

INTRODUCTION

Appreciation of the topological aspects of band structure has fundamentally changed the way we understand the electronic properties of solids. Band insulators with time reversal symmetry can be classified into normal insulators (NIs), weak topological insulators (WTIs), and strong topological insulators (STIs) based on their Z2 topological indices (–). Changing Z2 indices requires closing and reopening the bandgap, and topologically distinct insulating phases are separated by a gapless pan> class="Chemical">Dirac or Weyl semimetal phase. The relationship between these phases is summarized in the general phase diagram proposed by Murakami and Kuga (, ), as shown in Fig. 1A. These topological phases have been intensively studied in the past decade (–). In contrast, the transition between these phases is less explored because it requires changing the band parameters of the solid. It has been demonstrated that topological phase transitions can be induced either by chemical doping or by thermal lattice expansion (–). Nevertheless, the precise in situ control of topological phase transition in a three-dimensional (3D) system is still an outstanding challenge.

Fig. 1

Topological phase diagram and band structures of ZrTe5.

(A) Universal phase diagram of topological insulators proposed by Murakami for a 3D system (, , ). The control parameter 𝛿 describes the breaking of inversion symmetry. The control parameter ϵ does not break inversion symmetry. (B) Crystal structure of ZrTe5. Chains of ZrTe3 prisms (consisting of Tea and Ted atoms) extend along the a axis. These chains are connected by Tez atoms along the c axis to form layers. These layers are vdW bonded in the b axis direction. (C) Size of bandgap Eg at the Γ point as functions of strains in the a and c lattice directions. The dashed gray arrow indicates the anisotropic strain induced by a uniaxial stress along the a axis direction, as governed by the calculated Poisson’s ratio ϵ = −4.0ϵ. (D) Band structures for different strain states taken at points along the Poisson’s ratio path. These points (from left to right) correspond to STI, Dirac semimetal, and WTI, respectively. Fermi level is defined as the zero energy, and the k-point labeling is based on the primitive unit cell. A band inversion involving Ted and Tez p orbitals (shown as red and green colors, respectively) is seen in the STI phase.

(A) Universal phase diagram of topological insulators proposed by Murakami for a 3D system (, , ). The control parameter 𝛿 describes the breaking of inversion symmetry. The control parameter ϵ does not break inversion symmetry. (B) Crystal structure of ZrTe5. Chainpan>s of ZrTe3 prisms (consistinpan>g of Tea anpan>d pan> class="Chemical">Ted atoms) extend along the a axis. These chains are connected by Tez atoms along the c axis to form layers. These layers are vdW bonded in the b axis direction. (C) Size of bandgap Eg at the Γ point as functions of strains in the a and c lattice directions. The dashed gray arrow indicates the anisotropic strain induced by a uniaxial stress along the a axis direction, as governed by the calculated Poisson’s ratio ϵ = −4.0ϵ. (D) Band structures for different strain states taken at points along the Poisson’s ratio path. These points (from left to right) correspond to STI, Dirac semimetal, and WTI, respectively. Fermi level is defined as the zero energy, and the k-point labeling is based on the primitive unit cell. A band inversion involving Ted and Tez p orbitals (shown as red and green colors, respectively) is seen in the STI phase.

The transition-metal pentatellurides pan> class="Chemical">ZrTe5 is an ideal material to realize in situ control of topological phase transitions. ZrTe5 is a van der Waals (vdW) layered material crystallized in the Cmcm orthorhombic space group. Each layer consists of ZrTe3 chains extending along the a lattice direction, and the layers are stacked along the b lattice direction (Fig. 1B). The material has received substantial interest because its monolayer form is predicted to be a large bandgap quantum spin Hall insulator (). It was also suggested that the 3D bulk band structure is very close to the phase boundary between WTI and STI (–), indicated by the red line in the universal phase diagram shown in Fig. 1A (, ). Early studies of optical conductivity (), quantum oscillations (, ), photoemission, and negative longitudinal magnetoresistance (NLMR) () have supported the predicted Dirac semimetal-like band structure in the bulk, with a single Dirac point at the center of the Brillouin zone, Γ. Unlike topological Dirac semimetals such as Na3Bi or Cd3As2, there is no additional crystalline symmetry to protect the Dirac point in ZrTe5, and more recent spectroscopy measurements revealed that its band structure is better described by a massive Dirac dispersion, where mass plays the role of bandgap. The bandgap size measured by different experiments varies, ranging from 10 to 80 meV, and there are conflicting reports on whether the material is a WTI or STI (–), or changes with temperature (). These experimental findings suggest that the electronic structure of ZrTe5 may be very sensitive to external perturbation, such as lattice distortion.

