Resolution of two apparent paradoxes concerning quantum oscillations in underdoped high- Tc superconductors
ABSTRACT Recent quantum-oscillation experiments in underdoped high-temperature superconductors seem to imply two paradoxes. The first paradox concerns the apparent nonexistence of the signature of the electron pockets in angle-resolved photoemission spectroscopy (ARPES). The second paradox is a clear signature of a small electron pocket in quantum-oscillation experiments, but no evidence as yet of the corresponding hole pockets of approximately double the frequency of the electron pocket. This hole pockets should be present if the Fermi-surface reconstruction is due to a commensurate density wave, assuming that Luttinger sum rule relating the area of the pockets and the total number of charge carriers holds. Here we provide possible resolutions of these apparent paradoxes from the commensurate d -density wave theory. To address the first paradox we have computed the ARPES spectral function subject to correlated disorder, natural to a class of experiments relevant to the materials studied in quantum oscillations. The intensity of the spectral function is significantly reduced for the electron pockets for an intermediate range of disorder correlation length, and typically less than half the hole pocket is visible, mimicking Fermi arcs. Next we show from an exact transfer-matrix calculation of the Shubnikov-de Haas oscillation that the usual disorder affects the electron pocket more significantly than the hole pocket. However, when, in addition, the scattering from vortices in the mixed state is included, it wipes out the frequency corresponding to the hole pocket. Thus, if we are correct, it will be necessary to do measurements at higher magnetic fields and even higher-quality samples to recover the hole-pocket frequency.
- Physics. 01/2010; 3.
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ABSTRACT: We report on the direct probing of the Fermi surface in the bulk of the electron-doped superconductor Nd$_{2-x}$Ce$_x$CuO$_4$ at different doping levels by means of magnetoresistance quantum oscillations. Our data reveal a sharp qualitative change in the Fermi surface topology, due to translational symmetry breaking in the electronic system which occurs at a critical doping level significantly exceeding the optimal doping. This result implies that the $(\pi/a,\pi/a)$ ordering, known to exist at low doping levels, survives up to the overdoped superconducting regime.06/2009; - [Show abstract] [Hide abstract]
ABSTRACT: We consider quantum oscillation experiments in $\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$ from the perspective of an incommensurate Fermi surface reconstruction using an exact transfer matrix method and the Pichard-Landauer formula for the conductivity. The specific density wave order considered is a period-8 $d$-density wave in which the current density is unidirectionally modulated. The current modulation is also naturally accompanied by a period-4 site charge modulation in the same direction, which is consistent with recent magnetic resonance measurements. In principle Landau theory also allows for a period-4 bond charge modulation, which is not discussed, but should be simple to incorporate in the future. This scenario leads to a natural, but not a unique, explanation of why only oscillations from a single electron pocket is observed, and a hole pocket of roughly twice the frequency as dictated by two-fold commensurate order, and the corresponding Luttinger sum rule, is not observed. However, it is possible that even higher magnetic fields will reveal a hole pocket of half the frequency of the electron pocket or smaller. This may be at the borderline of achievable high field measurements because at least a few complete oscillations have to be clearly resolved.03/2012;
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arXiv:0811.1056v4 [cond-mat.supr-con] 11 Sep 2009
Resolution of two apparent paradoxes concerning quantum oscillations in underdoped
high-Tcsuperconductors
Xun Jia,1Pallab Goswami,2and Sudip Chakravarty1
1Department of Physics and Astronomy, University of California Los Angeles
Los Angeles, CA 90095-1547
2Department of Physics and Astronomy, Rice University, TX 777005
(Dated: September 11, 2009)
Recent quantum oscillation experiments in underdoped high temperature superconductors seem
to imply two paradoxes. The first paradox concerns the apparent non-existence of the signature of
the electron pockets in angle resolved photoemission spectroscopy (ARPES). The second paradox is
a clear signature of a small electron pocket in quantum oscillation experiments, but no evidence as
yet of the corresponding hole pockets of approximately double the frequency of the electron pocket.
