Correlating off-stoichiometric doping with nanoscale electronic disorder and quasiparticle interference pattern in high-$T_c$ superconductor Bi$_2$Sr$_2$CaCu$_2$O$_{8+δ}$

0.8 MB·

by  Sen Zhou

Correlating off-stoichiometric doping with nanoscale electronic disorder and
quasiparticle interference pattern in high-T superconductor Bi Sr CaCu O
c 2 2 2 8+δ
Sen Zhou, Hong Ding, and Ziqiang Wang
Department of Physics, Boston College, Chestnut Hill, MA 02467
(Dated: February 6, 2008)
A microscopic theory is presented for the observed electronic disorder in superconducting
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Bi2Sr2CaCu2O8+δ. Theessentialphenomenologyisshowntobeconsistentwiththeexistenceoftwo
0
typesof interstitial oxygen dopants: those servingprimarily as charge reserviors and thoseclose to
0
2 theapicalplanecontributingbothcarriersandelectrostatic potentialtotheCuO2 plane. Thenon-
linearscreeningofthelatterproducesnanoscalevariationsinthedopedholeconcentration,leading

n toelectronicinhomogeneity. BasedonanunrestrictedGutzwillerapproximationoftheextendedt-J
a model,weprovideaconsistentexplanationofthecorrelation betweentheobserveddopantlocation
J
and the pairing gap and its spatial evolutions. We show that the oxygen dopants are the primary

5 cause of both thepairing gap disorder and thequasiparticle interference pattern.
1
PACSnumbers: 71.27.+a,71.18.+y,74.25.Jb,74.70.-b

]
l
e
- Remarkable electronic inhomogeneities have been ob- has remained a central unresolved issue.
r
st servedbyscanningtunneling microscopy(STM)inhigh- In this paper, we show that, while dopant induced
. Tc superconductors Bi2Sr2CaCu2O8+δ[1, 2, 3, 4, 5], structural effects are present, the main cause for the
t
a Ca2−xNaxCuO2Cl2[6, 7], and Bi2−xPbxSr2CuOy[8] over electronic disorder is the dopant potential induced LDC
m a wide range of doping. The hallmark of the inhomo-
modulations. A clue as to how the latter can be con-
- geneity is the disordered, nanometer scale variation of sistent with the dopant locations comes from the follow-
d
the pairing energy gap and its anti-correlationwith that
n ing observations made on the experiments of McElroy
ofthecoherencepeakheightinthelocaldensityofstates
o et.al. [4]: (1) The observed dopants associated with the
c (LDOS). The origin of the electronic disorder has been −0.96eV peak cannot account for the total number of
[ the focus of severaltheoreticalstudies [9, 10, 11, 12, 13].
oxygendopants. Forexample,inthe underdopedsample
2 It was emphasized [1, 9, 10] that off-stoichiometric dop- with ∆¯ = 65meV, the observed dopant density is 2.7%.
v ing a Mott insulator, such as the high-Tc cuprates, cre- Evenif every oxygendopant is fully ionized, the average
6 ates inherently inhomogeneous electronic states associ-
doping would only be 5.5%, which is much smaller than
2 atedwiththeinterstitialorsubstitutionaldopantatoms.
the expected value around11% [14]. Thus, a substantial
4
Ionized, these off-plane dopants act as disordered cen-
4 numberofthedopantshasnotbeenlocated,toolargeto
ters of nonlinearly screened electrostatic potential, lead-
0 be accounted for by the uncertainty in the doping pro-
6 ingtoaspatiallyinhomogeneousdistributionofthelocal cess. (2) The correlation between the observed dopants
0 doping concentration (LDC) and hence that of the local
and the pairing gap is weak and well defined only in the
/ pairing gap. In a short coherence length superconductor
t statistical sense. An appreciable number of them can be
a describedbytheshort-rangeresonancevalencebondthe-
m foundinthesmallergapregionsortostraddlethebound-
ory [15, 16], the local pairing gap is anti-correlated with
ariesbetweenthesmallandlargegapregions. Thus,itis
- the LDC [9, 10].
d unlikely that the observeddopants strongly and directly
n affect the local electronic structure. (3) The in-gap, low
The notion of dopant induced electronic disorder is
o energystatesintheLDOShavetheirweightconcentrated
supported and further advanced by the recent STM ex-
c
in regions away from the identified dopants. This sug-
: periments of McElroy et.al. [4]. Identifying the spectral
v geststhatthese electronicstateswhichareencodedwith
peak around −0.96eVin the LDOS with the presence of
i
X a local excess oxygen atom, the correlation between the the quasiparticle interference modulations [4] are likely
localized or pinned by additional confining potentials in
r dopant location and the gap inhomogeneity has been es-
a regions away from the identified dopants.
