Des études de la surface de Fermi des cuprates dopés aux trous, par photoémission ou via des mesures d’oscillations quantiques, ont montré qu’un changement majeur s’opère de par et d’autre du point critique quantique, entre la phase métallique à haut dopage et l’isolant de Mott à bas dopage. La surface de Fermi des cuprates à droite du diagramme de phase consiste en une large bande de trous contenant(1+p)porteurs, alors qu’à gauche des oscillations quantiques ont montré l’existence de petites poches d’électrons [31]. La figure1.17résume les deux fréquences observées dans les oscillations quantiques.
Des calculs théoriques sur le type de reconstruction qui pourrait être causée par un ordre de rayures tel que celui décrit plus haut prédisent que la surface de Fermi reconstruite comporterait non seulement les poches d’électrons observées, mais qu’elle présenterait aussi des poches de trous [32]. L’observation d’une fréquence lente associée à une poche de trous rapportée dans [33] vient appuyer ces prédictions.
J. Phys.: Condens. Matter 21 (2009) 164212 L Taillefer
Figure 1. Phase diagram of hole-doped high-Tcsuperconductors.
(a) Schematic doping dependence of the antiferromagnetic (TN) and
superconducting (Tc) transition temperatures and the pseudogap
crossover temperature T∗. The fact that the large hole-like Fermi
surface characteristic of the overdoped metallic state, sketched in panel (c), is modified in the underdoped region (see text) implies that there is a critical doping p∗where Fermi surface reconstruction
occurs. (b) Schematic drawing of one possible reconstruction, that would result from an order with (π, π) wavevector, as in the antiferromagnetic state.
with increased disorder, was the high degree of ortho-II oxygen order in single crystals of YBa2Cu3Oy(YBCO) [11]. Quantum oscillations result from the Landau quantization of states in a magnetic field and the orbiting motion of quasiparticles around the various pockets of the Fermi surface in a metal. Their very observation confirms the existence of a coherent closed Fermi surface and their frequency F is a direct measure of the Fermi surface area, via the relation F = n"0, where n is the carrier density enclosed by the particular Fermi surface associated with a given frequency, and "0 is the quantum of flux.
First observed in the electrical resistance (both Hall and longitudinal; the Shubnikov–de Haas effect) [10], the same oscillations were soon also detected in the de Haas-van Alphen effect (magnetization) [12]. The Fourier transform of the oscillatory spectrum in YBa2Cu3O6.5, reproduced in figure 2 (from [13]), reveals a single frequency at F= 540 T [10,12]. In 2008, quantum oscillations were observed in strongly overdoped Tl2Ba2CuO6+δ(Tl-2201), which also reveal a single frequency, but now at F = 18 kT [14] (see figure 2). This large value matches the area derived previously from ADMR and ARPES measurements on the same material at a similar doping and agrees with n= 1 + p.
Figure 2. Fourier transform of the quantum oscillations detected in YBCO at p= 0.1 and Tl-2201 at p ≃ 0.25. Each reveals a single frequency F, but with vastly different values, as indicated. This shows that the Fermi surface in the underdoped regime includes a pocket which is much smaller than that in the overdoped regime, as sketched in the inset. Courtesy of Cyril Proust; reproduced from [13] with permission.
The contrast between Tl-2201 at p ≈ 0.25 and YBCO at p = 0.1 is dramatic: the Fermi surface area differs by a factor 30 (see figure2). Note that the small pockets detected in underdoped YBCO are not a special feature of the band structure of that particular material, since similar quantum oscillations were observed in the stoichiometric underdoped cuprate YBa2Cu4O8 [15, 16], whose band structure is significantly different [17]. Therefore, this transformation of the Fermi surface from large cylinder to small pockets is a robust signature of the pseudogap phase, which must occur at a T = 0 critical doping p∗ somewhere between 0.1 and 0.25 (see figure1).
3. Electron Fermi surface
A second important fact is that the low-frequency oscillations in YBa2Cu3O6.5 and YBa2Cu4O8 are observed on the background of a negative Hall coefficient RH at low temperature [18] (figure 3(a)). As a function of temperature,
RH(T ) goes from small and positive at high temperature to large and negative as T → 0 (figure 3(b)). Given that RH ∼ 1/n and F ∼ n, this is consistent with the transition from large to small Fermi surface revealed by quantum oscillations, but the fact that RH(T → 0) < 0 implies that the small Fermi surface seen in the underdoped regime must in fact be an electron-like pocket.
