Origin of electron-hole asymmetry in the scanning tunneling spectrum of B i 2 S r 2 CaC u 2 O 8+δ
OOrigin of electron-hole asymmetry in the scanning tunneling spectrum of Bi Sr CaCu O δ Jouko Nieminen , , ∗ Hsin Lin , R.S. Markiewicz , and A. Bansil Institute of Physics, Tampere University of Technology, P.O. Box 692, 33101 Tampere, Finland and Physics Department, Northeastern University, Boston, Massachusetts 02115 (Dated: Version of October 25, 2018)We have developed a material specific theoretical framework for modelling scanning tunnelingspectroscopy (STS) of high temperature superconducting materials in the normal as well as thesuperconducting state. Results for Bi Sr CaCu O δ (Bi2212) show clearly that the tunnelingprocess strongly modifies the STS spectrum from the local density of states (LDOS) of the d x − y orbital of Cu. The dominant tunneling channel to the surface Bi involves the d x − y orbitals of thefour neighbouring Cu atoms. In accord with experimental observations, the computed spectrumdisplays a remarkable asymmetry between the processes of electron injection and extraction, whicharises from contributions of Cu d z and other orbitals to the tunneling current. PACS numbers: 68.37.Ef 71.20.-b 74.50.+r 74.72.-h
Scanning tunneling spectroscopy (STS) has entered therealm of high-temperature superconductors powerfullyby offering atomic scale spatial resolution in combina-tion with high energy resolution. The physics of thesematerials is dominated by the cuprate layers, which areusually not exposed to the tip of the apparatus. Much ofthe existing interpretation of the spectra is based how-ever on the assumption that the STS spectrum is directlyproportional to the LDOS of the CuO layer, neglectingthe effects of the tunneling process in modifying the spec-trum in the presence of the insulating overlayers. Here,we focus on the Bi2212 system, which has been the sub-ject of an overwhelming amount of experimental work[1–5], although our results bear more generally on the STSspectra of the cuprates.Our analysis takes into account the fact that the cur-rent originating in the CuO layers reaches the tip afterbeing ‘filtered’ through the overlayers of SrO and BiO,and shows that instead of being a simple reflection ofLDOS of the CuO layers, the STS signal represents acomplex mapping of the electronic structure of the sys-tem. In particular, we find that the spectrum develops astriking asymmetry between positive and negative biasesbecause d z and other orbitals begin to contribute to thetunneling current with increasing bias voltage. Althoughthis asymmetry has often been taken to be the hallmarkof strong correlation effects[6], our results indicate thatthe nature of the tunneling process itself induces signif-icant electron-hole asymmetry even within the conven-tional picture, so that strong correlation effects on theSTS spectrum will be more subtle than has been realizedso far.In order to construct a realistic framework capable ofdescribing the STS spectrum of the normal as well as thesuperconducting state, we start with the normal stateHamiltonianˆ H = (cid:88) αβσ (cid:2) ε α c † ασ c ασ + V αβ c † ασ c βσ (cid:3) , (1) c ondu c t i ng l a y e r s f il t e r i ng l a y e r s Bi Sr
Ca Cu O (b) Top view(a) Side view
BiOSrOCuO CuO CaSrOBiO tip
FIG. 1. (a) Side view of tip placed schematically on topof seven layers used to compute the tunneling spectrum ofBi2212, where the surface terminates in the BiO layer. Tun-neling signal from the conducting CuO layers reaches thetip after passing through SrO and BiO layers. (b) Top viewof the surface showing arrangement of various atoms. Eighttwo-dimensional real space primitive unit cells used in thecomputations are marked by dashed lines. which describes a system of tight-binding orbitals created(or annihilated) via the real-space operators c † ασ (or c ασ ).Here α is a composite index denoting both the type oforbital (e.g. Cu d x − y ) and the site on which this or-bital is placed, and σ is the spin index. (cid:15) α is the on-siteenergy of the α th orbital. α and β orbitals interact witheach other through the potential V αβ to create the energyeigenstates of the system.Superconductivity is included by adding a pairing in-teraction term ∆ in the Hamiltonian:ˆ H = ˆ H + (cid:88) αβσ (cid:104) ∆ αβ c † ασ c † β − σ + ∆ † βα c β − σ c ασ (cid:105) (2)We take ∆ to be non-zero only between d x − y orbitalsof the nearest neighbor Cu atoms, and to possess ad-wave form, i.e. ∆ is given in momentum space by a r X i v : . [ c ond - m a t . s up r- c on ] J un (b) Superconducting(a) Normal VHS-b VHS-aVHS-bSC gap
