Abstract
The Hall coefficient R_{H} of Sr_{2}RuO_{4} exhibits a nonmonotonic temperature dependence with two sign reversals. We show that this puzzling behavior is the signature of two crossovers, which are key to the physics of this material. The increase of R_{H} and the first sign change upon cooling are associated with a crossover into a regime of coherent quasiparticles with strong orbital differentiation of the inelastic scattering rates. The eventual decrease and the second sign change at lower temperature are driven by the crossover from inelastic to impuritydominated scattering. This qualitative picture is supported by quantitative calculations of R_{H}(T) using the Boltzmann transport theory in combination with dynamical meanfield theory, taking into account the effect of spin–orbit coupling. Our insights shed new light on the temperature dependence of the Hall coefficient in materials with strong orbital differentiation, as observed in Hund’s metals.
Introduction
Measuring the Hall coefficient R_{H} is a standard way of characterizing charge carriers in quantum materials. For free carriers of a single type, the Hall coefficient R_{H} is simply given by the inverse of the density of carriers n and their charge e. In principle, R_{H} is negative for an electronlike Fermi surface (FS) and positive for a holelike FS, respectively. Sign changes of R_{H} can occur, for example, if the FS evolves from an electronlike to a holelike one with temperature.^{1} However, in complex materials with a FS composed of multiple sheets, interpreting R_{H} can be more complicated and also provides richer information when both electronlike and holelike carriers are present simultaneously. For instance, in the case of one holelike and one electronlike FS sheet, the corresponding Hall coefficient is given by an average of R_{H,e} <0 and R_{H,h} >0:
weighted by the squares of the individual hole and electron conductivities, σ_{h} and σ_{e}, respectively. Hence, the ratio of scattering rates between the two types of carriers enters in a key manner to determine both the overall sign and magnitude of the Hall coefficient.
The 4d transition metal oxide Sr_{2}RuO_{4} is such a complex material: with lowenergy bands built out of three Rut_{2g} orbitals (d_{xy}, d_{yz}, d_{xz}) hybridized with O–2p states, it has a FS comprising two electronlike sheets, β and γ, and one hole pocket, α.^{2,3,4} Indeed, experiments^{5,6,7} have observed a particularly intriguing temperature dependence of R_{H} in Sr_{2}RuO_{4}, as depicted in Fig. 1. R_{H} increases from a negative value of about −1 × 10^{–10} m^{3} C^{−1} at low temperatures (values between −1.37 × 10^{−10} and −0.7 × 10^{−10} m^{3} C^{−1} for T → 0 have been reported,^{5,6,7,8}) exhibit a sign change at T_{1} = 30 K (in the cleanest samples), reaches a positive maximum at about 80 K, changes sign a second time around T_{2} = 120 K, and eventually saturates to a slightly negative value for T > 200 K.
Shirakawa et al.^{5} suggested early on that the rich temperature dependence of R_{H} in Sr_{2}RuO_{4} points to the multicarrier nature of this material. This conclusion, reached by considering a Drude model with two types of carriers, was later refined in several works^{9,10,11} using Boltzmann transport theory calculations for tightbinding models assuming scattering rates 1/τ_{ν} = A_{ν} + B_{ν}T^{2} for the different FS sheets ν = {α, β, γ}, with adjustable parameters A_{ν} and B_{ν}. The overall takehome message of these phenomenological models is that R_{H} is highly sensitive to the precise details of the FS sheets and also to the temperature and sheet dependence of the scattering rates.
Another remarkable experimental finding provides insight in interpreting the temperature dependence of R_{H}:^{7} adding small amounts of Al impurities has a drastic impact on the intermediate temperature regime such that R_{H} no longer turns positive and instead increases monotonically from the lowT to the highT limit, as indicated by dotted lines in Fig. 1. Arguably, the similarity of the lowT values of R_{H} for different impurity concentrations provides evidence that the elasticscattering regime has been reached where R_{H} is mainly determined by FS properties (see also ref. ^{6}). In contrast, the temperature dependence itself must be due to inelastic scattering, possibly associated with electronic correlations.^{7}
In this work, we address this rich temperature dependence of R_{H} in Sr_{2}RuO_{4} and provide a clear interpretation of its physical meaning. We show that the two sign changes of R_{H}(T) in clean samples are the signatures of two important crossovers in the physics of this material. The increase of R_{H} upon cooling from high temperature signals the gradual formation of coherent quasiparticles, which is associated with a strong temperature dependence of the ratio of inelastic scattering rates between the xy and xz/yz orbitals. At low temperatures the decrease of R_{H} is due to the crossover from inelastic to impuritydominated scattering. These qualitative insights have relevance to a wide class of materials with orbital differentiation.
