Coherent quasi-particles-to-incoherent hole-carriers crossover in underdoped cuprates
M. Hashimoto, T. Yoshida, K. Tanaka, A. Fujimori, M. Okusawa, S. Wakimoto, K. Yamada, T. Kakeshita, H. Eisaki, S. Uchida
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Coherent quasi-particles-to-incoherent hole-carriers crossover in underdoped cuprates
M. Hashimoto, T. Yoshida, K. Tanaka, A. Fujimori, M. Okusawa, S. Wakimoto, K. Yamada, T. Kakeshita, H. Eisaki, and S. Uchida Department of Physics, University of Tokyo, Tokyo 113-8656, Japan Department of Physics, Faculty of Education, Gunma University, Maebashi, Gunma 371-8510, Japan QuBS, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japan Institute of Materials Research, Tohoku University, Sendai 980-8577, Japan SRL- ISTEC, Tokyo, 135-0062, Japan AIST, 1-1-1 Central 2, Umezono, Tsukuba, Ibaraki, 305-8568, Japan (Dated: November 1, 2018)In underdoped cuprates, only a portion of the Fermi surface survives as Fermi arcs due to pseu-dogap opening. In hole-doped La CuO , we have deduced the “coherence temperature” T coh ofquasi-particles on the Fermi arc above which the broadened leading edge position in angle-integratedphotoemission spectra is shifted away from the Fermi level and the quasi-particle concept starts tolose its meaning. T coh is found to rapidly increase with hole doping, an opposite behavior to thepseudogap temperature T ∗ . The superconducting dome is thus located below both T ∗ and T coh ,indicating that the superconductivity emerges out of the coherent Fermionic quasi-particles on theFermi arc. T coh remains small in the underdoped region, indicating that incoherent charge car-riers originating from the Fermi arc are responsible for the apparently metallic transport at hightemperatures. In order to understand the variety of interesting phe-nomena of doped Mott insulators [1, 2], it is necessaryto reveal how the electrons behave as a function of car-rier concentration [3]. High- T c superconductivity in thecuprates is one of the most spectacular examples of dopedMott insulators in which superconductivity [4] emergesfrom an unconventional normal state. Photoemissionstudies have revealed a wealth of its electronic struc-ture [5]. Recent angle-resolved photoemission (ARPES)studies of underdoped cuprates have revealed that theFermi arc in the nodal region, which remains without apseudogap above T c and governs the normal-state trans-port, plays important roles on the superconductivity, too[6, 7, 8, 9, 10]. Below T c , a d -wave superconducting gapopens on the Fermi arc, while the pseudogap in the antin-odal region, which opens below pseudogap temperature T ∗ ( > T c ), does not show a strong temperature depen-dence across T c . Thus, it has been suggested that thehigh- T c superconductivity is realized on the Fermi arcand that the pseudogap is not directly connected to thesuperconductivity [6, 7, 8, 11]. It has been also reportedthat the Fermi arc shrinks with underdoping [7, 9, 10]corresponding to the carrier number which decreases like n ∼ x and the arc length becomes longer with tempera-ture [10] until the Fermi surface recovers above T ∗ .On the other hand, it has not been obvious how thequasi-particles on the Fermi arc change with tempera-ture as well as doping. If one defines the Fermi energy ǫ F ∝ n/m ∗ of the doped holes, it should decrease withunderdoping, since the carrier mobility µ ∝ /m ∗ de-creases only slowly [12]. As the temperature increasesfrom below T F ≡ ǫ F /k B to above it, the doped holeswould change their character from a degenerate Fermiliquid (on the Fermi arc) consisting of coherent quasi- particles obeying the Fermi statistics to a classical gasof (incoherent) holes obeying the Boltzmann statistics.Therefore, if T F < T ∗ , it is expected that charge carri-ers will lose its quasi-particle properties before the entireFermi surface is recovered by the collapse of the pseu-dogap. So far, there has not been a quantitative ex-perimental estimate of T F in the under doped cuprates.Therefore, in order to observe such a crossover, we haveperformed systematic temperature and doping dependentangle-integrated photoemission (AIPES) measurementsof the single-layer cuprates La − x Sr x CuO (LSCO) andLa CuO . , and derived the crossover temperature orthe “coherence temperature” T coh , which should essen-tially follow T F .We measured LSCO samples with x = 0.03, 0.10 ( T c = 25 K), 0.15 ( T c = 38 K), 0.22 ( T c = 28 K), 0.30, andLa CuO . (LCO) with hole concentration p ∼ T c ∼
