On the possibility that STS "gap-maps" of cuprate single crystals are dominated by k-space anisotropy and not by nano-scale inhomogeneity

0.4 MB·English

by  J. R. Cooper

On the possibility that STS “gap-maps” of cuprate single crystals are dominated by k
space anisotropy and not by nano-scale inhomogeneity.
J.R. Cooper
Department of Physics, University of Cambridge,
J. J. Thomson Avenue, Cambridge CB3 OHE,U.K.∗
7
(Dated: February 6, 2008)
0
0
The results of a computer analysis of a simple 2D quantum mechanical tunnelling model are
2
reported. These suggest that the spatial dependence of the superconducting energy gap observed

n byScanningTunnellingSpectroscopy(STS)studiesofsinglecrystalsofthehighTc superconductor
a Bi2Sr2CaCu2O8+x isnotnecessarilycausedbynanoscaleinhomogeneity. Insteadthespatialdepen-
J denceof theSTS datacould arise from themomentum (k) dependenceof the energy gap, which is

6 a definingfeature of a d-wavesuperconductor. It is possible that this viewpoint could beexploited
2 toobtain k dependentinformation from STS studies.

]
n I. INTRODUCTION reasonablylargeFermisurface,P(k)isnotconstant,be-
o cause final states with larger values of |k | will not be
c T
- accessible. Butforplanartunnellingperpendiculartothe
r In recent scanning tunnelling spectroscopy (STS) ex- conductinglayersofaquasi-2DmaterialsuchasBi:2212,
p
periments, differential conductance (G ≡ dI/dV) ver- P(k)isconstantifthequasi-2DFermisurfaceiselectron-
u
s sus voltage (V) curves are measured at many thousands like and exactly circular6, or if the “tunnelling cone” is
. of points on the surface of single crystals of the highly wide enough, i.e. |k | is large enough, so that P does
t T
a anisotropic cuprate superconductor Bi2Sr2CaCu2O8+x not vary much over the Fermi surface6. At first sight,
m orBi:22121,2. TheformofG(V)ateverypointisroughly
spatial confinement of the initial electron states in the
- that expected from a tunnel junction between a normal sharp STS tip would be expected to give a larger spread
d metalandad-wavesuperconductor. However,thesuper- in k , so there would be an even greater tendency for
n T
conductinggapparameter,∆,determinedfromthepeaks
o P to be constant for c-axis tunnelling into Bi:2212 using
in G(V), varies strongly and continuously with position,
c STS. This seems to be the prevailing view in the field.
[ from 20 to 70 meV with a typical length scale of 1−2 Whilethemodelcalculationsreportedheredoshowthat
nm. It is thought that G(V) is a direct measure of the forrealisticSTStipsizesandtipmaterials,k isnotcon-
1 T
localquasi-particledensityofstates(DOS)andtherefore
v served,i.e. there is no well-defined refracted wave in the
these“gapmaps”arewidelyacceptedasbeingassociated
7 quasi-2D metal, they also show that P is not constant.
6 with nanoscale inhomogeneity, implying that the super- Insteaditfallsdrasticallyasthein-planewavenumberof
6 conducting gapandallothersuperconductingproperties Bi:2212,representedby the parameter|k | in the model,
y
1 are spatially inhomogeneous. It is difficult to reconcile
is increased.
0 this picture with the results of other experiments such
7 as NMR3 and heat capacity4, where such gross inhomo- Recent STS studies give clear experimental evidence
0 geneity is not detected. thatbiggergapsareobservedinareasofthecrystalwhere
/
t there are more off-stoichiometric oxygen atoms in the
a The energy (E) dependent quasi-particle DOS, N(E) BiO-SrO layer2. In a naive semi-classical picture these
m inad-wavesuperconductorwithak-dependentorderpa- O2− ions would scatter the tunnelling electrons, via the
- rameter∆(k)canbethoughtofasthesumofs-wavelike Coulomb interaction, giving them extra transverse mo-
d contributions from different directions in k-space, i.e.,
n mentum. So there could be a tendency for a particular
o N(E)= (2π2)3 R h¯|vd(Sk)|[E2−∆E(k)2]1/2, where dS is a small in-planedirectionofkto haveextraweightinG(V)ata
c element of a constant energy surface in the normal state certain separation between the STS tip and an O2− ion
v: withelectronvelocityv(k). ThemeasuredquantityG(V) and hence a particular value of ∆(k) could be favored
Xi is proportional to (2π2)3 R dER h¯|vd(Sk)|P(k)[E2E−δ∆(E(k−)V2])1/2, tpheearkes.inTGhi(sVp)roavsitdheesSaTmSetciphainsimsmovfeodravcarroisastitohnessuinrfathcee
where P(k) is the probability of tunnelling into a state
r ofthecrystalwithoutinvokingnanoscaleinhomogeneity.