This hole pockets should be present if the Fermi surface reconstruction is due to a commensurate
density wave, assuming that Luttinger sum rule relating the area of the pockets and the total number
of charge carriers holds. Here we provide possible resolutions of these apparent paradoxes from the
commensurate d-density wave theory. To address the first paradox we have computed the ARPES
spectral function subject to correlated disorder, natural to a class of experiments relevant to the
materials studied in quantum oscillations. The intensity of the spectral function is significantly re-
duced for the electron pockets for an intermediate range of disorder correlation length, and typically
less than half the hole pocket is visible, mimicking Fermi arcs. Next we show from an exact transfer
matrix calculation of the Shubnikov-de Haas oscillation that the usual disorder affects the electron
pocket more significantly than the hole pocket. However, when, in addition, the scattering from
vortices in the mixed state is included, it wipes out the frequency corresponding to the hole pocket.
Thus, if we are correct, it will be necessary to do measurements at higher magnetic fields and even
higher quality samples to recover the hole pocket frequency.
I.INTRODUCTION
High temperature superconductors have been ad-
dressed from a remarkable number of vantage points.
Nonetheless many of the basic questions still remain un-
resolved, and the notion of broken symmetries arising
from a Fermi liquid has been traditionally discarded in
favor of many exotic ideas. Here we revisit the Fermi
liquid concept and a particular broken symmetry in re-
sponse to a class of recent quantum oscillation exper-
iments.1,2,3,4,5,6We and others7,8,9,10,11,12,13have had
some success in this respect. If these theories and ex-
periments are correct, one will have to radically alter our
twenty year-old view of these superconductors.14The to-
tality of phenomenology cannot of course be explained
without serious Fermi liquid corrections. But as long as
the quasiparticle residue is finite, we hope that the low
energy properties can be understood from our perspec-
tive.
Two paradoxes have arisen in the context of the quan-
tum oscillation measurements. The first is the contrast
between Fermi arcs observed in angle resolved photoe-
mission (ARPES) experiments15on one hand and recent
quantum oscillation experiments suggesting small Fermi
pockets in underdoped YBa2Cu3O6+δ(YBCO),1,2,3,4,5,6
on the other. This is particularly clear in recent ARPES
experiments where an effort was made to examine YBCO
with similar doping as in the quantum oscillation mea-
surements.16
The second paradox is the non-existence of any evi-
dence of the hole pockets in quantum oscillation mea-
surements. Within the density wave scenario of wave
vector Q = (π,π) (the lattice spacing set to unity), there
should be two dominant frequencies in quantum oscilla-
tions. One corresponding to the electron pocket at about
500T and the other corresponding to the hole pocket at
around 900 T. These are of course rough numbers cor-
responding to approximately 10% doping, assuming that
the Luttinger sum rule is satisfied in the mixed state,
that is, the quantum oscillations reflect the normal state
even if the measurements may lie within the mixed state.
The concept of a broken symmetry is very powerful
because deep inside a phase a physically correct effec-
tive Hamiltonian can address many important questions,
whereas our inability to reliably predict properties of
even a single band Hubbard model, while widely pursued,
has been a limiting factor. This is not an empty exer-
cise, if new phenomena can be predicted or striking facts
can be explained with some degree of simplicity. That
broken symmetry both dictates and protects the nature
of elementary excitations, determining the properties of
matter, is important to emphasize.
The suggested form of order, the d-density wave
(DDW),17explains numerous properties of these super-
conductors, including the concomitant suppression of the
superfluid density18, Hall number,19and more recently
the large enhancement of Nernst effect in the pseudogap
state,20in addition to the existence of a single-particle
gap of dx2−y2 form above Tc. There are also theoretical
reasons why DDW is a possibility. It competes favorably
with other ordering tendencies in variational studies of
extended Hubbard models with nearest-neighbor repul-
Page 2
2
sion and pair-hopping terms.21,22It is also realized in
a class of two-leg ladder models with nearest-neighbor
repulsion.23However, whatever form the correct Hamil-
tonian takes, we know that it must favor d-wave super-
conductivity (DSC). Such a Hamiltonian will almost cer-
tainly favor DDW order as well, in light of the abundance
of local Hamiltonians which do not discriminate between
DSC and DDW order. In fact, two carefully designed dif-
ficult polarized neutron scattering experiments have pro-
vided tantalizing direct evidence of DDW order,24,25al-
though other experiments have claimed otherwise.26,27,28
A natural enemy of the pristine properties of matter
is disorder that is unavoidable in complex systems such
as high temperature superconductors. The role of dis-
order was emphasized in the original proposal of DDW
order as a relevant competing order in the phase dia-
gram of high-Tc superconductors,17although our views
of disorder have greatly evolved during the intervening
years. It is this DDW order combined with disorder that
would be the focus of the present manuscript in resolving
the paradoxes stated above. The disorder considered are
of two different types: (1) scattering due to impurities
and defects and (2) scattering from vortices in the mixed
states.
