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Last time: BCS and heavy-fermion superconductors• Bardeen-Cooper Schrieffer (conventional) superconductors
• Slave particles and gauge fields• Mean field theory• U(1) gauge theory• Confinement physics
III. Phenomenology of the underdoped cuprates
• Magnetic properties
• NMR/Knight shift on YBCO (Tc = 79 K)
• χs is T-independent from 300 K to 700 K
• χs drops below Heisenberg model expectation before Tc
• Strongly points to singlet formation as origin of pseudogap
A. The pseudogap phenomenon in the normal state
III. Phenomenology of the underdoped cuprates
• Specific heat• Linear T-dependence of specific heat
coefficient γ above Tc
• γ for YBa2Cu3O6+y for different y; optimally doped curves in the inset
• γ for La2-xSrxCuO4 for different x; overdoped curves in the inset
• γ at Tc reduces with decreasing doping
A. The pseudogap phenomenon in the normal state
III. Phenomenology of the underdoped cuprates
• DC Conductivity• Anomalous linear-T “normal” state resistivity
• AC Conductivity
• In-plane (CuO2 plane) conductivity (σa) only gapped below Tc
• Perpendicular conductivity (σc) gapped in the pseudogap phase
A. The pseudogap phenomenon in the normal state
III. Phenomenology of the underdoped cuprates
• ARPES• Superconducting gap exhibits nodes
• Pseudogap opens at (π/a, 0)
• Luttinger’s theorem Fermi surface volume = 1 – x
• Spectral weight in coherence peak vanishes with decreasing hole doping
A. The pseudogap phenomenon in the normal state
III. Phenomenology of the underdoped cuprates
• STM• Surface inhomogeneity in the gap function
• STM sees two dips first dip is indication of pseudogap state
A. The pseudogap phenomenon in the normal state
Coincidenceof Checkerboard ChargeOrder and Antinodal State Decoherence in StronglyUnderdoped Superconducting Bi 2Sr 2CaCu 2O8
K. McElroy, 1,2,3 D.-H. Lee, 1,2 J.E. Hoffman, 4 K.M. Lang, 5 J. Lee,3 E.W. Hudson, 6 H. Eisaki, 7
S. Uchida, 8 and J.C. Davis3,*1Physics Department , Universit y of California, Berkeley, Californi a 94720, USA
2Material Sciences Division , Lawrence Berkeley National Lab., Berkeley, Californi a 94720, USA3LASSP, Departmen t of Physics, Cornell Universit y, Ithaca, New York 14850, USA
4Departmen t of Applied Physics, Stanford Universit y, Stanford, Californi a 94305, USA5Departmen t of Physics, Colorado College, Colorado 80305, USA
6Departmen t of Physics,MI T, Cambridg e Massachusetts 02139, USA7AIST, 1-1-1 Central 2, Umezono , Tsukuba, Ibaraki, 305-8568 Japan8Departmen t of Physics, Universit y of Tokyo, Tokyo, 113-8656 Japan
(Received 4 June 2004; publishe d 18 May 2005)The doping dependenc e of nanoscal e electroni c structure in superconductingBi 2Sr2CaCu2O8 is
studied by scanning tunneling microscopy. At all dopings, the low energy density-of-state s modulationsare analyzed according to a simple model of quasiparticl e interference and found to be consisten t withFermi-arc superconduct ivity. The superconductin g coherence peaks, ubiquitou s in near-optimal tunnelingspectra, are destroyed with strong underdopin g and a new spectral type appears. Exclusively in regionsexhibitin g thisnew spectrum, we find local ‘‘checkerboard ’’ charge ordering of high energy states, with awave vector of ~Q 2 4:5a0;0 ; 0; 2 4:5a0 15%. Surprisingl y, this spatial ordering of highenergy states coexists harmoniousl y with the low energy Bogoliubov quasiparticl e states.
How theelectroni c structure evolveswith doping fromaMott insulator into ad-wave superconducto r isakey issuein understandin g the cuprate phase diagram. Recently ithas become clear that states in different parts of momen-tum space exhibit quite different doping dependences . TheFermi arc [1] (near nodal) states of superconductin g cup-rates retain their coherence asdoping is reduced, while theantinodal (near the edge of the 1st Brillouin zone) statesdiminish in coherence, eventually becoming completelyincoheren t at strong underdoping . Photoemissio n angle-resolved photoemissio n spectroscop y (ARPES) revealsthis directly because the nodal states persist almost intothe insulator [2,3] while the antinodal states rapidly be-come incoheren t [4–8]. Bulk probes like thermal conduc-tivity [9] and c-axis penetratio n depth [10] also show thatFermi-arc states survive down to the lowest superconduct-ing dopings. Other probes such as optical transient gratingspectroscop y [11], Raman scattering [12], and NMR [13]show very different scattering processes of antinodal ver-sus nodal states throughout the underdoped regime.
Here we report on doping-dependen t STM studies ofBi 2Sr2CaCu2O8 Bi 2212 to explore the nature ofantinodal decoherence . The local density of states(LDOS) is mapped by measuring the STM tip-sampledifferential tunneling conductanceg ~r;V dI=dVjr;V ateach location ~r and bias voltageV. Since LDOS ~r;EeV g ~r;V ,an energy-resol ved~r-spaceelectronic struc-ture map is attained. The magnitud e of the energy-gapcan also be mapped (gap map) [14,15].
Fourier transform scanning tunneling spectroscop y (FT-STS)wasrecentlyintroduced to cupratestudies[16–20].It
allows the ~q vectors of spatial modulation s ing ~r;V to bedetermined from the locations of peaks ing ~q;V , theFourier transform (FT) magnitude ofg ~r;V . This tech-nique has the unique capability to relate the nanoscale~r-space electronic structure to that in~k space [17].
For thisstudyweused single Bi-2212crystalsgrown bythe floating zone method with the doping controlled byoxygen depletion. The samples were cleaved in cryogenicultrahigh vacuum before immediat e insertion into theSTM. We acquired more than106 spectra for this study.
20 mV
70 mV
(a)
(d)(c)
(b)
50 nm
∆
FIG. 1 (color). (a)–(d) Measured ~r , gap maps, for the fourdifferent hole-dopin g levels listed in I. Color scales identical.
PRL 94, 197005 (2005) PHYSICA L REVIE W LETTERS week ending20 MAY 2005
• Slave particles and gauge fields• Mean field theory• U(1) gauge theory• Confinement physics
VI. Projected trial wavefunctions and other numerical results
• Anderson’s original RVB proposal
• The Gutzwiller projection operator
• Projection operator too complicated to treat analytically
• Properties of the trial wave function evaluated using Monte Carlo sampling
• Wave function ansatz
SC: superconducting without antiferromagnetismSC+AF: superconducting with antiferromagnetismSF: staggered-flux without antiferromagnetismSF+AF: staggered-flux with antiferromagnetismZF: zero-flux
VI. Projected trial wavefunctions and other numerical results
• d-wave BCS trial wavefunctionA. The half-filled case