Deviation from the Fermi-Liquid Transport Behavior in the Vicinity of a Van Hove Singularity
František Herman, Jonathan Buhmann, Mark H Fischer, Manfred Sigrist
DDeviation from the Fermi-Liquid Transport Behavior in the Vicinity of a Van Hove Singularity
František Herman, Jonathan Buhmann, Mark H Fischer,
2, 1 and Manfred Sigrist Institute for Theoretical Physics, ETH Zurich, 8093 Zurich, Switzerland Department of Physics, University of Zurich, 8057 Zurich, Switzerland
Recent experiments revealed non-Fermi-liquid resistivity in the unconventional superconductor Sr RuO when strain pushes one of the Fermi surfaces close to a van Hove singularity. The origin of this behavior andwhether it can be understood from a picture of well defined quasiparticles is unclear. We employ a Boltzmanntransport analysis beyond the single relaxation-time approximation based on a single band which undergoes aLifshitz transition, where the Fermi surface crosses a van Hove singularity, either due to uni-axial or epitaxialstrain. First analytically investigating impurity scattering, we clarify the role of the diverging density of statestogether with the locally flat band at the point of the Lifshitz transition. Additionally including electron-electronscattering numerically, we find good qualitative agreement with resistivity measurements on uni-axially strainedSr RuO , including the temperature scaling and the temperature dependence of the resistivity peak. Our resultsimply that even close to the Lifshitz transition, a description starting from well-defined quasiparticles holds. Totest the validity of Boltzmann transport theory near a van Hove singularity, we provide further experimentallyaccessible parameters, such as thermal transport, the Seebeck coefficient, and Hall resistivity and comparedifferent strain scenarios. I. INTRODUCTION
Fermi liquid theory, which establishes a one-to-one map-ping of electrons to well-defined quasiparticles, is a basis ofour understanding of metals. Its validity as well as its break-down are often characterized through transport properties. Inparticular, the quadratic temperature dependence of the resis-tivity ρ = ρ + AT due to electron-electron scattering andthe linear-in- T Seebeck coefficient Q are hallmarks of a Fermiliquid. Despite the theory’s great success in describing mostmetals, some classes of systems are known to violate these ex-pectations, most notably interacting fermions in one dimensionand systems close to a quantum critical point.Bringing the Fermi surface close to a van Hove singular-ity (vHS), i.e., a point of diverging density of states, providesanother example, where non-Fermi-liquid behavior can be ob-served. Whether such behavior is associated with a breakdownof Fermi liquid theory and the disappearance of well-definedquasiparticles is, however, not well established. Indeed, Buh-mann showed how non-generic transport behavior can beobserved within a picture of well-defined quasiparticles sub-ject to electron-electron (Umklapp) scattering close to a vHS.Experimentally, modifying the Fermi energy through dop-ing and thus moving the Fermi surface close to a vHS, on theone hand, is straight-forward. However, this introduces dis-order into the system, making comprehensive transport stud-ies impossible. Fermi surface engineering through tensile orcompressive strain, on the other hand, provides a non-invasivemethod for tuning the electronic structure of a material .Indeed, recent experiments have demonstrated both routes onthe single-layer perovskite Sr RuO , which at low temperatureexhibits almost perfect Fermi-liquid behavior below T ≈ K before entering a superconducting state at T c ≈ . K. Inter-estingly, the so-called γ band stemming mostly from Ru d xy orbitals, nearly touches the Brillouin zone (BZ) boundary,where the vHS is located and is, therefore, most interestingin connection with Fermi surface tuning.Barber et al. showed that uni-axial stress can be used as atuning knob for the normal state resistivity. Here, DFT calcu- lations indicate that the γ Fermi surface undergoes a Lifshitztransition [see Fig. 1 b)] at a critical stress that in resistiv-ity measurements coincides with a pronounced peak at lowtemperatures and T -linear scaling above. An alternative routefor Fermi surface engineering was demonstrated by Burganov et al. , who epitaxially grew thin films on lattice-mismatchedsubstrates. The resulting strain leads to a redistribution of elec-trons within the t g manifold effectively ‘doping’ the γ band.This effective doping can induce a Lifshitz transition as wellwith associated crossing of a vHS as indicated in Fig. 1 a) anddeviations from T behavior were observed in the resistivityon the samples close to the transition. Having access to theexposed surface, this setup additionally allows for characteri-zation of the electronic structure using STM and ARPES.While previous work on the effect of a Lifshitz transitionon the transport properties used approximations to theelectron-electron scattering within a Green’s function method,or through weighted scattering probabilities, our approach isbased on an analysis (both analytic and numerical) of the gen-eral transport coefficients within a