tablished ubiquitously. However, the observed dopants
reside close to the regionos of large pairing gap. This Based on these observations, we conjecture that there
is counter-intuitive since the negatively charged oxygen are two types (A and B) of interstitial oxygen dopants
ions are expected to attract nearby holes and create a in Bi Sr CaCu O . The type-B dopants serve pri-
2 2 2 8+δ
higher LDC with a smaller pairing gap [9, 10]. This dis- marily as charge reserviors. They only couple weakly
crepancy promotes the idea that dopant induced local to the CuO plane. These near nonbonding oxygen or-
2
structural distortions may play a more important role bitals give rise to the −0.96eV peak in the LDOS [4].
thanpotentialdisorder[4,17]andphenomenologicalthe- Recent ARPES measurements show that the dopant in-
ories in which the dopants serve as large pairing centers ducedstatesnear−0.96eVhaveB symmetryanddonot
1
[13]. Theoriginofthedopantinducedelectronicdisorder mix with the doped holes residing in the planar orbitals
2
1
[m18er].wTithhistfhuertthyepresBup-dpooprtasnttsh.eTidheenttiyfipcea-tAiondoopfatnhtes,foorn- − 4J giJj[∆∗ijǫσσ′ciσcjσ′ +χ∗ijc†iσcjσ +h.c.]
hXi,ji
the other hand, strongly affect the localelectronic struc-
1
ture in the CuO2 plane. Their electrostaticpotential en- + 4J giJj(|∆ij|2+|χij|2)− λini. (4)
hances the LDC which in turn pushes the orbital energy hXi,ji Xi
of type-A dopants into the broad valence band spectra
Thelocalenergyfortheelectronsisgivenbyε =V (i)+
below −1.2eV not yet accessible by STM probes. One i sc
λ −µ , where µ is the chemical potential and λ is a
possibility is that the type-B dopants sit above the BiO i f f i
fugacity that, together with the last term, ensures the
plane while type-Adopants comecloseto the apicalSrO
equilibrium condition under local occupation hc† c i =
layer. Wewilldiscussthemicroscopicoriginofthelatter iσ iσ
n [21]. V (i) is the screened Coulomb potential,
at the end. i sc
To support this physical picture, we extend the t-J x −δ
V (i)=VA+VB +V i . (5)
model to include the oxygen dopant potential, sc i i c |r −r |
i j
j6=i
1 X
H = − t Pc† c P +J (S ·S − nˆ nˆ )
ij iσ jσ i j 4 i j Thisisthedrivingforceoftheelectronicdisorderthrough
Xi6=j hXi,ji local doping variations. The correlation of the latter to
+ nˆ Vcnˆ − (VA+VB)(1−nˆ ). (1) thelocalpairinggapdisorderiscausedbytheGutzwiller
i ij j i i i
factorgt inEq.(3) thatmodulates the kinetic energylo-
i6=j i
X X
cally. Minimizing the ground state energy of Eq. (4), we
Here c†iσ creates an electron that hops between near obtainthe self-consistentequations for the set of param-
neighborsofthe Cu squarelattice via tij. Repeatedspin eters(∆ij,χij,λi,xi,µf),whicharesolvediterativelyfor
indices aresummedandnˆi =c†iσciσ isthe density opera- a given averagedoping δ.
tor. The LDCis givenbyx =1−n ,n =hc† c i. The We present our results for systems of 32 × 32 sites.
i i i iσ iσ
averagedoping will be denoted as δ =(1/N ) x on a We use t = (0.48,−0.16,0.05,0.05,−0.05)eV and J =
s i i ij
lattice of N sites. The second line in Eq. (1) describes 0.08eV such that the quasiparticle dispersion in Eq. (4)
s
P
the long-range Coulomb interactions between the elec- agrees with that measured by angle-resolved photoemis-
trons in the plane and between the in-plane doped holes sion [19]. For simplicity, half of the dopants are taken
andthe two-typesofoff-planedopants,Vc =V /|r −r | to be B-type and the other half A-type distributed ran-
ij c i j
and domly with d = 1a. The bare electrostatic poten-
A
NA(B) 2V tials are set to VA = Vc = 0.5eV, and VB = 0 for
VA(B) = A(B) (2) the weak coupling between B-dopants and doped holes.
