- Email: [email protected]
Spectroscopic imaging STM studies of broken electronic symmetries in underdoped cuprates
Physica B 407 (2012) 1859–1863
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Spectroscopic imaging STM studies of broken electronic symmetries in underdoped cuprates K. Fujita a,b,c,n, A. Mesaros d, M.J. Lawler e,a, S. Sachdev f, J. Zaanen d, H. Eisaki g, S. Uchida c, E.-A. Kim a, J.C. Davis a,b,h a
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973, USA c Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan d Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, The Netherlands e Department of Physics, Applied Physics and Astronomy, Binghamton University, Binghamton, NY 13902-6000, USA f Department of Physics, Harvard University, Boston, MA 02138, USA g Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan h School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland b
a r t i c l e i n f o
abstract
Available online 8 January 2012
We use spectroscopic imaging scanning tunneling microscopy (SI-STM) to visualize the spatial symmetries of the electronic states that occur at the pseudogap energy scale in underdoped cuprates. We find evidence for the local intra-unit-cell electronic nematicity—by which we mean the disordered breaking of C4v symmetry within each CuO2 unit cell [1]. We also find that the coexisting incommensurate (smectic) electronic modulations couple to the intra-unit-cell nematicity through their 2p topological defects [2]. & 2012 Elsevier B.V. All rights reserved.
In general for underdoped cuprates, the electronic excitations at the pseudogap energies E D1 are observed to be highly anomalous. They are associated with a strong antinodal pseudogap in k-space [3,4], they exhibit slow dynamics without recombination to form Cooper pairs [5], their Raman characteristics appear distinct from expectations for a d-wave superconductor [6], and they appear not to contribute to superfluid density [7]. The spatial symmetries of these states can be explored using high resolution SI-STM. This approach allows the simultaneous visualization of incommensurate electronic-structure modulations that break translational and rotational symmetry (smectic), and intraunit-cell electronic structure that does not break translational symmetry but does break rotational symmetry (nematic). The relationship between these electronic phenomena which, because they break different symmetries are distinct, can also be explored directly using this approach. To pursue these objectives, we have introduced several new SI-STM techniques [1,2]. We measure the differential conductance gðr,EÞ ¼ dI=dV ðr,E ¼ eVÞ simultaneously with topograph TðrÞ on the strongly
n Corresponding author at: LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA. E-mail address: [email protected] (K. Fujita).
0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.01.049
underdoped Bi2 Sr2 CaCu2 O8 þ d (Tc ¼50 K) at p 0:08 of hole density. Tip-sample junction for the spectroscopic measurement of gðr,EÞ is formed with Is ¼ 100 pA, V s ¼ 200 mV and gðr,EÞ spectra are taken from 200 mV to 200 mV at all the locations in a field of view at T ¼ 4.2 K. In order to characterize the spatial symmetry of the electronic structure, we use Zðr,EÞ gðr,E ¼ þ eVÞ= gðr,E ¼ eVÞ in which potentially severe systematic errors are effectively cancelled [9,10]. Atomically resolved r-space images of the static phenomena in Zðr,EÞ show highly similar spatial patterns at all energies near D1 but with variations of intensity due to the D1 disorder [10]. By changing to reduced energy variables E-e E=D1 ðrÞ and imaging Zðr,eÞ it becomes clear that these spatial patterns associated with E ¼ D1 ðrÞ appear altogether at e¼1 with a strong maximum in intensity [9]. Thus the pseudogap states of underdoped cuprates locally break translational symmetry, and reduce the expected 901-rotational (C4) symmetry of states within the unit cell to at least 1801-rotational (C2) symmetry [9,10], and possibly to an even lower symmetry. To find which spatial symmetries are actually broken by the cuprate pseudogap states, we use sub-unit-cell resolution imaging performed on multiple different underdoped Bi2 Sr2 CaCu2 O8 þ d samples with Tc between 20 K and 55 K. The required registry of the Cu sites in each Zðr,eÞ is carried out by a picometer scale transformation which renders the topographic image TðrÞ perfectly a0-periodic [1]; the same transformation is then applied