In this work, we present evidence for a strain-tuned topological phase transition corresponding to a sign change of the Dirac banpan>dgap inpan> pan> class="Chemical">ZrTe5. This conclusion is a result of extensive measurements of bulk electrical transport of ZrTe5 as a function of in situ tunable anisotropic strain. Although the measurement of surface states is often regarded as the smoking gun evidence of topological insulators, close to a topological phase transition the small bandgap and the Dirac-like dispersion in the bulk impose severe challenges for identifying surface states (). Here, we focus on the bulk transport properties, which are highly sensitive to the mass of bulk Dirac fermions near a topological phase transition. We observe that the resistivity exhibits a nonmonotonic strain dependence and reaches a minimum at a critical strain. The nonmonotonic strain dependence is distinct from the expected behavior of a conventional semiconductor, yet it is consistent with the sign switching of the Dirac mass of a gapped Dirac semimetal. We further found that the NLMR also shows a nonmonotonic strain dependence and becomes maximal at the critical strain. The extracted helicity relaxation time increases by 10-fold as the strain approaches the critical value. Our study presents ZrTe5 as a promising platform for on-demand control of nontrivial topological properties of materials.

RESULTS

Figure 1C shows a contour map of the size of the bandgap Eg at Γ as a function of ϵ and ϵ, i.e., strain (%) along the a and c lattice directions, calculated by density functional theory (DFT). It shows a V-shaped valley, with the minpan>imum of the valley (correspondinpan>g to Eg = 0) extendinpan>g along the diagonal direction. Z2 topological inpan>dices were also compupan> class="Chemical">ted, and the Eg = 0 line is the phase boundary between STI and WTI (see fig. S7). Figure 1C reveals a highly anisotropic strain dependence of Eg: The steepest gradient of Eg is along the direction in which ϵ and ϵ have opposite signs, as the case when a uniaxial stress is applied along the a lattice direction. In contrast, the strain induced by applying hydrostatic pressure corresponds to a trajectory that is almost parallel to the contour lines. The Poisson’s ratio of the anisotropic strain, ϵ/ϵ = −0.25, induced by a-axis uniaxial stress, was obtained using a fully relaxed vdW-DFT calculation, and it is indicated as the gray arrow in Fig. 1C. It requires less than a percent of ϵ to reach the WTI-STI phase boundary. We note that there is uncertainty in the DFT bandgap size for a given set of lattice constants. For example, several spectroscopy measurements reported Eg as low as 10 meV, which is significantly lower than the calculated 60-meV bandgap at zero strain (, ). Hence, the actual strain required to pass through Eg = 0 could be smaller (see Fig. 1D).

We use the bulk electrical transport to study the putative strain-tuned topological phase transition in ZrTe5. As the bulk energy banpan>dgap closes anpan>d reopens, the electrical tranpan>sport properties, such as resistivity, will fall anpan>d rise againpan>. The resistivity is determinpan>ed by the properties of quasiparticles near the chemical potential; hence, the strainpan> effect on resistivity will be larger if the dopinpan>g is lower so that the chemical potential is closer to the banpan>d edge. Therefore, we used the flux method to grow sinpan>gle crystals of pan> class="Chemical">ZrTe5, which is known to yield crystals closer to perfect chemical stoichiometry with ultralow carrier density (p-type, 1015 cm−3) (). The resistivity of ZrTe5 depends on both strain and temperature (see Figure 2). Figure 2C shows the resistivity as a function of temperature for several freestanding ZrTe5 single crystals before they were mounted on the strain apparatus. The insulating temperature dependence is consistent with other flux-grown crystals in the literature, indicating that, at base temperature, the chemical potential of our samples lies just slightly below the valence band maximum (, ).

Fig. 2

Temperature and strain dependence of resistivity of ZrTe5.

(A) Strain dependence of resistivity of ZrTe5 at T = 2 K for five samples S1 to S5. A clear minimum in resistivity can be seen for each sample. The resistivity is normalized by its minimum value ρmin, which varies between 1 and 16 milli-Ohm⋅cm. The x axis is the strain along the a lattice direction, which is estimated on the basis of the method described in Materials and Methods. (B) Resistivity versus strain for temperatures between 2 and 100 K. A clear minimum can be seen for the entire temperature range. (C) Resistivity versus temperature for three ZrTe5 crystals S1 to S3, as measured before gluing onto the three-piezo strain apparatus. (D) Three-piezostack apparatus used to deliver strain. (E) Quadratic coefficient Q obtained from fitting . The sensitivity of the response to strain shows a nonmonotonic temperature dependence, as discussed in the main text. Inset: Coefficient Q computed using Boltzmann transport equations (Materials and Methods). The main calculated features agree with the experimental data: A local minimum then maximum is seen with increasing temperature. The relative strength and temperature of these features depend on the carrier density input into the model (see Materials and Methods for more information).

