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Pseudogap in Bi2Sr2CaCu2O8+δ studied by measuring anisotropic

susceptibilities and out-of-plane transport

Takao Watanabe,1 Takenori Fujii,2 and Azusa Matsuda1,2

1NTT Basic Research Laboratories, 3-1, Morinosato Wakamiya, Atsugi-Shi, Kanagawa 243-0198,

Japan2Department of Applied Physics, Faculty of Science, Science University of Tokyo, 1-3

Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

(February 1, 2008)

Abstract

We find in the Bi2Sr2CaCu2O8+δ system that the characteristic temperatures

T ∗

χ (below which the uniform susceptibilities χab(T ) (H⊥c) and χc(T ) (H‖c)

decrease) and T ∗

ρc

(below which the out-of-plane resistivity ρc(T ) shows typical

upturn) coincide for all doping levels. We attribute the T dependence of χ’s

and ρc to the anomalous (pseudogapped) density-of-states (DOS) in high-Tc

cuprates. Furthermore, the anisotropy in the T dependence of χ’s is universal,

i.e., χc ∝ 1.6χab, showing that there is only a single T -dependent component

in the χ’s. This implies that the Curie-like behavior (dχ/dT < 0) observed in

overdoped samples is also caused by a DOS effect.

Typeset using REVTEX

1

Although in recent years a lot of evidence for the pseudogap phenomenon in high-Tc

cuprates has been obtained, there is still no consensus about when (at what temperatures) itopens [1]. This is mainly because of our incomplete understanding of the T dependence of theuniform susceptibility χ. For example, in the La2−xSrxCuO4 system, it is known that χ firstincreases, takes a broad maximun, and then decreases from some characteristic temperature(here defined as T ∗

χ) with decreasing temperature and T ∗

χ decreases with increasing doping (Srcontent) [2]. D. C. Johnston has interpreted this as the sum of two-dimensional (2D) S=1/2square-lattice Heisenberg antiferromagnetism and T -independent Pauli paramagnetism [2].Recently, there has been proposed a model based on the mode-mode coupling theory ofspin fluctuations in 2D metals with a technically nested Fermi surface, which explains theT dependence as a crossover from the localized to itinerant spin fluctuations [3]. In thosepictures, the gradual decrease in χ is not attributed to the formation of the pseudogap.On the other hand, J. W. Loram et al. have shown from susceptibility and high-resolutionspecific heat measurements that the pseudogap is developed below T ∗

χ in the YBa2Cu3O6+δ

[4] and La2−xSrxCuO4 systems [5]. Similar results have been reported from NMR Knightshift measurements [6]. Up to now, we have not had a unified description of the T/dopingdependence of χ, that is the pseudogap-like behavior at low dopings and Curie-like behaviorat high dopings.

Several characteristic temperatures are thought to indicate the opening of the pseudogap.In the case of YBa2Cu3O6+δ, T ∗

ρc

coincides with T ∗

ρa

[7] as well as T ∗

χ , where T ∗

ρc

and T ∗

ρa

are thetemperatures below which out-of-plane resistivity ρc(T ) shows typical semiconductive upturnand in-plane resistivity ρa(T ) shows a deviation from high-temperature T -linear behavior.This supports the view that the characteristic change in ρa(T ), ρc(T ), and χ(T ) comes solelyfrom the opening of the pseudogap. However, in Bi2Sr2CaCu2O8+δ, we have previouslypointed out that T ∗

ρc

and T ∗

ρa

are different [8]. Therefore, examining the above hypothesiswith materials other than YBa2Cu3O6+δ is important for understanding the meaning ofthe T ∗’s. Recently, K. Ishida et al. have found from their NMR measurement [9] thatT ∗ (the peak in 1/T1T ), T ∗

K (the steep decrease in the 63Cu Knight shift) and T ∗

ρa

followthe same doping dependence. They have also noticed that TmK (the gradual decrease inthe Knight shift from a constant and which thus corresponds to our T ∗

χ) follows a differentdoping dependence. They attribute T ∗

K to the pseudogap and TmK to a signature for thedevelopment of AF-spin correlations [3]. There has been up to now, however, no systematicstudy on the anisotropic susceptibilities χab(T ) (H ⊥ c), χc(T ) (H ‖ c) and ρc(T ) for theBi2Sr2CaCu2O8+δ system.

