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arXiv:0912.0728v3 [condmat.suprcon] 8 Jan 2010
The surprising quantum oscillations in electron doped high temperature
superconductors
Jonghyoun Eun, Xun Jia, and Sudip Chakravarty
Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 900951547, USA
(Dated: January 8, 2010)
Quantum oscillations in hole doped high temperature superconductors are difficult to understand
within the prevailing views. There are no viable explanations except that of a putative normal
ground state, which appears to be a Fermi liquid with a reconstructed Fermi surface. The oscillations
are due to formation of Landau levels. Recently the same oscillations were found in the electron
doped cuprate, Nd2−xCexCuO4, in the optimal to overdoped regime.
doped nonstoichiometric materials are naturally more disordered, they strikingly complement the
hole doped cuprates. Here we provide an explanation of these observations from the perspective of
density waves using a powerful transfer matrix method to compute the conductance as a function
of the magnetic field.
Although these electron
Periodically new experiments tend to disturb the sta
tus quo of the prevailing “dogmas” [1] in the area
of high temperature cuprate superconductors.
cent quantum oscillation (QO) experiments [2–9] fall
into this category [10].The first set of experiments
were carried out in underdoped high quality crystals
of wellordered YBa2Cu3O6+δ (YBCO), stoichiomet
ric YBa2Cu4O8 (Y124) and the overdoped single layer
Tl2Ba2CuO6+δ [11]. More recently oscillations are also
observed in electron doped Nd2−xCexCuO4(NCCO) [12].
The measurements in NCCO for 15%, 16%, and 17%
doping [12] are spectacular. The salient features are: (1)
The experiments are performed in the range 30 − 64T,
far above the upper critical field, which is about 10T
or less; (2) the material involves single CuO plane, and
therefore complications involving chains, bilayers, Ortho
II potential [13], etc. are absent; (3) stripes [14] may
not be germane, as they lead to far too many inconsis
tencies with angle resolved photoemission spectroscopy
(ARPES) [15]. It is true, however, that neither spin den
sity wave (SDW) nor ddensity wave (DDW) [16] are yet
directly observed in NCCO in the relevant doping range,
but QOs seem to require their existence, at least the
field induced variety (see, however Ref. [17]); (4) these
experiments are a tour de force because the sample is
nonstoichiometric with naturally greater intrinsic disor
der.The effect is therefore no longer confined to a limited
class of high quality single crystals; (5) The authors have
also succeeded in seeing the transition from low to high
frequency oscillations [18] in NCCO as a function of dop
ing.
Here we focus on NCCO. We shall see that disorder
plays an important role. Without it it is impossible to
understand why the slow oscillations damp out below
30T for 15% and 16% doping, and below 60T for 17%
doping, even though the field range is very high. For
17% doping, where a large hole pocket is observed corre
sponding to very fast oscillations (inconsistent with any
kind of density wave order), the necessity of such high
fields can have only one explanation, namely to achieve
Re
a sufficiently large ωcτ, where ωc = eB/m∗c, τ is the
scattering lifetime of the putative normal phase, m∗the
effective mass, and B the magnetic field. We are fully
aware of many complications, such as dislocations, mag
netic field inhomogenity, or the mosaic structure that can
extinguish QOs, but such details should be less important
than the gross features of a generic disorder that can be
studied rigorously using an exact transfer matrix method
and the Landauer formula for the conductance to bring
out the striking aspects of the problem. We have seen
previously [19] that the effect of longranged correlated
disorder is qualitatively similar to white noise insofar as
the QOs are concerned.
Here we show that the oscillation experiments reflect
a broken translational symmetry [20] that reconstructs
the Fermi surface in terms of electron and hole pock
ets [10]. We favor DDW for a number of reasons but will
also explore SDW. The DDW order [16] explains numer
ous properties of these superconductors, as pointed out
elsewhere [19], and we do not wish to repeat them here.
The mean field Hamiltonian for DDW in real space (its
microscopic origin is discussed elsewhere [19]), in terms
of the sitebased fermion annihilation and creation oper
ators ciand c†
i, is
HDDW=
?
i
ǫic†
ici+
?
i,j
ti,jeiai,jc†
icj+ h.c.,(1)
where the nearest neighbor hopping matrix elements are
ti,i+ˆ x = −t +iW0
4
(−1)(ix+iy),(2)
ti,i+ˆ y = −t −iW0
4
(−1)(ix+iy),(3)
Here W0 is the DDW gap. We also include the next
nearest neighbor hopping t′, whereas the third neighbor
hopping t′′is ignored. The parameters t and t′are cho
sen to closely approximate the more conventional band
structure, as shown Fig. 1