We consider two kinds of intrinsic disoredr: (a) Gaus-
sian white noise and (b) correlated disorder with a finite
correlation length. In the momentum space the scatter-
ing rate for correlated disorder will decay as exp(−q2l2
where q is the momentum transfer between the initial and
the final states, lD being the correlation length. There-
fore, because of its smaller size, the states correspond-
ing to the electron pockets are scattered more than on
the hole pockets. This is an interpretation of the phe-
nomenon and is based on intrapocket scattering. An al-
ternative interpretation involves the density of states on
the Fermi surface of the electron pockets. In contrast, for
white noise, scattering is independent of momentum and
affects both pockets similarly. Disorder naturally has a
strong effect on ARPES spectral function, which is sensi-
tive to the coherence factors that are analogs of Wannier
functions. In Shubnikov-de Haas oscillations it is only
the averaged effect of disorder that enters by determin-
ing the effective lifetime on the Fermi surface. Therefore,
the role of disorder is quite different, as we shall explicitly
see.
Correlated disorder is also experimentally relevant.
Unlike quantum oscillation experiments which probes
bulk properties, ARPES is inherently a surface probe.
In the relevant case of YBCO, as cleaved surfaces show
that CuO and BaO terminations give different contribu-
tions to the total photoemission intensity, with a hole
doping nh = 30%, almost irrespective of the nominal
bulk doping. This self-doping was controlled by evapo-
rating potassium in situ on the cleaved surface, so as to
reduce the hole content down to the value of underdoped
bulk YBCO (δ ≈ 0.5),16the doping level for which many
quantum oscillation experiments are carried out. The
potassium overlayer is likely to produce an effective cor-
D),
related disorder in the CuO plane.
To explain the second paradox we shall adapt an anal-
ysis of Stephen29to include a normal state that exhibits
DDW order. We shall see that the relativistic charac-
ter of the nodal fermions of the hole pocket, as opposed
to the nonrelativistic nature of the charge carriers of
the electron pocket provides a possible explanation. If
we denote the Dingle factor of the electron pocket by
De = e−π/ωcτv, the Dingle factor of the hole pocket is
Dh≈ D4.4
of the missing frequency from the hole pocket.
ωcis the cyclotron frequency corresponding to the elec-
tron pocket and 1/τv is the scattering rate of the elec-
trons from the vortices in the mixed state. The analysis
of Stephen also leads to a tiny shift of the relative fre-
quency of the quantum oscillations in the mixed state,
of the order of 10−6. Thus, there is enough leeway that
even a very large error in this estimate will not affect our
conclusions.
The organization of the manuscript is as follows. In
Sec. II we compute the ARPES spectral function and
show that disorder can destroy the evidence of electron
pockets. Section III is devoted to an exact transfer ma-
trix computation of Shubnikov-de Haas oscillations and
the effect of disorder on it. Section IV contains a discus-
sion of scattering of quasiparticles of the putative normal
state from the vortices in the mixed state. In Sec. V we
briefly summarize the salient features of our work.
e. This huge suppression may be the resolution
Here,
II. SPECTRAL FUNCTION
A.Hamiltonian
The Hamiltonian of commensurate DDW order in
terms of the fermion creation and destruction operators,
c†
kand ck, in the momentum space is
H1=
?
k∈RBZ
?
ǫkc†
kck+ ǫk+Qc†
k+Qck+Q
?