Our qualitative picture is supported by a quantitative calculation of R_{H}(T) using Boltzmann transport theory in combination with dynamical meanfield theory (DMFT),^{12} taking into account the electronic structure of the material. The spin–orbit coupling (SOC) is found to play a key role,^{13,14} because it has a strong influence on the shape of the FS and also controls the manner in which the scattering rates associated with the different orbitals combine into kdependent quasiparticle scattering rates at a given point on the FS.^{13,14,15}
Results
Dependence on scattering rate ratios
The orbital dependence of scattering rates is crucial for the understanding of the Hall effect. Therefore, we introduce a localized basis set of t_{2g}like orbitals χ_{m}〉 with basis functions labeled as m = {xy, xz, yz}, using maximally localized Wannier orbitals^{16,17} constructed from the Kohn–Sham eigenbasis of a nonSOC density functional theory (DFT) calculation. We treat SOC by adding an atomic SOC term for the t_{2g} subspace with an effective strength of λ = 200 meV, which takes already the correlation enhancement of the SOC by a factor of about two into account.^{15,18,19,20} We assign scattering rates η_{xy}, η_{xz}, and η_{yz} (due to crystal symmetries η_{xz} = η_{yz}) to each orbital, irrespective of the microscopic details of the underlying scattering mechanisms, which will be addressed at a later stage. Then, these scattering rates are converted into kdependent scattering rates for each band ν:
The overlap elements 〈χ_{m}(k)ψ_{ν}(k)〉^{2} correspond to the orbital character of the eigenstates ψ_{ν}(k)〉 of the Hamiltonian at a given momentum k. The orbital character for points on the FS is shown in the inset of Fig. 2. With the Hamiltonian and the scattering rates η_{ν}(k), we calculate R_{H} within Boltzmann transport theory using the BoltzTraP2 package.^{21,22} Further details on Eq. (2), the Hamiltonian construction and the transport calculations, are provided in the Methods section.
In the Boltzmann transport theory, R_{H} only depends on the scattering rates through their ratio ξ = η_{xy}/η_{xz/yz} and not through their absolute magnitude; a point we verified in our calculations. This also implies that within the constant isotropic scattering rate approximation, that is, ξ = 1, the full temperature dependence of R_{H} cannot be explained. The calculated R_{H} as a function of the scattering rate ratio ξ is displayed in Fig. 2. Without SOC R_{H} remains negative for all values of ξ and approaches zero as \(\xi \gg 1\). In this limit the γ sheet drops out and the contributions of the holelike α sheet and electronlike β sheet compensate each other. This means that it is not possible to explain the positive value of R_{H} observed experimentally in clean samples for T_{1} < T < T_{2} (Fig. 1) without taking SOC into account. With SOC we observe a very different behavior of R_{H}(ξ); it turns from negative to positive at \(\xi \simeq 2.6\). This is a result of two effects^{13,14,15} (see Fig. 2, inset): First, SOC changes the shape and size of the FS sheets, and second, it induces a mixing between different orbital characters, which varies for each point on the FS. Thus, the manner in which the scattering rates associated with the different orbitals combine into kdependent quasiparticle scattering rates (Eq. (2) is controlled by the SOC. From the calculated dependence of R_{H}(ξ) in the presence of SOC, we deduce that agreement with experiments would require ξ to be smaller than 2.6 at high temperatures, increase above this value at ~T_{2}, and then decrease again to reach a value close to unity at low temperatures.
Inelastic electron–electron scattering
We turn now to microscopic calculations by first considering inelastic electron–electron scattering ratios calculated with DMFT (see Methods). These calculations consider the t_{2g} subspace of states with Hubbard–Kanamori interactions of U = 2.3 eV and J = 0.4 eV.^{23} The calculated ξ(T) from inelastic scattering only is displayed in Fig. 3a. In agreement with previous studies,^{23,24} we find that the xy orbital is less coherent than xz/yz at all temperatures and η_{xy} > η_{xz/yz}. In Sr_{2}RuO_{4} the crossover from the lowT coherent Fermi liquid regime with \(\eta \sim T^2\) to an incoherent regime with a quasilinear temperature dependence of the scattering rate is well documented^{23,25} and also manifested in deviations of the resistivity from a lowtemperature quadratic behavior to a linear one.^{26} Importantly, this coherencetoincoherence crossover as well as the corresponding coherence scales are strongly orbital dependent. When approaching the Fermi liquid regime (T_{FL} ≈ 25 K^{26,27,28}) the scattering rate ratio reaches a value as large as \(\xi ^{\mathrm{FL}}\sim 3\) (Fig. 3a), but decreases rapidly upon heating with ξ = 1.8 at 300 K. We do not find a substantial change for even higher temperatures; at 500 K the scattering rate ratio is ξ = 1.6.