Boltzmann-equation ap-proach including effects of Umklapp scattering. The aim ofour work is then twofold: First, our numerical approach tak- FIG. 1. Fermi surface tuned through the vHS at ( π, ) or ( , π ) by a) adjusting the chemical potential (‘doping’) and b) applyinguniaxial stress. The dark areas [white diamond] denote regions, whereUmklapp scattering by ( π, ) or ( , π ) [ ( π, π ) ] are possible. a r X i v : . [ c ond - m a t . s t r- e l ] M a r ing both impurity scattering and interaction effects includingUmklapp scattering into account allows for a careful compari-son to existing experimental results of electrical transport uponFermi surface tuning. The observed agreement with experi-ment establishes that well-defined quasi-particles are indeedcapable of capturing all observed trends. Second, we pre-dict further transport signatures near the vHS, such as thermaltransport, Seebeck coefficient, Hall effect, a violation of theWiedemann-Franz law, and the Kadowaki-Woods ratio. Here,we compare the two scenarios of crossing the vHS at all fourboundaries of the Brillouin zone and the Lifshitz transitiononly at two points in the direction parallel or orthogonal to anapplied electrical field, see Figs. 1 a) and b). These predictionsallow for additional comparison of the vHS scenario within apicture of well-defined quasiparticles to experiment.In the following we first introduce the general Boltzmannformalism and model-specific details for both types of scat-terings. An analytical discussion of electrical transport forimpurity scattering which is mainly relevant for the low tem-perature regime explains some of the main features connectedwith the crossing of the Fermi surface through the vHS. Thenwe will turn to the numerical discussion and examine the differ-ent transport properties showing how deviations from standardFermi liquid behavior occurs in connection with the Lifshitztransitions within both scenarios of Fermi surface tuning. Aswe can reproduce many of the experimental findings within ourcalculation we conclude that the non-Fermi liquid behavior canbe accounted for within the picture of intact quasi-particles.Many of the calculated quantities have not been investigatedon real systems so far, such that our results could be furthertested in experiment. II. BOLTZMANN APPROACH
We investigate general transport properties by solving theBoltzmann transport equation (cid:0) ∂ t + (cid:219) r · ∇ r + (cid:219) p · ∇ p (cid:1) f ( u ) = (cid:2) ∂ t f ( u ) (cid:3) imp + (cid:2) ∂ t f ( u ) (cid:3) el − el , (1)where u = { t , r , p } denotes extended phase-space coordinatesand f ( u ) is the (spin-independent) distribution function. Inthe following, we are interested in the homogeneous, stationarysolution, such that f ( u ) ≡ f ( p ) . The left-hand side of Eq. (1)includes the effect of temperature gradients and fields throughthe substantial time derivative. The right-hand side includes inour case the input from impurity and electron-electron scatter-ing. Assuming Matthiesen’s rule to apply, we can treat the twocontributions separately. Impurity scattering is formulated as (cid:2) ∂ t f ( p ) (cid:3) imp = − Ω ∫ ( d p (cid:48) ) Γ imp p , p (cid:48) × (cid:8) f ( p ) (cid:2) − f ( p (cid:48) ) (cid:3) − (cid:2) − f ( p ) (cid:3) f ( p (cid:48) ) (cid:9) , (2)where ( d p ) = d p /( π (cid:126) ) , Ω is the sample volume and scat-tering rates are determined by the Fermi Golden rule, Γ imp p , p (cid:48) = π (cid:126) n imp | v imp | δ ( ε p − ε p (cid:48) ) . (3) Here, n imp is the density of impurities and v imp = (cid:104) p | ˆ V imp | p (cid:48) (cid:105) is the scattering matrix element, which we assume tobe isotropic, considering only s -wave (contact) scattering.We introduce the dimensionless impurity scattering strength n imp | v imp | / t = . , where t is the characteristic energyscale of the electron dispersion ε p as described in App. A.The effect of collision between electrons is included in thesecond term on the right hand side of Eq. (1), (cid:2) ∂ t f ( p ) (cid:3) el − el = − Ω ∫ ( d p )( d p )( d p ) Γ el − el p , p , p , p × (cid:110) f ( p ) f ( p ) (cid:2) − f ( p ) (cid:3) (cid:2) − f ( p ) (cid:3) − (cid:2) − f ( p ) (cid:3) (cid:2) − f ( p ) (cid:3) f ( p ) f ( p ) (cid:111) (4)whereby the two electrons change their momenta, ( p , p ) ↔( p , p ) . Scattering rates are computed via the Fermi Goldenrule taking a repulsive on-site Hubbard-U-type coupling (wefix U = t ), which yields Γ el − el p , p , p , p ∝ U δ ( ε p + ε p − ε p − ε p )× δ ( p + p − p − p ) (5)and satisfies energy as well as momentum conservation. Whilethe scattering rate is isotropic ( s -wave scattering) we find ahighly anisotropic contribution to Eq. (4) because of Umk-lapp scattering–the fact that momentum conservation allowsfor momentum transfer of reciprocal lattice vectors. Indeed,Umklapp scattering is the only way of momentum relaxationfor the electron-electron collision.The final ingredient for our calculations is the electron dis-persion, which we model after the γ band of Sr RuO withina tight-binding description, see App. A. In the dispersion,we include the effect of doping and uniaxial strain throughthe chemical potential and the hopping