+
?
k∈RBZ
(iWkc†
kck+Q+ h.c.),
(1)
where the ordering wave vector Q = (π,π), and ǫkis the
single particle spectra. The lattice constant is set to be
unity for simplicity. The reduced Brillouin zone (RBZ)
is bounded by ky± kx= ±π. We define ǫkby
ǫk= − 2t(coskx+ cosky) + 4t′coskxcosky
− 2t′′(cos2kx+ cos2ky) (2)
and the DDW gap by
Wk=W0
2(coskx− cosky).(3)
Page 3
3
B.Disorder
Potential disorder in real space with a finite correlation
length lDis modeled by
V (r) =
gV
2πl2
D
?
dx e
−|r−x|2
2l2
D G(x), (4)
where the disorder averages are ?G(x)?
?G(x)G(y)? = δ(x − y); the disorder intensity is set by
gV. This disorder Hamiltonian in the momentum space
is then
= 0 and
H2=
?
k1,k2∈BZ
V (k1,k2)c†
k1ck2+ h.c.,(5)
where the matrix elements are
V (k,k + q) =gV
2πe−
q2l2
2 u(q),
D
(6)
and u(q) is
u(q) =
1
2π
?
dy G(y)e−iq·y, (7)
satisfying the conditions of ?u(q)? = 0 and ?u(q)u(q′)? =
δ(q + q′). In practice, we generate u(q) directly with
the desired statistical properties and then compute the
matrix elements in Eq. (5).
C.Computation of ARPES spectral function
Once the full Hamiltonian H = H1+ H2is generated,
it is diagonalized by the transformation ck=?
where γl is the annihilation operator of quasiparticles
with energy El. The coefficients Pk,land energy Elare
obtained through an exact numerical diagonalization pro-
cedure. Finally, the ARPES spectral function A(k,ω) at
a temperature T is given by:
lPk,lγl,
A(k,ω) = 2π
?
l
|Pkl|2nlδ(ω − El),(8)
where nl = 1/[1 + exp((El− µ)/kBT)] is the fermion
occupation number. Note that the numerical implemen-
tation of Eq. (8) requires an approximation of the delta
function by, for example, a Lorentzian distribution.
We discretize the BZ with a mesh of size 80 × 80, and
diagonalize the corresponding Hamiltonian. The param-
eters we choose for YBCO at 10% doping are: t = 0.3 eV,
t′= 0.3t, t′′= t′/9.0, and W0= 0.0825 eV, same as be-
fore.7The chemical potential µ is set to be −0.2627 eV.
These parameters yield a hole doping of nh∼ 10%. The
temperature T = 10 K is chosen, where a typical ARPES
experiment is performed. For gV/(2π) = 0.1t, the quasi-
particle life time for lD= 0 is of the order τ ∼ 10−12s
from Fermi’s golden rule, which is a reasonable value.3,6
Note that the band width is 8t. The scattering rate for
FIG. 1:
related disorder corresponding to lD = 4.
parameters are stated in the text.
The spectral function A(k,ω) at ω = µ with cor-
The remaining
correlated disorder for a finite lD ∼ 4 is considerably
smaller, as can be seen from Eq. 6. The final spectral
function is obtained by averaging over 20 - 50 disorder
configurations until no difference is detected upon fur-
ther averaging. A typical result with disorder correlation
length lD= 4, in units of the lattice constant, is plotted
in Fig. 1. The electron pockets are barely visible, resem-
bling experimental observations. From Fermi’s golden
rule, the scattering rate is proportional to the square
of the matrix element between the initial and the final
states, V (k,k + q) ∼ exp(−q2l2
q is the momentum transfer in the scattering process. On
average, the scattering rate is therefore proportional to
exp(−q2
In particular, qtis roughly the size of a pocket. Because
electron pockets are much smaller than hole pockets, the
scattering rate is greater for electron pockets. The hole
pockets centered at (±π/2,±π/2) have vanishingly small
spectral function on the back side due to the coherence
factors and appear as Fermi arcs instead,30whose lengths
are further reduced by disorder.
