Zinc Impurities in the 2D Hubbard model

4 Pages·0.13 MB·

by  Th. A. Maier

Zinc Impurities in the 2D Hubbard model
Th. A. Maier and M. Jarrell1
1University of Cincinnati, Cincinnati OH 45221, USA
We study the two-dimensional Hubbard model with nonmagnetic Zn impurities modeled by bi-
2 narydiagonal disorder using QuantumMonteCarlo within the DynamicalCluster Approximation.
0 With increasing Zn content we find a strong suppression of d-wave superconductivity concomitant
0 toareductionofantiferromagnetic spinfluctuations. Tc vanisheslinearlywithZnimpurityconcen-
2 tration. ThespinsusceptibilitychangesfrompseudogaptoCurie-Weisslikebehaviorindicatingthe
existence of free magnetic moments in the Zn doped system. We interpret these results within the
n
RVBpicture.
a
J

3
Introduction Chemical substitution in high-T super- proximation (CPA) [8]. The CPA shares the same mi-
c
conductors provides a powerful probe into the complex croscopic definition [9] as its equivalent for correlated
]
n nature of both the superconducting and normal state of clean systems, the Dynamical Mean Field Approxima-
o these materials. Experiments substituting different im- tion (DMFA) [10]. Both approachesmap the lattice sys-
c
purities for Cu show that nonmagnetic impurities (Zn, tem onto an effective impurity problem embedded in a
-
r Al) are just as effective in suppressing superconductiv- host that represents the remaining degrees of freedom.
p
ity as magnetic dopants (Ni, Fe) [1]. Based on Ander- This single-site approximation neglects interference ef-
u
s son’s theorem [2] these results are taken as strong evi- fects of the scattering off different impurity sites (cross-
t. dence for an unconventional pairing state described by ingdiagrams)andcorrelationeffectsbecomepurelylocal.
a
an anisotropic order parameter with nodes on the Fermi Therefore,it inhibits a transitionto a state describedby
m
surface. Indeed, recently the characteristicfourfoldsym- a nonlocal (d-wave) order parameter and thus is not ap-
-
d metry of the dx2−y2-wave order parameter was observed propriate for our investigations here.
n in the spatialvariationofthe localdensity ofstates near The Dynamical Cluster Approximation (DCA) [9, 11,
o theZnimpurityusingscanningtunnelingmicroscopy[3]. 12, 13] systematically incorporates nonlocal corrections
c
[ Nuclear magnetic resonance experiments show that to the CPA/DMFA by mapping the lattice system onto
even a nonmagnetic impurity substituted for Cu induces an embedded periodic cluster of size Nc. For Nc =1 the
1
aneffectivemagneticmomentresidingontheneighboring DCA is equivalentto the CPA/DMFAand by increasing
v
7 Cu sites [4]. In addition, the bulk susceptibility in im- theclustersizeNc thelength-scaleofpossibledynamical
3 purity substituted underdoped cuprate superconductors correlationscanbegraduallyincreasedwhiletheDCAso-
0 shows Curie-Weiss like behavior irrespective of the mag- lution remains in the thermodynamic limit. In the clean
1 netic structure of the dopants [1]. This indicates the ex- limit the DCA applied to the Hubbard model has been
0
istence of free magnetic moments in the impurity doped shownto describe the essentiallow energyphysics of the
2
CuO planes. Theformationofthesemomentswithsub- cuprates [14, 15, 16]: It captures the antiferromagnetic
0 2
/ stitution of nonmagnetic impurities is explained to arise phase near half filling and the transition to a supercon-
t
a from breaking singlets[1, 5] in an antiferromagnetically ductingphasewithdx2−y2-waveorderparameteratfinite
m correlated host by removing Cu spins. doping. In the normal state it exhibits non-Fermi liquid
- In this letter we focus on the suppression of supercon- behavior in form of a pseudogap in the density of states
d andasuppressionofspinexcitationsatlowtemperatures
ductivity by Zn impurities and the change of the bulk
n
in the underdoped regime.
o magnetic susceptibility to a Curie-Weiss like behavior
c in underdoped systems. Despite a wide variety of the- In this letter we study the two-dimensional(2D) Hub-
v: oretical studies addressing these issues, a complete un- bardmodelincludingapotentialscatteringtermaccord-
i derstanding of the effects of Zn doping has not been ing to the chemistry of Zn impurities using the DCA.