BiO Cu d x -y VHS-a
Cu d x -y weight electron weight E n er gy ( e V ) E n er gy ( e V ) FIG. 2. (a) Normal state electronic spectrum of Bi2212. Cu d x − y weight of states is given by the color of dots. Inset showsthe energy region of the VHS’s on an expanded scale. (b) Electronic spectrum of the superconducting state. Electron weightof states is given by the color of dots. Inset highlights the region of the superconducting gap near the antinodal point. ∆ k = ∆2 [cos k x a − cos k y a ] , where a is the in-plane lat-tice constant. This interaction allows electrons of op-posite spin to combine into superconducting pairs suchthat the resulting superconducting gap is zero along thenodal directions k x = ± k y , and is maximum along theantinodal directions.The Bi2212 sample is modeled as a slab of seven layersin which the topmost layer is BiO, followed by layersof SrO, CuO , Ca, CuO , SrO, and BiO, as shown inFig. 1(a). The tunneling computations are based on a2 √ × √ s, p x , p y , p z ) for Bi and O; s for Sr;and (4 s, d z − r , d xy , d xz , d yz , d x − y ) for Cu atoms. Thisyields 58 orbitals in a primitive cell, used in band calcu-lations, and a total of 464 orbitals for Green functionsupercell calculations in 256 evenly distributed k-points.Finally a gap parameter value of | ∆ | = 0 . eV is cho-sen to model a typical experimental spectrum[1] for the generic purposes of this study.The LDOS and tunneling computations are based onGreen function formalism. At first, the normal-stateGreen function is constructed via Dyson’s equation usingmethodology described in Ref. 11. At this stage a self-energy for orbital α , Σ ± α = Σ (cid:48) α ± i Σ (cid:48)(cid:48) α is embedded inDyson’s equation for possible effects of various bosoniccouplings and correlation effects [12, 13]. For simplic-ity, we have assumed the self-energy to be diagonal inthe chosen basis [14]. In building up the Green func-tion in the superconducting state, we utilize the conven-tional BCS-type self-energy Σ BCS = ∆ G h ∆ † (see, e.g.,Ref. 15), where G h is the hole part of normal state Greenfunction.Fig. 2 shows the calculated band structure of Bi2212 inthe normal and the superconducting state from Hamil-tonians of Eqs. 1 and 2. The normal state is seen toproperly display the major features such as: The pair ofCuO bands crossing the Fermi energy ( E F ) with the as-sociated van Hove singularities (VHS’s) marked VHS-a(antibonding) and VHS-b (bonding), split by 250 meVat the ( π,
0) point; BiO bands lying about 1 eV above E F ; and the ‘spaghetti’ of bands involving various Cuand O orbitals starting at a binding energy of around 1eV below E F . Although states near E F are mainly ofCu d x − y and O p x,y character, they also contain someBi and Cu d z admixture. In the superconducting statein Fig. 2(b), a quasiparticle spectrum mirrored through E F is obtained with a doubled number of bands due tothe pairing interaction. A d-wave superconducting gapopens up in both CuO bands near E F . The quasiparti-cles have a mixed electron/hole character only near the CuCu O OBi (c) (a)
VHS-aVHS-b (d)
O Bi
Coherencepeaks (b)
Cu d x -y LDOS
Bias Voltage (V) Bias Voltage (V) d I / d V ( a r b . un i t s ) d I / d V ( a r b . un i t s ) E-E F (eV) E n er gy ( e V ) ExperimentTheory ExperimentTheory d x -y other Σ ' Σ '' FIG. 3. (a) Computed (black line) and experimental (red line) differential tunneling spectra, dI/dV , as a function of biasvoltage (in volts). LDOS of Cu d x − y electrons is shown (green line) as a function of energy, where the horizontal scale is ineV. (b) Computed topographic image of the BiO surface in which Bi atoms are bright and O atoms are dark. (c) The maintunneling channel from Cu d x − y electrons on nearest neighbor Cu atoms to the tip through overlap with the central