(A) Strain dependence of resistivity of ZrTe5 at T = 2 K for five samples S1 to S5. A clear minpan>imum inpan> resistivity canpan> be seen for each sample. The resistivity is normalized by its minpan>imum value ρminpan>, which varies between 1 anpan>d 16 milli-Ohm⋅cm. The x axis is the strainpan> along the a lattice direction, which is estimapan> class="Chemical">ted on the basis of the method described in Materials and Methods. (B) Resistivity versus strain for temperatures between 2 and 100 K. A clear minimum can be seen for the entire temperature range. (C) Resistivity versus temperature for three ZrTe5 crystals S1 to S3, as measured before gluing onto the three-piezo strain apparatus. (D) Three-piezostack apparatus used to deliver strain. (E) Quadratic coefficient Q obtained from fitting . The sensitivity of the response to strain shows a nonmonotonic temperature dependence, as discussed in the main text. Inset: Coefficient Q computed using Boltzmann transport equations (Materials and Methods). The main calculated features agree with the experimental data: A local minimum then maximum is seen with increasing temperature. The relative strength and temperature of these features depend on the carrier density input into the model (see Materials and Methods for more information).

Uniaxial stress was applied along the a axis of the pan> class="Chemical">ZrTe5 crystals using a piezoelectric apparatus introduced by Hicks et al. (, ), as shown in Fig. 2D. Resistivity (ρ) and strain (ϵ) were measured along the a axis (Materials and Methods). The strain dependence of the resistivity, i.e., the elastoresistivity, is nonmonotonic at T = 2 K, as shown in Fig. 2A. For all samples measured, the resistivity shows a minimum at a critical strain ϵmin and increases as the sample is strained away from ϵmin. There is an uncertainty in determining the zero-strain state, i.e., the absolute value of ϵmin, due to the mismatch of thermal contraction between the sample and apparatus. Nevertheless, on the basis of a detailed analysis, we estimated that ϵmin < 0.12% (Materials and Methods). For all the data shown here, ϵ is measured from ϵmin. We note that, although the size of the strain response varies from sample to sample, the appearance of a resistivity minimum is a robust phenomenon seen in every case. It is well known that the resistivity of semiconductors can have a large linear response to strain due to its sensitivity to the position of band edges as a function of strain. Such a nonmonotonic strain dependence is very unusual but is precisely what is expected for the STI-WTI topological phase transition as described above.

The nonmonotonic elastoresistivity is quadratic in the vicinity of ϵmin. This is consistent with expectations based on the massive Dirac Hamiltonianpan>, which has been shownpan> to successfully describe the magnpan>eto-optical spectrum of pan> class="Chemical">ZrTe5 (, , ). In general, the k ∙ p Hamiltonian of a small bandgap semiconductor gives a dispersion , where m, which is half of the bandgap Eg = 2∣m∣, is analogous to the rest mass of a relativistic free particle, and vα and kα are the Fermi velocities and crystal momentum. Because the strain (ϵ, ϵ, and ϵ) induced by the uniaxial stress does not break the D2h point group symmetry of ZrTe5, the parameters of the Hamiltonian, including m, should vary linearly with strain. The band inversion process corresponds to the change in the sign of m as it is tuned through zero by strain. However, the physical observables that determine the resistivity, including bandgap, density of states (DOS), and velocity, do not depend on the sign of m and thus must vary as m2 to lowest order. As a result, the resistivity will also be parabolic in strain.

The linear elastoresistivity coefficient in nondegenerate semiconductors is proportional to the ratio of deformation potential (∂Eg/∂ϵ) to temperature (kBT) (). We derive a similar relation for the quadratic elastoresistivity coefficient measured in ZrTe5. At low temperature inpan> the quanpan>tum degenerate regime when kBT ≪ EF, the Fermi energy EF plays the same role as kBT. On the basis of dimensional anpan>alysis, the quadratic coefficient ∆ρ/ρ is determinpan>ed by m normalized by EFWe also performed a Boltzmanpan>n tranpan>sport calculation anpan>d obtainpan> the same expression. The calculation assumes the simplest situation where a fixed number of relativistic electrons are scattered from charged impurities (Materials anpan>d Methods). This expression agrees with the expectation that the sensitivity of resistivity to strainpan> decreases as the chemical potential moves away from the banpan>d edge. Usinpan>g the deformation potential, ∂m/∂ϵ, calculapan> class="Chemical">ted by DFT in the above formula, a fit of the quadratic coefficient yields a Fermi energy of 4 to 8 meV and a carrier density of 0.3 × 1015 to 2.4 × 1015cm−3. These values are comparable with angle-resolved photoemission spectroscopy and transport measurements of similar samples (, ).

We note that, in Fig. 2A, a slight deviation from perfectly quadratic behavior can be seen. Away from the critical strain, we do expect this deviation from quadratic (symmetric) behavior due to the higher-order strain dependence or the strain dependence of other band parameters such as vα, which we assumed to be a constant in the Boltzmann transport model. In addition, the Poisson ratio may also be different for large compressive and tensile strain states.