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to the simultaneously obtained Zðr,eÞ to register all the electronic structure data to this ideal lattice. To visualize the separate broken symmetries in the E D1 electronic structure, we consider Zðq,e ¼ 1Þ in Fig. 1a inset; this is the Fourier-space representation of electronic structure of the E D1 states. Taking into account only the Bragg peaks at Q x and Q y (red circles in Fig. 1a inset) the coarse grained image for disordered C4 symmetry breaking of intra-unit-cell electronic structure of Fig. 1a is revealed as shown in Fig. 1b. By contrast, if one focuses upon the incommensurate modulations S x and S y (blue circles in Fig. 1a), we find a disordered electronic structure which breaks both C4 and translational symmetry locally as shown in Fig. 1c. Although these two types of electronic phenomena represent clearly distinct broken symmetries, SI-STM reveals that they coexist in the E D1 pseudogap electronic structure of underdoped cuprates [1]. In order to investigate the intra-unit-cell C4 breaking for more detail, here we focus on Q ¼ Q x and Q y . The topograph TðrÞ is shown in Fig. 2a and the inset compares the Bragg peaks of its real (in-phase) Fourier components Re TðQ x Þ and Re TðQ y Þ demonstrating that Re TðQ x Þ=Re TðQ y Þ ¼ 1. Therefore, TðrÞ preserves the C4 symmetry of the crystal lattice. In contrast, Fig. 2b shows that the Zðr,e ¼ 1Þ determined simultaneously with Fig. 2a breaks various crystal symmetries [1]. The inset of Fig. 2b shows that since Re TðQ x Þ=Re TðQ y Þ a 1 the pseudogap states break C4 symmetry on the average throughout the field-of view (FOV) in Fig. 2b. We defined a normalized measure of intra-unit cell C2 symmetry as a function of e: OQN ðeÞ
Re ZðQ y ,eÞRe ZðQ x ,eÞ Z ðeÞ
Qy
Qx Sx
Sy
x
y 4 nm
0.47
1.61
ð1Þ
where Z ðeÞ is the spatial average of Zðr,eÞ. The plot of OQN ðeÞ in Fig. 2c shows that the magnitude of OQN ðeÞ is low for e5 D0 =D1 , begins to grow near e D0 =D1 , and becomes well defined as e 1 or E D1 (D0 is defined as a kink in local density of states [8]). Fig. 2d shows the topographic image of a representative region from Fig. 2a; the locations of each Cu site R and of the two O atoms within its unit cell are indicated. Fig. 2e shows Zðr,e ¼ 1Þ measured simultaneously with Fig. 2d with same Cu and O site labels. Next we define X Z x ðR,eÞZ y ðR,eÞ ORN ðeÞ ð2Þ Z ðeÞN R where Z x ðR,eÞ is the magnitude of Zðr,eÞ at the O site a0 =2 along the x-axis from R while Z y ðR,eÞ is the equivalent along the y-axis, and N is the number of unit cells. This is the r-space measure of C2 symmetry which is approximately equivalent of OQN ðeÞ in Eq. (1) but counting only O site contributions. Fig. 2f contains the calculated value of ORN ðeÞ from the same FOV as Fig. 2a and b revealing the good agreement with OQN ðeÞ. This independent quantitative measure of intra-unit-cell electronic nematicity again shows that C4 symmetry is strongly broken in the pseudogap states and, moreover, that the intra-unit-cell C4 symmetry breaking is due primarily to electronic inequivalence of the two O sites within each unit cell. For all samples studied, the low values of Zðr,eÞ found for at low e occur because these states are dispersive Bogoliubov quasiparticles [10,11] and cannot be analyzed in terms of any static electronic structure, smectic or otherwise. More importantly 9OQS ðeÞ9 shows no tendency to become well established at the pseudogap or any other energy [1]. The distinct properties of the smectic electronic structure modulations at E D1 can be examined independently of the intra-unit-cell symmetry breaking by focusing only upon the incommensurate modulation peaks S x and S y . A coarse grained image of the local smectic symmetry breaking (Fig. 1c) reveals the
