Reply to Comment on "Magnetotransport signatures of a single nodal electron pocket constructed from Fermi arcs"
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Reply to Comment on “Magnetotransport signatures of a single nodal electron pocketconstructed from Fermi arcs”
N. Harrison , S. E. Sebastian Mail Stop E536, Los Alamos National Labs.,Los Alamos, NM 87545 Cavendish Laboratory, Cambridge University, JJ Thomson Avenue, Cambridge CB3 OHE, U.K (Dated: February 26, 2018)
In a recent manuscript, we showed how an electronpocket in the shape of a diamond with concave sides (seefor example Fig. 1a) could potentially explain changes insign of the Hall coefficient R H in the underdoped high- T c cuprates as a function of magnetic field and tempera-ture. For simplicity, this Fermi surface is assumed to beconstructed from arcs of a circle connected at vertices(see Fig. 1b), which is an idea borrowed from Banikand Overhauser. Such a diamond-shaped pocket is pro-posed to be the product of biaxial charge-density waveorder, which was subsequently confirmed in x-ray scat-tering experiments. Since those x-ray scattering exper-iments were performed, the biaxial Fermi surface recon-struction scheme has garnered widespread support in thescientific literature.
It has been shown to accuratelyaccount for the cross-section of the Fermi surface pocketobserved in quantum oscillation measurements, thesign and behavior of the Hall coefficient, the size ofthe high magnetic field electronic contribution to theheat capacity and more recently the form of the angle-dependent magnetoresistance. In their comment, Chakravarty and Wang raise sev-eral important questions relating to the validity of theHall coefficient we calculated for such a diamond-shapedFermi surface pocket. These questions concern specifi-cally ( ) whether a change in sign of the Hall coefficient R H with magnetic field and temperature is dependent ona ‘special’ form for the rounding of the vertices in Fig. 1a,( ) whether a pocket of such a geometry can producequantum oscillations in R H in the absence of other Fermisurface sections and ( ) whether a reconstructed Fermisurface consisting of a single pocket is less ‘natural’ thanone consisting of multiple pockets. Below we considereach of these in turn.
1. Rounding of the diamond vertices
In our model, we assume the quasiparticle scatteringrate τ − to be uniform and consider a scenario in whichsharp corners on the Fermi surface are the product ofelectrons being Bragg reflected by the crystalline lat-tice, as is found to be the case in Al. In the cuprates,we assume that the sharp corners at the vertices of thediamond-shaped pocket result from the Bragg reflectionof quasiparticles by the periodic potential of the charge-density wave. While the finite size ∆ of the periodic po-tential causes the vertices to become rounded ( e.g. solidline in Fig. 2a), the precise form of the rounding depends v v α+π/2 ( , ) v b ( π /a, π /b) α k F a vertex FIG. 1: ( a ), Schematic diamond-shaped electron pocket fromRef. 1, with blue arrows indicating the direction of cyclotronmotion and v and v indicating the Fermi velocity direction.( b ), Schematic showing how the electron pocket is producedby connecting ‘arcs’ of a larger hole Fermi surface, with α being the angle subtended by the arc and the dotted linesindicating how they are connected. on whether the Bragg reflection is elastic or inelastic.The mutually consistent values of the zero field Hallcoefficient we obtained for the diamond using the Jones-Zener and Shockley-Chambers tube integral meth-ods are neither a coincidence nor a consequence of ushaving assumed a special form for the rounding. Rather,they are a consequence of us having assumed the Braggreflection to be an elastic process. In the elastic limit,the x component of the velocity v k orthogonal to theBragg plane reverses sign upon reflection while the y component tangential to the Bragg plane remains un-changed (see Fig. 2). In such a situation, the velocity ofthe quasiparticles evolves in a manner similar to that de-scribed by Equation (5) of Ref. 1. As long as the Braggreflection remains an elastic process, the Hall coefficientfor diamond-shaped pocket (with α > ◦ ) will changesign as a function of the magnetic field.Our conclusion in Ref. 1 is in agreement with thatof Banik and Overhauser, but is quite different fromthat reached by Ong. Ong shows that the sign of R H does not change when a uniform τ − is replaced by auniform magnitude | l ( k ) | of the mean free path vector l ( k ) = v k τ k . However, a uniform | l ( k ) | is incompati-ble with elastic Bragg reflection. To maintain a constant | l ( k ) | while traversing the vertices, either of two uncon-ventional scenarios would need to apply. In one scenario, Bragg plane b v v real-space k y k x k-space a y x FIG. 2: a , Solid lines showing the reconstructed Fermi sur-face in the vicinity of a vertex produced by Bragg reflection.Dotted lines indicate the Fermi surface in the absence of hy-bridization. v and v are velocities before and after a quasi-particle traverses the vertex. ( b ) Schematic of the Bragg re-flection in real-space assumed to be responsible for the sharpcorner. the scattering rate would need to be locally suppressed atthe vertices to compensate for the momentarily reducedmagnitude | v k | = v F cos( α/ π/
4) of the velocity at thevertices given by Equation (5) of Ref. 1. In the other, the y -component of quasiparticle velocity would need to mo-mentarily accelerate to a higher value at the vertices inorder to maintain both v k and τ k constant. Neither ofthese scenarios appear to be more realistic than that weassumed in Ref. 1.When interactions do accompany Bragg reflection, asin the case of ‘hot spots,’ it is more likely that thesewill suppress the contribution to R H from the vertices,causing a sign change in R H to occur for smaller valuesof the parameter α in Fig. 1. Possibilities include a localincrease in the effective mass at the hot spots, or anincrease in the quasiparticle scattering rate.