We also measured the elastoresistivity at higher temperatures, which shows quadratic behavior up to 100 K. The quadratic coefficient exhibits a nonmonotonic temperature dependence, as shown in Fig. 2E. This is a consequence of a crossover from the quantum degenerate to nondegenerate regime as temperature increases. By including the Fermi-Dirac distribution inpan> the Boltzmanpan>n tranpan>sport calculation, we are able to reproduce the nonmonotonic temperature dependence of the quadratic coefficient, as shownpan> inpan> the inpan>set of Fig. 2E. The overall qualitative agreement suggests that the tranpan>sport properties of pan> class="Chemical">ZrTe5 are well captured by the dynamics of massive Dirac fermions with a strain tunable mass across a wide temperature range.

If the bandgap is truly zero at ϵmin, then the band structure at that point is equivalent to a massless Dirac semimetal, which is knownpan> to host the pan> class="Disease">chiral anomaly when electric and magnetic fields are parallel. A manifestation of the chiral anomaly is NLMR, which has been observed in ZrTe5 previously (). This effect was initially considered in gapless Weyl or Dirac semimetals (, ), yet for a gapped Dirac semimetal, essentially the same mechanism could still apply in the semiclassical regime provided that Eg/EF << 1 (). In a gapped Dirac semimetal, electron helicity plays the same role as chirality in gapless semimetals. The helicity relaxation rate is proportional to (Eg/EF)2. As a result, we expect a suppression of NLMR (i.e., positive longitudinal magnetoconductivity) when the bandgap opens.

To investigate this, we measured the strain dependence of longitudinal magneto-transport (I ∥ B ∥ a; see Materials and Methods for field alignment). Figure 3 (A and B) shows the resistivity versus field for different strain states. An NLMR is clearly observable, and it disappears if the field is misaligned by more than 1° (fig. S6), consistent with the literature (, ). At ϵ = ϵmin (gray curves), the NLMR reaches its maximum, and it is progressively suppressed when the sample is strained away from this point. Two other samples were measured and show the same strain dependence (fig. S5). The magnetoresistance shows an upturn once the field surpasses the quantum limit (estimated to be 2 T for 1015 cm−3 carrier density) (, ). We notice that the upturnpan> is shifpan> class="Chemical">ted to higher fields on the STI side, which may be related to the unusual behavior of lowest Landau levels of topological insulators (). Nevertheless, we leave the study of high-field magnetoresistance for future works and focus on the weak field semiclassical regime (B < 0.5 T), in which the magnetoconductance Δσ = σ(B) − σ0 is positive except showing a small dip near zero field (Fig. 3C). The magnetoconductance was fitted with a quadratic field dependence, σ = σ0 + αB2 (see Materials and Methods for details of fitting). The quadratic coefficient α is proportional to the helicity relaxation time (), and it shows a 10-fold increase as ϵ approaches ϵmin (Fig. 3D). This diverging behavior is consistent with the suppression of helicity relaxation due to the vanishing of the bandgap at ϵ = ϵmin.

Fig. 3

Strain dependence of longitudinal magnetoresistance and magnetoconductance of ZrTe5 at T = 2 K.

NLMR for negative strains (A) and positive strains (B) measured relative to ϵmin, corresponding to the STI and WTI phases, respectively. The negative magnetoresistance is strongest at ϵmin, at which ZrTe5 is a gapless Dirac semimetal. Straining the crystal away from ϵmin suppresses the negative magnetoresistance. (C) Weak field positive magnetoconductance for several strain set points close to ϵmin. The conductance is fitted to the equation σ(B) = σ0 + αB2 (black solid curves), where α is a positive coefficient proportional to the helicity relaxation time. (D) Quadratic coefficient α as a function of strain measured relative to ϵmin.

NLMR for negative strains (A) and positive strains (B) measured relative to ϵmin, corresponding to the STI and WTI phases, respectively. The negative magnetoresistance is strongest at ϵmin, at which ZrTe5 is a gapless pan> class="Chemical">Dirac semimetal. Straining the crystal away from ϵmin suppresses the negative magnetoresistance. (C) Weak field positive magnetoconductance for several strain set points close to ϵmin. The conductance is fitted to the equation σ(B) = σ0 + αB2 (black solid curves), where α is a positive coefficient proportional to the helicity relaxation time. (D) Quadratic coefficient α as a function of strain measured relative to ϵmin.

DISCUSSION

Our extensive transport measurements and detailed data analysis have revealed an exceptionally delicate topological ground state of ZrTe5. The highly nonlinpan>ear elastoresistivity from a nonsymmetry breakinpan>g strainpan> is consistent with the signpan> swipan> class="Chemical">tching of the Dirac mass. This is further corroborated by the diverging strain dependence of positive longitudinal magnetoconductance near the transition point, which is a more profound consequence of the vanishing of the Dirac mass. In this sense, our study not only represents a new approach to characterize topological phase transitions but also demonstrates a method to precisely control phenomena associated with the chiral anomaly. We note that the applied strain does not break inversion symmetry. It is possible to apply an external electric field to exfoliated thin flakes to break inversion symmetry and induce the Weyl semimetal phase shown in Fig. 1A.