x
y
On ( r )
4 nm
-0.016
-0.002 π
2π
0
π/2
3π/2
0, 2π
x
y 4 nm
-0.02
Os ( r ) +0.02
Fig. 1. (a) Zðr,e ¼ 1Þ with its fourier transform in the inset. (b) Coarse grained image of C4 symmetry breaking at Q ¼ Q x ,Q y . (c) Coarse grained image of C4 symmetry breaking at Q ¼ S x ,S y . Spatial resolution of coarse graining average is set by the circle in the inset of a for each Q . (c inset) Schematic image of an edge dislocation in a crystalline solid (solid circles indicate atomic locations) and in the two dimensional smectic phase of a liquid crystal (solid white lines indicate modulation period). In both cases it is the spatial phase of periodic modulations which winds around the dislocation core by precisely 2p. Inset—schematic image of a superfluid or superconducting vortex overlapped with its phase field which winds by exactly 2p. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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0
0.030
0.002
X Y
e=1 ω ~ Δ1
1.0
1.2
0.8
q ( 2π / a0 )
Y
1.0
1.2
q ( 2π / a0 )
Y
X
ON
0.8
Q
0
Re Z
X Y
0
Re T
K. Fujita et al. / Physica B 407 (2012) 1859–1863
-0.010
X
0.2 2 nm
High
Low
2 nm
1.5
0.6 0.8 e ( = ω / Δ1)
1.0
1.2
f
Cu Ox Oy
0 e=1 ω ~ Δ1 ON
R
Cu Ox Oy
0.5
0.4
O
ON
Y
Y
X
Cu
ON
X -0.040
d 5Å
e Low
High
5Å
0.2 0.5
1.5
0.4
0.6 0.8 e ( = ω / Δ1)
1.0
1.2
Fig. 2. (a) Topographic image TðrÞ. Inset shows real part of Bragg peak intensity in the fourier transform of TðrÞ along x and y direction. (b) Pseudogap states Zðr,e ¼ 1Þ. Inset shows real part of Bragg peak intensity in the fourier transform of Zðr,e ¼ 1Þ along x and y direction. (c) The value of OQN ðeÞ in Eq. (1) computed from Zðr,eÞ data measured in the same FOV as a and b. Magnitude is low for all e o D0 =D1 and then rises rapidly to become well established at e 1 or E ¼ D1 . (d) TðrÞ in the box of Fig (a). (e) Zðr,e ¼ 1Þ in the box of Fig. b. (f) Same as (c) but the value of ORN ðeÞ in Eq. (2). As in (c), its magnitude is low for all e o D0 =D1 and then rises rapidly to become well established at e 1 or E ¼ D1 . If the function in Eq. (2) is evaluated using the Cu sites only the C4 symmetry breaking is zero (black diamonds) as it must be.
very short correlation length of the strongly disordered smectic modulations [9,10]. The amplitude and phase of two unidirectional modulation components (along x, y) within box in Fig. 1a can be further extracted as shown in Figs. 3a and b [2]. To do so, we denote the local contribution to the S x modulations at position r by a complex field C1 ðrÞ. This contributes to the Zðr,e ¼ 1Þ data as
C1 ðrÞeiSx r þ Cn1 ðrÞeiSx r 29C1 ðrÞ9 cosðS x r þ f1 ðrÞÞ
ð3Þ
thus allowing the local phase f1 ðrÞ of S x modulations to be mapped; similarly for the local phase f2 ðrÞ of the S y modulations. In Figs. 3c and d we show images of f1 ðrÞ and f2 ðrÞ derived from Zðr,e ¼ 1Þ . They reveal that the smectic phases f1 ðrÞ and f2 ðrÞ take on all values between 0 and 72p in a highly complex spatial pattern. More significant is the observation of large numbers of topological defects with 72p phase winding. These are indicated by black (þ 2p) and white (2p) circles in Figs. 3c and d and occur in approximately equal numbers. A typical example of an individual dislocation and its associated topological defect are seen in the solid box in Figs. 3a and c, respectively. The dislocation core and its associated 2p phase winding are clearly seen. Moreover the amplitude of C1 ðrÞ or C2 ðrÞ always goes to zero near each topological defect. These data are all in excellent agreement with the theoretical expectations for quantum smectic dislocations (Fig. 1c inset).