Without t′′, it is difficult to fit precisely the experimen
tal frequencies, but the approximate magnitudes and the
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2
(0,0)(0,0)(
)
FIG. 1: (Color online) The solid curve represents the t−t′−t′′
band structure ( t = 0.38eV, t′= 0.32t, t′′= 0.5t′), and
the dashed curve corresponds to t − t′band structure, (see
Table I). The quasiparticle energy is plotted in the Brillouin
zone along the triangle (0,0) → (π,0) → (π,π) → (0,0). In
the inset the chemical potential, µ, was adjusted to obtain
approximately 15% doping.
trends are correctly reproduced. In principle, the transfer
matrix program can be modified at a considerable com
putational cost to incorporate t′′. Similarly the SDW
mean field Hamiltonian is
HSDW =
?
i,σ
?ǫi+ σVS(−1)ix+iy?c†
i,σci,σ
+
?
i,j
ti,jeiai,jc†
icj+ h.c.
(4)
and the spin σ = ±1, while the magnitude of the SDW
amplitude is VS. In both cases a constant perpendicular
magnetic field B is included via the Peierls phase factor
ai,j=2πe
h
potential in the Landau gauge. We note that usually a
perpendicular magnetic field, even as large as 60T, has
little effect on the DDW order [21], except close to the
doping at which it collapses, where field induced order
may be important. The onsite energy is δcorrelated
white noise defined by the disorder average ǫi = 0 and
ǫiǫj = V2
0δi,j.For an explicit calculation we need to
choose the band structure parameters, W0, VS, and the
disorder magnitude V0. When considering the magni
tude of disorder one should keep in mind that the full
band width is 8t. The magnetic field ranges roughly be
tween 30T and 64T, representative of the experiments in
NCCO. The magnetic length is lB=
B = 30T is approximately 12a, where the lattice constant
a is equal to 3.95˚ A.
The transfer matrix method and the calculation of the
Lyapunov exponents are fully described elsewhere [19].
This is a very powerful method and the results obtained
are rigorous compared to ad hoc broadening of the Lan
dau levels, which will also require many more adjustable
parameters to explain the experiments. Once the distri
?i
jA · dl, where A = (0,−Bx,0) is the vector
??/eB, which for
bution of disorder is specified there are no further approx
imations. Here we merely note that the values of M were
chosen to be much larger than our previous work [19], at
least 128 (that is 128a in physical units) and sometimes
as large as 512. As mentioned before L is varied be
tween 105and 106. This easily led to an accuracy better
than 5% for the smallest Lyapunov exponent, γi, in all
cases. The conductance is calculated from the Landauer
formula [22]:
σxx(B) =e2
h
M
?
i=1
1
cosh2(Mγi).(5)
There are clues in the experiments [12] that disorder
is very important. For 15 and 16% doping the slow os
cillations in experiments, of frequency 290 − 280T, are
not observed until the field reaches above 30T, which
is much greater than Hc2 < 10T. For 17% doping the
onset of fast oscillations at a frequency of 10,700T are
strikingly not observable until the field reaches 60T. The
estimated scattering time from the Dingle factor at even
optimal doping and at 4K is quite short.
For 17% doping corresponding to µ = −0.322t and
the band structure given in Table I, a slight change in
disorder from V0= 0.7t to V0= 0.8t makes the difference
between a clear observation of a peak to simply noise
within the field sweep between 60 − 62T, as shown in
Fig. 2 and Fig. 3. Since in this case W0 = VS = 0,
there is little else to blame for the disappearance of the
oscillations for fields roughly below 60T. The results are
essentially identical for small values W0, such as 0.025t.
TABLE I: Parameters
Order
DDW 15%
DDW 16%
SDW 15%
SDW 16%
t (eV)
0.3
0.3
0.3
0.3
t′
W0
VS
*
*
0.05t −0.403t 0.8t
0.05t −0.366t 0.8t
µV0 F (T)
0.45t 0.1t
0.45t 0.1t
0.45t
0.45t
−0.40t 0.8t
−0.365t 0.8t
195
165
195
173
*
*
For 15% and 16% dopings we chose V0 to simulate
the fact that oscillations seem to disappear below 30T.
The field sweep was between 30 − 60T. The results for
DDW order are shown in Fig. 4 and Fig. 5.
Fig. 6 for which 350 points were sampled in this range, for
the remaining figures sampling was done for 201 points.
Therefore the widths of these peaks should only be taken
as a qualitative indicator, but not literally. The most
remarkable feature of these figures is that disorder has
completely wiped out the large electron pocket leaving
the small hole pocket visible. To emphasize this point
we also plot the results for 15% doping but with much
smaller disorder V0 = 0.2t; see Fig. 6. Now we can see
the fragmented remnants of the electron pocket. With
further lowering of disorder, the full electron pocket be
comes visible. It is clear that disorder has a significantly
Except for
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100001500020000 2500030000
0
20
40
60
80
0.01620.01630.01640.0165
1
0.0166
0.10
0.05
0.00
0.05
0.10
F ( T )
1/B ( T )
Intensity (arbitrary units)