One defining signature that distinguishes STI from WTI is the existence of surface states on all crystal facets, which is not addressed in the current study. On the other hand, all of our transport data on n class="Chemical">ZrTe5 present strong evidence of an inpan> situ strainpan>-tuned topological phase transition inpan> this material. We note that our experimental setup for applyinpan>g strainpan> is compatible with photoemission and scanninpan>g tunnelinpan>g spectroscopy measurements. Future studies inpan>corporatinpan>g spectroscopic techniques capable of measurinpan>g surface states and band topology should be able to provide a comprehensive understandinpan>g of topological phase transitions.

MATERIALS AND METHODS

Material growth and sample preparation

Single-crystal n class="Chemical">ZrTe5 was grown with the flux method (). Zr slugs (99.9% pure, Alfa Aesar) and Te shot (99.9999% pure, Alfa Aesar) were loaded inpan>to a quartz ampule inpan> a Zr:Te ratio of 1:100. The ampule was warmed to 900°C inpan> 9 hours, kept at 900°C for 72 hours, and then cooled to 505°C inpan> 48 hours. To promote large crystal growth, the ampule was repetitively cooled to 440°C and warmed to 505°C. Last, the ampule was cooled to 460°C and decann class="Chemical">ted in a centrifuge at this temperature.

For electrical transport measurements, single crystals of typical dimensions 1.5 mm × 0.1 mm × 0.02 mm were sputtered with gold, and then 25-μm-diameter gold wires were placed on the crystals and adhered with silver paint. The resistance was measured with an SRS830 lock-in amplifier and an SRS CS580 current source. Given the needle-like nature of the single crystals, the resistance was measured along the n class="Species">a axis of the crystal.

Strain apparatus

Uniaxial stress was applied to single crystals using a homebuilt three-piezostack device (shown in fig. S1). Three piezoelectric actuators are aligned in parallel with each other. A U-shaped titanium block was glued to the outer two piezoelectric actuators, and a small titanium block was glued to the middle actuator, forming a small gap between these blocks. Applying a voltage to the outer piezostacks while applying an equal and opposite voltage to the middle piezostack will strain the piezostacks and change the gap size.

A crystal is glued across this apparatus gap. Tuning this gap with the piezostack voltage will apply uniaxial stress to the crystal. For a similar apparatus, Hicks et al. () showed that gluing only the bottom surface of the crystal to these plates can lead to strain gradients between the top and bottom surfaces of the crystal. These gradients can be suppressed by submerging the crystal in glue, as we did. Hicks et al. showed that the strain gradients are small when the ratio of t/LG (sample thickness to gap size) is small. For our measurements, this ratio is small, ranging from 0.02 to 0.08. We simulated the strainpan> distribution with finpan>ite element anpan>alysis. Our finpan>ite element anpan>alysis does show that there are still some small strainpan> gradients along the vertical axis of the crystal, mostly confinpan>ed to the bottom quarter of the crystal. The strainpan> along the pan> class="Species">a axis ϵ is equal to α∆L/L, where the change of the gap size ∆L/L is estimated by a strain gauge glued on the piezostacks. The constant α takes into account a strain relaxation effect, which is estimated by finite element analysis and is typically about 0.8. The strain along the b axis and c axis are determined by the Poisson’s ratio. The resistance-strain dependence identified in this work is a smoothly varying function. Because of this, these small strain gradients have minimal impact on our interpretation of the spatially average resistance.

Care was taken during the construction of the three-piezo apparatus to ensure fine alignment of the piezostacks, minimizing any stress in the secondary axes. First, a “scaffolding” piece was machined with indents the exact dimensions of the piezoelectric actuators and the titanium blocks. The actuators anpan>d blocks were placed inpan> these inpan>dents anpan>d then glued together while secured inpan> precise alignpan>ment. The scaffoldinpan>g block was then removed after the glue dried. Second, the middle pan> class="Chemical">titanium block and the outer titanium block were machined with a thin flexor plate connecting them. This flexor plate restricts motion between the blocks in any axis except the primary strain axis.

Strain was measured by a foil strain gauge glued to one of the piezostacks, measuring ϵpiezo. The displacement strain of the device was estimated as this strainpan> multiplied by the mechanpan>ical advanpan>tage of the apparatus, , where pan> class="Chemical">Lp is the length of the piezostack, LG is the length of the gap the sample is glued across, and is the displacement of the apparatus. Because of a low signal-to-noise ratio associated with the strain gauge measurement, we presented data plotted against piezostack voltage rather than plotted directly against the measured strain. We then calibrated this with a strain per volt calibration fitted from the strain measurement. We carefully minimized the amplitude of the voltage sweeps applied to the piezostacks, staying centered around ϵmin to minimize any hysteresis effect in the piezostacks.