The emergence of an electron pocket in the Fermi surface of these hole-doped materials is of fundamental significance: it immediately suggests that the transformation of the Fermi surface is caused by the onset of a new periodicity, typically imposed by some density-wave order [19]. The simplest 2
F i g u r e 1.17 Transformées de Fourier de deux signaux d’oscillations quantiques (figure re-
produite de [34]). En mauve : un cuprate sur-dopé (Tl2Ba2CuO6+δ, p'0.25)et sa grosse surface de trous se traduisant par une fréquence élevée. En rouge : un cuprate sous-dopé (YBa2Cu3O7−δp =0.1) et la basse fréquence associée à de petites poches d’électrons. L’encart montre un schéma de la surface de Fermi générique des cuprates du côté sur-dopé (en mauve) et sous-dopé (en rouge).
22
Signatures de la reconstruction dans les mesures de transport
La reconstruction de la surface de Fermi avec l’apparition de poches d’électrons entraîne un changement brusque dans l’effet Hall, mais qui peut être soit positif soit négatif selon les mobilités des différentes bandes pouvant être impliquées. La figure1.18montre des données d’effet Hall pour trois cuprates, La2−xSrxCuO4, La1.6−xNd0.4SrxCuO4et YBa2Cu3O7−δ , à
un dopage d’environ p=0.12. La reconstruction de la surface de Fermi se manifeste comme une chute de l’effet Hall pour les trois composés à ce dopage faible ; l’amplitude de la chute est beaucoup plus marquée dans YBa2Cu3O7−δ que dans La2−xSrxCuO4. L’anomalie dans
le coefficient de Hall de La1.6−xNd0.4SrxCuO4autour de 75 K est associée à la transition structurale vers la phase tétragonale à basse température.
F i g u r e 1.18 Coefficient de Hall normalisé par sa valeur à 100 K en fonction de la température
pour La2−xSrxCuO4(p=0.13 en bleu, B = 61 T), La1.6−xNd0.4SrxCuO4(p=0.12 en rouge, B = 5 T) et YBa2Cu3O7−δ(p=0.12 en vert, B = 45 T). Les données sont respectivement tirées de [35], [18] et [36].
23 La signature de la reconstruction apparait à une température qui coincide avec l’appa- rition de l’ordre de charge à longue portée, TCO. La figure1.19compare des données de diffraction des rayons-X avec le comportement du coefficient de Hall. La température TCOà laquelle l’ordre de charge est détecté correspond avec la température à laquelle le coefficient de Hall de La1.8−xEu0.2SrxCuO4et YBa2Cu3O7−δ chute sous l’effet de la reconstruction de la
surface de Fermi.J. Phys.: Condens. Matter 21 (2009) 164212 L Taillefer
Figure 5. Stripe order and Hall coefficient in Eu–LSCO at p = 1/8. (a) Temperature dependence of charge order in Eu–LSCO at
p= 1/8, as detected by resonant soft x-ray diffraction (data
from [30]). (b) Hall coefficient versus temperature measured in
B= 15 T for Eu–LSCO (green; left axis; data from [29]) and YBCO (red; right axis; data from [18]) at p= 1/8.
a material isostructural to Eu–LSCO, which exhibits very similar charge and spin ordering. The onset of charge order in the two materials occurs at essentially the same temperature, TCO, as a function of doping. Two measures of
TCO, obtained respectively from x-ray diffraction [30,31] and
nuclear quadrupole resonance (NQR) [32], are plotted in the phase diagram of figure6. In particular, we have compared two samples of Nd–LSCO, respectively at p= 0.20 and 0.24 [33]. In figure 7, we show the in-plane resistivity ρ(T ) and Hall coefficient RH(T ), from [33], and the Seebeck coefficient (or
thermopower) S, plotted as S/T , from [34], as a function of temperature.