Connecting these results to the discussion of Fig. 2 above, the temperature dependence of ξ directly translates into that of R_{H}, as shown in Fig. 3b. Like in experiments, R_{H} is negative at high temperatures, but when the temperature is lowered it increases and crosses zero at 110 K. This demonstrates that electronic correlations are indeed able to turn R_{H} positive and suggests the following physical picture: the electronic transport in Sr_{2}RuO_{4} crosses from a regime governed by incoherent electrons at high temperatures, connected to a weaker orbital differentiation of scattering rates and a negative R_{H}, over to a coherent Fermi liquid regime, with a stronger orbital differentiation and positive R_{H}. The resulting sign change at 110 K can be seen as a direct consequence of this coherencetoincoherence crossover. We emphasize that this sign change is only observed when SOC is taken into account. Without SOC R_{H} is purely negative and shows only a weak temperature dependence (Fig. 3b, dashed line).
When moving along the FS from Γ–M (θ = 0°) to Γ−X (θ = 45°), the mixing of the orbital character induced by SOC (Eq. (2)) leads to angulardependent scattering rates η_{ν}(θ) (Fig. 3c). At θ = 0° the ratio of scattering rates between the γ and β sheets is large, because these bands still have mainly xy and xz/yz character, respectively (Fig. 2, inset). As expected from Fig. 3a, this sheet dependence decreases with increasing temperature. On the other hand, at θ = 45° the ratio is small, due to a very similar orbital composition of the γ and β sheets. The α pocket (being almost entirely xz/yz) has the lowest scattering rate and turns R_{H} positive when ξ becomes large enough at low temperatures. To shed more light on the interplay of the individual FS sheets, we can phenomenologically assign constant scattering rates to each FS sheet, as shown in Fig. 3d. We see that for R_{H} to be positive a necessary condition is η_{β} > η_{α}. This again highlights the importance of SOC, because without SOC the α and β sheets have entirely xz/yz orbital character, and thus η_{α} = η_{β}. Should one make this assumption also in the presence of SOC, it would not result in R_{H} > 0 for any ratio η_{γ}/η_{α} (Fig. 3d, dashed line).
Impuritydominated scattering
Considering inelastic scattering only would yield a positive R_{H} at even lower temperatures deep in the Fermi liquid regime. However, at such low temperatures elastic scattering is expected to dominate over inelastic scattering. The extracted DMFT scattering rates at 29 K with 5.5 meV for the xy and 1.9 meV for the xz/yz orbitals are of the order of the impurity scattering for “clean” samples with residual resistivities of \(\sim {\hskip 1pt} 0.5\,{\mathrm{\mu}}\Omega \,{\mathrm{cm}}\). Therefore, we add a constant elastic scattering η^{el.} to the orbitaldependent inelastic scattering \(\eta _m^{{\mathrm{inel}}{\mathrm{.}}}\). This elastic term is assumed to be isotropic: \(\eta _{xy}^{{\mathrm{el}}{\mathrm{.}}} = \eta _{xz/yz}^{{\mathrm{el}}{\mathrm{.}}}\). The resulting temperature dependence of R_{H} for values of η^{el.} ranging from 0.1 to 10 meV is shown in Fig. 4. The dashed lines are calculated with the Fermi liquid form \(\eta _m^{{\mathrm{inel}}{\mathrm{.}}} = A_mT^2\) and parameters A_{m} determined from the calculated inelastic scattering rates at 29 K.