X
achieved. These approaches are mostly based on the de- We will show that potential scattering by Zn impuri-
ar scription of impurities embedded in a BCS host [6] or tiesstronglysuppressessuperconductivityaswellasspin
ongeneralizationsoftheAbrikosov-Gorkovequationsfor fluctuationsandchangesthemagneticsusceptibility toa
nonmagnetic impurities in unconventional superconduc- Curie-Weiss like temperature dependence in the under-
tors using phenomenological pairing interactions [7]. In doped region.
ordertocaptureeffectslikemomentformationbysubsti- Formalism The correlated electrons in a CuO plane
2
tution of nonmagnetic impurities, it seems necessary to doped with Zn impurities are described by the 2D Hub-
describe correlations and the scattering from impurities bard model with diagonal disorder
on the same footing within a microscopic approach.
The most widely applied technique developed to de- H =−t X c†iσcjσ +UXni↑ni↓+Xǫiniσ, (1)
scribe disordered systems is the Coherent Potential Ap-
<ij>,σ i iσ
2
where we used standard notation. The disorder induced QMC result for the cluster Green function Gc =
ǫ1,...,ǫN,ij
bytheZnsitesoccursinthelocalorbitalenergiesǫ which Gc [U,G ]thusdependsontheparticulardisorder
i ij ǫ1,...,ǫN,ij
are independent quenched random variables distributed configuration {ǫ }. The disorder averaged cluster Green
i
according to some specified distribution P(ǫ ,...,ǫ )= function is then obtained from the individual results by
1 N
N P(ǫ ) where N is the number of lattice sites. For a
Qi=1 i Gc =hGc i, (6)
concentration x of Zn impurities we use the binary alloy ij ǫ1,...,ǫN,ij
distribution
where the average < ... >= N dǫ P(ǫ )(...) is to
R Qi=1 i i
P(ǫ )=xδ(ǫ +V/2)+(1−x)δ(ǫ −V/2), (2) be taken for a system of N sites. Note that in principal
i i i c
QMCcalculationsforallpossibledisorderconfigurations
where V is the energy difference between the Cu and Zn {ǫ } have to be carried through. However, contributions
i
3dorbitals. Forsimplicityweusesite-independentvalues from configurations with m Zn impurities on the cluster
for the hopping integralt and the Coulomb repulsion U. (m ≤ Nc) are weighted by a factor of xm(1−x)(Nc−m)
The microscopic derivation of the DCA algorithm was to the integral Eq.(6). Therefore it seems reasonable for
discussed in detail for correlated systems in [12, 16] and small concentrations x ≪ 1 and cluster sizes N to con-
c
for disordered systems in [9]. The DCA has a simple sideronlythoseconfigurationswithnoneoronlyasingle
physical interpretation for systems where the intersite Zn impurity on the cluster and to neglect configurations
correlations have only short spatial range. The corre- withmorethanoneZnimpurity. Thenthedisorderaver-
sponding self-energy may then be calculated on a coarse agedclusterGreenfunctionGc isobtainedasaweighted
ij
grid of Nc =LDc selected K points only, where Lc is the sum from the two configurations as
linear dimension of the cluster of K points. Knowledge
ofthemomentumdependenceonafinergridmaybedis- Gc =xN Gc +(1−xN )Gc , (7)
ij c 1,ij c 0,ij
carded to reduce the complexity of the problem. To this
endthefirstBrillouinzoneisdividedintoN cellsofsize where Gc denotes the QMC result for the cluster
c 1/0,ij
(2π/L )D around the cluster momenta K. The propa- Greenfunction for the configurationwith one or zeroZn
c
gators used to form the self-energy are coarse grainedor impuritiesonthecluster,respectively. Theprefactorsfol-
averagedoverthemomentaK+k˜surroundingthecluster lowwith(1−x)Nc ≈1−xNc,xNc(1−x)Nc−1 ≈xNc for
momentum K x << 1 in linear approximation in the impurity concen-
tration x. The disorder averaged cluster Green function
N
G¯(K,iωn) = c XG(K+k˜,iωn) Gcij is then transformed back to cluster reciprocal space
N
to calculate a new estimate of the cluster self-energy
k˜
N 1
= c X . (3) Σ(K)=G(K)−1−Gc(K)−1, (8)
N iω −ǫ −Σ(K,iω )
k˜ n K+k˜ n
which is then used to repeat the steps starting from
In Eq.(3) we make use of the fact that the approxima- Eq.(3) until convergence is reached.