2. Oscillations in the Hall coefficient
As Chakravarty and Wang correctly point out, theBoltzmann transport equation in the presence of Landauquantization is an intractable problem, requiring someform of approximation to be made. In our Hall effectcalculations, we chose to treat magnetic quantum os-cillations in the transport using an oscillatory scatteringrate ˜ τ − . Such an approach is appealing for several rea-sons. First, the transport scattering rate is generally re-lated to the number of available states for scattering inaccordance with Fermi’s golden rule, causing it to ap-proximately resemble the oscillatory electronic density-of-states. Second, the use of an oscillatory scattering ratehas been shown to enable reasonably accurate modelingof quantum oscillations in the transport of several differ- ent quasi-two-dimensional metals. Third, it correctlyreproduces a non-oscillatory Hall coefficient in the case ofa single Fermi surface pocket of ideal circular geometry.Chakravarty and Wang are correct in stating that thereare no oscillations in the Hall coefficient of the diamond-shaped pocket in the limits ω c τ ≪ ω c τ ≫
1. Infact, the oscillations of the Hall coefficient vanish in bothlimits ω c τ ≪ ω c τ ≫ The verticesand concave sides of the diamond give opposing electron-and hole-like contributions to the Hall coefficient, withthe concave sides dominating over the vertices in weakmagnetic fields (when α > ◦ ). If R and R are theindividual Hall coefficients in the two band model, then R H = R R / ( R + R ) in the limit ω c τ → ∞ , which isnon-oscillatory owing to the contributions from quantumoscillatory diagonal terms ( σ xx and σ yy ) containing τ − having vanished.However, contrary to Chakravarty and Wang, we ar-gue that magnetic quantum oscillations in the under-doped high- T c superconductors are in fact observed inthe intermediate regime in which ω c τ ≈ In sucha regime, the quantum oscillatory diagonal conductiv-ity terms containing τ − do not vanish in a two bandmetal, leading to quantum oscillations in R H . In fact,Chakravarty has recently advocated such a scenario. Ina very similar way to a two band metal, the Hall coef-ficient of a diamond-shaped pocket also contains non-vanishing contributions from τ − in the intermediateregime ω c τ ≈
1, as demonstrated algebraically by Banikand Overhauser. We therefore argue that in a very sim-ilar way to a two band metal, a diamond-shaped pocketwill also exhibit quantum oscillations in R H .
3. Single versus multiple pockets
The occurrence of multiple pockets in the majorityof Fermi surface reconstruction scenarios doesnot make these scenarios more likely, as argued byChakravarty and Wang. Other considerations such asthe small value of the electronic heat capacity at highmagnetic field in fact constrain the number of pock-ets per CuO plane to unity, making such multiplepocket scenarios less likely. There are at least two othermaterials in which Fermi surface reconstruction byincommensurate spin- and or charge-density wave orderhas been shown experimentally to yield only a single re-constructed pocket. With this in mind, two Fermi surfacereconstruction scenarios based on biaxial charge-densitywave order have been shown to be capable of pro-ducing a reconstructed Fermi surface consisting of onlya single pocket. A. Acknowledgements
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