At p = 0.24, RH(T ) is flat at low temperature (see
figure7(c)) and equal to the value expected of a single large cylinder containing 1+ p holes, namely RH = +1/e(1 + p),
just as found in Tl-2201 at a similar doping [3]. The other two coefficients, ρ(T ) and S/T , are equally monotonic and featureless. By contrast, at p = 0.20, all three transport coefficients exhibit a pronounced upturn below 40 K, which shows that the Fermi surface has undergone a significant modification. These simultaneous upturns coincide with the onset of charge order detected by NQR at TCO = 40 K [30],
Figure 6. Temperature-doping phase diagram of Nd–LSCO showing the superconducting phase below Tc(open black circles) and the pseudogap region delineated by the crossover temperature T∗
ρ (blue squares). Also shown is the region where static magnetism is observed below TM(full red circles) and charge order is detected below TCO(black diamonds and green circles). These onset temperatures are respectively defined as the temperature below which: (1) the resistance is zero; (2) the in-plane resistivity ρ(T ) deviates from its linear dependence at high temperature; (3) an internal magnetic field is detected by zero-field muon spin relaxation (µSR); (4) charge order is detected by either x-ray diffraction or NQR. All lines are a guide to the eye. The red dashed line shows the onset of spin modulation as detected by neutron diffraction [39]. The blue line (T∗
ρ) above p= 0.20 is made to end at p = 0.24, thereby defining the critical doping where T∗
ρ goes to zero as p∗= 0.24. Experimentally, this point must lie in the range 0.20 < p∗! 0.24, since ρ(T ) remains linear down to the lowest temperature at
p= 0.24 [33]. TMis obtained from the µSR measurements of [36]. The red line is made to end below p= 0.20, as no static magnetism was detected at p= 0.20 down to T = 2 K. TCOis obtained from hard x-ray diffraction on Nd–LSCO (full black diamonds [31]) and from resonant soft x-ray diffraction on Eu–LSCO (open
diamonds [30]). The onset of charge order has been found to coincide with the wipe-out anomaly in NQR, reproduced here from [32] for Nd–LSCO (closed green circles) and Eu–LSCO (open green circles).
as reproduced in figure 7(a). So as in the case of Eu–LSCO at p= 1/8, there is little doubt that stripe order causes Fermi surface reconstruction in Nd-LSCO.
An intriguing difference is that RH(T ) rises below TCOat
p= 0.20 (figure7(c)), while it drops at p= 1/8 (figure5(b)). If the drop in RH(T ) near p= 1/8 is caused by a high-mobility
electron pocket, then the rise at p = 0.20 suggests that this electron pocket is absent at higher doping. Calculations of the Hall coefficient in the stripe-ordered phase [35] reveal that a negative RH requires a finite spin-stripe potential, the cause
of a robust electron pocket in the Fermi surface [23] (as in figure4). The observed evolution from a rise in RHjust below
p∗ to a drop in RH (in some cases to negative values) near p = 1/8, may therefore reflect an increase in the spin-stripe
potential relative to the charge-stripe potential. This would seem consistent with the fact that static spin order detected by muon spin relaxation, whose onset at TMis plotted in figure6,
is absent at p= 0.20 and strongest at p = 0.12 [36]. 4
F i g u r e 1.19 Intensité de la diffraction de rayons-X associée à l’ordre de charge à p=0.125
dans La1.8−xEu0.2SrxCuO4en fonction de la température et coefficient de Hall en fonction de la température à p=0.125 dans La1.8−xEu0.2SrxCuO4(en vert, axe de gauche) et YBa2Cu3O7−δ(en rouge, axe de droite) en fonction de la température (figure reproduite de [37]).
24 L’effet de la reconstruction de la surface de Fermi sur le transport évolue avec le dopage ; alors qu’on observe une chute dans le coefficient de Hall à TCO autour de p = 0.12, la reconstruction cause une remontée dans RHà p =0.20, comme le montre la figure1.20. On y voit des données sur La1.8−xEu0.2SrxCuO4à p=0.12 et La1.6−xNd0.4SrxCuO4à p=0.20.
