aa r X i v : . [ c ond - m a t . s up r- c on ] N ov Echolocation by Quasiparticles
Sumiran Pujari and C. L. Henley
Department of Physics, Cornell University, Ithaca, New York 14853-2501
It is shown that the local density of states (LDOS), measured in an Scanning Tunneling Microscopy (STM)experiment, at a single tip position contains oscillations as a function of energy , due to quasiparticle interference,which is related to the positions of nearby scatterers. We propose a method of STM data analysis based on thisidea, which can be used to locate the scatterers. In the case of a superconductor, the method can potentiallydistinguish the nature of the scattering by a particular impurity.
PACS numbers: 74.55.+v,72.10.Fk,73.20.At,74.72.ah
Scanning Tunneling Microscopy (STM), which measuresthe “local density of states” (LDOS) as a function of positionand energy set by the bias voltage, has opened the door toimaging the sub-nanoscale topography and electronic struc-ture of materials, including normal metals [1] and especiallycuprate superconductors [2, 3, 4, 5, 6, 7, 8, 9].The dispersion relations of (Landau or Bogoliubov) quasi-particles may be extracted from STM data on normal met-als [10, 11] and superconductors [13], via the inverse methodcalled Fourier transform scanning tunneling spectroscopy(FT-STS) [10, 13], or directly in real space [11]. This tech-nique is based on the fact that impurities produce spatial mod-ulations of the LDOS in their vicinity – standing waves inthe electronic structure that generalize the Friedel oscillationsfound in metals at the Fermi energy. In the cuprates BSCCOand CaCuNaOCl [13], experiments showed these quasipar-ticle oscillations were dominated by eight wavevectors thatconnect the tips of “banana” shaped energy contours in recip-rocal space, the so-called Octet model as explained theoret-ically [12]. For optimally doped samples, the dispersion in-ferred from these wavevectors agrees well with d -wave BCStheory indicating the existence of well-defined BCS quasipar-ticles in this regime.The central observation of this paper is that the sameFriedel-like oscillations of the LDOS, analyzed in thespace/momentum domain by FT-STS, are also manifested inthe energy/time domain. Our analysis shows that the smallimpurity-dependent modulations of the LDOS have a period,in energy, inversely proportional to the time required by aquasiparticle wavepacket to travel to the nearby impurities andback – hence we call it “quasiparticle echo”. From this, inprinciple, one can determine the location and (in a supercon-ductor) the nature of the point scatterers in a particular sample. Quasiparticle echo — The basic idea of the LDOS mod-ulations may be understood semiclassically. The LDOS N ( ~r ; ω ) is defined as − (1 /π )Im G ( ~r, ~r ; ω ) , the time Fouriertransform of the local (retarded) Green’s function G ( ~r, ~r ; t ) .Imagine a bare electron wavepacket (centered on energy ω ) isinjected at time t = 0 at point ~r in a two-dimensional mate-rial: the Green’s function expresses its subsequent evolution.Assuming there are well-defined quasiparticles at this energywith dispersion E ( ~k ) ; then for every wavevector ~k on the en-ergy contour E ( ~k ) = ω , the wavepacket has a component spreading outwards at the group velocity ~v g ( ~k ) ≡ ∇ ~k E ( ~k ) / ~ .When this ring reaches an impurity at ~r imp , it serves as asecondary source and the reflected wavepacket arrives at the“echo time” T e ≡ | ~R || ~v g ( ~k ) | (1)for the ~k such that ~v g ( ~k ) k ~R ≡ ~r imp − ~r . This creates a sharppeak at t = T e in G ( ~r, ~r ; t ) [see Fig. 1 (d)], and hence modu-lations as a function of ω in its Fourier transform N ( ~r ; ω ) withperiod ∆ ω = 2 π ~ /T e [14]. Generically, for a particular im-purity direction, | ~v g | varies with energy, so the the modulationin δN ( ω ) due to the impurity is “chirped” correspondingly.We illustrate the quasiparticle echo first by a numerical cal-culation for a normal metal, defined by the lattice Schrodingerequation for the wavefunction u i on site i : X j ( t ij + µ i δ ij ) u i = Eu i . (2)Here the t ′ ij s are intersite hoppings and the µ i ’s are on-sitepotentials (including the chemical potential); in this paper,we assume they are translationally invariant except at dis-crete (and dilute) impurity sites. We take the specific caseof nearest-neighbor hopping t at half-filling, so the the dis-persion is ǫ ( k x , k y ) = − t (cos k x + cos k y ) , and we placeone (repulsive site potential) impurity at the origin. To nu-merically calculate the LDOS, we used the Recursion method[16], which is well-suited for cases without translational sym-metry.Fig. 1(a) shows the impurity case LDOS which has echooscillations on top of what otherwise would have been cleancase LDOS, visible along the sides of the peak. Note that,for us to see more than one oscillation within the bandwidth,the impurity must be at least several sites away; hence theoscillations always have small amplitude and are best viewedby subtracting the clean LDOS. Throughout the paper, energyis in units of t and time in units of t − with t = 1 and ~ = 1 .For a given energy ω , we define ∆ ω ( ω ) / as the separa-tion of the zeroes that bracket ω in the (subtracted) δN ( ω ) trace, and let T e ( ω ) ≡ π ~ / ∆ ω ( ω ) . We chose E = 0 . t and ~R in the [1,1] direction, for which the group velocity is v g = 2 . t/ ~ . Then, using δN (20 , ω ) , δN (30 , ω ) , a) - - Ω N H , ; Ω L b) - - - - - Ω ∆ N H , ; Ω L c) - - - - - - - Ω ∆ N H , ; Ω L d) G H , ; T L FIG. 1: LDOS as a function of energy, showing oscillations due toquasiparticle echoes. (a) LDOS at a point √ away from a pointimpurity along the [1 , direction(lattice constant = 1 ). (b) corre-sponding LDOS after subtracting the clean LDOS : δN (20 , ω ) ,(c) δN (40 , ω ) , (d) Magnitude of Local Green’s function as afunction of time : | G (20 , T ) | . The singularity appears at time T e / , where T e is given by (1). As we change the distance along thisdirection, the shortest echotime changes in proportion in accordancewith our semiclassical expectations. and δN (40 , ω ) [the first and last trace of these are shownin Fig. 1(c,d)], we read off ∆ ω/ . , . , and . ,from which v g T e / . √ , . √ , and . √ , re-spectively. The proportionality between the oscillation rateand the actual distance confirms the semiclassical explanationof these modulations. Echolocation — Using these quasiparticle echoes, we canlocate the position of impurities by measuring the LDOS wig-gles at a few points in the vicinity. At each point, we ex-tract the wiggle period ∆ ω and hence the echo time T e ≡ π/ ∆ ω . Then (1) defines a locus of possible impurity lo-cations, { ~v group ( ~k ) T e / ǫ ( ~k ) = ω . The intersection ofthe loci from STM spectra taken at multiple points ~r will lo-cate ~r imp uniquely. Furthermore, via a more exact derivationof the LDOS modulations (see below), the amplitude of theLDOS modulations tells the scattering strength of the impu-rities (in Born approximation they are proportional to eachother). Once an impurity has been pin-pointed, the higher-energy STM spectrum at that point may independently iden-tify the chemical nature of the impurity, e.g. in cuprates [15]and thus may reveal which kinds of impurities are importantfor the scattering of quasiparticles.As a test, we evaluated the subtracted LDOS at three points ~r A = ( − , , ~r B = ( − , , and ~r C = (15 , , withthe impurity at ~r = 0 . From the half-periods of the wiggles atenergy = 0 . t , (extracted as before) we found the respectiveecho times T A = 39 . , T B = 20 . and T C = 36 . . The threescaled loci(scaled by half the respective echotimes), shown inFig. 2 e), intersect at (0,0) as can be seen graphically, therebydemonstrating the idea of echolocation. A more careful nu-merical analysis can be done to extract errors in echolocationas well. a) - - -
20 0 20 40 - - b) - - - - - Ω ∆ N H r A ; Ω L c) - - - - - Ω ∆ N H r B ; Ω L d) - - - - Ω ∆ N H r C ; Ω L FIG. 2: (a) Schematic of few measurements around an impurity. Thearrowheads represent the STM Tip postions. After measurement, weget (b) δN ( ~r A ; ω ) , (c) δN ( ~r B ; ω ) and (d) δN ( ~r C ; ω ) . Extractingthe echotimes for each measurement at ω = 0 . t , we locate the im-purity, shown as a black dot, in the first panel. Note, that the locus ofimpurity locations changes with ω , and is of the shape shown only at ω = 0 . t . Analytic derivation — Adopting the T-matrix formalism,we can obtain an analytic form for the LDOS modulations.Formally, the difference in dirty LDOS and clean LDOS for asingle point impurity is given by δN ( ~r ; ω ) = − π Im h G ( ~r − ~r imp ; ω ) T ( ω ) G ( ~r imp − ~r ; ω ) i (3)where G ( ~r, ~r imp ; ω ) ≡ G ( ~r − ~r imp ; ω ) ≡ G ( ~R ; ω ) is thefree propagator; LDOS modulations are due to interferencebetween the two G factors. G ( ~R ; ω ) = lim δ → + Z B.Z. dk