Imaging the locations of topological defects (Figs. 3c and d) with the intra-unit-cell nematicity (Fig. 2b) reveals another key result. Fig. 3e shows the locations of all topological defects in Figs. 3c and d plotted as black dots on the simultaneously obtained dON ¼ ON ðrÞ/ON S representing the fluctuations of the intra-unit-cell nematicity. By eye, nearly all the topological defects appear located in white regions of vanishing dON ðrÞ ¼ 0. This can be quantified by plotting the distribution of distances of topological defects from the nearest zero of dON ðrÞ ¼ 0, thereby showing that they are far smaller than expected if the topological defects were uncorrelated with dON ðrÞ(Fig. 3e inset). These data provide empirical evidence for a coupling between the smectic topological defects and the fluctuations of the intra-unit-cell nematicity at E D1 . Thus, when Zðr,EÞ images of the intra-unit-cell electronic structure in underdoped Bi2 Sr2 CaCu2 O8 þ d are analyzed using two independent techniques, compelling evidence for intraunit-cell electronic symmetry breaking is detected specifically of the states at the E D1 pseudogap energy. Moreover, this intraunit-cell symmetry breaking coexists with finite Q ¼ S x ,S y smectic electronic modulations, and the coupling between these states are achieved through the 2p topological defects creating spatial fluctuation of the nematicity. To study the doping dependence of these phenomena and the relationship to superconductivity is next challenge.
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K. Fujita et al. / Physica B 407 (2012) 1859–1863
→
→
Ψ1 ( r )
Ψ2 ( r )
2 nm low
2 nm high
4 nm
Counts
0 10 8 6 4 2 0
4 nm
2π
-
dex drand
δOn > 0
0 20 40 60 80 d ( pix )
+
→
l
δOn < 0
→
→
∇ ϕ( r ) 4 nm -0.0043
0
2π
+0.0043
Fig. 3. (a and b) C1 ðrÞ and C2 ðrÞ. (c and d) f1 ðrÞ and f2 ðrÞ. (e) dON ðrÞ with topological defects (black dots). Inset shows histogram of distance between topological defects and the nearest point on the dON ðrÞ ¼ 0 contour for experimental (red) and randomly generated defects (blue). (f) Schematic image of how topological defects create dON ðrÞ fluctuations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
K. Fujita et al. / Physica B 407 (2012) 1859–1863
Acknowledgments We are grateful to P. Abbamonte, D. Bonn, J.C. Campuzano, D.M. Eigler, E. Fradkin, T. Hanaguri, W. Hardy, J.E. Hoffman, S. Kivelson, A.P. Mackenzie, M. Norman, B. Ramshaw, G. Sawatzky, J.P. Sethna, H. Takagi, and J. Tranquada, for helpful discussions and communications. References [1] M.J. Lawler, K. Fujita, et al., Nature 466 (2010) 347. [2] A. Mesaros, K. Fujita, et al., Science 333 (2011) 426.
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[3] A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 71 (2003) 473. [4] J.C. Campuzano, M.R. Norman, M. Randeria, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Superconductors, vol. II, Springer, New York, 2004, pp. 167–273. [5] N. Gedik, J. Orenstein, R. Liang, D.A. Bonn, W.N. Hardy, Science 300 (2003) 1410. [6] M.L. Tacon, et al., Nat. Phys. 2 (2006) 537. [7] R. Khasanov, et al., Phys. Rev. B 82 (2010) 020511. (R). [8] J.W. Alldredge, et al., Nat. Phys. 4 (2008) 319. [9] Y. Kohsaka, et al., Science 315 (2007) 1380. [10] Y. Kohsaka, et al., Nature 454 (2009) 1072. [11] K. McElroy, et al., Nature 422 (2003) 592.