Cu atom,followed by hopping to the p z orbitals of the apical O-atom and the Bi atoms. (d) Computed and measured spectra in thelow energy region marked by gray shading in (a). Inset gives the real (solid lines) and imaginary (dashed lines) parts of theself-energy applied to the Cu d x − y states and to all other states [14]. edges of the gap.To compute the tunneling spectra we apply theTodorov-Pendry expression [16, 17] for the differentialconductance σ between orbitals of the tip ( t, t (cid:48) ) and thesample( s, s (cid:48) ), which in our case yields σ = dIdV = 2 πe ¯ h (cid:88) tt (cid:48) ss (cid:48) ρ tt (cid:48) ( E F ) V t (cid:48) s ρ ss (cid:48) ( E F + eV ) V † s (cid:48) t , (3)where the density matrix ρ ss (cid:48) = − π (cid:80) α G + sα Σ (cid:48)(cid:48) α G − αs (cid:48) is, in fact, the spectral function written in terms ofretarded/advanced Green function and the self-energy.Eq. 3 differs from the more commonly used Tersoff-Hamann approach[18] in that it takes into account thedetails of the symmetry of the tip orbitals and their over-lap with the surface orbitals.The use of the spectral function recasts Eq. 3 into theform σ = (cid:88) tα T tα , (4)where T tα = − e ¯ h (cid:88) t (cid:48) ss (cid:48) ρ tt (cid:48) ( E F ) V t (cid:48) s G + sα Σ (cid:48)(cid:48) α G − αs (cid:48) V † s (cid:48) t , (5)and the Green functions and the self-energy are evalu-ated at energy E = E F + eV b . Eq. 5 is similar to the Landauer-B¨uttiker formula for tunneling across nanos-tructures (see, e.g., Ref. 19), and represents a slight re-formulation of Refs. 20 and 21.The nature of Eq. 5 can be understood straightfor-wardly: G sα gives the amplitude with which electronsresiding on the α th orbital in the solid propagate to thesurface at energy E broadened by Σ (cid:48)(cid:48) α . The term V st is the overlap between the surface orbital and the tip,while ρ tt (cid:48) gives the available states at the tip. Hence, T tα gives the contribution of the α th orbital to the cur-rent, and the summation in Eq. 4 collects these individualcontributions to yield the total tunneling current whichreaches the tip. Thus, selecting individual terms in Eq. 5provides a transparent scheme to define tunneling pathsbetween the sample and the microscope tip.Fig. 3(a) shows the tunneling spectra over the broadenergy range of ± d z and other orbitals contributingto the ‘spaghetti’ of bands starting around 1 eV bindingenergy (see Fig. 2(a)). We emphasize that the LDOSof the Cu d x − y (green line in Fig. 3(a)) does not pro-vide a good description of the spectrum. In particular,the Cu d x − y LDOS possesses an asymmetry which isopposite to that of the tunneling spectrum.Fig. 3(d) gives a blow up of the low energy regionof ± . eV , shown by gray shading in Fig. 3(a). Thecomputed spectrum is seen to reproduce the coherencepeaks and the characteristic peak-dip-hump feature. Thegeneric form of the real and imaginary parts of the self-energies applied to the Cu d x − y orbitals (solid anddashed blue lines, respectively) and the rest of the or-bitals are given in the inset [14]. Fig. 3(b) shows thecomputed ‘topographic map’ of the BiO surface in con-stant current mode. Bi atoms appear as bright spots inaccord with experimental observations, while O atoms sitat the centers of dark regions. Bias Voltage (V)
Bias Voltage (V) d I / d V ( a r b . un i t s ) d I / d V ( a r b . un i t s ) Cu d z Cu d x -y L1-nnL1-nnnL2-total
Layer 2 (Cu d x -y ) totalnnnnn FIG. 4.
Main frame : Partial contributions to the tunnelingcurrent from various orbitals in the two cuprate layers. TheCuO layer closest to the tip is identified as layer 1 or L1,while the second layer is denoted by L2. Specific contributionsare: d x − y orbitals of the four nearest neighbor Cu atoms(L1-nn, blue line); d x − y orbitals of the four next nearestneighbor Cu atoms of the first layer (L1-nnn, green line) ; d x − y orbitals of the Cu atoms of the second layer (L2, redline); d z of the central Cu atom of L1 (magenta line). Inset :Decomposition of the current from the second cuprate layer:Total contribution (red line); contribution of the four nearestneighbours (blue line); and the next nearest neighbours (greenline).