Single crystals were grown using the traveling solvent floating zone (TSFZ) method witha feed rod of Bi2.1Sr1.9CaCu2O8+δ. To control δ precisely, we first made the PO2 − T equi-librium phase diagram of a Bi2.1Sr1.9CaCu2O8+δ single crystal for various δ using a thermo-gravimetric measurment. The samples were then annealed under the corresponding oxygenpartial pressures at 600 C for 20∼30 h, after which, in order to suppress high-temperaturedisorder that causes the Curie-Weiss behavior, they were slowly cooled to 300∼400 C whilekeeping the equilibrium oxygen pressures. Finally, they were rapidly quenched [8]. Heavilyoverdoped samples with Tc ≈60 K were made by high O2 pressure (400 atm) annealing at500 C for 50 h using a hot isostatic pressing (HIP) furnace. The δ for the HIP-treatedsample was estimated as ≈0.3 from our equilibrium phase diagram [8]. We attained highaccuracy in the measurements by using the reciprocating sample option (RSO: absolute sen-

2

sibility ≥5 x 10−8 emu) of a Quantum Design superconducting quantum interference device(SQUID) magnetometer as well as by using large crystals (sample weight ≈20 mg).

The magnetic susceptibilities χc(T ) and χab(T ) under H=5 T of the Bi2Sr2CaCu2O8+δ

single crystals with various oxygen contents (δ) are shown in Figs. 1(a) and (b), respectively.The overall magnitude of χc(T ) and χab(T ) increases with increasing δ. We interpret thisin terms of an increase in the density-of-states (DOS) near the Fermi level with carrierdoping. The susceptibilities in the underdoped (δ=0.22, Tc=82 K) and optimum-doped(δ=0.25, Tc=89 K) samples monotonically decreases with decreasing temperature. For theslightly overdoped sample (δ=0.26, Tc=86 K), they are almost constant at high temperaturesand gradually decrease from a characteristic temperature T ∗

χ [shown by the arrow in Fig.1(a)] upon cooling. A further increase in δ causes a shift of T ∗

χ to lower temperatures,whereas above T ∗

χ the susceptibilities prominently decrease with increasing temperature.T ∗

χ is estimated as 200, 175, and 135 K for δ=0.26, 0.27, and 0.28, respectively. Here, T ∗

χ

was determined as the temperature at which χc deviates 2 % from the high temperatureT -linear behavior. The difference in the absolute value between χc and χab may come fromthe anisotropic Van Vleck paramagnetism of the Cu2+ ions, which is almost completelyT -independent [10].

The out-of-plane resistivity ρc(T ) for optimum and overdoped (0.26≤δ≤0.3, 86 K≥Tc≥60K) Bi2Sr2CaCu2O8+δ single crystals is shown in Fig. 2. The ρc in a optimum-doped sample(δ=0.25, Tc=89 K) shows semiconductive behavior in all temperature regions measured.With increasing δ, the overall magnitude of ρc decreases and the characteristic temperatureT ∗

ρc

(shown by the arrow in Fig. 2) for the onset of upturn appears. If T ∗

ρc

is defined in thesame manner as T ∗

χ , it can be estimated as 185, 165, and 130 K for δ=0.26, 0.27, and 0.28,respectively. For the heavily overdoped sample (δ≈0.3, Tc=60 K), ρc(T ) shows no upturnbehavior, suggesting the pseudogap completely vanishes at this doping level.

The above obtained T ∗

χ and T ∗

ρc

together with several other characteristic temperatures,including Tc, are plotted as a function of carrier concentration p in Fig. 3. Here p wasestimated from Tc using the empirical relation Tc/Tc,max = 1 − 82.6(p − 0.16)2 [11]. T ∗

χ andT ∗

ρc

coincides for each p. Unlike YBa2Cu3O6+δ, which may change its dimensionality from two(2D) to three (3D) upon doping, the Bi2Sr2CaCu2O8+δ stays in 2D for all doping levels. Wehave previously shown that the ρc in the Bi2Sr2CaCu2O8+δ system is governed by a simpletunneling process [12]. Then ρc should reflect the in-plane DOS. Since the magnitude of spinsusceptibility also reflects the DOS, the coincidence of T ∗