Contents lists available at SciVerse ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Spectroscopic imaging STM studies of broken electronic symmetries in underdoped cuprates K. Fujita a,b,c,n, A. Mesaros d, M.J. Lawler e,a, S. Sachdev f, J. Zaanen d, H. Eisaki g, S. Uchida c, E.-A. Kim a, J.C. Davis a,b,h a
LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA CMPMS Department, Brookhaven National Laboratory, Upton, NY 11973, USA c Department of Physics, The University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan d Instituut-Lorentz for Theoretical Physics, Universiteit Leiden, The Netherlands e Department of Physics, Applied Physics and Astronomy, Binghamton University, Binghamton, NY 13902-6000, USA f Department of Physics, Harvard University, Boston, MA 02138, USA g Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki 305-8568, Japan h School of Physics and Astronomy, University of St. Andrews, North Haugh, St. Andrews, Fife KY16 9SS, Scotland b
a r t i c l e i n f o
abstract
Available online 8 January 2012
We use spectroscopic imaging scanning tunneling microscopy (SI-STM) to visualize the spatial symmetries of the electronic states that occur at the pseudogap energy scale in underdoped cuprates. We find evidence for the local intra-unit-cell electronic nematicity—by which we mean the disordered breaking of C4v symmetry within each CuO2 unit cell [1]. We also find that the coexisting incommensurate (smectic) electronic modulations couple to the intra-unit-cell nematicity through their 2p topological defects [2]. & 2012 Elsevier B.V. All rights reserved.
In general for underdoped cuprates, the electronic excitations at the pseudogap energies E D1 are observed to be highly anomalous. They are associated with a strong antinodal pseudogap in k-space [3,4], they exhibit slow dynamics without recombination to form Cooper pairs [5], their Raman characteristics appear distinct from expectations for a d-wave superconductor [6], and they appear not to contribute to superfluid density [7]. The spatial symmetries of these states can be explored using high resolution SI-STM. This approach allows the simultaneous visualization of incommensurate electronic-structure modulations that break translational and rotational symmetry (smectic), and intraunit-cell electronic structure that does not break translational symmetry but does break rotational symmetry (nematic). The relationship between these electronic phenomena which, because they break different symmetries are distinct, can also be explored directly using this approach. To pursue these objectives, we have introduced several new SI-STM techniques [1,2]. We measure the differential conductance gðr,EÞ ¼ dI=dV ðr,E ¼ eVÞ simultaneously with topograph TðrÞ on the strongly
n Corresponding author at: LASSP, Department of Physics, Cornell University, Ithaca, NY 14853, USA. E-mail address: [email protected] (K. Fujita).
0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2012.01.049
underdoped Bi2 Sr2 CaCu2 O8 þ d (Tc ¼50 K) at p 0:08 of hole density. Tip-sample junction for the spectroscopic measurement of gðr,EÞ is formed with Is ¼ 100 pA, V s ¼ 200 mV and gðr,EÞ spectra are taken from 200 mV to 200 mV at all the locations in a field of view at T ¼ 4.2 K. In order to characterize the spatial symmetry of the electronic structure, we use Zðr,EÞ gðr,E ¼ þ eVÞ= gðr,E ¼ eVÞ in which potentially severe systematic errors are effectively cancelled [9,10]. Atomically resolved r-space images of the static phenomena in Zðr,EÞ show highly similar spatial patterns at all energies near D1 but with variations of intensity due to the D1 disorder [10]. By changing to reduced energy variables E-e E=D1 ðrÞ and imaging Zðr,eÞ it becomes clear that these spatial patterns associated with E ¼ D1 ðrÞ appear altogether at e¼1 with a strong maximum in intensity [9]. Thus the pseudogap states of underdoped cuprates locally break translational symmetry, and reduce the expected 901-rotational (C4) symmetry of states within the unit cell to at least 1801-rotational (C2) symmetry [9,10], and possibly to an even lower symmetry. To find which spatial symmetries are actually broken by the cuprate pseudogap states, we use sub-unit-cell resolution imaging performed on multiple different underdoped Bi2 Sr2 CaCu2 O8 þ d samples with Tc between 20 K and 55 K. The required registry of the Cu sites in each Zðr,eÞ is carried out by a picometer scale transformation which renders the topographic image TðrÞ perfectly a0-periodic [1]; the same transformation is then applied