FIG. 2: (Color online)The main plot shows the Fourier trans
form of the field sweep shown in the inset.
at 10,695T. The inset is a smooth background subtracted
Shubnikovde Haas oscillations, as calculated from the Lan
dauer formula for 17% doping as a function of 1/B. The dis
order parameter is V0 = 0.7t.The band structure parameters
are given in Table I.
The peak is
0.01620.01640.0166
0.1
0.08
0.06
0.04
0.02
0
0.02
0.04
0.06
0.08
0.1
1/B ( T )
1





FIG. 3: (Color online)The same parameters as in Fig. 2 but
V0 = 0.8t. The background subtracted conductance is simply
noise to an excellent approximation.
stronger effect on the electron pockets than on the hole
pockets. This, as we noted earlier, is largely due to higher
density of states around the antinodal points, which sig
nificantly accentuates the effect of disorder [19].
have done parallel calculations with SDW order as well.
The results are essentially identical.
We
They are shown
0 5001000 150020002500
0
20
40
60
0.020 0.0250.030
0.6
0.4
0.2
0.0
0.2
0.4
T )
Intensity (arbitrary units)
80
0.6
F (
1/B ( T )
1



FIG. 4: (Color online)The same plot as in Fig. 2, except for
15% doping and DDW order. The parameters are given in
Table I.
0500 100015002000 2500
0
20
40
60
80
0.018 0.022
1/B ( T )
0.0260.03
0.6
0.3
0
0.3
0.6
Intensity (arbitrary units)
F ( T )
1


FIG. 5: (Color online)The same plot as in Fig. 2, except for
16% doping and DDW order. The parameters are given in
Table I.
05001000150020002500
0
20
40
60
80
0.0200.025 0.030
6
4
2
0
2
4
Intensity (arbitrary units)
F (T)
1/B (T )
1



FIG. 6: (Color online) The same plot as in Fig 4, except that
V0 = 0.2t instead of 0.8t. There is now a fragmented electron
pocket centered around 2100T and the main peak is at 183T.
The rest of the parameters are given in Table I.
again for 15 and 16% doping in Fig. 7 and Fig. 8. We have
kept all parameters fixed, while adjusting the the SDW
gap to achieve as best an approximation to experiments
as possible.
For NCCO it is no longer a mystery as to why the
frequency corresponding to the larger electron pocket is
not observed. As we have shown, disorder is the culprit.
Neither is the comparison with ARPES controversial[15],
0 50010001500 20002500
0
20
40
60
80
0.0200.025 0.030
1.0
0.5
0.0
0.5
1.0
Intensity (arbitrary units)
F ( T )
1/B ( T )
1


FIG. 7: (Color online) The same plot as in Fig. 4 for 15%
doping but using SDW order. The main peak is at 195T.
The rest of the parameters are given in Table I.
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4
05001000150020002500
0
20
40
60
80
0.0200.0250.030
1.5
1.0
0.5
0.0
0.5
1.0
1.5
Intensity (arbitrary units)
1/B ( T )
1
F ( T )