χ and T ∗

ρc

simply indicates that thepseudogap opens at these temperatures. This is different from the explanation where thegradual decrease in χ is attributed to the development of AF-spin correlations. It should benoted, however, that the pseudogap behavior at the doping level of δ=0.28 (p=0.193) may notbe distinguishable from the superconductive DOS fluctuation effect [13]. We have confirmedthis pseudogap opening at T ∗

χ/T ∗

ρc

by scanning tunneling spectroscopy (STS) (shown asT ∗

tunnel in Fig. 3) [14,15], although we recently recognized that there was a little shift in thedoping level due to oxygen depletion from the surface in the experiment in [14]. Our resultshere may conflict with the NMR measurement [9] and the angle-resolved photoemissionspectroscopy (ARPES) [16], which show that the pseudogap opens at around T ∗

ρa

,(shown asT ∗

K and T ∗

ARPES, respectively, in Fig. 3). We could not observe any discontinuous changein temperature dependence or anisotropy of the susceptibility at around T ∗

ρa

. We point out,however, that if T ∗ in [9] is defined as a temperature at which 1/T1T departs from high-

3

temperature Curie-Weiss behavior, it will come close to TmK and thus will come close to ourpseudogap phase boundary. A slight dip structure in the DOS, which causes the decreasein the susceptibility, may not have been assigned as a gap opening by the ARPES in termsof the leading edge analysis. In order to understand the anomaly seen at T ∗

ρa

, one may needto consider the strongly k-dependent quasiparticle lifetime, or it may be simply unrelatedto the pseudogap phenomena.

Next, in order to analyze the T dependence of anisotropic susceptibilities quantitatively,we plotted χc(T ) vs. χab(T ) with an implicit parameter T for several doping levels (Fig. 4).Here, we assume that the T dependence comes from spin susceptibilities of the doped CuO2

plane. Each of the χc(T ) vs. χab(T ) plots [Figs. 4(a), (b) and (c)] shows the same linearrelation, i. e., χc ∝ 1.6χab, except near Tc (≤140 K). This implies that the spin susceptibilityhas a single component with the ratio of anisotropic g-factors, (gc/ga)

2=1.6, and thus theCurie-like explanation for the χ of overdoped samples may be excluded. This is because itis unlikely that the localized paramagnetic centers, which could have appeared additivelyin the overdoped state, have accidentally the same anisotropy as the main spin component.This single-component hypothesis is consistent with NMR experiments [17] and ensuresthat we are actually measuring the intrinsic spin susceptibility. In Fig. 3, we included thetemperatures at which the deviation from the linear scaling becomes evident, Tscf . Sincethe Tscf scales with Tc, it is considered to be the onset temperature of Aslamasov-Larkin(AL)-type superconductive fluctuation, which is extremely anisotropic in high-Tc cuprates[10].

As already mentioned, we consider that the T dependence of χ for underdoped samplesis caused by the pseudogap opening, which has been evidenced by STS [14,15,18,19]. Then,it is natural to consider that the Curie-like behavior of χ for overdoped samples is alsocaused by a DOS effect, since the system is expected to be more Fermi-liquid like upondoping. One such possibility is an van-Hove singularity (vHs), whose existence has beenconfirmed by ARPES at somewhat below the Fermi energy (εF ) for a slightly overdopedsample [20]. The vHs is considered to grow and come near the εF with increasing doping.Here, we assume simply the Pauli paramagnetism for the single-component spin suscepti-bility, χpauli = µ2

B

∫

∞

−∞N(ε)(−∂f(ε)/∂ε)dε, where N(ε) is the DOS and f(ε) is the Fermi

function. Since χpauli reflects the thermally averaged DOS near εF , when one takes intoaccount the anomalous energy-dependent DOS (pseudogap and vHs) and its doping evolu-tion, it will explain the χ observed. This kind of interpretation was first proposed by J. W.Loram et al. [5] and later asserted by G. V. M. Williams et al. [21]. Our data here supporttheir interpretation from a different point of view.