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K. Fujita et al. / Physica B 407 (2012) 1859–1863
to the simultaneously obtained Zðr,eÞ to register all the electronic structure data to this ideal lattice. To visualize the separate broken symmetries in the E D1 electronic structure, we consider Zðq,e ¼ 1Þ in Fig. 1a inset; this is the Fourier-space representation of electronic structure of the E D1 states. Taking into account only the Bragg peaks at Q x and Q y (red circles in Fig. 1a inset) the coarse grained image for disordered C4 symmetry breaking of intra-unit-cell electronic structure of Fig. 1a is revealed as shown in Fig. 1b. By contrast, if one focuses upon the incommensurate modulations S x and S y (blue circles in Fig. 1a), we find a disordered electronic structure which breaks both C4 and translational symmetry locally as shown in Fig. 1c. Although these two types of electronic phenomena represent clearly distinct broken symmetries, SI-STM reveals that they coexist in the E D1 pseudogap electronic structure of underdoped cuprates [1]. In order to investigate the intra-unit-cell C4 breaking for more detail, here we focus on Q ¼ Q x and Q y . The topograph TðrÞ is shown in Fig. 2a and the inset compares the Bragg peaks of its real (in-phase) Fourier components Re TðQ x Þ and Re TðQ y Þ demonstrating that Re TðQ x Þ=Re TðQ y Þ ¼ 1. Therefore, TðrÞ preserves the C4 symmetry of the crystal lattice. In contrast, Fig. 2b shows that the Zðr,e ¼ 1Þ determined simultaneously with Fig. 2a breaks various crystal symmetries [1]. The inset of Fig. 2b shows that since Re TðQ x Þ=Re TðQ y Þ a 1 the pseudogap states break C4 symmetry on the average throughout the field-of view (FOV) in Fig. 2b. We defined a normalized measure of intra-unit cell C2 symmetry as a function of e: OQN ðeÞ
Re ZðQ y ,eÞRe ZðQ x ,eÞ Z ðeÞ
Qy
Qx Sx
Sy
x
y 4 nm
0.47
1.61
ð1Þ
where Z ðeÞ is the spatial average of Zðr,eÞ. The plot of OQN ðeÞ in Fig. 2c shows that the magnitude of OQN ðeÞ is low for e5 D0 =D1 , begins to grow near e D0 =D1 , and becomes well defined as e 1 or E D1 (D0 is defined as a kink in local density of states [8]). Fig. 2d shows the topographic image of a representative region from Fig. 2a; the locations of each Cu site R and of the two O atoms within its unit cell are indicated. Fig. 2e shows Zðr,e ¼ 1Þ measured simultaneously with Fig. 2d with same Cu and O site labels. Next we define X Z x ðR,eÞZ y ðR,eÞ ORN ðeÞ ð2Þ Z ðeÞN R where Z x ðR,eÞ is the magnitude of Zðr,eÞ at the O site a0 =2 along the x-axis from R while Z y ðR,eÞ is the equivalent along the y-axis, and N is the number of unit cells. This is the r-space measure of C2 symmetry which is approximately equivalent of OQN ðeÞ in Eq. (1) but counting only O site contributions. Fig. 2f contains the calculated value of ORN ðeÞ from the same FOV as Fig. 2a and b revealing the good agreement with OQN ðeÞ. This independent quantitative measure of intra-unit-cell electronic nematicity again shows that C4 symmetry is strongly broken in the pseudogap states and, moreover, that the intra-unit-cell C4 symmetry breaking is due primarily to electronic inequivalence of the two O sites within each unit cell. For all samples studied, the low values of Zðr,eÞ found for at low e occur because these states are dispersive Bogoliubov quasiparticles [10,11] and cannot be analyzed in terms of any static electronic structure, smectic or otherwise. More importantly 9OQS ðeÞ9 shows no tendency to become well established at the pseudogap or any other energy [1]. The distinct properties of the smectic electronic structure modulations at E D1 can be examined independently of the intra-unit-cell symmetry breaking by focusing only upon the incommensurate modulation peaks S x and S y . A coarse grained image of the local smectic symmetry breaking (Fig. 1c) reveals the