integrals, respectively.Note that we use a non-trivial Poisson ratio in our calcula-tions, which leads to an asymmetric response to positive andnegative uniaxial strains.To simplify Eq. (1), we use the parametrization f ( p ) = (cid:20) + exp (cid:18) ε ( p ) − µ T − φ ( p ) (cid:19)(cid:21) − , (6)which yields in linearized form f ( p ) ≈ f ( p ) + f ( p ) [ − f ( p )] (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) scattering phase space φ ( p ) , (7)with f ( p ) the Fermi-Dirac distribution. The correction φ ( p ) to be determined, thus, directly relates to the scattering phasespace. This correction contains the necessary information tocalculate transport coefficients such as the longitudinal elec-tronic conductivity for an electric field E along the x axis, σ xx = eE ∫ ( d p ) f ( p ) (cid:2) − f ( p ) (cid:3) φ ( p ) v x ( p ) (8)and the longitudinal thermal conductivity for a temperaturegradient T (cid:48) = ( ∇ T ) x along the x axis, κ xx = T (cid:48) ∫ ( d p ) f ( p ) (cid:2) − f ( p ) (cid:3) φ ( p ) v x ( p )( ε p − µ ) . (9)Before solving the full Boltzmann equation including electron-electron scattering numerically in Sec. IV, we first analyze thelow-temperature limit where impurity scattering dominates. III. IMPURITY SCATTERING
Focusing on impurity scattering only we obtain the lin-earized Boltzmann equation for φ ( p ) up to linear order in aconstant electric field E and a temperature gradients ∇ r T . Asshown in App. B, the resulting corrections reads φ ( p ) = − (cid:104)(cid:16) ε p − µ T (cid:17) ∇ r T + e E (cid:105) · v p T τ ( ε p ) . (10)Here, v p ≡ (cid:219) r = ∂ p ε p is the velocity and the effect ofimpurity scattering appears in the scattering time τ ( ε ) = (cid:126) / (cid:2) π n imp | v imp | Ω N ( ε ) (cid:3) with N ( ε ) the density of states atenergy ε . Note that for s -wave scatterers the scattering time isnot direction dependent.Sufficiently far from the vHS, the density of states N ( ε ) ≈ N is only weakly depending on energy, resulting in an essentiallyconstant scattering time τ . Therefore, we recover the well-known scattering-time approximation f E ( p ) ≈ f (cid:16) p + e τ (cid:126) E (cid:17) , (11) f ∇ T ( p ) ≈ + e ε p − µ T − τ v · ∇ T . (12)As we approach the vHS, we find that φ ( p ) vanishes forall directions of p , since τ ( ε ) ∝ / N ( ε ) → . Thus, weexpect for both types of Lifshitz transitions a qualitativelysimilar behavior, namely a suppression in all directions due tothe vanishing scattering time. Physically, the large scatteringphase space available at energies near the vHS leads to a fastmomentum relaxation such that an applied field or temperaturegradient can shift the Fermi distribution only weakly. Note thatthis effect is naturally weaker in the case of a Lifshitz transitionat only one van Hove point as realized by uniaxial strain.For a qualitative understanding of electrical transport at lowtemperatures and Fermi energy near a vHS, we use Eq. (10) toobtain the conductivity through Eq. (8), σ xx = e T ∫ ( d p ) f ( p ) (cid:2) − f ( p ) (cid:3) τ ( ε p ) (cid:0) v x p (cid:1) (13) = e T ∫ d ε f ( ε ) (cid:2) − f ( ε ) (cid:3) τ ( ε ) (cid:68) (cid:0) v x p (cid:1) | ∇ p ε p | (cid:69) ε , (14)where we have introduced the equal-energy-contour average (cid:104) A (cid:105) ε = ∫ ε p = ε ( d p ) A . For T → , this integral is dominated by ε = ε F , the Fermi energy, such that σ xx ≈ e τ ( ε F ) (cid:68) (cid:0) v x p (cid:1) | v p | (cid:69) ε F , (15) FIG. 2. Conductivity dip due to enhanced impurity scatteringtogether with the switching sign of the Seebeck coefficient near theLifshitz point normalized to the undoped resistivity µ = . t . Inseta) shows the density of states together with the energy intervals of thecontributing states for µ / t = . (green), µ / t = . (red), µ / t = . (blue). Inset b) shows the contributing states for these three cases forthe scattering phase space f ( p ) (cid:0) − f ( p ) (cid:1) ≤ − . where (cid:104)·(cid:105) ε F denotes a Fermi surface average. As noted above,the suppression of the scattering time τ ( ε F ) close to the vHSleads to a reduction of the conductivity. Figure 2 shows the cor-responding dip in the longitudinal conductivity by evaluatingEq. (13) together with the Seebeck coefficient approximatedthrough Mott’s formula Q = − π k B Te σ (cid:48) ( ε ) σ ( ε ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ε = µ , (16)considering the doping scenario at a low temperatures ( T / t ∼ − ) . Note that the shown conductivity is already in goodqualitative agreement with the experimental findings of Refs. 8and 10.An interesting feature arises for the case where uniaxialstrain pushes the Fermi surface through van Hove points at k = ( , π ) corresponding to negative ε xx in Fig. 1 b). Inthis case, the averages over the energy contours in Eq. (14) aredominated by the momentum direction along [ , ] . For T = ,the conductivity vanishes for ε xx = ε vH according to Eq.