a k, so G(V) only reflects the behavior of N(E) precisely
when P(k) is constant. For a defect-free planar metal- The model calculations reported here are consistent
insulator-superconductortunneljunction,transversemo- with this semi-classical picture in that the strong k
y
mentum (k ) will be conserved because of translational dependence of P mentioned above becomes much less
T
invariance. Also, because of the exponential decay of marked when there are O2− ions near the STS tip. Al-
the evanescent wave in the barrier, the tunnelling cur- though they do not give a completely adequate descrip-
rent will be dominated by states with smaller values of tion of the observed G(V) curves, they seem to capture
|k |, that lie within a “tunnelling cone” whose angular the important physics and future microscopic calcula-
T
widthdependsonthestrengthandwidthofthebarrier5. tions of the G(V) curves should take the present results
So, for planar tunnelling into a superconductor with a into account.
2
V
4 4
l = 1.4 nm l = 0.7 nm
x
II 2 eV 3 3
L 3eV
V L +A
x III 1 eV 2 2
E
F
I IV' V Ly 5.5eVI 0.8 nm 0.(7b-3) eVV y (nm) 01 y (nm) 01 +A/2
IV -1 -1 0
2L
T
II y G X G -2 -2 -A/2
x -3 (a) -3 (b)
III
-4 -4 -A
(a) Y a Y
0 1 2 3 4 0 1 2 3 4
L
A x(nm) x(nm)
G X (c) G FIG. 2: (Color on-line). Contour plots in the (x,y) plane of
real part of wave function ψR in region V for wavelengths of
FIG. 1: (a) Spatial regions used for 2D tunnelling calcula- (a)1.4nmand(b)0.7nm,usingtheparametersgiveninFig.
tions, I STS tip, II vacuum, III BiO-SrO layer, IV and IV’ 1(b)andwithoutanyO2− ions. Herexismeasuredfromthe
oxygen ions, V Bi:2212. (b) Corresponding potential ener- start of region V.The linear color scale is also shown.
gies versus distance (x) at y = 0. (c) Typical Fermi surface
cross-section7, with solid arrows marking the minimum and
maximumkvectorsconsideredhereanddashedarrowsshow-
takento be 0.56nm wide9. In most of the calculations a
ing theangle α with the(−π,0) direction (see text).
barrierheightof1eV,consistentwithbandtheory7,was
used. Thislayerisnowbelievedtobe insulatinginview
of (a) the extremely large and non-metallic c-axis resis-
II. THE MODEL tivityofBi:2212crystals10and(b)variousexperimentson
intrinsicc-axistunnellinginmesastructures,forexample
The spatial dimensions and energy barriers of the 2D Ref. 11. RegionV, where the potentialenergyV(x,y) is
quantummechanicaltunnellingmodelaredefinedinFig- constant, corresponds to the bulk of the cuprate super-
ures 1(a) and 1(b) respectively. Region I corresponds to conductor. Representing the electronic structure of the
the STS tip, which is the projection of a truncated cone cuprates by a free electron model, which is effectively
withsidesat30degreestothesurfacenormal,i.e. tothe done here, is a drastic approximation. However includ-
c-axisofthe cupratecrystal. TheFermienergyof5.5eV ing anextrabarrierto representthe c-axislattice poten-
inthetipcorrespondstoafreeelectronwavevectorof12 tial,i.e. the next BiO-SrOlayer,does notalter the main
nm−1 which is close to the Fermi wave vector (k Au) of conclusionsbecauseadefect-freeplanarbarrierwouldre-
F
typical tip materials such as Au or a Pt alloy. The vac- flect semi-circular waves such as those shown in Figure
uumgapinregionIIwasgenerallytakentohaveawidth 2 without altering k . This point was checked explic-
y
L = 0.24 nm and a barrier height of 3 eV as deduced itly by introducing an appropriate extra planar barrier
A
from STS work8. The boundary condition on the small in region V. This had no effect on the behavior shown
line of length 2L represents an incoming electron wave later for P(k ) in the absence of O2− ions, but because
T y
of the form exp[ikAu(xcosθ + ysinθ)] travelling at an of standing waves there was more scatter in the results.