FIG. 8: (Color online) The same plot as in Fig. 7, except for
16% doping and using SDW order. The main peak is at 173T.
The rest of the parameters are given in Table I.
as in the case of YBCO, since there is good evidence of
Fermi surface crossing in the direction (π,0) → (π,π),
which is a signature of the electron pocket. The cross
ing along (π,π) → (0,0) can be easily construed as an
evidence of a small hole pocket for which half of it is
made invisible both from the coherence factors and dis
order effects [19]. For electron doped materials, such as
NCCO and PCCO, it is known [15] that the Hall co
efficient changes sign around 17% doping and therefore
the picture of reconnection of the Fermi pockets is en
tirely plausible, with some likely magnetic breakdown ef
fects. The real question is what is the evidence of SDW
or DDW in the relevant doping range between 15% and
17% . From neutron measurements we know that there
is no long range SDW order for doping above 13.4% [23].
We cannot rule out field induced SDW at about 30T,
but the experimental proof is required to show its real
ity. For DDW, there are no corresponding neutron mea
surements. Given that DDW is considerably more hid
den [16, 24] from common experiments, it is more chal
lenging to establish it directly. NMR experiments in high
fields can be valuable.
In the absence of disorder or thermal broadening, the
oscillation waveforms are never sinusoidal in two dimen
sions and contain many Fourier harmonics. At zero tem
perature moderate disorder converts the oscillations to
sinusoidal waveform with rapidly decreasing amplitudes
of the harmonics. Further increase of disorder ultimately
destroys the amplitudes altogether. Many experiments
exhibit roughly sinusoidal waveform at even ultra low
temperatures, implying that disorder is important. The
remarkably small electronic dispersion in the direction
perpendicular to the CuOplanes cannot alone account
for the waveform.
It is unquestionable that the QO experiments are likely
to change the widespread views in the field of high
temperature superconductivity. Although, the measure
ments in YBCO are not fully explained, the measure
ments in NCCO have a clear and simple explanation, as
shown here. However, given the similarity of the phe
nomenon in both hole and electron doped cuprates, it is
likely that the quantum oscillations have the same origin
in both and no particularly exotic mechanism is required.
This work is supported by NSF under the Grant DMR
0705092. All calculations were performed at Hoffman 2
Cluster. We thank E. Abrahams and N. P. Armitage for
a critical reading of the manuscript.
[1] P. W. Anderson, The theory of superconductivity in the
highTc cuprates (Princeton University Press, Princeton,
New Jersey, 1997).
[2] N. DoironLeyraud, C. Proust, D. LeBoeuf, J. Levallois,
J.B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy,
and L. Taillefer, Nature 447, 565 (2007).
[3] A. F. Bangura, J. D. Fletcher, A. Carrington, J. Lev
allois, M. Nardone, B. Vignolle, P. J. Heard, N. Doiron
Leyraud, D. LeBoeuf, L. Taillefer, et al., Phys. Rev. Lett.
100, 047004 (2008).
[4] D. LeBoeuf, N. DoironLeyraud, J. Levallois, R. Daou,
J. B. Bonnemaison, N. E. Hussey, L. Balicas, B. J.
Ramshaw, R. Liang, D. A. Bonn, et al., Nature 450,
533 (2007).
[5] C. Jaudet, D. Vignolles, A. Audouard, J. Levallois,
D. LeBoeuf, N. DoironLeyraud, B. Vignolle, M. Nar
done, A. Zitouni, R. Liang, et al., Phys. Rev. Lett. 100,
187005 (2008).
[6] E. A. Yelland, J. Singleton, C. H. Mielke, N. Harrison,
F. F. Balakirev, B. Dabrowski, and J. R. Cooper, Phys.
Rev. Lett. 100, 047003 (2008).
[7] S. E. Sebastian, N. Harrison, E. Palm, T. P. Murphy,
C. H. Mielke, R. Liang, D. A. Bonn, W. N. Hardy, and
G. G. Lonzarich, Nature 454, 200 (2008).
[8] A. Audouard, C. Jaudet,
D. Bonn, W. Hardy, L. Taillefer, and C. Proust, Phys.
Rev. Lett. 103, 157003 (2009).
[9] J. Singleton, C. De La Cruz, R. D. McDonald, S. Li,
M. Altarawneh, P. Goddard, I. Franke, D. Rickel, C. H.
Mielke, X. Yao, et al., arXiv:0911.2745 (2009).
[10] S. Chakravarty, Science 319, 735 (2008).
[11] B. Vignolle, A. Carrington, R. A. Cooper, M. M. J.
French, A. P. Mackenzie, C. Jaudet, D. Vignolles,
C. Proust, and N. E. Hussey, Nature 455, 952 (2008).
[12] T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner,
M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys.
Rev. Lett. 103, 157002 (2009).
[13] D. Podolsky and H.Y. Kee, Phys. Rev. B 78, 224516
(2008).
[14] A. J. Millis and M. R. Norman, Phys. Rev. B 76, 220503
(2007).
[15] N. P. Armitage,P. Fournier,
arXiv:0906.2931 (2009).
[16] S. Chakravarty, R. B. Laughlin, D. K. Morr, and
C. Nayak, Phys. Rev. B 63, 094503 (2001).
[17] P. M. C. Rourke et al., arXiv:0912.0175.
[18] C. Kusko, R. S. Markiewicz, M. Lindroos, and A. Bansil,
Phys. Rev. B 66, 140513(R) (2002).
[19] X. Jia, P. Goswami, and S. Chakravarty, Phys. Rev. B
80, 134503 (2009).
[20] S. Chakravarty and H.Y. Kee, Proc. Natl. Acad. Sci.
D. Vignolles,R. Liang,
and R. L. Green,
Page 5
5
USA 105, 8835 (2008).
[21] H. K. Nguyen and S. Chakravarty, Phys. Rev. B 65,
180519 (2002).
[22] D. S. Fisher and P. A. Lee, Phys. Rev. B 23, 6851 (1981).
[23] E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K.
Mang, and M. Greven, Nature 445, 186 (2007).
[24] C. Nayak, Phys. Rev. B 62, 4880 (2000).