(15)due to the vanishing scattering time. Once the temperatureincreases, the range of energy averaging grows such that with τ also the conductivity becomes finite and grows with increas-ing temperature. While upon uniaxial deformation ( ε xx < )the Fermi surface along [ , ] passes through the vHS, it re-treats from the van Hove point along [ , ] due to Luttinger’stheorem. Therefore, the Fermi velocity v x F along [ , ] in-creases monotonically with increasing deformation. Since thescattering time τ is an approximately symmetric function of ε xx − ε vH with its minimum at ε xx = ε vH , the ε xx -dependenceof the average of (cid:104) (cid:0) v x p (cid:1) /| v p |(cid:105) ε leads to a shift of the minimumtowards ε xx < ε vH for increasing temperature. This behavioragrees well with the numerical solution of the linearized Boltz-mann equations presented in the next section and reproducesalso the qualitative behavior observed in experiments . Forthe opposite situation of positive uniaxial strain, ε xx > , thisargument does not apply and we will see below that there isno temperature-dependent shift of the conductance minimum. IV. NUMERICAL RESULTS
We now turn to the numerical solution of the linearizedBoltzmann Eq. (1) including both impurity and electron-electron scattering. The most important part of momentumspace for this calculation is close to the Fermi surface repre-senting the scattering phase space. Thus, we adopt the dis-cretization scheme from Ref. 16 shown in Fig. 3, with mo-mentum space patches following band energy equipotentiallines distributed between ε F − T and ε F + T together withequally distributed angular ϑ coordinates. An advantages ofour technique is that the patched discretization of the BZ isadaptive to the temperature. In other words, we always workwith the same number of patches regardless of the temperature.Note that we normalize the temperature scale with respect to t = . eV in order to compare with experimental results.We use a sufficiently dense set of angular ( ϑ ) and energy( ε ) contours, namely 160 angular and 30 energy contours, toensure high accuracy. A high-resolution discretization is par-ticularly important for the calculation of the electron-electroncollision integral, Eq. (4), and its anisotropy due to Umklappprocesses , where the relevant phase space is located aroundthe crossing points of the Fermi surfaces and the Umklappzones denoted in Fig. 1. We distinguish two Umklapp pro-cesses, one for reciprocal lattice vectors ( π, ) and ( , π ) corresponding to the boundaries of the dark blue zones inFig. 1 and the ( π, π ) indicated by the white diamond-shapedboundary. Only such high resolution allows us to analyzethe subtle low-temperature dependence of transport quantitieswhich are strongly influenced by the position of the Fermisurface crossings. Only when we generate maps (and lowtemperature details) of quantities such as shown in Fig.4 e),f) do we reduce the angular resolution to 40 contours for per-formance reasons. This resolution is sufficient in these casesto provide information on temperature dependence of the dis-played quantities. A. Electrical transport
Figure 4 provides an overview of our main numerical resultsfor the electrical conductivity, σ xx , for the two cases of Fermisurface tuning using the full solution to Eq. (1). The upper sixpanels provide scans of the tuning parameter µ for the case ofdoping on the left-hand side and ε xx for uniaxial deformationon the right-hand side. The Lifshitz transitions occur at µ = . t and ε xx = ε vH ∼ ± . . Panels a) and b) displaythe conductivity with dips, which become more pronouncedwith decreasing temperature, at the Lifshitz transitions. Theshift of the minimum away from the Lifshitz-transition pointin the case ε xx < is marked by dots in panel b). Panels c)and d) show the relative change of the resistivity ∆ ρ xx ( µ ) = ρ xx ( µ ) − ρ xx ( µ F ) normalized with respect to ρ xx ( µ F ) , where µ F = . t is the chemical potential of the undoped case, FIG. 3. Discretization of the first BZ adapted to the scattering phasespace. The red curve corresponds to the Fermi surface crossingthe VHS. Blue curves denote the boundary equipotentials of thediscretization, i.e., ε F − T and ε F + T . Note that for demonstrationpurposes, a large temperature ∼ K is shown. and ∆ ρ xx ( ε ) = ρ xx ( ε ) − ρ xx ( ) normalized with respect to ρ xx ( ε xx = ) . In both cases, the dips in σ xx translate topeaks whose maxima grow with decreasing temperature. Notethat for ε xx < the resistivity, including the shift of themaxima, resembles qualitatively very well the experimentalresults found in Ref. 8.We turn to the low-temperature behavior of ρ xx , which weuse as our first tool to identify deviations from the standardFermi-liquid picture. The temperature dependence of the re-sistivity, as displayed in the panels g) and h) can be fittedby ρ xx ( T ) = ρ + AT α , (17)with the parameters A and α besides the residual resistivity ρ . The exponent α = denotes a Fermi liquid, while α smaller than 2 is considered as non-Fermi liquid behavior. Wedetermine the exponent from the numerical results using α = ∂ ln (cid:2) ρ ( T ) − ρ ( ) (cid:3) ∂ ln T , (18)which yields the α -maps in panels e) and f). In both cases, wesee triangle-shaped regions with α ≈ far enough from theLifshitz points. On the other hand, at the Lifshitz transition afan of values α clearly smaller than 2 opens. The two regimesare separated by bright stripes ( α > ), which result fromEq. (18) when ρ xx has a kink-like feature as the vHS entersthe scattering phase space upon increasing the temperature.These kinks are visible by eye in the plots of ρ xx in panel g)for µ / t = . and . . B. Kadowaki-Woods ratio