F
angle θ to the normal, i.e. on this line (where x=0) the The present model is therefore a useful starting point
real and imaginary parts of the wave function, ψ and fordiscussingtheeffectofoxygenionsinthebarrierlayer
R
ψI were set equal to cos(kFAuysinθ) and sin(kFAuysinθ) andfordiscussingwhethertheSTStechniquedoesindeed
respectively. The STS tip width 2L was often fixed at give a true k-space average of the electronic DOS. The
T
0.3 nm but calculations were made for a range of val- value of the Fermi energy E in region V was varied
F
ues between 0.15 and 5 nm. For 2L = 5 nm (i.e. 10 over the range 0.76 to 3.04 eV because this corresponds
T
times the wavelength of the electrons in the Au tip) the to free electron wave numbers from 4.47 to 8.94 nm−1
result expected for a planar junction, namely conserva- and wavelengths between 1.4 and 0.7 nm. These wave
tionoftransversemomentumwasobtained. Howeverfor numbers are approximately equal to the minimum and
2L ≤0.6nmtherewasnoevidencefork conservation, maximumin-plane valuesofk for electronstatesatthe
T T F
i.e. no signofa wavepropagatingatthe appropriatean- Fermi energy in the cuprates shown in Figure 1(c). In
gleinregionV.Theangleθ wasvariedbetween0and30 tunnelling problems one always considers the “physical
degrees, but since the number of initial states in the tip electron”as emphasizedby Andersoninconnectionwith
increases as sinθ, Ref. 5, it was decided that 25 degrees inter-layer tunnelling in the cuprates.
was appropriate for comparison with experiment. This The small circular regions IV and IV’ in Figure 1(a)
choice does not affect our conclusions. correspondtosphericallysymmetricO2− ionswithafull
Region III corresponds to the BiO-SrO layer which is outer shell situated 0.1 nm from the CuO layer. They
2
3
0.4 0 0.04
0.3 RIme yy ttkk1155 No O2- ions -0.5 0.03 RIme yy ssuu1155 O0. 2-,T -i0p. 6d insmtances --21.5
-1
y)m( 0.2 -1.5 ym()0.02 -2.5
yRe(), I 0.1 --22.5 yLog||10 yRe(), I0.01 --33.5 yLog||10
0 0
-4
-3
-0.1 l = 1.4 nm (a) -3.5 -0.01 l = 1.4 nm (c) --54.5
-0.2 -4 -0.02
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
y (nm) y (nm)
0.15 -0.5 0.04 -1
0.1 RIme yy ttkk113355 No O2- ions -1 0.03 RIme yy ssuu113355 O0. 2-,T -i0p. 6d insmtances -1.5
-1.5 -2
)0.05 )0.02
ym( -2 ym( -2.5
yRe(), I 0 --32.5 yLog||10 yRe(), I0.01 --33.5 yLog||10
-0.05 0
-3.5 -4
-0.1 l = 0.7 nm -4 -0.01 l = 0.7 nm -4.5
(b) (d)
-0.15 -4.5 -0.02 -5
-4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4
y (nm) y (nm)
FIG. 3: Plots of the electron wave function (ψR, ψI and |ψ|) in a narrow strip of width x = 0.05 nm extending from y = -3.8
to3.8nm,justinsideregionV,i.e. thefirstCuO2 layer. Computeddataforthetwolimitingwavelengthsareshownin(a)and
(b)intheabsenceofO2− ionsandin(c)and(d)withO2− ionsat+0.2and-0.6nmfromthecenteroftheSTStip. Parameter
values are those in Figure 1 (b),with 2LT = 0.3 nm, LA=0.24 nm and θ (see text) = 25 degrees.