F i g u r e 1.20 Coefficient de Hall en fonction de la température pour La1.8−xEu0.2SrxCuO4à p=0.125 et La1.6−xNd0.4SrxCuO4à p=0.20 (données sur La1.8−xEu0.2SrxCuO4 tirées de [26], données sur La1.6−xNd0.4SrxCuO4tirées de [25]). On voit que la reconstruction de la surface de Fermi à bas dopage se traduit par une chute de l’effet Hall tandis qu’à haut dopage elle se manifeste plutôt comme une remontée. Au-delà de p = 0.24, aucune remontée (ou autre changement) n’est observée dans l’effet Hall, tel qu’attendu si la reconstruction est associée à l’ordre de charge puisque ce dernier disparait au point critique quantique de p∗ = 0.235 ± 0.005. La différence marquée de comportement sous et au-dessus de p∗est illustrée à la figure1.21. L’effet Hall dans un modèle à une bande dépend du nombre de porteurs. Pour une bande de trous contenant(1+p)
porteurs, le coefficient de Hall attendu à température nulle est de RH =V/ne=V/(1+p)e. Les données reproduites à la figure1.21, montrent qu’à p = 0.24, le coefficient de Hall converge à basse température vers une valeur de RH = 0.45±0.05 mm3C−1. Le nombre de porteurs associé est nH = 1.3±0.15 ce qui est très proche de la valeur attendue de
(1+p) =1.24. Une étude sur l’effet Hall dans le cuprate YBa2Cu3O7−δ montre bien que la
25 La résistivité, qui dévie d’une dépendance linéaire en température sous T∗, subit ensuite dans La1.6−xNd0.4SrxCuO4une forte remontée à des températures plus basses qui coincident avec l’ordre de charge détecté par la résonance quadripolaire [25], tel qu’identifié sur la figure1.13présentée en début de section. La surface de Fermi passe d’une grosse poche de trous à quelques poches d’électrons et de trous. Cette perte de porteurs diminue le nombre de charges disponibles au transport, ce qui pourrait augmenter la résistivité. Elle diminue aussi le nombre de diffuseurs, ce qui pourrait diminuer la résistivité. Selon le composé, le bilan de l’effet de la reconstruction sur la résistivité sera positif ou négatif. Dans YBa2Cu3O7−δ , le désordre est faible et la diffusion est dominée par les collisions entre
porteurs. La reconstruction fait donc chuter la résistivité en diminuant la diffusion. Dans La1.6−xNd0.4SrxCuO4, où le désordre est plus grand, la perte de porteurs est plus importante
que la perte de diffuseurs ce qui augmente la résistivité [
NATURE PHYSICS
DOI: 10.1038/NPHYS1109
25].LETTERS
ρ TNQR RH (mm 3 C ¬1) 0 1 2 0 25 50 75 100 T (K) 0 100 200 (µ Ω c m) Nd-LSCO p = 0.20 p = 0.20 p = 0.24
Figure 4|Normal-state Hall coefficient. Hall coefficient R
H(T) of
Nd-LSCO as a function of temperature for p = 0.20 and 0.24, measured in
a magnetic field of 15 T. Below 12 K, the 0.20 data are in 33 T, a magnetic
field strong enough to fully suppress superconductivity (see Supplementary
Information). The dashed blue horizontal line is the value of R
Hcalculated
for a large cylindrical Fermi surface enclosing 1+p holes, namely R
H= V/e
(1+p), at p = 0.24. At p = 0.20, the rise in R
H(T) at low temperature
signals a modification of this large Fermi surface. The upturn is seen to
coincide with a simultaneous upturn in ⇢(T) (reproduced in black from
Fig. 1) and with the onset of charge order at T
NQRas detected by NQR
(see the text and ref. 21).
cot✓
H(T ) ⇠ ⇢
ab(T )/R
H(T ), is also linear at low temperature. We
infer that a single anomalous scattering process dominates the
electron–electron correlations at low temperature at p
⇤(or just
above). This shows that the Fermi-liquid behaviour observed at
p = 0.3 (in LSCO), where ⇢
ab(T ) ⇠ T
2below T ⇡ 50 K (ref. 11),
breaks down just before the onset of the pseudogap phase at p
⇤.
This kind of ‘non-Fermi-liquid’ behaviour, whereby ⇢(T ) ⇠ T
as T ! 0, has typically been observed in heavy-fermion metals
at the quantum critical point where the onset temperature for
antiferromagnetic order goes to zero
3. It is also consistent with the
marginal-Fermi-liquid description of cuprates
15.
In summary, our experimental findings offer compelling
evidence that the pseudogap phase ends at a T = 0 critical point
p
⇤located below the onset of superconductivity (at p
c
⇡ 0.27), in
agreement with previous but more indirect evidence from other
hole-doped copper oxides
16. Moreover, they impose two strong
new constraints on theories of the pseudogap phase: (1) its onset
below p
⇤modifies the large Fermi surface characteristic of the
overdoped metallic state; (2) quasiparticle scattering at p
⇤is linear
in temperature as T ! 0.
The existence of a quantum critical point is consistent with two
kinds of theory of the pseudogap phase. The first kind invokes
the onset of an order, with some associated broken symmetry
6–8.