In the Fermi-liquid regime ( α = ), the fitting parameter A inEq. (17) can be used together with the Sommerfeld coefficient FIG. 4. Summary of numerical results for the electronic transport: Temperature evolution of the conductivity [a), b)], normalized resistivity[c), d)], Temperature scaling coefficient α [e), f)], as well as resistivity curves for various values of µ / t and ε xx comparing band filling (leftpanel) and uniaxial strain (right panel) scenarios. Subfigures g) and h) correspond to temperature cuts of the resistivity at values of interestingband fillings and values of uniaxial strains. Notice the strongly nonlinear temperature dependence of ρ xx ( T ) considering µ / t = . [greencurve in subfigure g)] and similar dependence at the both Lifshitz points ε vHS ± [orange and red curves in subfigure h)]. FIG. 5. Coefficient A of ρ ( T ) = ρ + AT [subfigures a) and b)] together with the Kadowaki-Woods ratio R KW = A / γ [subfigures c) and d)]with as a function of band filling (left panel) and uniaxial strain (right panel). Colored regions correspond to the non-Fermi liquid behavior ofthe resistivity ( α (cid:44) ), outside of which the density of states (gray lines) remains almost constant. γ ∝ N ( ε F ) to define the Kadowaki-Woods ratio R KW = A / γ .This ratio is empirically a material-independent constant forseveral material classes and has been suggested to remainconstant, if the strength of the electron correlations varies fora fixed bare band structures . In their experiments with uni-axially deformed Sr RuO , Barber et al. observed an increaseof A faster than that of γ , anticipated from DFT calculations,upon tuning the Fermi surface towards the vHS for ε xx < .In order to analyze this behavior, we determine A and R KW within our approach, where we calculate γ from our tight-binding band structure. Figure 5 shows both A and R KW inthe low-temperature regime, where α = indicates Fermi-liquid-like behavior. We find that A increases as the Fermisurface shifts towards the vHS and indeed faster than γ suchthat the Kadowaki-Woods ratio also increases in a similar way.Note that the behavior we find for A considering ε xx < inpanel b) is in good qualitative agreement with the experimentalfindings . C. Thermal transport & Seebeck coefficient
The electronic contribution to the thermal transport can becalculated numerically in a manner analogues to the electricalconductivity in the previous section. Figures 6 a) and b) showthe thermal conductivity κ xx , which exhibits similar features asthe conductivity σ xx presented in Fig. 4, including, in particu-lar, the dips at the Lifshitz transitions. These dips are naturallywider, because the integral for the thermal conductivity κ xx = T (cid:48) ∫ ( d p ) f ( p ) v x p (cid:0) ε ( p ) − µ (cid:1) , (19) contains also the electron dispersion which is essentially flat atthe vHS. A further difference is the weaker temperature depen-dence of the minima. However, there is again a temperaturedependent shift of the minimum position for uniaxial strain ε xx < as observed and discussed for σ xx [see Fig. 4 b)].The Seebeck coefficient Q relates a temperature gradient to aresulting electric field, E = Q ∇ T under open circuit conditionand follows from the solution to the linearized Boltzmannequation. It is given by Q = | E | e | ∇ T | ∫ ( d p ) v x p f ( p ) (cid:0) − f ( p ) (cid:1) ˜ φ ∇ T ( p ) ∫ ( d p ) v x p f ( p ) (cid:0) − f ( p ) (cid:1) ˜ φ E ( p ) , (20)where ˜ φ E and ˜ φ ∇ T correspond to the solution of the Boltz-mann equation with only external electric field E or temper-ature gradient ∇ T . We can use the Seebeck coefficient as asecond tool to identify deviations from Fermi-liquid behavior,since for a Fermi liquid, Q has linear temperature dependence.For this reason, we plot the ratio Q / T in Figs. 6 c) and d) as amap of temperature versus chemical potential and strain. Weagain recognize different regimes, where the Umklapp pro-cesses are clearly observable in both panels. In the panelse) and f), we focus on the low temperature regime and show Q / T scans for fixed temperatures. We find little temperaturedependence away from the Lifshitz points in accordance withexpectations for Fermi liquids. Around the Lifshitz points,on the other hand, anomalies emerge with lowering temper-ature including a sign change. Interestingly, this agrees wellwith the Mott formula given by Eq. (16) considering the caseof tuning by chemical potential. The dip in conductance re-sults in a sign change with the anomalies corresponding tothe inflection points of σ xx ( µ ) . The reason for this agreement FIG. 6. Temperature evolution of the thermal conductivity [a) and b)], as well as Seebeck coefficient divided by the temperature Q / T [c) andd)] together with the detail at the low temperature Q / T [e) and f)] comparing the band-filling and the uniaxial-strain scenario. lies in the disappearance of the highly anisotropic contribu-tion of electron-electron scattering to momentum relaxationat low temperature. At higher temperature these anisotropieschange the behavior of Q / T rather profoundly . The sameargument applies also to the case of Fermi-surface tuning byuniaxial strain, where, near the Lifshitz point, the Mott formulacan be approximated by Q ∝ − σ ( ε xx ) ∂σ ( ε xx ) ∂ | ε xx | . (21)Thus, a change of ε xx has an analogous effect as a variation ofthe chemical potential, justifying this approximation. D. Wiedemann-Franz law