For small enough η^{el.} we observe a second zero crossing of R_{H}(T) and a regime with R_{H} < 0 at low temperatures, which is consistent with R_{H}(T) depicted in Fig. 1. For T → 0 the fully elasticscattering regime is reached, and thus R_{H} is not influenced by the magnitude of the (isotropic) scattering rate, but rather by the shape of the FS only. This regime corresponds to ξ = 1 in Fig. 2, for which we obtain R_{H} = −0.94 × 10^{−10} m^{3} C^{−1}, in good quantitative agreement with experiments.^{5,6,7,8} With increasing temperature the influence of elastic scattering fades away and the precise interplay with inelastic scattering shapes the overall temperature dependence of R_{H}. Hence, we see that also the lowtemperature zero crossing has a simple physical interpretation: it signals the crossover between the regime dominated by elastic scattering at low temperatures and the regime dominated by inelastic scattering at higher temperatures. Matching the two terms in the scattering rate, a simple estimate of the corresponding crossover scale is \(T_1\sim \sqrt {\eta ^{{\mathrm{el}}{\mathrm{.}}}/A_{xy}} \sim \sqrt {\eta ^{{\mathrm{el}}{\mathrm{.}}}T_{{\mathrm{FL}}}}\). This scale obviously depends on the elasticscattering rate, and coincides approximately with the Fermi liquid coherence scale T_{FL} only for the cleanest samples reported in which \(\eta ^{{\mathrm{el}}{\mathrm{.}}}\sim T_{{\mathrm{FL}}}\). For even cleaner samples we predict T_{1} < T_{FL}.
On the contrary, for larger η^{el.} we find that R_{H}(T) ceases to exhibit any zero crossing and is negative in the whole temperature range. Only in very clean samples can the inelastic scattering rate sufficiently exceed the elastic one for the sign changes of R_{H} to occur. This is further substantiated by experimental Hall measurements for samples where the residual resistivity was altered by introducing different amounts of Al impurities, cf. the dependence of R_{H} on η^{el.} in Fig. 4 and the inset with experimental data from ref. ^{7}
In the highT limit, we obtain a value of R_{H}, which is more negative and temperature dependent than the one reported in experiments.^{5,6,7} Within the Boltzmann transport theory this would imply that a larger ratio η_{xy}/η_{xz/yz} is needed. Likewise, resistivities are significantly underestimated in DMFT transport calculations for T > 300 K in this material.^{24} A possible explanation is that other sources of inelastic scattering, for example, electron–phonon scattering, could play an important role in the highT regime. We emphasize, however, that all experimental evidence points towards negligible magnetic contribution (due to processes like skew scattering) and a standard orbitaldominated Hall effect in Sr_{2}RuO_{4}.^{6,8,10}
Discussion
In summary, our quantitative calculations and qualitative interpretations explain the highly unusual temperature dependence of the Hall coefficient of Sr_{2}RuO_{4}. The highT sign change of R_{H}(T) in clean samples is the direct consequence of the crossover from a highT incoherent regime to a coherent regime with orbital differentiation. The orbital composition of each quasiparticle state on the FS, as well as the distinct scattering rates of the different orbitals, is crucial to this phenomenon and are properly captured by DMFT. This is in line with recent insights from angleresolved photoemission spectroscopy.^{15} In turn, the lowT sign change is due to the crossover from inelastic to impuritydominated scattering, which is further substantiated by comparing our results to experimental data on samples with a higher impurity concentration. Because it directly affects the shape of the FS sheets and strongly mixes their orbital character, SOC is found to be essential in explaining R_{H}(T).
Orbital differentiation is actually a general feature common to Hund’s metals,^{23,29,30,31,32} a broad class of materials in which the electronic correlations are governed by the Hund’s coupling, comprising for example transition metal oxides of the 4d series as well as ironbased superconductors.^{33,34,35,36} We note that a nonmonotonic temperature dependence of the Hall coefficient has also been reported for Sr_{3}Ru_{2}O_{7}.^{37} Beyond ruthenates, LiFeAs and FeSe are two compounds without FS reconstruction due to longrange magnetic order, which display striking similarities to Sr_{2}RuO_{4} in many regards. The FS of these superconductors is also composed of multiple electron and holelike sheets with distinct orbital composition and strong orbital differentiation.^{31,32} Indeed, the Hall coefficient of LiFeAs has a strong temperature dependence^{38} and that of FeSe displays two sign changes in the tetragonal phase.^{39,40} These examples show that strongly correlated materials with multiple FS sheets of different or mixed orbital character and a orbitaldifferentiated coherencetoincoherence crossover are expected to show a pronounced temperature dependence of the Hall coefficient. Sign changes then emerge in materials with balanced electron and holelike contributions. These observations point to a wide relevance of our findings beyond the specific case of Sr_{2}RuO_{4}.