tion of the lattice self-energy Σ(k) by the cluster self- Results Westudythesuperconductinginstabilityand
energy Σ(K) optimizes the free energy [12, 13]. Σ(K) magnetic properties of the Hamiltonian Eq.(1)in the in-
summarizes the single-particle effects of the interaction termediatecouplingregimeat5%holedoping(n=0.95)
term (second term) and the disorder term (third term) for different Zn impurity concentrations x. We perform
oftheHamiltonian,Eq.(1). ThedispersionǫK+k˜ denotes calculations for Nc = 4, the smallest cluster size that
thespectrumofthekineticpart(firstterm)ofEq.(1). In allows for a transition to a dx2−y2-wave superconduct-
order to avoid overcounting of self-energy diagrams on ing state while preserving the lattice translational and
the cluster, the cluster excluded propagator point group symmetries. We set the hopping integral
t = 0.25eV and the Coulomb repulsion U = W = 2eV,
G(K)−1 =G¯(K)−1+Σ(K) (4)
where W = 8t is the bandwidth of the non-interacting
system. We choose V = 5eV > W +U to simulate the
is used as bare propagator for the cluster problem. To
nonmagnetic closed shell (d10) configuration of the im-
diagonalize the disorder part it is convenient to per-
purity site.
form a Fourier transform to cluster real space G =
1/NcPKeiK·(Xi−Xj)G(K) where the disorder is diiajgo- tenWdeedsesa-rwchavede,fopr- asunpderdc-ownadvuectoivrditeyr wpiatrhams-ewtearvse., eAxs-
nal. The inverse bare cluster propagatorfor a particular
we found for the clean system (x=0) [14, 16], only the
disorder configuration {ǫ } is then given by
i
dx2−y2-wave pair field susceptibility for zero center of
[G−1 ] =[G−1] −ǫ δ , (5) mass momentum diverges at low temperatures. Fig.1
ǫ1,...,ǫN ij ij i ij illustrates the inverse pair field susceptibility P−1 as a
d
and is used to initialize a QMC simulation to calcu- function of temperature T for the impurity concentra-
late the effects of the Coulomb interaction U. The tions x = 0 (circles), x = 0.05 (squares) and x = 0.10
3
15 5
x=0.00
0.02 χ(T)=χ -χ T+C/(T-Θ) x=0.05
c 4 0 1 x=0.10
T
0.01
10
3
0
1 0.05 0.1
-Pd x χ
P -1∝ (T-T)γ 2
d c
5
x=0.00, T=0.020, γ=0.72
c
x=0.05, T=0.012, γ=0.82 1
c
x=0.10, T=0.002, γ=0.86
c
00 0.05 0.1 0.15 0.2 00 0.05 0.1 0.15 0.2
T T
FIG. 1: The inverse pair-field susceptibility versus tempera- FIG.2: The bulkmagneticsusceptibility versustemperature
turewhenNc =4,U =W =2eV,V =5eV at5%dopingfor when Nc = 4, U = W = 2eV, V = 5eV at 5% doping for
impurity concentrations x = 0 (circles), x = 0.05 (squares) impurity concentrations x = 0 (circles), x = 0.05 (squares)
andx=0.10(diamonds). Thesolidlinesrepresentfitstothe and x = 0.10 (diamonds). The solid lines are guides to the
function Pd−1 ∝(T −Tc)γ. Inset: Critical temperature Tc as eye. The dashed line represents a fit to the function χ(T)=
a function of impurity concentration x. χ0−χ1T +C/(T −Θ).