The assumption of the Pauli paramagnetism for the spin susceptibility enables us toestimate the pseudogap value. Here in the underdoped state, the energy independent back-ground DOS, N0, was assumed (the vHs may sit far below εF and be smeared out). Forthe pseudogap, the BCS-type DOS (Dynes formula) with the d-wave symmetry gap was

assumed, N(ε)/N0 =∫

2π0

(dθ/2π)Re[| ε − iΓ | /√

(ε − iΓ)2 − (∆ cos(2θ))2], where Γ is thequasiparticle scattering rate, which was assumed here as 2kBT in a consistent way with thetransport data, and ∆ is a maximum d-wave gap, the temperature dependence of whichwas neglected. Then we can fit the experimental spin susceptibility χiso

spin(T ) [22] by thecalculated ones with a free parameter ∆ as shown in Fig. 5. The pseudogap value ∆ was es-timated as 65±3 and 48±3 meV for the underdoped (δ=0.22) and optimum doped (δ=0.25)

4

samples, respectively. These pseudogap values coincide with our direct STS measurements[14,15] and, possibly, with the ”high energy gap” recently suggested by photoemission spec-troscopy [23], but are larger than the superconducting gap (20 meV≤∆≤45 meV) obtainedby ARPES/STS [14,15,16,18,19].

In conclusion, the experimentally obtained pseudogap phase boundary (T ∗

χ and T ∗

ρc

) isnot a smooth extrapolation of the Tc boundary of the heavily overdoped state, rather, itseems to cross the Tc boundary at a slightly overdoped level (Fig. 3). This result togetherwith the estimated larger pseudogap compared to superconducting gap suggests the twogaps are different in origin, although there are many arguments that the pseudogap is aprecursor of the d-wave superconducting gap [18].

We would like to thank Professors M. Kohmoto and M. Suzuki for helpfull discussionsand Dr. H. Shibata for extending the use of the HIP facility.

5

REFERENCES

[1] B. Batlogg and V. J. Emery, Nature 382, 20 (1996).[2] D. C. Johnston, Phys. Rev. Lett. 62, 957 (1989).[3] K. Miyake and O. Narikiyo, J. Phys. Soc. Jpn. 63, 3821 (1994).[4] J. W. Loram et al., Physica C 235-240, 134 (1994).[5] J. W. Loram et al., Proceedings of the 10th Anniversary HTS Workshop on Physics

Materials and Applications (World Scientific, Singapore, 1996), p. 341.[6] G. V. M. Williams et al., Phys. Rev. Lett. 79, 721 (1997).[7] K. Takenaka et al., Phys. Rev. B 50, 6534 (1994). T. Shimizu. et al., J. Phys. Soc. Jpn.

62, 3710 (1993).[8] T. Watanabe, T. Fujii, and A. Matsuda, Phys. Rev. Lett. 79, 2113 (1997).[9] K. Ishida et al., Phys. Rev. B 58, R5960 (1998).

[10] D. C. Johnston et al., Phys. Rev. B 42, 8710 (1990).[11] J. L. Tallon et al., Phys. Rev. Lett. 75, 4114 (1995).[12] T. Watanabe and A. Matsuda, Phys. Rev. B54, R6881 (1996).[13] A. A. Varlamov, Physica C 282-287, 248 (1997).[14] A. Matsuda, S. Sugita, and T. Watanabe, Phys. Rev. B 60, 1377 (1999). A. Matsuda, S.

Sugita, and T. Watanabe, Proceedings of the 11th International Symposium on Super-conductivity (ISS ’98) (Springer-Verlag Tokyo 1999), p. 151.

[15] A. Matsuda, S. Sugita, T. Fujii and T. Watanabe (unpublished).[16] H. Ding et al., Nature 382, 51 (1996); D.S. Marshall et al., Phys. Rev. Lett. 76, 4841

(1996).[17] H. Alloul et al., Phys. Rev. Lett. 63, 1700 (1989). H. Alloul et al., Phys. Rev. Lett. 70,

1171 (1993).[18] C. Renner et al., Phys. Rev. Lett. 80, 149 (1998).[19] M. Miyakawa et al., Phys. Rev. Lett. 80, 157 (1998).[20] Z.-X. Shen et al., Science 267, 343 (1995).[21] G. V. M. Williams et al., Phys. Rev. B 57, 8696 (1998).[22] The isotropic component of the spin susceptibilities, χiso

spin, was obtained by averaging theanisotropic spin susceptibilities, i. e., χiso

spin=(2/3)χabspin+(1/3)χc

spin, where χabspin and χc

spin

were obtained using the values in the literature for χV Vab and χV V

c [10].[23] F. Ronning et al., Science 282, 2067 (1998). A. Ino et al., Phys. Rev. Lett. 81, 2124 (1998).