give rise to a Coulomb field outside a typical atomic III. METHOD AND RESULTS
radius of 0.06 nm. In the metallic region V, this field
will be strongly screened by the mobile electrons (on a
A commercially available partial differential equa-
length scale of order k−1). As often done in tunnelling
F tion (PDE) PC package12 was used to solve the 2D
problems15 this screening is representedby anequal and
Schr¨odingerequations for ψ andψ with the above po-
opposite image charge in region V giving a dipole field R I
tentials. AnexampleprogramattributedtoBackstrom12
from each non-stoichiometric O2− ion in the BiO-SrO
and initial tests for the textbook problem of tunnelling
layer. For simplicity, dielectric screening in the BiO-SrO
through a 1D potential barrier13 were helpful in setting
layeris ignoredsince positive polarizationchargesatthe
theboundaryconditions(BCs),detailsofwhicharegiven
II-III boundary can then be neglected. However in or-
in footnote 14. The BCs on the semicircular bounding
der to compare with experiments, where the mean spac-
surfaces shown in Figure 1(a) were chosen so that out-
ing between O2− ions is typically 0.8 nm, the Coulomb
wardly propagating waves are favored and the absence
field was multiplied by a Gaussian attenuation factor,
of significant standing waves was verified by comparing
exp(−(y−y )2/σ2) where y is the y co-ordinate of the
1 1 plots of |ψ| andψ alongvariouslines in regionV. Most
O2− ion and σ = 0.4 nm. This was done because the R
of the calculations were done for the geometry shown in
tunnelling electrons will be scattered by local increases
Figure 1(a)with L = L = 4 nm but checks weremade
in the Coulomb potential above a uniform background x y
for larger values of L , for a rectangular bounding re-
potential given by more distant O2− ions. The Coulomb x
gionandalsofor a cylindricallysymmetric 3Dcase. The
potential inside region IV was set equal to a constant,
graphical output given by the PDE program was typi-
namely its value at the III-IV boundary. For simplicity
cally in the form of contour plots of ψ such as those
it was also assumed that the dipole electric field does R
in Figures 2(a) and (b) or plots of ψ and log|ψ| along
notextendintothe vacuumregionII orinto themetallic R
certain lines (not shown). Arrays of 101 x 101 ψ and
region V. R
ψ valuesovervarioususer-definedregionsofFigure1(a)
I
werealsogeneratedand these wereprocessedfurther us-
ing other commercially available PC packages.
Figures 2(a) and (b) show examples of the contour
4
plots for the two extreme kF values mentioned above in IV. COMPARISON WITH EXPERIMENT
the absence of any impurities in the barrier. Note that
the shorter wavelength waves are more concentrated in
In order make an initial comparison with experiment
the forward direction. As justified in detail later, this
the following procedure was used. The ψ(y) data such
leadstothekeyresultofthepresentwork,namelyP(k )
y as those shown in Figures 3(a) to 3(d) were multiplied
is not constant for tunnelling between a sharp STS tip
by exp(−ı2πy/λ) or exp(ı2πy/λ) and integrated over y
and Bi:2212 and therefore the G(V) curves will give a
to give the (complex) numbers a(k ) and a(−k ). These
y y
distorted image of the DOS. The barriers are relatively
are the quantum mechanical overlap integrals between
weaksothatthewidthofthe“source”alongyjustinside
an initial electron state in the STS tip and plane waves
region V is considerably wider than that of the STS tip
of wavelength λ propagating in the ±y directions in re-
(0.3 nm). For the parameters given in Figure 1(b) and
gion V. Hence the quantity |a(k )|2 + |a(−k )|2 is the
y y
used in Figure 2, the regions of high intensity below the
required probability P(k ) of electron tunnelling from
y
STS tip and 0.2 nm inside the conducting regionV have
the tip into a state in region V with |k | = 2π/λ. P(k )
y y
FWHM in |ψ|, of δy = 1.16 nm and 0.84 nm, for λ =
was calculated for a range of λ values between 1.4 and
1.4 and 0.7 nm respectively. The ratios λ/δy differ by a
0.7 nm. Since the Schr¨odinger equation is linear in ψ
factor 1.5 and this, together with the fact that λ ≃ δy
andthesolutionsareobtainedbymatchingψ and∇ψ at
andδx,mustbeberesponsiblefordifferenceinspreading
the boundaries it is plausible, but not proved, that the
out of the waves shown Figures 2(a) and 2(b). Another
same probability function P(k ) would also apply to the
y
important point is that, as mentioned in the previous
physically realistic case of the Bi:2212 bands where, as
section, there is little asymmetry between positive and shown in Figure 1(c), there is a range of |k| values with
negative y values, despite the fact that the wave coming
the same (Fermi) energy. Typical behavior of P(k ) is
y
in to the STS tip is at 25 degrees to the surface nor-
shown in Figures 4(a) and 4(b) for various positions of
mal, and therefore has significant transverse momentum
the STS tip relative to one or two O2− ions respectively.