x
y
On ( r )
4 nm
-0.016
-0.002 π
2π
0
π/2
3π/2
0, 2π
x
y 4 nm
-0.02
Os ( r ) +0.02
Fig. 1. (a) Zðr,e ¼ 1Þ with its fourier transform in the inset. (b) Coarse grained image of C4 symmetry breaking at Q ¼ Q x ,Q y . (c) Coarse grained image of C4 symmetry breaking at Q ¼ S x ,S y . Spatial resolution of coarse graining average is set by the circle in the inset of a for each Q . (c inset) Schematic image of an edge dislocation in a crystalline solid (solid circles indicate atomic locations) and in the two dimensional smectic phase of a liquid crystal (solid white lines indicate modulation period). In both cases it is the spatial phase of periodic modulations which winds around the dislocation core by precisely 2p. Inset—schematic image of a superfluid or superconducting vortex overlapped with its phase field which winds by exactly 2p. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1861
0
0.030
0.002
X Y
e=1 ω ~ Δ1
1.0
1.2
0.8
q ( 2π / a0 )
Y
1.0
1.2
q ( 2π / a0 )
Y
X
ON
0.8
Q
0
Re Z
X Y
0
Re T
K. Fujita et al. / Physica B 407 (2012) 1859–1863
-0.010
X
0.2 2 nm
High
Low
2 nm
1.5
0.6 0.8 e ( = ω / Δ1)
1.0
1.2
f
Cu Ox Oy
0 e=1 ω ~ Δ1 ON
R
Cu Ox Oy
0.5
0.4
O
ON
Y
Y
X
Cu
ON
X -0.040
d 5Å
e Low
High
5Å
0.2 0.5
1.5
0.4
0.6 0.8 e ( = ω / Δ1)
1.0
1.2
Fig. 2. (a) Topographic image TðrÞ. Inset shows real part of Bragg peak intensity in the fourier transform of TðrÞ along x and y direction. (b) Pseudogap states Zðr,e ¼ 1Þ. Inset shows real part of Bragg peak intensity in the fourier transform of Zðr,e ¼ 1Þ along x and y direction. (c) The value of OQN ðeÞ in Eq. (1) computed from Zðr,eÞ data measured in the same FOV as a and b. Magnitude is low for all e o D0 =D1 and then rises rapidly to become well established at e 1 or E ¼ D1 . (d) TðrÞ in the box of Fig (a). (e) Zðr,e ¼ 1Þ in the box of Fig. b. (f) Same as (c) but the value of ORN ðeÞ in Eq. (2). As in (c), its magnitude is low for all e o D0 =D1 and then rises rapidly to become well established at e 1 or E ¼ D1 . If the function in Eq. (2) is evaluated using the Cu sites only the C4 symmetry breaking is zero (black diamonds) as it must be.
very short correlation length of the strongly disordered smectic modulations [9,10]. The amplitude and phase of two unidirectional modulation components (along x, y) within box in Fig. 1a can be further extracted as shown in Figs. 3a and b [2]. To do so, we denote the local contribution to the S x modulations at position r by a complex field C1 ðrÞ. This contributes to the Zðr,e ¼ 1Þ data as
C1 ðrÞeiSx r þ Cn1 ðrÞeiSx r 29C1 ðrÞ9 cosðS x r þ f1 ðrÞÞ
ð3Þ
thus allowing the local phase f1 ðrÞ of S x modulations to be mapped; similarly for the local phase f2 ðrÞ of the S y modulations. In Figs. 3c and d we show images of f1 ðrÞ and f2 ðrÞ derived from Zðr,e ¼ 1Þ . They reveal that the smectic phases f1 ðrÞ and f2 ðrÞ take on all values between 0 and 72p in a highly complex spatial pattern. More significant is the observation of large numbers of topological defects with 72p phase winding. These are indicated by black (þ 2p) and white (2p) circles in Figs. 3c and d and occur in approximately equal numbers. A typical example of an individual dislocation and its associated topological defect are seen in the solid box in Figs. 3a and c, respectively. The dislocation core and its associated 2p phase winding are clearly seen. Moreover the amplitude of C1 ðrÞ or C2 ðrÞ always goes to zero near each topological defect. These data are all in excellent agreement with the theoretical expectations for quantum smectic dislocations (Fig. 1c inset).