An analysis of the partial terms of Eq. 5 reveals thatthe d x − y orbital of the Cu atom lying right underthe Bi atom gives zero contribution to the current [24].The dominant contribution to the spectrum comes fromthe four nearest neighbor (NN) Cu atoms as indicated schematically in Fig. 3(c). A detailed decomposition ofEq. 4 is shown in Fig. 4, where paths starting from theCuO layer closest to the tip (L1), as well as from the sec-ond cuprate layer (L2) are considered. The signal fromcuprate layers is dominated by the d x − y orbitals on thefour nearest neighbour (nn) Cu atoms in L1 up to about-0.7 eV (blue line). At higher binding energies, the con-tribution from the d z electrons from the Cu atom rightbelow the Bi atom or the tip grows rapidly (magentaline).A smaller but still significant contribution comes fromthe four next nearest neighbour (nnn) d x − y orbitals inL1 spread over a wide energy range (green line, main fig-ure), while the total current originating from the d x − y orbitals of L2 is quite localized over zero to -0.6 eV bias(red line, main figure). Fig. 4 emphasizes the nature ofthe current associated with the cuprate layers and pointsout an intrinsic electron-hole asymmetry originating fromthe d z orbitals. We note however that the Bi and O or-bitals in the surface Bi-O layer can also play a role inproducing an asymmetric background current.In conclusion, we find that STS spectrum for Bi2212 isstrongly modified from the LDOS of d x − y by the effectof the tunneling process or what we may call the tunnel-ing matrix element. Much of the observed asymmetry ofthe spectrum can be explained within the conventionalpicture due to the turning on of Cu d z and other chan-nels with increasing (negative) bias voltage. This indi-cates that the effects of strong electronic correlations onthe tunneling spectrum are more subtle than has beenthought previously. However, we should note that wehave not analyzed spectra associated with the deeply un-derdoped regime where charge order has been reported[25]. The present method naturally allows an analysisof the tunneling signal in terms of the possible tunnel-ing channels and the related selection rules. Our schemecan be extended to incorporate effects of impurities andvarious nanoscale inhomogeneities by using appropriatelylarger basis sets in the computations. Acknowledgments
This work is supported by the US Department of En-ergy, Office of Science, Basic Energy Sciences contractDE-FG02-07ER46352, and benefited from the allocationof supercomputer time at NERSC, Northeastern Univer-sity’s Advanced Scientific Computation Center (ASCC),and the Institute of Advanced Computing, Tampere. ∗ jouko.nieminen@tut.fi[1] K. McElroy et al. , Science , 1048 (2005).[2] E.W. Hudson et al. , Nature , 920 (2001).[3] A.N. Pasupathy et al. , Science , 196 (2008).[4] A.V. Balatsky et al. , Rev. Mod. Phys. , 373 (2006).[5] Ø. Fischer et al. , Rev. Mod. Phys. , 353(2007).[6] T. Hanaguri et al. , Nature , 1001 (2004). [7] V. Bellini, F. Manghi, T. Thonhauser, and C. Ambrosch-Draxl,
Phys. Rev. B , 184508(2004).[8] H. Lin, S. Sahrakorpi, R.S. Markiewicz, and A. Bansil, Phys. Rev. Lett. , 097001 (2006).[9] J.C. Slater and G.F. Koster, Phys. Rev. , 1498 (1954).[10] W.A. Harrison, Electronic Structure and Properties ofSolids.
Dover, New York (1980).[11] J.A. Nieminen and S. Paavilainen,
Phys. Rev. B , 2921(1999).[12] B.W. Hoogenboom, C. Berthod, M. Peter, Ø. Fischer,and A.A. Kordyuk, Phys. Rev. B , 224502(2003).[13] G. Levy de Castro et al. , cond-mat/0703131 (2007).[14] The specific self-energy used is shown in the inset to Fig.3(d). This includes a Fermi liquid broadening ( E ) of allthe levels. The peak-dip-hump feature in the d x − y or-bitals is modeled via coupling to a generic bosonic mode.[15] A.L. Fetter and J.D. Walecka, Quantum Theory of Many-Particle Systems.
Dover (2003).[16] T.N. Todorov et al. , J.Phys.: Condens. Matter , 2389(1993).[17] J.B. Pendry et al. , J.Phys.: Condens. Matter , 4313 (1991).[18] J. Tersoff and D.R. Hamann, Phys. Rev. B , 805(1985).[19] Y. Meir and N.S. Wingreen, Phys. Rev. Lett. , 2512(1992).[20] H. Ness and A.J. Fisher, Phys. Rev. B , 12469 (1997).[21] T. Frederiksen, M. Paulsson, M. Brandbyge, and A.-P. Jauho, Phys. Rev. B , 205413 (2007).[22] The distinct VHS peaks in the computed spectrum areexpected to be broadened to yield a broad hump muchlike the experimental spectrum due to self-energy cor-rections resulting from magnetic response of the electrongas in the -400 meV range (see, e.g., Ref. 23). Theseself-energy corrections are not included in the presentcalculations.[23] R.S. Markiewicz, S. Sahrakorpi, and A. Bansil, Phys.Rev. B , 174514 (2007).[24] I. Martin, A.V. Balatsky, and J. Zaanen, Phys. Rev. Lett. , 097003 (2002).[25] Y. Kohsaka et al. , Science315