x dk y (2 π ) e i~k. ~R ω + iδ − ǫ ( ~k ) (4)The integrand is singular all along the energy contour ǫ ( ~k ) = ω , which we also parametrize as ~k ǫ ( s ) , where s is the arc-length in reciprocal space. By the change of variables z ≡ e ik y we convert the inner ( k y ) integral to a complex contourintegral in the z plane (rewriting ǫ ( k x , k y ) as an analytic func-tion of z ); for k x values found on the energy contour, the z path encounters two poles, one inside and one outside, de-pending on the sign of δ . Extracting the residue and absorbingfactors, we get G ( ~R ; ω ) = 12 πi I η ( s ) ds e i~k ǫ ( s ) · ~R ~ v g ( ~k ǫ ( s )) + G non-singular (5)where η ( s ) = 1 on the half of the energy contour where sgn( δ ) = sgn( | ~v g ( ω, s ) | ) and zero on the other half. The non-singular term G non-singular comes from the integrals over k y which do not cross the energy contour.At large ~R , the two-dimensional BZ integration will bedominated by those ~k [18] on the energy contour where thephase in the numerator is stationary, i.e. ~v g ( ~k ) k ~R : let us callsuch a point ~k ~R (so it is a function of the direction ˆ R and of ω ). Using standard formulas of the stationary phase approxi-mation [19] we get asymptotically G ( ~R ; ω ) = − ie iπ/ v g s πκ | ~R | e i~k ~R ( ~R,ω ) · ~R . (6)Here κ − is the curvature d ~k ǫ /ds of the energy contour at ~k ~R .Using (1) and (3), we finally get δN ( ω ) = T π v g κR cos (cid:16) ~k ~R ( ~R, ω ) · ~R (cid:17) . (7)valid in the limit of a distant impurity. (All factors are actuallyfunctions of ~R and ω : these arguments are shown only in therapidly varying factors.) As we change ω to ω + δω keeping ~R fixed, the chain rule gives ~k ~R ( ω + δω ) − ~k ~R ( ~R, ω ) = v − g δω ˆ R so, with φ = ~k ~R · ~R , we get cos (cid:16) ~k ~R ( ~R, ω + δω ) · ~R (cid:17) → cos( φ + T e δω ) . (8)This confirms the simple semiclassical prediction ∆ ω =2 π/T e (see Eq. (1)) for the modulation period due to echoes.The same quasiparticle interference is responsible for the spa-tial oscillations evident in (7) and the energy oscillations in(8). Echoes in cuprate superconductors — Additional relevantissues arise in case of superconductors. To discuss these, weuse a mean-field Bogoliubov-DeGennes(BDG) Hamiltonianwith/without a single point impurity as shown below. X j (cid:20) t ij + µ i δ ij ∆ ij ∆ ∗ ij − t ij − µ i δ ij (cid:21) (cid:20) u i v i (cid:21) = E (cid:20) u i v i (cid:21) (9)where we are using a lattice formulation of BDG equations.The u i s and v i s represent particle and hole amplitudes on site i , t ij s and µ i s represent the intersite hoppings and site chemi-cal potentials respectively, and ∆ ij represent the off-diagonalorder parameter amplitude. We discuss d -wave superconduc-tors (dSC’s) to highlight this method’s application to cuprates.For dSCs, ∆ ij is nonzero only on nearest-neighbor bonds and ∆ ˆ i, ˆ i ± ˆ x = − ∆ ˆ i, ˆ i ± ˆ y because of the d -wave nature. Our normalstate is the same nearest neighbor tight binding model on thesquare lattice with t = 1 and off-diagonal hopping amplitudesset to | ∆ | = 0 . . The Recursion method was extended to su-perconductors in [17] and is used for our numerics. In Fig.3 c) and d), we show the LDOS(after subtracting the cleanLDOS shown in Fig. 3 a)) at √ distance from an impurityalong the (1 , direction for the case of a potential scattererand an anomalous pair potential scatterer (which scatters anelectron into a hole and vice versa) respectively.In contrast to the normal case, there are two different wig-gles : a fast one and a slow one. The reason for this is that thedSC quasiparticle dispersion gives rise to two different group a) - - Ω N H Ω L b) - - - - - - k x k y c) - - - - - Ω ∆ N O R D H , ; Ω L d) - - - - Ω ∆ N ANO H , ; Ω L FIG. 3: Quasiparticle echoes in a d -wave SC. (a) no impurity N ( ω ) showing the d-wave gap, (b) A caricature of the two dif-ferent group velocities along (1,1) direction for d-wave Bogoliubovdispersion, (c) δN ORD (20 , ω ) for an ordinary impurity and (d) δN ANO (20 , ω ) for an anomalous impurity. velocities in the (1 , direction [20]. We also note that thefast wiggles exist only within the gap while the slow wigglesare both inside and outside the gap. In Fig. 3 b), we showthe constant energy contours for the quasiparticle dispersiongiven by E ( ~k ) = q ǫ ( ~k ) + ∆( ~k ) , the gradient of which isthe quasiparticle group velocity. From Fig. 3 b), we see