We also demonstrate the dependence of the spectral
function A(k,ω = µ) on lDin Fig. 2. The disorder corre-
lation lengths are lD= 0,2,8,16 for panels (a) through
(d), respectively. There are three distinct regimes de-
pending on lD.For small lD, the electron and hole
pockets will be almost equally scattered. As a conse-
quence, the spectral function is smeared out everywhere;
see Fig. 2(a). For intermediate values of lD, for example
lD = 2 in Fig. 2(b) and lD = 4 in Fig. 1, scattering is
more prominent for electron pockets, resulting in a pic-
ture consisting of only four Fermi arcs. Finally, as the
correlation length lDincreases further, Fig. 2(c) and (d),
the electron pockets reappear. Indeed, though more dis-
order scattering occurs on the electron pockets, the spa-
tial variation of disorder, hence the net effect of disorder,
D/2) (see Eq. (6)), where
tl2
D/2), where qtis a typical momentum transfer.
Page 4
4
FIG. 2: The spectral function A(k,ω = µ) for gV/(2π) =
0.1t. The correlation lengths are lD = 0,2,8,16 from panels
(a) through (d), respectively. The remaining parameters are
given in the text.
becomes weaker.
?
?
?
?
??
?
FIG. 3:
A(k,ω) at two k points in the Brillouin zone, as indicated
in the insert.
The energy dependence of the spectral function
To characterize the energy dependence of the spectral
function, we compute A(k,ω) as a function of ω at two k
points in the Brillouin zone. One of them is at the inter-
section ΓY line with the inner side of the Fermi surface,
the other is situated on the electron pocket along the di-
rection ΓM; see Fig. 3. The disorder correlation length
was chosen to be lD = 4. Although there are peaks at
ω ∼ µ for both, the peak corresponding to the electron
pocket is significantly suppressed by disorder; the sec-
ond peak at ω − µ ∼ −0.2eV is clearly an artifact of our
simple theory and such high energy states would surely
decay once correlation effects are taken into account by
the creation of particle-hole pairs.
In Fig. 4, A(k,ω) is plotted as a function of both k and
ω. The horizontal axis is along the path Γ → Y → M →
Γ in the Brillouin zone. The spectral function is negli-
gibly small outside the reduced Brillouin zone bounded
by kx±ky= ±π,30and consequently there are no peaks
in the central region of Fig. 4. Close to k = (π,0) and
ω = µ, A(k,ω) has very small intensity due to long range
correlated disorder, consistent with our previous observa-
tion that the electron pockets are most likely unobserv-
able.
??
FIG. 4: (Color online)The gray scale plot of the spectral func-
tion A(k,ω) as a function of both k and ω. Red dashed curves
indicate the quasi-particle dispersion relation. A horizontal
dashed line shows the chemical potential µ. The remaining
parameters are given in the text.
III. SHUBNIKOV-DE HAAS OSCILLATIONS
A.The transfer matrix method
Let us now consider the effect of disorder on
Shubnikov-de Haas (SdH) oscillations of the conductivity,
σxx. The tight-binding Hamiltonian on a square lattice
in a sample of dimension N×M, with the lattice constant
set to unity, is
H =
?
i
ǫic†
ici+
?
i,j
ti,jeiai,jc†
icj+ h.c.,(9)
where ciis the fermionic annihilation operator at the site
i. The spin degrees of freedom are omitted for simplicity.
The hopping amplitude ti,j vanishes except for nearest
and next nearest neighbors. To include two-fold commen-
surate DDW order, the nearest neighbor hopping ampli-
tudes are chosen to be
ti,i+ˆ x= −t +iW0
ti,i+ˆ y= −t −iW0
4
(−1)(n+m),
4
(−1)(n+m),
(10)
where (n,m) are a pair of integers labeling a site: i =
nˆ x + mˆ y, and W0 is the DDW gap; for the next near-
est hopping ti,j = t′. The on-site impurity energy ǫi is
defined by
ǫi=V0
Z
?
r
Gre
−|r−i|2
2l2
D ,(11)
Page 5
5
which is analogous to Eq. (4).
disorder, we set the disorder averages ?Gr? = 0 and
?GrGr′? = δr,r′, and Z =?