Finite element analysis

We used the ANSYS Academic Research Mechanical 19.1 finite element analysis software to calculate the strain transmission in the primary axis and deformation in the secondary axis for all of our ZrTe5 samples. The model is shownpan> inpan> fig. S2. The stiffness tensor for pan> class="Chemical">ZrTe5 was taken from a previously calculated value by the Materials Project (). The sample is modeled as mounted to the strain apparatus in a puddle of epoxy. The Young’s modulus and Poisson’s ratio of the epoxy were given the same values as the work done by Hicks et al. (). For each crystal measured, the model took the exact crystal dimensions and gap size of the apparatus as inputs and computed an average relaxation constant α, defined as for , where ϵ is the strain delivered to the crystal. Our results are shown in table S1.

Experimental procedure

After cooling to 2 K, the sample was warmed by incremental temperature set points. In situ stress was applied to the crystal at each temperature set point by applying a triangle voltage waveform across the outer two piezo actuators and an equal and opposite sign waveform across the middle actuator. For each temperature set point, the voltage waveform was allowed to loop several times to inspect any hysteresis effects. The magnitude and offset of this waveform were adjusn class="Chemical">ted durinpan>g warminpan>g to stay centered around the resistanpan>ce minpan>imum. At temperatures 80 K anpan>d higher, it took anpan> inpan>creasinpan>g amount of compressive strainpan> to stay centered on the resistanpan>ce minpan>imum. At these temperatures, the samples often buckled as negative strainpan> was applied. This bucklinpan>g led to a large hysteresis developinpan>g inpan> the resistanpan>ce-strainpan> relation, anpan>d the experiment was terminpan>apan> class="Chemical">ted.

Calibration to zero-strain state

One of the advantages of the three-piezo apparatus compared to direct gluing to a single piezostack is that the thermal strain is minimized. This is because the large thermal expansion of the outer two piezostacks was compensated by the expanpan>sion of the middle piezostack. However, there was still a non-negligible thermal strainpan> resultinpan>g from the mismapan> class="Chemical">tch of thermal expansions between the crystal and the titanium pieces of the apparatus. Because titanium is known to have a smaller thermal expansion compared to most materials, it is expected that cooling the apparatus will impart a tensile strain to a mounted crystal. By tuning the controllable strain of the apparatus, this thermal strain can be compensated if a reference calibration is available.

A zero-strain calibration was constructed by measurinpan>g the resistance of crystals before gluinpan>g to the strainpan> apparatus. After measurinpan>g the zero-strainpan> resistance, crystals were glued to the apparatus and the resistance was measured while coolinpan>g from 300 to 2 K. The apparatus strainpan> could then be adjusted to tune the strained resistance to the calibration resistance, keeping the crystal in the zero-strain state. For most crystals, we were able to track the zero-strain state of the crystal down to about 60 K. Between 60 and 300 K, crystals always had a positive gauge factor (GF) (tensile strain increased resistance). This indicates that, in this temperature range, the zero-strain state resides at higher strains compared to ϵmin. Below 60 K, two things occurred that made tracking zero-strain state difficult. The first was that the resistance sensitivity to strain became weak—the GF approached zero. This indicates that the crystal either approaches or passes through ϵmin below 60 K. The second difficulty is that the resistance of the mounted crystal measured a slightly higher value than the calibration resistance for temperatures below 60 K. This occurred even when the strained crystal was tuned to ϵmin.

The higher resistance measured on the strain apparatus compared to the unstrained crystal can be attributed to a slight aginpan>g effect inpan> the crystals. Figure S3 (A anpan>d B) shows two pan> class="Chemical">ZrTe5 crystals, with the resistance versus temperature measured twice. The second measurement was performed after the crystals were left sitting in atmosphere for 18 days. Between measurements, the 2 K resistance increased 2.4 and 3.4% for each crystal. The difference in resistance between crystals is negligible above 60 K, the same approximate temperature below which we are unable to tune the strained resistance to the unstrained resistance. Because of this slight aging effect, we were unable to locate the zero-strain state of crystals glued to the apparatus at low temperatures. Figure S3C shows the resistance of a strained crystal to its unstrained resistance, measured 1 day apart. Even with only 1 day between measuring the unstrained crystal and the strained crystal, there is still a slight aging effect that prohibits tracking the zero-strain state.

To estimate the location of the zero-strain state, i.e., the absolute value of ϵmin, we performed the following experiment. Crystals were glued directly on the side wall of a single piezostack, as shown in fig. S4 (B and C). The piezostack has unusual highly anisotropic thermal expansion properties; it expands by about 0.1% along the polling direction and contracts along the transverse direction as it is cooled to liquid helium temperatures. Gluing crystals oriented parallel anpan>d pan> class="Disease">perpendicular to the piezostack polling axis imparts a very different strain during cooling, mimicking scenarios where samples were glued on substrates with different thermal expansion coefficients. Tuning the piezostack voltage at 2 K adds a much smaller tunable strain (~ 0.01 to 0.02%) on top of this thermal strain. Using this tunable strain, we were able to measure a linear elastoresistance. The slope of this linear response, defined as , measures the local derivative of the nonlinear resistivity versus strain curves. A positive or negative GF indicates which side of ϵmin the thermally strained crystal resides on. As seen in fig. S4, the parallel (perpendicular) orientations measure a positive (negative) GF at 2 K. This indicates the sensitivity of the thermal strain to sample preparation and allows us to make an estimate of ϵmin. On the basis of published data for the thermal expansion of similar piezostacks and ZrTe5, cooling to 2 K strains the perpendicular glued sample by about +0.08% (, ). The parallel glued sample is strained even more than this. We measured GF (2 K) = −73 for the perpendicular glued sample. This indicates that the parallel glued thermal strain is between −0.04 and −0.01% with respect to ϵmin, as calibrated by the quadratic response we measured for samples in this work. Combining this with our estimate for the thermal strain, we estimated that ϵmin is at most +0.12% at 2 K. With the DFT estimate of dEG/dε~6000 meV, this estimates an upper bound of the EG in the zero-strain state of 72 meV, which is consistent with most reported values of the EG.