i
|r −r )|2+d2 Note that the locations of type-A and type-B dopants
ℓAX(B)=1 i ℓA(B) A(B) are naturally anticorrelated overthe averageinterdopant
q
where dA, dB are the setback distances and NA, NB the distance a 2/δ. The spatial distribution of the LDC
number of type-A and type-B dopants respectively, δ = x and the d-wave pairing order parameter ∆ (i) are
i d
p
2(NA+NB)/Ns. shown as 2D maps in Figs. 1a and 1b for a typical sys-
Eq.(1)describesdopedMottinsulatorsbecauseofthe tem at δ = 10.2%. The projected dopant locations are
projection operator P that removes double occupation. superimposed. The A-dopants are ubiquitously corre-
The projection is most conveniently implemented using lated with the LDC. Due to the local pairing nature of
theGutzwillerapproximationbythestatisticalweighting the short-range RVB state, x is in turn strongly anti-
i
factors multiplying the coherent states, thus renormaliz- correlated with ∆ (i) [9]. Notice that the modulations
d
ingthe hoppingandthe exchangeparameters[20]. Since induced by the A-dopants leaves the B-dopants unwit-
the dopants break translation symmetry, we extend the tingly correlated overall with the low doping and strong
approach to the disordered case by the renormalization pairing regions. The LDOS is calculated from the pro-
tij →gitjtij and J →giJjJ where the Gutzwiller factors jected retarded Green’s function in the Gutzwiller ap-
proximation [21], LDOS(i,ω) = −2ImgtG(i,i,ω+i0+).
ii
4x x 4
gt = i j , gJ = (3) To reduce finite size effects, we average over different
ij s(1+xi)(1+xj) ij (1+xi)(1+xj) boundaryconditionscorrespondingto20×20supercells.
As in the STM experiments, the local tunneling gap ∆
depend on the local doping at the sites connected by T
is extractedfromthe coherencepeak positionatpositive
the hopping and the exchange processes. The exchange
bias in the LDOS. In Fig. 1c, the tunneling gap map is
term is decoupled in terms of the bond χ = hc† c i
ij iσ jσ shownwith the dopantlocations. Evidently, ∆ is small
and the pairing ∆ij = hǫσσ′ciσcjσ′i fields, leading to a near the A-dopants where the LDC is high. NoTtice how-
renormalized mean-field Hamiltonian,
ever,thattheB-dopantsarefoundwithhighstatisticsin
H = − gt t c† c + ε c† c andaroundthe regionswhere the tunneling gapis large,
GA ij ij iσ jσ i iσ iσ
consistent with their identification with those observed
i6=j i
X X
3
FIG. 2: Evolutions of the LDOS along two line cuts (shown
inFig.1c) from small toaverage(a) andaveragetolarge (b)
gap regions. Dashedlines are guide to eye.
FIG. 1: Doping and pairing disorder on a 32×32 system
at δ = 10.2%. 2D maps are shown for the LDC xi (a), di-
mensionlessd-wavepairingorderparameter∆d (b),andtun-
neling gap ∆T in meV (c) with the projected type-A (open Clearly, larger (smaller) gap regions are associated with
black circle) and type-B (white solid circle) dopant locations smaller(larger)coherencepeaksinagreementwithSTM
superimposed. (d) Correlation functions among the dopant [1, 2, 3, 4, 5]. Note that the line-cut passes through lo-
location, ∆T, xi, and ∆d. cally highly overdoped regions with x >25%. In a uni-
i
formsystem, the pairing gapwould be vanishingly small
and the system essentially in the normal state at such
by STM [4]. To elucidate the correlation between the high doping levels. Fig. 2a and Fig. 1c show, however,
dopants and electronic disorder, we calculate the nor- thatthesmallerlocalgapisstillsizable. Thatoverdoped
malized cross-correlation function between the dopants regions of sizes not exceeding the coherence length have
and the tunneling gap, gapvalues considerablyabovethosein a cleansampleat
thecorrespondingdopingswasfirstpointedoutinRef.[9],
δ∆ (r )δO (r +r )
C (r )= i T i A(B) i j andislikelyamanifestationoftheproximityeffectdueto
∆T−OA(B) j Pi[δ∆T(ri)]2 j[δOA(B)(rj)]2 the surrounding largergapregions. Indeed, Fig. 2 shows
that as we move awayfrom the small (large) gapregion,
q
where δf(r ) = f(r )−Phfi. The doPpant locations are a larger (smaller) gap emerges and evolves progressively
i i
modeled by Lorentzians of width 2a [4]. Fig. 1d shows stronger until it becomes the dominant gap as the line
that, while ∆ is strongly anticorrelated with A-dopant cutenters the lower(higher) dopingregion. This feature
T
locations(C (0)∼−0.6),the B-dopants(observed hasbeenobservedrecentlybySTMwithhighenergyres-
∆T−OA
by STM) are positively correlated with the local tun- olutions [5].