factor. V0parameterizes the disorder intensity. Note that
Eq.(11) reduces to ǫi= V0Giin the limit lD→ 0, and ǫi
becomes uncorrelated random variables. A constant per-
pendicular magnetic field B is included via the Peierls
phase factor ai,j=2πe
h
is the vector potential in the Landau gauge. We note
that a perpendicular magnetic field even as large as 60 T
has little effect on DDW order.31
In this section we choose t = 0.29 eV, t′= 0.1 eV,
and W0= 0.065 eV. The chemical potential is set to be
µ = −0.28 eV. Note that these parameters are slightly
different from those in the previous section, although the
hole doping is again ∼ 10%. We have left out the third
nearest neighbor hopping, which greatly complicates the
transfer matrix calculation without offering any particu-
lar insight. The disorder intensity V0 = 0.4t leads to a
quasi-particle life time of the order of ∼ 10−12s in the
limit of lD= 0. The magnetic field ranges from B = 20 T
to B = 75 T, representative of the quantum oscillation
experiments. The only relevant length scale here is the
magnetic length lB =
??/eB, which for B = 20 T is
∼ 15a, a being lattice constant equal to 3.85˚ A.
Now consider a quasi-1D system, N ≫ M, with
a periodic boundary condition along y-direction.
Ψn = (ψn,1,ψn,2,...,ψn,M)Tbe the amplitudes on the
slice n for an eigenstate with a given energy E, then the
amplitudes on three successive slices satisfy the relation
To model correlated
re−|r|2/2l2
Dis a normalization
?i
jA · dl, where A = (0,−Bx,0)
Let
?Ψn+1
Ψn
?
=
?T−1
n(E − Hn) −T−1
1
nTn−1
0
??
Ψn
Ψn−1
?
(12)
,
where Hn is the Hamiltonian within the slice n, and
the matrix Tn corresponds to the hopping between the
slices n and n + 1. Tn is tridiagonal, as electrons can
hop from a site on slice n to three sites on the slice
n + 1. All postive Lyapunov exponents of the transfer
matrix,32γ1> γ2> ... > γM, are computed by iterating
Eq. (12) and performing orthonormalization regularly.
The convergence of this algorithm is guaranteed by the
well known Oseledec theorem.33For the above parame-
ters a transverse dimension corresponding to M = 40 is
sufficient. Equation (12) was iterated 105to 106times
until the relative errors of less than 1% of all the Lya-
punov exponents were achieved.
B.Computation of σxx
The conductivity σxxat zero temperature is obtained
from the Landauer formula:34,35,36
σxx(B) =e2
h
M
?
i=1
1
cosh2(Mγi).(13)
The SdH oscillations of σxx at zero temperature for
?
?
?
?
FIG. 5: Fourier transform of SdH oscillations at zero temper-
ature for lD = 0 in arbitrary units. Insert shows oscillations
as a function of the inverse magnetic field. Note the presence
of higher harmonics. The parameters are described in the
text.
lD= 0 is shown in the insert of Fig. 5. A third order poly-
nomial was used to subtract the background. The Fourier
transform is shown in the main panel of Fig. 5 . Clearly,
there are two main oscillation frequencies F1= 490±30T
and F2= 915 ± 40T, corresponding to electron and hole
pockets respectively, although a few harmonics are also
visible. Though the first peak at F1agrees with experi-
mental observations, the second peak at F2has not yet
been observed.
For correlated disorder, the Fourier transform of σxxis
plotted in Fig. 6. In all the cases, two main frequencies of
F1∼ 500T and F2∼ 900T are prominent. As for A(k,ω),
an increase of lDincreases the amplitudes of F1and F2,
because the effective disorder becomes weaker with the
increased correlation length. Also the higher harmonics
are more visible. There is, however, a sharp distinction
between the dependence of the two physical quantities,
which becomes clear when we consider the white noise
case: A(k,ω) is completely smeared out in the momen-
tum space because white noise scatters between all pos-
sible wave vectors; see Fig. 2 (a). The coherence factors
are of crucial importance for the spectral function. In
contrast, the SdH oscillations are damped by the Dingle
factor, which is parametrized by a single lifetime and dis-
order enters in an averaged sense. This striking contrast
is clear in Fig. 5. The surprise is that impurity scatter-
ing affects the electron pocket more than the hole pocket,
which is remarkably robust even for white noise. There
must be a piece of physics missing, if we are to explain
why the hole pocket is not observed in SdH. This miss-
ing physics we argue is the vortex scattering rate which
affects the two pockets very differently.
IV.VORTEX SCATTERING RATE IN THE
MIXED STATE
We have shown previously9that the scattering rate of
the DDW quasiparticles, corresponding to the electron