Reply to Comment on "Magnetotransport signatures of a single nodal electron pocket constructed from Fermi arcs"

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by  N. Harrison

Reply to Comment on “Magnetotransport signatures of a single nodal electron pocket
constructed from Fermi arcs”
N. Harrison1, S. E. Sebastian2
1Mail Stop E536, Los Alamos National Labs.,Los Alamos, NM 87545
2Cavendish Laboratory, Cambridge University, JJ Thomson Avenue, Cambridge CB3 OHE, U.K
(Dated: January 17, 2017)

In a recent manuscript,1 we showed how an electron a b
7 vertex
pocketinthe shapeofadiamondwithconcavesides(see
1
0 forexampleFig.1a)couldpotentiallyexplainchangesin k
2 signoftheHallcoefficientRH intheunderdopedhigh-Tc F
cuprates as a function of magnetic field and tempera-
n α
ture. For simplicity, this Fermi surface is assumed to be
a
6 J c(soenestFruigc.te1db)f,r1omwhaircchsiosfaanciidrcelaebcoornrnoewcetdedfraotmveBrtainceiks v2 α+π/2 v1 (π/a,π/b) v
1 and Overhauser.2 Such a diamond-shaped pocket is pro-
posed to be the product of biaxial charge-density wave

] order,3 which was subsequently confirmed in x-ray scat-
l
e teringexperiments.4,5 Sincethosex-rayscatteringexper-
-
iments were performed, the biaxial Fermi surface recon-
r (0,0)
t structionschemehasgarneredwidespreadsupportinthe
s
. scientific literature.6–8 It has been shown to accurately
t
a account for the cross-sectionof the Fermi surface pocket FIG.1: (a),Schematicdiamond-shapedelectronpocketfrom
m observed in quantum oscillation measurements,9–11 the Ref. 1, with blue arrows indicating the direction of cyclotron
- sign and behavior of the Hall coefficient,1,12 the size of motionandv1andv2indicatingtheFermivelocitydirection.
d (b), Schematic showing how the electron pocket is produced
the high magnetic field electronic contribution to the
n by connecting ‘arcs’ of a larger hole Fermi surface, with α
heat capacity13 andmore recently the form ofthe angle-
o being the angle subtended by the arc and the dotted lines
dependent magnetoresistance.14
c indicating how they are connected.
[ In their comment,15 Chakravartyand Wang raise sev-