Imaging the locations of topological defects (Figs. 3c and d) with the intra-unit-cell nematicity (Fig. 2b) reveals another key result. Fig. 3e shows the locations of all topological defects in Figs. 3c and d plotted as black dots on the simultaneously obtained dON ¼ ON ðrÞ/ON S representing the fluctuations of the intra-unit-cell nematicity. By eye, nearly all the topological defects appear located in white regions of vanishing dON ðrÞ ¼ 0. This can be quantified by plotting the distribution of distances of topological defects from the nearest zero of dON ðrÞ ¼ 0, thereby showing that they are far smaller than expected if the topological defects were uncorrelated with dON ðrÞ(Fig. 3e inset). These data provide empirical evidence for a coupling between the smectic topological defects and the fluctuations of the intra-unit-cell nematicity at E D1 . Thus, when Zðr,EÞ images of the intra-unit-cell electronic structure in underdoped Bi2 Sr2 CaCu2 O8 þ d are analyzed using two independent techniques, compelling evidence for intraunit-cell electronic symmetry breaking is detected specifically of the states at the E D1 pseudogap energy. Moreover, this intraunit-cell symmetry breaking coexists with finite Q ¼ S x ,S y smectic electronic modulations, and the coupling between these states are achieved through the 2p topological defects creating spatial fluctuation of the nematicity. To study the doping dependence of these phenomena and the relationship to superconductivity is next challenge.
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→
→
Ψ1 ( r )
Ψ2 ( r )
2 nm low
2 nm high
4 nm
Counts
0 10 8 6 4 2 0
4 nm
2π
-
dex drand
δOn > 0
0 20 40 60 80 d ( pix )
+
→
l
δOn < 0
→
→
∇ ϕ( r ) 4 nm -0.0043
0
2π
+0.0043
Fig. 3. (a and b) C1 ðrÞ and C2 ðrÞ. (c and d) f1 ðrÞ and f2 ðrÞ. (e) dON ðrÞ with topological defects (black dots). Inset shows histogram of distance between topological defects and the nearest point on the dON ðrÞ ¼ 0 contour for experimental (red) and randomly generated defects (blue). (f) Schematic image of how topological defects create dON ðrÞ fluctuations. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
K. Fujita et al. / Physica B 407 (2012) 1859–1863
Acknowledgments We are grateful to P. Abbamonte, D. Bonn, J.C. Campuzano, D.M. Eigler, E. Fradkin, T. Hanaguri, W. Hardy, J.E. Hoffman, S. Kivelson, A.P. Mackenzie, M. Norman, B. Ramshaw, G. Sawatzky, J.P. Sethna, H. Takagi, and J. Tranquada, for helpful discussions and communications. References [1] M.J. Lawler, K. Fujita, et al., Nature 466 (2010) 347. [2] A. Mesaros, K. Fujita, et al., Science 333 (2011) 426.
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[3] A. Damascelli, Z. Hussain, Z.-X. Shen, Rev. Mod. Phys. 71 (2003) 473. [4] J.C. Campuzano, M.R. Norman, M. Randeria, in: K.H. Bennemann, J.B. Ketterson (Eds.), The Physics of Superconductors, vol. II, Springer, New York, 2004, pp. 167–273. [5] N. Gedik, J. Orenstein, R. Liang, D.A. Bonn, W.N. Hardy, Science 300 (2003) 1410. [6] M.L. Tacon, et al., Nat. Phys. 2 (2006) 537. [7] R. Khasanov, et al., Phys. Rev. B 82 (2010) 020511. (R). [8] J.W. Alldredge, et al., Nat. Phys. 4 (2008) 319. [9] Y. Kohsaka, et al., Science 315 (2007) 1380. [10] Y. Kohsaka, et al., Nature 454 (2009) 1072. [11] K. McElroy, et al., Nature 422 (2003) 592.