k =kAusin(25)=5.1 nm−1 along y. In contrast to the
y F Generally P varies as Aexp(−B|k |), where A and B
y
planar tunnelling case, no refracted wave that conserves
areconstants,butespeciallywhenthespacingofthetwo
k isproducedbecausetheSTStipwidth,2L =0.3nm
y T ionsis∼λ,thereissomeextracurvatureinplotsoflogP
is too small, as shown later in Figure 4(c).
vs. k whichprobablyarisesfromtwo-beaminterference
y
effects. As shown by the curve ta in Figure 4(b), two
nearby O2− ions reduce P by 2-3 orders of magnitude,
Computed data are shown in more detail in Figures thisisunderstandablebecausetheheightoftheBiO-SrO
3(a) to 3(d) as plots of ψ , ψ and log|ψ| versus y at barrier is increased locally from 1 eV to approximately
R I
a mean distance along x of 0.25 nm into region V. The 5 eV. Figure 4(c) shows the behavior of P(ky) for differ-
data shown are in fact the sum of 100 ψ values between ent STS tip widths (2LT) ranging from 0.15 to 3 nm in
x = 0 and 0.5 nm, now measured from the beginning the absenceofO2− ions. The exponentialbehaviormen-
of region V, for fixed values of y. A strip of this width tioned above persists up to 2LT = 0.6 nm, after which
corresponds to the first CuO bi-layer in Bi:2212,9 but there is a sharper fall associated with the tendency to-
2
the same results are obtained for different strips e.g. be- wards conservation of ky for a planar junction.
tween x = 0 and 0.3 nm. The |ψ| plots are shown on TheseP(k )valueswerethenusedtocalculatethecor-
y
a semi-logarithmic scale in order to see the behavior at respondingG(V)curvesexpectedinthesuperconducting
largervalues ofy. There arenosignificantoscillationsin state. Referring back to Figure 1(c), the contribution to
|ψ|showingthattheBCsdoindeedproducepropagating G(V)froma k-vectoronthe Fermisurfacewasweighted
andnotstandingwaves. Theperiodsoftheoscillationsin byP(|k|)where|k|isthe distancefromthe Γ point. Be-
ψ andψ along y arevery closeto the twowavelengths cause of the simplicity of the model, calculations were
R I
used, 0.7 and 1.4 nm. Figures 3(a) and 3(b) show data limited to the region between the two solid arrows in
with no O2− ions in the BiO-SrO layer. Comparison of Figure 1(c). The behavior of P(k ) for different O2− en-
y
the two figures shows that the shorter wavelength has vironments shown in Figures 4(a) and (b) means that
loweramplitude oscillations,by at leasta factor 10,over small gap regions near the d-wave nodes are heavily fa-
most of the range of y confirming the differences shown voredintheabsenceofO2− ionsbutthatscatteringfrom
in Figures 2(a) and 2(b). Figures 3(c) and 3(d) show O2−ionsallowshighergapregionstobeaccessed. Figure
the effect of having one O2− ion at y = +0.2 nm and 4(d)showsthatthisconclusionisnotstronglydependent
another at y = −0.6 nm, a spacing typically observedin on parameter values such as the STS tip width 2L , the
T
the experiments2, the center of the STS tip being at y BiO-SrO barrier height or the spacing between the STS
= 0. The key point is that in the presence of O2− ions tipandtheBi:2212surface(L ). Sometypicalcalculated
A
theamplitudeismuchlessdependentonthewavelength. G(V) curves are shown as un-normalized plots in Figure
This suggests that STS does not give a true DOS aver- 5(a)andasnormalizedonesinFigure5(b). Ingenerating
age for quasi-2D superconductors and that the presence thesecurvesthe standardweak-couplingd-waveformfor
of O2− ions allows states with larger values of k to be the gapparameter,∆(k)=∆cos(2α) wasused, whereα
y
accessed. is the angle between k and the (−π,0) direction shown
5
2 104 4
10 0n oA O u b tk 46 AA uuij 10 +n7o -O7 t gtk ++53 --95 tsfx n+o 9O -29- tskz ++ 75 --79 ttgf 3.5 +3 -3 ta