eral important questions relating to the validity of the
1
v Hall coefficientwe calculated for sucha diamond-shaped on whether the Bragg reflection is elastic or inelastic.
5 Fermi surface pocket. These questions concern specifi- The mutually consistent values of the zero field Hall
9 cally (1 ) whether a changein signof the Hall coefficient coefficient we obtained for the diamond using the Jones-
2 RH withmagneticfieldandtemperatureisdependenton Zener16 andShockley-Chamberstubeintegral17,18 meth-
4 a‘special’formfortheroundingoftheverticesinFig.1a,
ods are neither a coincidence nor a consequence of us
0 (2 ) whether a pocket of such a geometry can produce
havingassumedaspecialformforthe rounding. Rather,
.
1 quantumoscillationsinR intheabsenceofotherFermi
H they are a consequence of us having assumed the Bragg
0 surface sections and (3 ) whether a reconstructed Fermi
reflection to be an elastic process. In the elastic limit,
17 surfaceconsisting ofa single pocketis less ‘natural’than the x component of the velocity vk orthogonal to the
one consisting of multiple pockets. Below we consider
: Bragg plane reverses sign upon reflection while the y
v each of these in turn.
component tangential to the Bragg plane remains un-
i
X changed (see Fig. 2). In such a situation, the velocity of
thequasiparticlesevolvesinamannersimilartothatde-
r
a 1. Rounding of the diamond vertices scribed by Equation (5) of Ref. 1. As long as the Bragg
reflection remains an elastic process, the Hall coefficient
◦
In our model, we assume the quasiparticle scattering for diamond-shaped pocket (with α> 42.3 ) will change
rate τ−1 to be uniform and consider a scenario in which sign as a function of the magnetic field.
sharp corners on the Fermi surface are the product of Our conclusion in Ref. 1 is in agreement with that
electrons being Bragg reflected by the crystalline lat- of Banik and Overhauser,2 but is quite different from
tice, as is found to be the case in Al.2 In the cuprates, that reached by Ong.19 Ong shows that the sign of R
H
we assume that the sharp corners at the vertices of the does not change when a uniform τ−1 is replaced by a
diamond-shaped pocket result from the Bragg reflection uniform magnitude |l(k)| of the mean free path vector
of quasiparticles by the periodic potential of the charge- l(k) = vkτk.19 However, a uniform |l(k)| is incompati-
densitywave.1 While the finite size∆oftheperiodic po- ble with elastic Braggreflection. To maintaina constant
tential causes the vertices to become rounded (e.g. solid |l(k)| while traversing the vertices, either of two uncon-
lineinFig.2a),thepreciseformoftheroundingdepends ventionalscenarioswouldneedtoapply. Inonescenario,
2
entquasi-two-dimensionalmetals.23–25Third,itcorrectly
reproducesanon-oscillatoryHallcoefficientinthecaseof
k-spa ce real-space
a single Fermi surface pocket of ideal circular geometry.
a b
ChakravartyandWangarecorrectinstatingthatthere
v2 v1 areno oscillationsin the Hall coefficientofthe diamond-
shaped pocket in the limits ω τ ≪ 1 and ω τ ≫ 1. In
c c
ky Bragg plane fact,theoscillationsoftheHallcoefficientvanishinboth
y limitsωcτ ≪1andωcτ ≫1forawidevarietyofdifferent
Fermi surface models (in the absence of a quantum Hall
effect). To illustrate generality, it is instructive to con-
x
sidertheexampleofatwobandmetal,which,whencom-
posedofelectronandholepocketswithdifferentsizesand
kx mobilities,yieldssimilarmagneticfield-dependentbehav-
ior to that of our diamond-shaped pocket.2 The vertices
andconcavesidesofthediamondgiveopposingelectron-
and hole-like contributions to the Hall coefficient, with
FIG. 2: a, Solid lines showing the reconstructed Fermi sur- the concave sides dominating over the vertices in weak
◦
face in the vicinity of a vertex produced by Bragg reflection. magnetic fields (when α > 42.3 ). If R1 and R2 are the
Dotted lines indicate the Fermi surface in the absence of hy- individual Hall coefficients in the two band model, then
bridization. v1 andv2 arevelocitiesbeforeandafteraquasi- RH = R1R2/(R1+R2) in the limit ωcτ → ∞, which is
particle traverses the vertex. (b) Schematic of the Bragg re- non-oscillatoryowingtothecontributionsfromquantum
flection in real-space assumed toberesponsible for thesharp oscillatory diagonal terms (σ and σ ) containing τ−1
xx yy
corner.
having vanished.
However,contrarytoChakravartyandWang,15 wear-
thescatteringratewouldneedtobelocallysuppressedat gue that magnetic quantum oscillations in the under-
doped high-T superconductors are in fact observed in
the vertices to compensate for the momentarily reduced c
magnitude|vk|=vFcos(α/2+π/4)ofthevelocityatthe the intermediate regime in which ωcτ ≈ 1.27,28 In such
a regime, the quantum oscillatory diagonal conductiv-
verticesgivenbyEquation(5)ofRef.1. Intheother,the
ity terms containing τ−1 do not vanish in a two band
y-componentofquasiparticlevelocitywouldneedto mo-
metal, leading to quantum oscillations in R .23 In fact,
mentarily accelerate to a higher value at the vertices in H
order to maintain both vk and τk constant. Neither of Chakravartyhasrecentlyadvocatedsuchascenario.26In
a very similar way to a two band metal, the Hall coef-
these scenarios appear to be more realistic than that we
ficient of a diamond-shaped pocket also contains non-
assumed in Ref. 1.
When interactions do accompany Bragg reflection, as vanishing contributions from τ−1 in the intermediate
in the case of ‘hot spots,’ it is more likely that these regime ωcτ ≈ 1, as demonstrated algebraically by Banik
will suppress the contribution to R from the vertices, andOverhauser.2 We therefore arguethat in a verysim-
H
causing a sign change in R to occur for smaller values ilar way to a two band metal, a diamond-shaped pocket
H
of the parameterα in Fig.1. Possibilitiesinclude a local will also exhibit quantum oscillations in RH.1
increase in the effective mass at the hot spots,20 or an
increase in the quasiparticle scattering rate.21
3. Single versus multiple pockets
2. Oscillations in the Hall coefficient The occurrence of multiple pockets in the majority
of Fermi surface reconstruction scenarios6,7,26,29,30 does
As Chakravarty and Wang correctly point out,15 the not make these scenarios more likely, as argued by
BoltzmanntransportequationinthepresenceofLandau Chakravarty and Wang.15 Other considerations such as
quantization is an intractable problem, requiring some the small value of the electronic heat capacity at high
form of approximation to be made. In our Hall effect magnetic field in fact constrain the number of pock-
calculations,1 we chose to treat magnetic quantum os- ets per CuO plane to unity,28 making such multiple
2
cillations in the transport using an oscillatory scattering pocket scenarios less likely. There are at least two other
rate τ˜−1. Such an approach is appealing for several rea- materials31,32 in which Fermi surface reconstruction by
sons. First, the transport scattering rate is generally re- incommensurate spin- and or charge-density wave order
lated to the number of available states for scattering in has been shown experimentally to yield only a single re-
accordance with Fermi’s golden rule,22 causing it to ap- constructedpocket. Withthisinmind,twoFermisurface
proximately resemble the oscillatory electronic density- reconstruction scenarios based on biaxial charge-density
of-states. Second,theuseofanoscillatoryscatteringrate wave order11,33 have been shown to be capable of pro-
has been shown to enable reasonably accurate modeling ducing a reconstructed Fermi surface consisting of only
ofquantum oscillationsinthe transportofseveraldiffer- a single pocket.
3
A. Acknowledgements cussions that inspired this work took place at the Aspen
Center for Physics, which is supported by the National
This work was supported by the US Department of Science Foundation grant PHY-1066293.
Energy ”Science of 100 tesla” BES program. The dis-
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