35 K). The sample temperature was varied between 10K and 300 K. The total energy resolution was set at ∼ E F was within1 meV. Experimental details were described before [11].In Fig. 1(a)-(g), we have reproduced the temperaturedependent photoemission spectra of LSCO, LCO andgold near E F from Ref. [11]. One can see that the Fermiedge and its temperature dependence are most clearly ob-served for gold as well as in the overdoped samples, butbecomes blurred with underdoping at elevated tempera-tures. In the underdoped region, the edge becomes lessclear with temperature than that in the overdoped region,indicating that the Fermi-Dirac statistics lose its mean-ing with temperature in the underdoped region. In orderto evaluate the disappearance of the Fermi edge quanti-tatively, that is, crossover from coherent quasi-particles F (eV) -0.1 0.0 0.1 - d I/ d ε I ( a r b . un i t s ) (a) x = 0.03 (b) x = 0.10 (c) LCO ( p = 0.12) (d) x = 0.15 (e) x = 0.22 (f) x = 0.30
300 K250200150100 50 30 10 (g) Au(h) (i) (j) (k) (l) (m) (n)
FIG. 1: (Color online) Doping and temperature dependences of the AIPES spectra near E F for La − x Sr x CuO (LSCO),La CuO . (LCO, p ∼ to incoherent hole carriers, we have analyzed the spectrausing first derivative and scaling relationship, as we shalldescribe below.Figure 1(h)-(n) shows the (smoothed) first derivativecurves of the spectra. The peak positions indicated byred symbols can be regarded as the leading edge mid-point of the raw spectra. For the gold spectra [Fig. 1(n)],the peak of the first derivative curves appears exactlyat E F at any temperature although the peak becomesbroader with temperature, as expected from the Fermi-Dirac (FD) distribution function. In the case of the heav-ily overdoped LSCO of x = 0.30 too, one can observe apeak up to 300 K as in the case of gold. However, thereis a slight shift of the peak position toward below E F andthe peak shows an asymmetric tail extending below E F ,reflecting the strong slope of the density of states (DOS).With decreasing hole concentration, the shift of the peakstarts at a lower temperature and becomes stronger. Theasymmetric broadening of the peak with increasing tem-perature also becomes more pronounced. For the mostunderdoped x = 0.03 sample, only at low temperatures,one can barely see a small peak near E F arising fromthe tiny Fermi cut-off. With increasing temperature, thepeak is rapidly shifted away from E F , and becomes am-biguous and difficult to define, corresponding to the dis-appearance of the Fermi edge in the raw spectra [Fig.1(a)].In Fig. 2(a), the peak position E peak ( x, T ) of thefirst derivative curves is plotted as a function of tem-perature for various hole concentrations. With decreas-ing hole concentration, the deviation of the peak po-sition from E F occurs at lower temperatures as men-tioned above. If one defines T coh by the temperatureabove which the deviation of the peak position from E F becomes significant, Fig. 2(a) indicates that T coh in- creases with decreasing hole concentration. T coh may beregarded as the crossover temperature from the Fermi-liquid-like metallic state to a non-metallic (or incoherent-metallic) one which may be considered as a collection ofhole carriers (polarons?) that show incoherent hopping.In order to deduce the x dependence of the coherencetemperature T coh ( x ) from the set of experimental data,we have performed a scaling analysis of the peak po-sition E peak ( x, T ). By assuming that E peak ( x, T ) obeysthe scaling relation E peak ( x, T ) /E coh ( x ) = f ( T /T coh ( x )),where E coh ( x ) is the x -dependent “coherence energyscale” ( ∼ ǫ F /k B as we shall see below), all the data pointsfall onto a single curve as shown in Fig. 2(b) and (c)[13]. In this plot, one can see a temperature-dependentcrossover at T /T coh ( x ) ∼
1, from the weakly tempera-ture dependent E coh ( x, T ) to the strongly temperaturedependent E coh ( x, T ). In the case of the Fermi liquid,a simulation for a DOS with a finite slope multiplied bythe FD function has shown that the leading edge show asmall shift approximately proportional to T . One cansee that the small shifts for T /T coh < T behavior as shown in Fig. 2(c). At highertemperatures T /T coh >
1, the leading edge becomes lesswell-defined and show more rapid shift than in the low-temperature region. In fact, the peak shift is practicallylinear in T . We note that a T ln T behavior is expectedfor the chemical potential of a classical hole gas obeyingthe Boltzmann statistics, which is almost linear in T . T coh ( x ) and E coh ( x ) thus deduced are plotted in Fig.3. For x = 0.03, T coh is lower than 100 K , but increasesquickly with increasing hole concentration. It exceeds300 K for optimally doped x = 0.15 and becomes evenhigher for x = 0.22 and 0.30. Note that E coh ( x ) and T coh ( x ) are mutually consistent as they satisfy E coh ( x ) ∼ k B T coh ( x ). The present results are in accordance with2 f ( T / T c oh ) T/T coh x = 0.03 0.10 0.12 0.15 ∝ ( T/T coh ) P ea k P o s i t i on ( m e V ) Temperature (K) x = 0.03 0.10 0.12 0.15 0.22 0.30 Au (a)(b) f ( T / T c oh ) T/T coh (c)