thatalong (1 , , the banana-shaped energy contours in the firstand third quadrants give one velocity (which corresponds tothe slow wiggles), while the contours in the second and fourthquadrants give a slower velocity (which corresponds to thefast wiggles). For E > | ∆ | , there are no longer “banana” con-tours, so we get only one group velocity (similar to the normalcase) and hence only one kind of wiggle is seen in Fig. 3(c,d)outside the cusps.Once the impurity is located using the loci intersectionmethod desribed before, one can study the LDOS data aroundthe impurity to infer the impurity’s strength and whether it isordinary (magnetic/nonmagnetic) (cf. Ref. 21 and referencestherein) or anomalous [22]. This distinction is already visiblein individual spectra: provided the normal state is particle-hole symmetric, one gets particle-hole symmetric echo oscil-lations δN ANO from an anomalous impurity, since it scatterselectrons into holes and vice versa [Fig. 3(d)]; this is not thecase for δN ORD from an ordinary impurity [Fig. 3(c)].A second diagnostic distingushing (nonmagnetic) ordinaryscatterers from anomalous ones is the real-space pattern of thesurrounding standing waves in the LDOS, which is best seenin Born Approximation. In this limit, the impurity T-matrix isof the form (in the × Nambu notation) U imp τ or ∆ imp τ for the ordinary or anomalous cases, respectively. Then theecho oscillations take the respective forms δN ORD ∝ U imp ( G − G ); δN ANO ∝ ∆ imp (2 G G ) . (10)Here, the G ij s are the matrix elements of the usual free prop-agator G ( ~k ; ω ) = ( ω − E ( ~k ) ) − (cid:2) ω + ǫ ( ~k ) τ + ∆( ~k ) τ (cid:3) thus in real space G ( ~R ; ω ) = πi (2 π ) I η ( s ) ds g ( ~k ( s, ω )) + G non-singular (11)where g ( ~p ; ~R, ω ) ≡ ω ( ǫ ( ~p ) τ + ∆( ~p ) τ ) .We can carry out the stationary phase approximation as be-fore, but instead we numerically calculated the propagator us-ing Eq. (11), since we are interested in LDOS informationaround(close) to the impurity. In Fig. 4, we show δN aroundan impurity over a grid of 20x20 lattice points(shown onequadrant with others related by symmetry). a) - - b) - FIG. 4: Shown are the δN ( ~R ; ω = 0 . t ) around an impurity overa grid of (0,20)x(0,20) with other quadrants related by symmetry.(a) δN ORD , (b) δN ANO . A subgap value of ω = 0 . t was chosenarbitrarily. We see that certain of the real-space oscillations, present incase of the ordinary impurity, are suppressed in the case ofa d-wave anomalous impurity. This is the same effect as thesuppression of certain “octet” vectors [12, 13] for the case ofd-wave anomalous impurity as argued in [22]’s Eq. 10 and thefollowing paragraph. Our real-space analysis qualitatively du-plicates that of Ref.22 illustrating how the real-space QPI andour energy-domain echoes are complementary manifestationsof the same phenomenon.
Conclusion and Discussion — In conclusion, we have in-troduced a method of STM data analysis in the energy do-main as a phenomenological tool for the study of real materi-als, complementary to FT-STS. Since it is based on the same quasiparticle interference effects already used successfully inFT-STS, we have confidence that the signals will be observ-able. They should be particularly strong in materials with anenergy-dependent group velocity in some range of energies,such as d-wave superconductors and also graphene [23].Since the echo analysis can be done in local patches of thesample (unlike FT-STS which fourier transforms over a largerregion), we can locally verify the existence of quasiparticlesat various energies through QPI. In particular, in cuprates,echoes might be used to check the hypothesis of quasiparti-cle extinction [24] above a certain energy. Furthermore, wehave argued that echo analysis might reveal the nature of spe-cific impurities [25] in a sample, information which hitherto was (at best) known statistically.We thank S. C. Davis, A. V. Balatsky, and P. J. Hirschfeldfor discussions. S.P. thanks Stefan Baur for helping withMathematica. This work was supported by NSF Grant No.DMR-0552461. [1] M. F. Crommie, C. P. Lutz, and D. M. Eigler, Science 262, 218(1993); M. F. Crommie, C. P. Lutz, and D. M. Eigler, Nature(London) 363, 524 (1993).[2] A. Yazdani et al. , Phys. Rev. Lett. 83, 176 (1999).[3] E. W. Hudson et al. , Science 285, 88 (1999).[4] S. H. Pan, et al , Nature 413, 282 (2001).[5] T. Cren et al
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