Because T
⇤marks a crossover and not a sharp transition, this
order is presumably short range or fluctuating. In the electron-
doped copper oxides, for example, the pseudogap phase has
been interpreted as a fluctuating precursor of the long-range
antiferromagnetic order that sets in at lower temperature
17, and
the signatures of the pseudogap critical point in transport are
similar to those found here: a linear-T resistivity as T ! 0 (ref. 18)
and a sharp change in R
H(T = 0) (ref. 19). For Nd-LSCO and
LSCO, an analogous scenario would be ‘stripe’ fluctuations, as a
precursor to the static spin and charge modulations observed at
lower temperature
20. Note that in Nd-LSCO at p = 0.20 the onset
of the upturn in ⇢(T ) and R
H(T ) at T
min= 37 K coincides with the
loss of NQR intensity at T
NQR= 40 ± 6 K (ref. 21) (see Fig. 4). In
Nd-LSCO at p = 0.15, this so-called ‘wipe-out’ anomaly in NQR at
T
NQR= 60 ± 6 K (ref. 21) was shown to coincide with the onset of
charge order measured via hard-X-ray diffraction, at T
ch= 62±5 K
(ref. 22) (see Fig. 3). Direct evidence of a charge modulation via
resonant soft-X-ray diffraction was reported recently for the closely
related material Eu-LSCO, with T
ch= 70±10 K at p = 0.15 (ref. 23),
whereas T
NQR= 60 ± 6 K (ref. 21) in Eu-LSCO at p = 0.16 (see
Fig. 3). Clearly, the upturn in ⇢(T ) is correlated with the onset of
charge order in these two materials. While the correlation between
T
NQRand T
minhas been noted previously
20, the mechanism causing
the upturn in ⇢(T ) remained unclear. Our data show that the
mechanism is a change in Fermi surface, and the positive rise in
R
H(T ) imposes a strong constraint on the topology of the resulting
Fermi surface. An additional constraint comes from the fact that
R
H(T ) drops to negative values near p = 1/8, not only in Nd-LSCO
(ref. 13) and other materials with ‘stripe’ order
24,25, but also in
YBa
2Cu
3O
y(ref. 26).
Recent calculations of the Fermi-surface reconstruction caused
by stripe order are consistent with a negative R
Hnear p = 1/8 in
that spin stripes tend to generate an electron pocket in the Fermi
surface
27. Interestingly, charge stripes do not
27, and this might
explain the positive rise in R
Hseen at higher doping, provided that
stripe order involves predominantly charge order at high doping (in
line with the fact that charge order sets in at a higher temperature
than spin order
20,21).
In the other kind of theory of the pseudogap phase, the
critical point reflects a T = 0 transition from small hole pockets,
characteristic of a doped Mott insulator, to a large hole pocket,
without symmetry breaking
4,5. Recent work suggests that the
quasiparticle scattering rate above such a critical point may indeed
grow linearly with temperature
28. Although calculations are needed
to confirm this, a change in carrier density from n ⇡ p to n = 1+p
would seem natural in this kind of scenario. However, it is more
difficult to see what could cause the negative values of R
H(T ! 0)
near p = 1/8. It seems that stripe order or fluctuations would have
to be invoked as a secondary instability inside the pseudogap phase,
with an onset in doping that would be essentially simultaneous with
p
⇤in the case of Nd-LSCO.
We end by comparing our results qualitatively with those of
previous high-field studies on LSCO. The resistivity shows very
similar features at high temperature: linear T above T
⇤(ref. 14) and
an upturn below T
⇤(ref. 29). The Hall coefficient of LSCO (ref. 30),
on the other hand, has a more subtle and complex evolution with
doping than that presented here for Nd-LSCO, which makes it
harder to pinpoint p
⇤using the same criteria as we have used above.
Nonetheless, it seems likely that the same fundamental mechanisms
are responsible for both the linear-T resistivity and the resistivity
upturns, and for the onset of the pseudogap at T
⇤, in both LSCO
and Nd-LSCO.
Methods
Single crystals of La
2 y xNd
ySr
xCuO
4(Nd-LSCO) were grown with a Nd content
y = 0.4 using a travelling-float-zone technique and cut from boules with nominal
Sr concentrations x = 0.20 and 0.25. The actual doping p of each crystal was
estimated from its T
cand ⇢(250 K) values compared with published data, giving
p = 0.20±0.005 and 0.24±0.005, respectively. The resistivity ⇢ and Hall coefficient
R
Hwere measured at the National High Magnetic Field Laboratory (NHMFL)
in Tallahassee in steady magnetic fields up to 35 T and in Sherbrooke in steady
fields up to 15 T. The field was always applied along the c axis. Neither ⇢ nor R
Hshowed any field dependence up to the highest fields. More details are available in
Supplementary Information.
Received 5 June 2008; accepted 24 September 2008;
published online 2 November 2008
References
1. Lee, P. A., Nagaosa, N. & Wen, X.-G. Doping a Mott insulator: Physics of
high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006).
2. Timusk, T. & Statt, B. The pseudogap in high-temperature superconductors:
An experimental survey. Rep. Prog. Phys. 62, 61–122 (1999).
NATURE PHYSICS| VOL 5 | JANUARY 2009 | www.nature.com/naturephysics 33
F i g u r e 1.21 Coefficient de Hall mesuré avec un champ magnétique suffisamment fort pour