neling gap, and moreover,the correlation is significantly Nextweshowthe A-dopants,the culpritofstronggap
weaker, C (0) ∼ 0.3, in excellent agreement with disorder, also serve as the primary cause of the low-
∆T−OA
experiments [4]. energy quasiparticle interference modulations [22]. In
The large LDC variation is not inconsistent with the Fig.3a,the2DmapsoftheLDOS(i,ω)isplottedatfixed
STM integrated LDOS variation that diminishes only energies ω, showing different interference patterns. The
when integrated up to −0.6eV to −0.9eV, where large maximaofthespatialmodulationsarepreferentiallycen-
incoherent background and the spectral weight associ- tered around the A-dopant sites, thus away from the B-
ated with the dopants contribute appreciably [4]. It is a dopants, consistent with STM observations. This turns
subtle task to relate quantitatively the integratedLDOS outtobetrueforalllowenergystates. InFig.3c,weplot
or the topography to the LDC since the former are ob- the local correlation between the LDOS and the dopant
tainedinthe constanttunneling currentmode wherethe locations as a function of bias voltage. For both positive
tip to sample distance changes significantly, leading to and negative energies, the correlations are positive and
reduced spatial variations of the spectral weight [1, 9]. strong (reaching 0.9) with A-dopants and negative (an-
Thisformofdopantinducedelectronicdisordercande- ticorrelated) and weak (reaching −0.3) with B-dopants.
scribe the basic properties of the inhomogeneouslow en- Thepeakanddiparound±70meVarerelatedtotheVan
ergystatesobservedexperimentally. InFig.2,wepresent Hove-likefeatureintheLDOSspectraseeninFig.2. Just
the LDOS along the two line-cuts indicated in Fig. 1c. as in the case of dopant-gapcorrelations,the strong cor-
4
studiedsystematicallybycontrolledtrivalent(Ln3+)sub-
stitution of Sr2+ in Bi Sr Ln CuO where Ln=La,
2 2−y y 6+δ
Pr, Nd, Sm, Eu, Gd, and Bi [17, 25]. We propose that
the STM experiments be carried out on Bi2212 samples
with controlled trivalent substitution of Sr in the apical
plane. Our theory predicts that the density of the ob-
servable type-B dopants would decrease with increasing
Ln3+ substitution, while that of the type-A dopants in-
creases, leading to stronger electronic disorder.
We have shown that the electrostatic potential of the
off-stoichiometric A-dopants can be the primary cause
of the electronic disorder and quasiparticle interference
modulations observed in Bi Sr CaCu O . The elec-
2 2 2 8+δ
tronic inhomogeneity in our theory is driven by that of
the kinetic energy or the coherence of doped holes in a
dopedMottinsulator. Incoherentexcitationsbeyondthe
Gutzwillerapproximationandthe dopantinducedstruc-
turaldistortionscanalsocontributetotheelectronicdis-
order. Weexpectsuchdopantinducedelectronicdisorder
inalldopedcupratesuperconductors,withspecificprop-
erties dependent onthe dopant locationsand the crystal
FIG. 3: Electron LDOSmaps (a) and theFouriertransform
field environment, interstitial or substitutional, ordered
of the quasiparticle interference pattern (b) at low energies.
(c)LocalcorrelationsofLDOS(i)andthedopantlocationsas or disordered. The off-plane dopant structure together
a function of bias energy. with the role of the apical oxygen may account for the
varying properties of the cuprates that share, otherwise,
identical CuO planes.
2
WethankJ.C.Davis,P.Hirschfeld,W.Ku,D.-H.Lee,
relation of the interference modulations with A-dopants
P. A. Lee, C. Li, S. H. Pan, and S. Uchida for discus-
producesaweakanticorrelationwiththeB-dopantsasin
sions. This work is supported by DOE grant DE-FG02-
STM experiments [4]. To further illustrate the interfer-
99ER45747,ACS39498-AC5M,andNSFDMR-0353108.
ence pattern, the Fourier transform of the quasiparticle
ZW thanks the KITP at UCSB for hospitality and ac-
LDOS are shown in Fig. 3b at the corresponding ener- knowledges the support of NSF grant PHY94-07194.
gies. The dominant interference wavevector q (marked
1
by arrows)connecting the tips of the Fermi arcs [23] are
clearlyseentodispersewithenergynear(0,±2π/4a)and
(±2π/4a,0),whiletheq connectingtwotipsofthesame
7
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5
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