Measurement of the longitudinal magnetoresistance

Longitudinal magnetoresistance was measured at strain set points for field parallel the current along the a axis. In addition to the results reporpan> class="Chemical">ted in Fig. 3 (A and B), two other crystals were measured (shown in fig. S5). For all crystals, we measured a positive magnetoconductance after a small dip in the magnetoconductance for very low fields. The small dip in longitudinal magnetoconductance near zero field is commonly observed in other materials that exhibit chiral anomaly. Its origin is not completely understood. Two possibilities are weak antilocalization effect and the classical Lorentz longitudinal magnetoresistance in anisotropic metals that saturates when ωcτ ~ 1. The positive magnetoconductance was suppressed for strains as low as 0.02% away from ϵmin for both compressive and tensile strains.

NLMR is only observed for magnetic fields finely aligned to the current, I ∥ B ∥ a. We aligned our apparatus such that the magnetic field is parallel to the n class="Species">a axis of the sample. The apparatus itself was machinpan>ed to withinpan> 20-μm precision so that the plane of the edge of apparatus can be very precisely aligned to the magnetic field (<1°). The primary opportunity for misalignment is the crystal beinpan>g misaligned withinpan> the plane of the apparatus. To minpan>imize misalignment along this axis, we used an optical microscope to inpan>spect and adjust the alignment of the crystal after it was placed inpan> glue on the apparatus. The glue adherinpan>g the crystal dried slowly enough to allow time for alignment adjustment under the optical microscope. We found that we can frequently align crystals withinpan> 1° of alignment usinpan>g this simple technique.

To verify that the crystal and apparatus were fully aligned with magnetic fields, we constructed a strainpan> apparatus on a Quanpan>tum Designpan> Dynpan>aCool sinpan>gle-pan> class="Disease">axis rotation puck (as shown in fig. S6A). We tuned ϵ to ϵmin, which we know from Fig. 3 will have the strongest NLMR response. We then measured longitudinal magnetoresistance for several angles between field and a axis, from −1.0° to +0.5° in increments of 0.25° (shown in fig. S6B). We found that the NLMR was only observed for alignments better than 1.0°, which is consistent with other results (). Because NLMR is not observed for misalignments 1° or greater even for ϵ = ϵmin, the observation of NLMR is evidence that our crystals are aligned to within 1° of accuracy or less.

Further details about the fitting of longitudinal magnetoconductance

Positive magnetoconductance was fitted to the equation σ(B) = σ0 + αB2, where α is a positive coefficient that is proportional to helicity relaxation time. To eliminpan>ate errors to the fit associapan> class="Chemical">ted with the dip of magnetoconductance near zero field, the data were fitted only for the magnetic field bounded between a lower bound and 0.5T, BLB <∣B∣ < 0.5T. Figure S7A shows representative data and fitted curves for BLB = 0.2T. The coefficient α as a function of strain is plotted in fig. S7B. The coefficient α peaks at ϵmin, as discussed in the main text. α is plotted for several choices of BLB, and the choice of BLB is found to not significantly influence the strain dependence of α. For the data shown in Fig. 3C, BLB = 0.2T, and the error bars are determined by varying the fitting range.

Boltzmann transport theory of massive Dirac semimetal

In the quantum degenerate regime, the conductivity of a 3D electron gas can be expressed aswhere kF is the Fermi wave vector determined by the carrier density and l is the mean free path. In ZrTe5, the carriers are most likely originpan>apan> class="Chemical">ted from the ionization of impurities, and the carrier density is determined by the number of impurities n. Close to zero gap, the ionization is practically insensitive to the modulation of bandgap due to strain. Therefore, we assumed that carrier density and kF are independent of the strain, and the strain dependence of conductivity is dominated by the change of mean free path. At low temperature, the mean free path is limited by scattering from the ionized impurity centers and can be expressed in terms of the impurity concentration n and the i transport cross-section σtranNear the Γ point, the band structure of ZrTe5 obeys a relativistic Dirac dispersion E2 = m2 + ℏ2k2v2. The scattering cross-section for this relativistic dispersion by a Coulomb potential obeys the relativistic Mott formulawhere θ0 is the small angle cutoff due to the screening of the Coulomb interaction, and is the dimensionless coupling constant, where ϵ is the static dielectric constant. β = v2ℏ2k2/(m2 + v2ℏ2k2) follows from applied to the dispersion E2 = m2 + ℏ2k2v2.