The anisotropy of scattering due to Umklapp processes af-fects the electrical and thermal conductance in a different way.This impacts the temperature dependence of both, as well asthe Wiedemann-Franz law. The Lorenz, number defined as L ( T ) = κ xx ( T ) T σ xx ( T ) , (22)is constant and given by L = π k B / e for isotropic scatter-ing. In Fig. 7, we display the ratio L ( T )/ L for a range of vary-ing band fillings [a)] and uniaxial deformation [b)] includingthe Lifshitz transitions. A general observation is that L < L ,signaling that the thermal transport is more efficiently impededby scattering here than the charge transport. In the regions farfrom the Lifshitz transitions, which we identified previously as the (triangle-shaped) Fermi liquid regime, L / L is largerand additionally increases as temperature decreases, such that L approaches L . This indicates that electron-electron scat-tering is responsible for the suppression of L , while isotropicimpurity scattering leads to a higher value of L . Notable isthe difference between the Fermi liquid regions for µ < . t and µ > . t in Fig. 7a). While in the former case both Umk-lapp processes, involving the reciprocal lattice vectors of thekind of ( π, ) and ( π, π ) , are allowed, in the latter case onlythe ( π, ) -type of Umklapp contributes and leads to a morepronounced anisotropy. At the same time, the electron den-sity, which increases with the chemical potential, increases thescattering probability. This leads to a stronger suppression of L than in the range µ < . t . In contrast, the Fermi liquidregimes for a uniaxial deformation are fairly comparable intheir behavior and the whole picture looks rather symmetricaround ε xx .In the vicinity of the Lifshitz points, we observe a pro-nounced drop of L / L , whose onset is temperature dependentand can be identified by the overlap of the scattering phasespace with the vHS. This enlarged phase space is accompa-nied by more electronic states contributing to energy transport.This is supported by the observation that, comparing Figs. 4a) and b) with Figs. 6 a) and b), the dips in the conductivity aresharper in the former than the latter case. This gives rise to thestrong reduction of L . Right at the Lifshitz point, on the otherhand, we find a very narrow region where L recovers for verylow temperatures. This effect is due to the weak temperaturedependence of the minima values for κ xx , in contrast to thestrong decrease of σ xx with lowering T . FIG. 7. Deviations from the Wiedemann-Franz law as a function of band filling (left panel) and uniaxial strain (right panel). The dashed linesmark the Lifshitz transitions.FIG. 8. Temperature dependence of the Hall resistivity for the band-filling as well as the uniaxial-strain scenario. The line at R H = denoteswhere the character of the charge carriers changes. In summary, the Lorenz number deviates rather stronglyfrom the Wiedemann-Franz value L as a consequence of therather complex scattering behavior encountered in our system.This deviation shows further pronounced features through thespecial scattering properties close to the Lifshitz transition. E. Hall resistivity
The Hall resistivity R H = σ xy / B z ( σ xx + σ xy ) –we set themagnetic field along the z and the current flow along the x axis–is used to characterize the nature of the charge carriers. It isinteresting to follow the evolution of R H ( = / ne ) in the twocases of Fermi surface change. In Fig. 8 we display the result for R H from the numerical solution of the Boltzmann equationsincluding an out-of-plane magnetic field . There is a largedifference between the Fermi surface tuning by band filling[a)] and uniaxial deformation [b)] considering the magnitude:For the latter tuning, the effect is an order of magnitude largerin the considered range of tuning parameters. The sign changeof R H , Fig. 8 a), reflects the change from an electron- to ahole-like Fermi surface with a strong temperature dependenceof the zero-crossing point. Even at the lowest temperature,however, the sign change of R H is not exactly at the Lifshitztransition.In fact, the current density along the x direction yields thedominant modification of the Fermi distribution along [ , ] .The sign of R H is determined by the curvature of the Fermisurface region, which dominantly carries the current. Lookingat Fig. 1 a), the curvature at the Lifshitz transition is convexand yields an electron-like behavior of R H . Only raising thechemical potential µ higher leads to emergence of dominantlyconcave curvature around the [ , ] direction and thus, to thesign change of R H at µ ≈ . t for T = K . For higher tem-perature, this feature shifts to lower µ due to thermal smearingof the Fermi surface region.For