2 A ug 8 A uk + 4.5 -4.5 sw V) 3 +3 -5 sx
1 1 G( V) +2 -6 su
P (k)y0.1 P (k)y0.1 1 104 alised G( 2.52 ++75 --79 ttgf
m +9 -9 sz
or 1.5
+2 -6 su +3 -5 sx +2 -6 su N no O2- tk
0.01 0.01 1
x 0.1 x 4
(a) +3 -3 ta (b) 0.5 (b)
x 4 (a)
0.001 0.001 0 100 0
0.4 0.5 0.6 0.7 0.8 0.9 0.4 0.5 0.6 0.7 0.8 0.9 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 3
k(1010m-1) k(1010m-1) eV/D eV/D
y y
1.5 tq -11
10 3.0 tk FIG.5: (a) Calculated G(V) curvesfor ad-waveDOSin the
6.0 tn -12 BiO-SrO same (arbitrary) units plotted versus eV/∆ for no O2− ions,
1 barrier (eV) curvetk,and pairs of O2− ionsat variousdistances, y in Ao,
-13 from the STS tip at y = 0. (b) Corresponding G(V) curves
P (k)y00.0.11 12 to -10B (10m) --1154 LT mnourlmtiaplliezsedoft0o.4u.nity at eV = 3∆ and displaced vertically by
18 tp x 0.01 L
0.001 30 tr -16 A
P=Aexp(Bk) where the gap appears to be larger. It also noteworthy
y (d)
0.0001 -17 that curves labelled tk, tg, tf and sz seem to show very
(c) x 0.001 0 1 2 3 4 5
L (10-10m) or V(eV) similar behavior at low voltages a feature that is also
0.4 0.5 0.6 0.7 0.8 0.9 present in the experimental data, for example Figure 3
ky(1010m-1) of Ref. 1. The peaks in G(V) vary from V = 0.7∆/e
to ∆/e . However even though P(k ) varies drastically
y
FIG. 4: Plots of P(ky), the probability weighting factor (see with ky when there is little scattering by the O2− ions,
text) versus ky, (a) with no O2− ions present (solid black the corresponding G(V) curves do appear to be more
circles) and with one ion present at different distances, in rounded than those observed experimentally, especially
Ao units (0.1nm) from the STS tip. (b) with pairs of ions recently2,18.
at different distances, in Ao, and (c) with no ions present Within the present approach even higher k- space se-
but for various tip widths, 2LT in Ao. (d) Effect of various lectivity may be needed to account for the experimental
parameters, the BiO-SrO barrier height in eV, the vacuum
gap L in Ao, and the STS tip half-width L on the mean STS data. There are several ways in which this might
A T
slope of log e(P) vs. ky in theabsence of O2− ions. occur. Firstlyinthepresentmodeltheband-structureof
theBiO-SrOlayerisignored. Inamicroscopictreatment
one would consider tunnelling to arise from virtual exci-
tations into the BiO-SrO conduction band. This band is
by dashed arrowsin Figure 1(c). Tunnelling data invari-
likelytohaveanarrowenergywidthandhenceforagiven
ably show some broadening of the G(V) curves that is
impact parameter the transverse momentum gained by
ascribed to a finite quasi-particle lifetime (h¯/Γ). In the
scattering from an O2− ion may well be better defined
Dynes formula16 used by Wei et al.6 E is replaced by
than in the present model. Secondly the band structure
E −ıΓ and N(E) is given by the real part of the usual
of the CuO layer is not included. Since this is a tight
expression. This formula should not be applied when 2
binding band structure, Fermi surface states nearer the
Γ ∼ ∆16and therefore, in the present calculations, mod-
Brillouin zone boundary, i.e. those with small values of
erate damping, namely Γ=0.1∆(k) was assumed for all
the angle α in Figure 1(c), will contain significant com-
k. In future it might be possible to derive experimental
ponents of states with higher momenta, namely k+G
values of Γ(E,k) by appropriate fits to the STS G(V)
whereGisareciprocallatticevector. Thiswillgiveeven
curves.
less weight to the states with higher values of |k|.
Thirdly the standard “semiconductor model” is used
to describe the tunnelling DOS of a superconductor.