(diamonds). The corresponding critical temperature T χ(T) shows Curie-Weiss like behavior; i.e. it can be fit
c
is then calculated by extrapolating P−1(T) to zero us- by the function χ(T) = χ − χ T + C/(T − Θ), with
d 0 1
ing the function P−1 ∝ (T −T )γ (represented by the Θ≈0. This clearly indicates the formation of free mag-
d c
solid lines in Fig.1). T is rapidly suppressed, and the netic moments due to the breaking of singlet bonds by
c
critical exponent γ increases,with increasing Zn content the substitution of nonmagnetic impurities.
x. Consistent with experiments [1] this falloff of Tc is The inverse antiferromagnetic spin susceptibility
linear in x resulting in a critical impurity concentration χ (T) for the same parameter set as in Fig.2 is illus-
AF
xc ≈ 0.10, beyond which the instability to the super- trated in Fig.3. For the clean system (x=0) 5% hole
conducting phase disappears. Compared to experiments doping marks the criticaldoping, beyond whichno tran-
(xc <∼0.05) our calculation predicts a result about twice sitiontoanantiferromagneticstatecanbefound[14]. As
as high for the critical doping. However,considering the canbe seenfromthe extrapolationof the circlesin Fig.3
simplifieddescriptionoftheproblemusinga2DHubbard the corresponding Ne´el temperature T = 0. As the
N
model with the complexity of the Zn impurities reduced concentration of impurities is increased we notice that
to a model of diagonal disorder, we expect agreement for T >∼ 0.05 χAF decreases with x. However at lower
only on a qualitative level. temperatures (T <∼0.05)χ−AF1 for finite impurity concen-
In order to study the correlation between the suppres- tration x falls below the curve of the clean system and
sionofsuperconductivityandpossiblemomentformation eventually goes through zero indicating a transition to
by Zn substitution we study the bulk (q = (0,0)) and the antiferromagnetic state. Therefore it can be inferred
the antiferromagnetic (q = (π,π)) spin susceptibilities. that a finite concentration of impurities, i.e. spin vacan-
Fig.2 shows the bulk magnetic susceptibility χ(T) as a cies,acts to enhance antiferromagneticspincorrelations.
function of temperature for the same impurity concen- Ourresultsforthesiteandtimeaveragedlocalmagnetic
trations x used in Fig.1. As we demonstrated in [14], moment Tχii shown in the inset to Fig.3 support this
the clean system exhibits an anomaly in χ(T) at a char- conjecture: At low temperatures this quantity increases
acteristic temperature T⋆, below which spin excitations with impurity concentration x indicating that local mo-
are suppressed and χ(T) falls with decreasing tempera- ments get stabilized. It is important to note that recent
ture. T⋆ thusmarkstheonsetofsingletformationdueto NMR measurements on YBCO [17] show a similar en-
strong short-ranged antiferromagnetic spin correlations. hancementofantiferromagneticspincorrelationsaround
In agreement with experiments [1], the substitution of Zn impurities at low temperatures.
impurities leadsto adramaticchangeofthe lowtemper- Similar results were also obtained in [18, 19] where an
ature behavior of the magnetic spin susceptibility. De- antiferromagneticHeisenberg model with a spin vacancy
spite the nonmagnetic nature of the impurity scattering wasstudied. Inthelatterreport,theenhancementofan-
term in our simulation, χ(T) displays an upturn at low tiferromagneticspincorrelationswasascribedtopruning
temperatures as x is increased. Moreover at x = 0.10, ofsingletsbythenonmagneticimpurityinanresonating-
4
00..44 tions that mediate pairing get suppressed. Consistent
with experiments, T decreases linearly with impurity
c
0.3 concentration. With increasing Zn content we further
00..33 χii find a change of the magnetic susceptibility to Curie-
T
Weiss like behavior indicative of the existence of free
0.2
1 magnetic moments.
-F00..22
χA 0.02 0.04 0.1 χAF-1∝(T-TN)γ ThOeunrornemsualtgsnceatincbimepuunrditeiresstobordeawkiltohcianltshinegRleVtsBapnidcttuhrues.
T
generate unpaired spins. A concomitant suppression of
00..11 x=0.00, γ=1.53
x=0.05, γ=1.42 antiferromagnetic spin fluctuations that mediate pairing
x=0.10, γ=1.35 results from the pruning of RVB states [19]. Conse-
quently,superconductivitygetsstronglysuppressedwith
00
00 0.05 0.1 0.15 0.2 increasing Zn content.
T
Acknowledgements We acknowledge useful conversa-
tions with W. Putikka, D.J. Scalapino, Y. Wang and M.