the Fermi surface tuned by uniaxial deformation, Fig. 8b), the sign of R H is opposite for positive and negative strain ifthe temperature is sufficiently high, a behavior which becomesmore pronounced with growing T . For the lowest tempera-ture displayed, however, R H is strictly negative. Again, wemay consider the curvature of the current-carrying parts forthe Fermi surface. For ε xx < , the Fermi surface around the[1,0] direction remains convex, in other words electron-like,such that R H < . For the other strain direction, the curvatureis concave only in a small part near the vHS, which, due toits orientation (with normal vector dominantly along the [0,1]direction), contributes only weakly to the current density. Themajor contribution to the current density originates from theconvex Fermi surface parts with Fermi velocities of sizable x component, as can be anticipated from Fig. 1 b). In thiscase, however, increasing temperature yields a Fermi surfacesmearing which yields an effective curvature analogous to thesituation we observed in the case of varying chemical poten-tial. The extrema of R H at the two Lifshitz transitions and theirincrease are caused by the dips of σ xx which appears quadrat-ically in the denominator of R H and the strong temperaturedependence.To our knowledge there are no Hall effect measurements forany of the investigated situation. It would indeed be a helpfultest for the quasiparticle picture and our approach to be able tohave a comparison with experiments. V. CONCLUSIONS
Our analytical and, in particular, numerical investigationof transport properties in a single-band model with a Fermisurface undergoing a Lifshitz transition displays a complexbehavior. Our numerical scheme allows us to take the highlyanisotropic structure of the electron-electron scattering withUmklapp processes into account. In combination with simpleisotropic impurity scattering, we find that the electrical resis-tivity strongly deviates from the standard Fermi-liquid picture, although our starting point relies on the integrity of the quasi-particle description.While our study is motivated by experiments on Sr RuO ,which possesses three bands at the Fermi level, we focused ona single-band picture including the γ band only. The α and β bands correspond to hybridized quasi-one-dimensional bandsand are only weakly affected by the tuning parameters we haveused. In particular, their Fermi surfaces never approach thevan Hove points in the BZ. Thus, the single-band approxima-tion can be justified on a qualitative level, since the γ bandis expected to dominate the anomalous transport propertiesnear the Lifshitz transitions. Nevertheless, in order to obtaina more quantitative picture, all bands should be considered.Moreover, scattering vertex corrections may yield importantcorrections which have not been taken into account here .These extensions are referred to future studies.While a standard perturbation picture of the shifted Fermisurface due to an external electric field fails in the vicinity ofthe Lifshitz point, our semi-analytical and numerical approachto the solution of the Boltzmann transport equation and thecomparison with experiment indicates that a quasiparticle de-scription may indeed be used throughout the whole range ofFermi surface tuning including the Lifshitz points. Analyz-ing different transport properties, we see that the electron-electron scattering yields strong modifications due to the highanisotropy introduced by Umklapp processes. These processesact in a restricted phase space, which is rather strongly temper-ature dependent. The considered (non-invasive) Fermi-surfacetuning allows to modify this Umklapp phase space and to probeits impact on transport properties. These include not only theelectrical conductivity, but also the thermal conductivity, See-beck coefficient, and in a more indirect way the Hall effect.Our predictions of these general transport properties for thetwo different Fermi-surface-tuning possibilities allow for fur-ther experimental scrutiny of the quasiparticle picture for bothcharge and heat transport and thus, of the electronic nature ofa system close to a van Hove singularity. ACKNOWLEDGMENTS
We would like to thank Clifford Hicks and Andy Mackenziefor many valuable discussions. We gratefully acknowledgefinancial support by the Swiss National Science Foundationthrough Division II (No. 163186). J. M. Buhmann, Phys. Rev. B , 245128 (2013). C. W. Hicks, D. O. Brodsky, E. A. Yelland, A. S. Gibbs, J. A. N.Bruin, M. E. Barber, S. D. Edkins, K. Nishimura, S. Yonezawa,Y. Maeno, and A. P. Mackenzie, Science , 283 (2014). D. R. Overcash, T. Davis, J. W. Cook, and M. J. Skove, Phys. Rev.Lett. , 287 (1981). C. L. Watlington, J. W. Cook, and M. J. Skove, Phys. Rev. B ,1370 (1977). U. Welp, M. Grimsditch, S. Fleshler, W. Nessler, J. Downey, G. W.Crabtree, and J. Guimpel, Phys. Rev. Lett. , 2130 (1992). Y. Maeno, K. Yoshida, H. Hashimoto, S. Nishizaki, S. ichi Ikeda,M. Nohara, T. Fujita, A. P. Mackenzie, N. E. Hussey, J. G. Bednorz,and F. Lichtenberg, J. Phys. Soc. Jpn. , 1405 (1997). C. Bergemann, A. P. Mackenzie, S. R. Julian, D. Forsythe, andE. Ohmichi, Adv. Phys. , 639 (2003). M. E. Barber, A. S. Gibbs, Y. Maeno, A. P. Mackenzie, and C. W.Hicks, Phys. Rev. Lett. , 076602 (2018). I. M. Lifshitz, Sov. Phys. JETP , 1130 (1960). B. Burganov, C. Adamo, A. Mulder, M. Uchida, P. D. C. King,J. W. Harter, D. E. Shai, A. S. Gibbs, A. P. Mackenzie, R. Uecker,M. Bruetzam, M. R. Beasley, C. J. Fennie, D. G. Schlom, andK. M. Shen, Phys. Rev. Lett. , 197003 (2016). A. Varlamov, V. Egorov, and A. Pantsulaya, Advances in Physics , 469 (1989). R. Hlubina, Phys. Rev. B , 11344 (1996). R. Markiewicz, Journal of Physics and Chemistry of Solids ,1179 (1997). J. M. Buhmann,