6

Figure captions

Fig. 1. Magnetic susceptibilities (a) χc(T ) (H ‖ c) and (b) χab(T ) (H ⊥ c) of theBi2Sr2CaCu2O8+δ single crystals with various oxygen contents (δ) under H=5 T. The solidstraight lines in (a), which are linear extraporations of χc at higher temperatures, are eyeguides for the overdoped (δ=0.26, 0.27, and 0.28) samples. Arrows indicate the temperaturesT ∗

χ at which χc starts to decrease from its linear high temperature behaviors.Fig. 2. Out-of-plane resistivity ρc(T ) of a Bi2Sr2CaCu2O8+δ single crystal for oxygen con-tent 0.26≤δ≤0.3. The solid straight lines, which are linear extraporations of ρc at highertemperatures, are eye guides for the overdoped (δ=0.26, 0.27, and 0.28) samples. Arrowsindicate the temperatures T ∗

ρc

below which ρc shows typical upturn.Fig. 3. Characteristic temperatures obtained by various measurements versus carrier con-centration p. The p values (including for the previous results) were systematically estimatedfrom the empirical Tc vs. p relation in the text.Fig. 4. χc(T ) versus χab(T ) plots for (a) underdoped (δ=0.22), (b) slightly overdoped(δ=0.26) and (c) heavily overdoped (δ≈0.3) Bi2Sr2CaCu2O8+δ single crystals. Some typicaltemperatures are shown by arrows.Fig. 5. Isotropic component of the spin susceptibilities, χiso

spin(T ), of a Bi2Sr2CaCu2O8+δ

single crystal for oxygen content δ=0.22 (underdoped) and 0.25 (optimum doped). Thesolid lines are numerical fits assuming the BCS-type DOS for the pseudogap.

7

0

1

2

3

4

50 100 150 200 250 300 350 400

χ c (10

-7em

u/g)

Temperature (K)

T*χHIP (δ≈0.3)

δ=0.28

δ=0.27δ=0.26

δ=0.25δ=0.22

T*χ

T*χ

(a)

T. Watanabe et al., Fig. 1(a)

-1

-0.5

0

0.5

1

1.5

50 100 150 200 250 300 350 400

χ ab (

10-7

emu/

g)

Temperature (K)

HIP (δ≈0.3)δ=0.28δ=0.27

δ=0.26δ=0.25δ=0.22

T. Watanabe et al., Fig. 1(b)

(b)

0

1

2

3

4

5

6

7

50 100 150 200 250 300

ρ c (Ω

cm)

Temperature (K)

T*ρc

δ=0.25

δ=0.26δ=0.27

δ=0.28

HIP (δ≈0.30)

T. Watanabe et al., Fig. 2

T*ρc

T*ρc

50

100

150

200

250

0.1 0.15 0.2 0.25

TcT*χT*ρ

c

T*tunnelT*ρ

a

T*K

T*ARPES

Tscf

Tem

pera

ture

(K

)

p

T. Watanabe et al., Fig. 3

H. Ding et al.

K. Ishida et al.

0

0.5

1.0

1.5

2.0

-8 -6 -4 -2 0 2

χ c (1

0-7em

u/g)

χa b

(10-8emu/g)

underdoped (δ=0.22)

95 K

300 K

135 K

χc ∝ 1.55χ

a b

T. Watanabe et al., Fig. 4(a)

(a)

1.8

1.9

2.0

2.1

2.2

2.3

2.4

2.5

2 3 4 5 6

χc ∝ 1.62χ

a b

140 K

300 K

110 K

χ c (1

0-7em

u/g)

χa b

(10-8emu/g)

slightly overdoped (δ=0.26)

200 K

T. Watanabe et al., Fig. 4(b)

(b)

3.1

3.2

3.3

3.4

3.5

3.6

3.7

0.95 1.05 1.15 1.25 1.35

χ c (1

0-7em

u/g)

χa b

(10-7emu/g)

heavily overdoped (δ≈0.30)

χc ∝ 1.61χ

a b

380 K

100 K

75 K

125 K

T. Watanabe et al., Fig. 4(c)

(c)

0

0.5

1.0

1.5

2.0

50 100 150 200 250 300

χ spin

iso (1

0-7em

u/g)

Temperature (K)

δ=0.25 (∆=48 meV)

δ=0.22 (∆=65 meV)

T. Watanabe et al., Fig. 5