FIG. 3: The inverse antiferromagnetic (q =(π,π)) spin sus-
Vojta. This work was supported by NSF grant DMR-
ceptibility versus temperature when Nc =4, U =W =2eV,
V = 5eV at 5% doping for impurity concentrations x = 0 0073308. This research was supported in part by NSF
(circles), x = 0.05 (squares) and x = 0.10 (diamonds). The cooperative agreement ACI-9619020 through computing
solid lines represent fits to the function χ−AF1 ∝ (T −TN)γ. resources provided by the National Partnership for Ad-
Inset: The site and time averaged magnetic moment versus vanced Computational Infrastructure at the Pittsburgh
temperature; spin vacancies reduce fluctuations. The solid
Supercomputer Center.
lines are guides to theeye.
valence-bond picture (RVB). In the RVB picture, the
short-ranged order in the doped model is described as a
[1] G. Xiao, et al., Phys.Rev.B 42, 8752 (1990).
collectionofnearest-neighborsingletbondswhichfluctu- [2] P.W. Anderson, J. Phys. Chem. Solids 11, 25 (1959).
ateasafunctionoftimeandspace. WhenaZnimpurity [3] S.H. Pan, et al., Nature403, 746 (2000).
is introduced into the system, the singlet fluctuations on [4] H. Alloul, et al., Phys. Rev. Lett. 67, 3140 (1991); A.V.
adjacent sites are suppressed since these sites now have Mahajan, et al.,Phys. Rev.Lett. 72, 3100 (1994).
one less neighbor to form short-ranged bonds with. The [5] A.M. Finkelstein, et al.,Physica C 168, 370 (1990).
[6] T. Xiang and J.M. Wheatley, Phys. Rev. B 51, 11721
suppression of singlet fluctuations on the sites adjacent
(1995); G. Haran´ and A.D.S. Nagi, Phys. Rev. B 58,
to the Zn dopant also enhances moment formation on
12441 (1998).
these sites. [7] R.J.Radtke,etal.,Phys.Rev.B48653(1993);S.Haas,
Wefindfurtherevidenceofthesuppressionofspinfluc- et al.,Phys.Rev. B 56, 5108 (1997).
tuations by Zn impurities in the decrease of the anti- [8] D.W.Taylor,Phys.Rev.1561017(1967);P.SovenPhys.
ferromagnetic exponent γ towards the mean-field result Rev.156809(1967);P.L.LeathandB.Goodman,Phys.
of one with increasing impurity content x. Concomi- Rev. 148 968 (1966).
[9] M. Jarrell and H.R. Krishnamurthy, Phys. Rev. B, 63
tantwiththisbehaviorthecorrespondingexponentγ for
125102 (2001).
the pair-field susceptibility (cf. Fig.1) increases towards
[10] W. Metzner and D. Vollhardt, Phys. Rev. Lett. 62, 324
one. Fromthisobservationweconcludethatthesuppres- (1989); E. Mu¨ller-Hartmann: Z. Phys. B 74, 507–512
sionofsuperconductivityinZndopedsystemsoriginates (1989);T.Pruschke,etal.,Adv.inPhys.42,187(1995);
in the suppression of antiferromagnetic spin fluctuations A. Georges, et al.,Rev.Mod. Phys.68, 13 (1996).
that mediate pairing. [11] M.H. Hettler et al., Phys.Rev.B 58, 7475 (1998).
Summary Inthisletterwehaveusedthe2DHubbard [12] M.H. Hettler, et al.,Phys. Rev.B 61, 12739 (2000).
[13] Th. Maier, et al.,Eur. Phys. J. B 13, 613 (2000).
model with binary diagonaldisorder to study the effects
[14] M. Jarrell, et al.,EuroPhy.Lett., 56 563 (2001).
of nonmagnetic (Zn) impurities on high-temperature su-
[15] Th. Maier, et al.,Phys. Rev.Lett. 85, 1524 (2000).
perconductivity. WefindthatZnimpuritiesstronglysup- [16] M. Jarrell, et al.,Phys.Rev. B 64, 195130 (2001).
press superconductivity. As evidenced by the increasing [17] M.-J. Julien et al.,Phys.Rev. Lett.84, 3422 (2000).
mean-field character of the pairfield and antiferromag- [18] N. Bulut, et al.,Phys. Rev.Lett. 62, 2192 (1989).
netic susceptibilities with Zn substitution, spin fluctua- [19] G. B. Martins, et al.,Phys. Rev.Lett. 78, 3563 (1997).