Methods
Hamiltonian and SOC
We use a maximally localized Wannier function construction^{16,17} to obtain an effective lowenergy Hamiltonian for the three t_{2g}like orbitals centered on the Ru atoms. This construction is based on a nonSOC DFT calculation, using the software packages WIEN2k^{41} with GGAPBE,^{42} wien2wannier,^{43} and wannier90.^{44} We incorporate the SOC as an additional local term, where we neglect the coupling to e_{g} orbitals, as these are well separated in energy. It has been shown that electronic correlations lead to an effective enhancement of the SOC in Sr_{2}RuO_{4} by nearly a factor of two.^{18,19,20} As the corresponding offdiagonal elements of the selfenergy (in the orbital basis) are approximately frequency independent,^{20} we model the effect of correlations by using a static effective SOC strength of λ = 200 meV, instead of the DFT value of about λ^{DFT} = 100 meV. This is crucial to obtain precise agreement with the FS recently measured with photoemission experiments.^{15} We point out that even when using λ^{DFT} we find the same qualitative conclusions for the sign changes of R_{H}. We refer to ref. ^{15} for further details on the DFT calculation and the Hamiltonian construction.
Dynamical meanfield theory
For the DMFT calculations we use the TRIQS library^{45} in combination with the TRIQS/DFTTools^{46} package and the TRIQS/CTHYB^{47} impurity solver (3.85 × 10^{9} measurements). Due to the fermionic sign problem of the TRIQS/CTHYB solver, low temperatures are only accessible without SOC. However, in Sr_{2}RuO_{4} the diagonal parts of the selfenergy are, to a good approximation, unchanged upon the inclusion of SOC.^{20,48} Therefore, we perform oneshot DMFT calculation using the Hamiltonian without SOC and added Hubbard–Kanamori interactions (including spinflip and pairhopping terms) with U = 2.3 eV and J = 0.4 eV from cRPA.^{23} Inelastic scattering rates η_{m} are extracted from nonSOC selfenergies Σ_{m}(iω_{n}) by fitting a polynomial of 4th order to the lowest 6 Matsubara points and extrapolating Im[Σ_{m}(iω_{n} → 0)], a procedure used in ref. ^{23} We calculate the standard deviation with nine consecutive DMFT iterations to obtain error bars for the inelastic scattering rate ratios. Note that Im[Σ_{m}(iω_{n})] is diagonal in the orbitals m, and thus η_{mm′} = 0 for m ≠ m′.
Transport
We calculate R_{H} using Boltzmann transport theory as implemented in the BoltzTraP2 package and described in the corresponding refs. ^{21,22} We use a 46 × 46 × 46 input kgrid, which is interpolated on a five times denser grid with BoltzTraP2. From Eq. (2) we obtain a scattering rate for each band and k point, which we set in BoltzTraP2 as scattering_model. Offdiagonal elements of the scattering rates, η_{νν′}(k) and η_{mm′}, are not captured by Eq. (2). In the case of Sr_{2}RuO_{4} the scattering rates in the orbital basis η_{mm′} are indeed diagonal (see above). The offdiagonal elements of η_{νν′}(k) and possible interband transitions are not considered in BoltzTraP2, but we verified with Kubo transport calculations (TRIQS/DFTTools^{46}) that these are negligible for the ordinary conductivity σ_{xx}. We also calculated R_{H} with the Kubo formula without SOC using the DMFT spectral functions A(k, ω) and neglecting interband transitions.^{1} The results at T = 464.2, 232.1 and 116.0 K differ from Boltzmann transport theory by <0.025 × 10^{−10} m^{3} C^{−1}, that is, <10%. Boltzmann transport theory results are accurate because the relatively sharp (resilient) quasiparticle peaks persist up to high temperatures,^{1,23,49,50} which was recently discussed for other ruthenate compounds,^{51} too. We note that recently interest has been devoted to the theoretical descriptions of the Hall effect in strongly correlated systems beyond Boltzmann transport theory.^{52,53,54}
Data availability
All data generated and analyzed during this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We gratefully acknowledge useful discussions with Gabriel Kotliar, Andrew Mackenzie, Hugo Strand, Andrea Damascelli, Reza Nourafkan, AndréMarie Tremblay. J.M. is supported by the Slovenian Research Agency (ARRS) under Program P10044. M.A. acknowledges support from the Austrian Science Fund (FWF), project Y746, and NAWI Graz. This work was supported in part by the European Research Council grant ERC319286QMAC. The Flatiron Institute is a division of the Simons Foundation.
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M.Z. performed all calculations and the results were analyzed by M.Z. and A.G. All authors discussed and interpreted the results at different stages. The whole project was initiated by A.G. The manuscript was written by M.Z. with the help of all authors.
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Correspondence to Manuel Zingl.
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