V. DISCUSSION ThetheoryofBlonder,KlapwijkandTinkham(BTK)19,
whichmakesuseoftheBogoliubovequationsratherthan
The un-normalized G(V) curves shown in Figure 5(a) the Schr¨odinger equation, often provides good fits to
are similar to some of the published experimental data, G(V) curves of superconductors obtained with either
e.g. thefirstpaperbyPanetal.17wherethereisclearlya metallic point contacts or small-area tunnel junctions,
strongreductioninthemagnitudeofG(V)intheregions despite fact that it uses a 1D model. BTK theory con-
6
tains some features that could be relevant here, for ex- symmetric situation where the O2− ion is positionedim-
amplethequasi-particlecurrenttransformsintoasuper- mediatelybelowtheSTStipandtheelectronwavecomes
current over the coherence length which in the present in at normal incidence.
case is highly k-dependent. A paper20 applying BTK
theory to a d-wave superconductor does conclude that
G(V) ∼ N(V), but because a 1D model is used, k is
T VI. SUMMARY AND CONCLUSIONS
conserved, in contrast to the present work.
Fourthly, the presence of a pseudogap would also re-
A new way of interpreting the “gap-maps” observed
duceoreliminatecontributionstoG(V)fromtheregions
ofthe Fermisurface withlargervalues of|k|. In amodel in STS studies of cuprate superconductorshas been pro-
used to account for heat capacity data22 there is a tri- posed that does not invoke nanoscale inhomogeneity. In
principle, perhaps when extended to include real atomic
angular non-states-conserving pseudogap that ARPES
orbitals and band structure, it can be used to obtain k-
experiments, e.g. Refs. 23 and 24, suggest has great-
resolved information from STS data.
est effect in the anti-nodal directions. Introduction of
a pseudogap would help in fitting the V-shaped, appar- Note added on 26th Jan. 2007. Since submitting this
ently non-states-conserving G(V) curves that are often paper I became awareof analytical theory27 of the scan-
observed experimentally by STS2,17,18,21. Reports from ningtunnellingmicroscopeforasphericalSTStip. Equa-
other STS groups21,25,26 suggesting that, especially for tions (4) and (9) of Ref. 27 lead to results that are con-
overdoped Bi:2212 crystals, the normalized G(V) curves sistent with the present work for a single barrier with-
can be uniform for linear scans over 10−20 nm do not out any O2− ions. Neglecting terms with reciprocal lat-
−→
necessarilycontradictthepicturepresentedhere,sinceas tice vectors, G 6= 0, gives a tunnelling probability P ∝
showninFigs. 4(a), 4(b) and5(b) the normalizedP(ky) exp[−2(κ2+|−→k|||2)1/2|−→r0|], where27 κ = h¯−1(2mφ)1/2
−→
curves and hence the normalized G(V) curves, can be is the minimum inverse decay length, k the in-plane
||
quite similar for a variety of O2− spacings. electronwave-number,φ the work function (i.e. the bar-
Finally and perhaps most importantly, the k- rier height relative to E ) and −→r the distance between
F 0
selectivity could be further increased by the interference the center of curvature of the STS tip and the sample
of scattered waves from different O2− ions. There are surface. For a barrierheight of3 eV and |−→r | = 0.8 nm,
0
someindicationsofthiseffectintheP(ky)curvelabelled fitting the above formula to P = Aexp(Bky) over the
tg in Figure 4(b) where the STS tip is placed symmet- range of k (≡−→k ) used here gives B = -9.6 10−10 m−1.
y ||
rically between two ions at a distance ∼ λ from each.
Bearinginmindthatthepresentcalculationsweremade
These effects could well be more marked in the realis-
for a flat STS tip and a 2D model, this agrees well with
tic 3D case where “multiple beam” interference is more the data point inFigure 4(d), whereB = -1210−10 m−1
likely, for example the scattered waves from 3 or 4 suit-
when the BiO-SrO and vacuum barrier heights are both
ably spaced O2− ions could interfere constructively for
3 eV, and the width of the combined barrier is 0.8 nm.
certain directions of k to give 9 or 16 times higher in-
tensity. Since the positions ofthe O2− ionscanbe found
usingSTS2itmightevenbepossibletolocatesuchstruc-
tures andhence obtain better k-resolutionin the experi- VII. ACKNOWLEDGEMENTS
ments. AfasterPDEprogramwouldbe neededto inves-
tigate these 3D aspects using the present model. How- This work forms part of a long-standing collaboration
ever the main result reported here, namely the strong withJ.W.LoramandJ.L.Tallonwhohaveprovidedkey
variation of P with k in the absence of O2− ions and insightsandsuggestionsatallstages. Helpfulsuggestions
y
the weaker dependence in their presence has been ver- werereceivedfromC. Bergemann,who alsosupplied the
ified for one special 3D case. This is the cylindrically PDE program,and T. Benseman.
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