Unconventional transport properties of correlatedtwo-dimensional Fermi liquids , Ph.D. thesis, ETH Zurich (2013). J. M. Ziman,
Principles of the Theory of Solids , 2nd ed. (Cam-bridge University Press, 1972). J. M. Buhmann, M. Ossadnik, T. M. Rice, and M. Sigrist, Phys.Rev. B , 035129 (2013). Y.-T. Hsu, W. Cho, A. F. Rebola, B. Burganov, C. Adamo, K. M.Shen, D. G. Schlom, C. J. Fennie, and E.-A. Kim, Phys. Rev. B , 045118 (2016). K. Kadowaki and S. B. Woods, Solid State Communications ,507 (1986). A. C. Jacko, J. O. Fjærestad, and B. J. Powell, Nature Physics ,422 EP (2009). J. M. Buhmann and M. Sigrist, Phys. Rev. B , 115128 (2013). Appendix A: Model for the γ band of Sr RuO We approximate the γ -band, including the effect of uni-axial stress, bu a two-dimensional nearest- and next-nearest-neighbor tight-binding model, ε k = − (cid:2) t x cos ( k x a ) + t y cos ( k y b ) (cid:3) − t (cid:48) (cid:2) cos ( k x a + k y b ) + cos ( k x a − k y b ) (cid:3) . (A1)Note that in the main text, we use the momentum p = (cid:126) k .For the implementation of the band tuning by uniaxial strainwe follow Ref. 8 to and describe the deformation of the unitcell for the lattice constants a and b along x - and y -direction,respectively, by a ( ε xx ) = a ( + ε xx ) , b ( ε yy ) = a ( + ε yy ) , where ε xx and ε yy are the strain components along the twomain axes. For given applied stress σ xx along the x -directionthe two strain components are connected via the Poisson ratio ν xy : ε yy = ν xy ε xx with ν xy < and σ xx = E ε xx ( E : Youngelasticity modulus). The lattice deformation modifies of thehopping integrals linearly in the strain, t x ( ε xx ) = t ( − αε xx ) , t y ( ε xx ) = t ( + αν xy ε xx ) t (cid:48) ( ε xx ) = t (cid:48) (cid:0) − α ( − ν xy ) ε xx / (cid:1) , where α is a constant parameter adjusting the scale of the effectof the strain.For numerical calculations, we use the hopping matrix el-ements t / t = . , t (cid:48) / t = . , the bare chemical potential µ / t = . , α = and ν xy = − . , where t = . eV is thenearest-neighbor-hopping strength of the α -band of Sr RuO grown on SrTiO . For the Hall resistivity, we use a mag-netic field in the dimensionless units B z = ( π ) ea (cid:126) B z , withthe value B z = . . Appendix B: Linearized Boltzmann equation for impuritiesscattering
Here, we derive the linearized Boltzmann equation for smallexternal electric fields E and temperature gradients ∇ r T ( r ) .
1. Collision integral
First, we calculate the impurity-scattering collision integralEq. (2) for isotropic point scattering centers, with Eq. (3), tolowest order in the correction to the Fermi-Dirac distribution f ( p ) . Using the expansion of Eq. (7) and f ( p ) = f ( ε p ) , wefind (cid:2) ∂ t f ( p ) (cid:3) imp = − Ω n imp v π (cid:126) ∫ ( d p (cid:48) ) δ ( ε p − ε p (cid:48) )× f ( ε p ) (cid:2) − f ( ε p ) (cid:3) (cid:2) φ ( p ) − φ ( p (cid:48) ) (cid:3) . (B1)We can further simplify this expression to (cid:2) ∂ t f ( p ) (cid:3) imp = − Ω n imp v π (cid:126) f ( ε p ) (cid:2) − f ( ε p ) (cid:3)∫ ε p = ε p (cid:48) ( d p (cid:48) )| ∇ ε p (cid:48) | (cid:2) φ ( p ) − φ ( p (cid:48) ) (cid:3) . (B2)Using the fact that φ ( p (cid:48) ) is an odd function of p (cid:48) only the firstcontribution in Eq. (B2) survives, and introducing the densityof states N ( ε ) = ∫ ε (cid:48) p = ε ( d p (cid:48) )/| ∇ ε p (cid:48) | , we find the linearizedcollision integral (cid:2) ∂ t f ( p ) (cid:3) imp = − π Ω n imp v imp N ( ε p ) (cid:126) (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) τ ( ε p ) − f ( ε p ) (cid:0) − f ( ε p ) (cid:1) φ ( p ) . (B3)As expected for point scattering centers the scattering time isangle independent (s-wave scattering).
2. Temperature gradient and external electric field
Introducing v p ≡ (cid:219) r = ∇ p ε p , we can write the second termof the left-hand side of Eq. (1) to lowest order as (cid:219) r · ∇ r f ( p ) = v p · ∇ r f ( u ) (B4) = f ( ε p ) (cid:2) − f ( ε p ) (cid:3) (cid:16) ε p − µ T (cid:17) ∇ r T ( r ) · v p . (B5)Further, we calculate the third term on the left-hand side ofEq. (1). Considering only an electric field, i.e., (cid:219) p = − e E , thisyields to lowest order in the applied field E (cid:219) p · ∇ p f ( p ) = − e E · ∇ p f ( p ) (B6) = − e E · v p T f ( p ) (cid:2) − f ( p ) (cid:3) . (B7)Using Eq. (B3), (B7) and (B4) we find the linearized Boltz-mann equation (cid:2) ∂ t + τ ( ε p ) − (cid:3) φ ( u ) = − (cid:104)(cid:16) ε p − µ T (cid:17) ∇ r T ( r ) + e E (cid:105) · v p T , (B8)1which finally leads to the stationary solution φ ( p ) = − (cid:104)(cid:16) ε p − µ T (cid:17) ∇ r T ( r ) + e E (cid:105) · v p T τ ( ε p ) ..