Because , it follows thatwhich is the result in the main text. The carrier density for each sample measured based on this computation is estiman class="Chemical">ted inpan> table S1.

Temperature dependence of resistivity/gap relation

A semiclassical Boltzmann transport model was used to compute the temperature dependence of the resistivity sensitivity to strain. The Dirac band at the Γ poinpan>t is assumed to dominpan>ate conduction, and the conductivity can be expressed aswhere The scatterinpan>g time τ can be expressed as , where l(E) is the energy-dependent scatterinpan>g length mentioned above, inpan>versely proportional to σtran. We assume that scatterinpan>g off of charged impurities is the dominpan>ant scatterinpan>g mechanism inpan> this simplistic model. Given the Dirac dispersion of E2 = m2 + ℏ2k2v2, the DOS can be written as

Combining these, the conductivity may be written as

Given the chemical potential μ as a function of temperature, can be compun class="Chemical">ted for a range of temperatures, and a quadratic relation between 1/σ and m can be fitn class="Chemical">ted.

All that is left in the model is to compute μ(T). This can easily be done by charge conservation. The impurity doping n is defined as n – p, where n and p are the electron and hole densities. n(T) and p(T) can be compun class="Chemical">ted as

Given n as the input parameter, μ(T) can be solved numerically by enforcing n = n(T) − p(T) for electron doping or n = p(T) − n(T) for hole doping. The only other input parameter for our computational model is the velocity v that appears in the n class="Chemical">Dirac dispersion. We used v, v, v = 1.7,5.2,2.2 × 105m/s, as measured by SdH oscillations ().

DFT calculations

The electronic structure and energy gap of ZrTe5 under strainpan> were calculapan> class="Chemical">ted by the Quantum Espresso package () based on DFT. The Perdew-Burke-Ernzerhof exchange-correlation functional (PBE-GGA) () with spin-orbit coupling and projector augmented wave () method were used. The unstrained lattice parameters (a = 3.97976 Å, c = 13.6762 Å) were determined by experimental data at 10 K (), and an 11 × 11 grid in the parameter ranges of 1.0a to 1.02a and 0.99c to 1.01c was considered for studying the gap behavior with lattice variation. The raw data of the zone-center gap are shown in fig. S8A, and the interpolated data with 2D cubic splines are shown in Fig. 1C. An 8 × 8 × 4 momentum grid was used in the self-consistent calculation, and the kinetic energy cutoff and convergence criterion were set to 30 and 10−7 rydberg, respectively. The DFT band structures for ZrTe5 in different strained states are shown in fig. S9. The labeling of the high-symmetry k-points was based on the Brillouin zone of the primitive unit cell.

The topological Z2 indices of different strained structures were also computed to identify their topological nature. With inpan>put from the electronic structure calculations of Quanpan>tum Espresso, the maximally localized Wanpan>nier functions were first compupan> class="Chemical">ted using the Wannier90 package (), which, in turn, allowed the determination of Z2 indices by tracking hybrid Wannier charge centers using the WannierTools package (). As shown in fig. S8B, the bottom right of the phase diagram is an STI with Z2 indices (1;110), and the upper left is a WTI [with Z2 indices (0;110)]. Phase transition between the STI and WTI states was directly controlled by closing the energy gap at the Brillouin zone center.

Additional structure relaxation calculations were performed with the vdW-DFT (, ), corrected usinpan>g the exchanpan>ge-hole dipole moment model (). The vdW-DFT fully relaxed lattice parameters of layer-structured pan> class="Chemical">ZrTe5 are within 1% error compared to the unstrained experimental data. To determine the Poisson ratio, conventional cells of different fixed lattice parameters along the a axis were considered, while the b and c lattice parameters were allowed to evolve freely in the structure relaxation calculations. The results are shown in fig. S8C.

We have computed the DOS around the Γ poinpan>t for different strainpan>ed structures shownpan> inpan> Fig. 1D. Specifically, we made a k-grid of 21 × 21 × 11 poinpan>ts around the Γ poinpan>t (±0.01 b1, ±0.01 b2, ±0.01 b3) anpan>d counpan> class="Chemical">ted the states within Efermi ± 𝝙, where 𝝙 ranges from 6 to 10 meV (corresponding to 70 to 120 K) to plot the DOS for different strained states. As shown in fig. S8D, the resulting DOS around the Γ point has a fairly parabolic strain dependence. We also performed a third-order polynomial fit y(x) = a3x3 + a2x2 + a1x + a0 for the curves [where y stands for the DOS and x stands for 𝝙a] and found that, within the range of interests, the anharmonic term a3x3 is roughly 1000 times smaller than a2x2. Therefore, to leading order, a parabolic strain dependence of DOS near the Γ point remains a valid assumption.