FIG. 2: (Color online) Temperature dependence of the peakposition E peak ( x, T ) in the first derivative of spectra for LSCOand LCO, together with gold reference. (a) Raw peak po-sition E peak ( x, T ). (b) Scaling plot of the peak position E peak ( x, T ) /E coh ( x ) = f ( T /T coh ( x )). One can see that, for T 100 K) below x ∼ . x dependence rather than x -linear doping de-pendence. In the case of the 2D doped Mott insulator,Imada [23] has indicated that, near the metal-insulatortransition (MIT), the doping dependence of T coh andchemical potential µ behave as T coh ∝ µ ∝ x z/d accord-ing to hyperscaling, where z is the dynamical exponentof the MIT and d is the spatial dimension ( d = 2). Fig-ure 3 shows that the doping dependence of T coh is moreconsistent with ∝ x than ∝ x , implying that z ∼ z = 2. Parcollet and Georges [24] havealso indicated the x doping dependence of T coh in thelow-doping regime of the t − J model. The x doping de-pendence of T coh is also consistent with the suppressionof the chemical potential shift in the underdoped region[25], which also indicates an anomalously large exponent z [26].Finally, we compare the present result with the mobil-ity µ of doped carriers reported by Ando et al [12]. While µ at a fixed temperature depends on doping, we find that µ at T coh is almost doping independent: For x = 0.03, µ at T coh ∼ 75 K is ∼ 10 cm /Vs and for x = 0.12, µ at T coh ∼ 210 K is ∼ /Vs. Here, µ ≡ σ/n has beenestimated under the assumption that n = x [12]. This“critical” µ corresponds to the mean-free path of ∼ ω = 0 or Drude weight in optical conduc-tivity [27, 28, 29, 30]. Depression of the Drude weightwith temperature has been observed as expected theo-retically [31]. At a fixed low temperature, Drude weight[27, 28, 29, 30], the mobility of carriers [12] and the nodalspectral weight at E F from ARPES measurements [32]increase with doping. These doping dependences are con-sistent with the doping dependence of T coh , although itis difficult to estimate the value of T coh from these mea-surements.In conclusion, in the underdoped region, the quasi-particles on the Fermi arc start to lose its coherencequickly above T coh , which exhibits opposite doping de-pendence to T ∗ . The result that T c Electron theory of metals (Cambridge Uni-versity Press, Cambridge, 2001).[4] M. Tinkham, Introduction to Superconductivity (McGrawHill Inc., New York, 1996).[5] A. Damascelli, Z. Hussain, and Z.-X. Shen, Rev. Mod.Phys. , 473 (2003).[6] K. Tanaka et al. , Science , 1910 (2006).[7] W. S. Lee et al. , Nature , 81 (2007).[8] T. Kondo, T. Takeuchi, A. Kaminski, S. Tsuda and S.Shin, Phys. Rev. Lett. , 267004 (2007).[9] T. Yoshida et al. , J. Phys. Condens. Matter , 125209(2007).[10] A. Kanigel et al. , Nature Phys. , 447 (2006).[11] M. Hashimoto et al. , Phys. Rev. B , 140503(R) (2007).[12] Y. Ando, A. N. Lavrov, S. Komiya, K. Segawa and X. F.Sun, Phys. Rev. Lett. , 017001 (2001).[13] T. effect of the finite resolution in the peak shift can beeliminated based on a simulation assuming a linear DOS.[14] A. Ino et al. , Phys. Rev. Lett. , 2124 (1998).[15] N. Momono et al. , Physica C , 603 (1999).[16] A. Kaminski et al. , Phys. Rev. Lett. , 207003 (2003).[17] M. L. Tacon et al. , Nature phys. , 573 (2006).[18] A. A. Kordyuk et al. , Phys. Rev. B , 020504(R) (2009).[19] A. A. Kordyuk et al. , Phys. Rev. B , 214513 (2005).[20] Y. Kohsaka et al. , Science , 1380 (2007).[21] H. Ding et al. , Phys. Rev. Lett. , 227001 (2001).[22] F. Zhang, C. Gros, T. Rice, and H. Shiba, Supercond.Sci. Technol. , 36 (1988).[23] M. Imada, J. Phys. Soc. Jpn. , 2954 (1995).[24] O. Parcollet and A. Georges, Phys. Rev. B , 5341(1999).[25] A. Ino et al. , Phys. Rev. Lett. , 2101 (1997).[26] N. Furukawa and M. Imada, J. Phys. Soc. Jpn. , 3331(1992).[27] M. Dumm, S. Komiya, Y. Ando, and D. N. Basov, Phys.Rev. Lett. , 077004 (2003).[28] K. Takenaka et al. , Phys. Rev. B , 092405 (2002).[29] K. Takenaka, J. Nohara, R. Shiozaki, and S. Sugai, Phys.Rev. B , 134501 (2003).[30] M. Ortolani, P. Calvani, and S. Lupi, Physical ReviewLetters , 067002 (2005). 31] J. Jaklic, P. Prelovsek, Phys. Rev. B , 6903 (1995).[32] T. Yoshida et al. , Phys. Rev. Lett. , 027001 (2003)., 027001 (2003).