Local density of states induced by anisotropic impurity scattering in a d-wave superconduct
a r X i v :c o n d -m a t /0307427v 1 [c o n d -m a t .s u p r -c o n ] 17 J u l 2003
Local density of states induced by anisotropic impurity scattering in a d-wave
superconductor
P.Pisarski and G.Hara′n
Institute of Physics,Politechnika Wroc l awska,Wybrze˙z e Wyspia′n skiego 27,50-370Wroc l aw,Poland
We study a single impurity e?ect on the local density of states in a d-wave superconductor
accounting for the momentum-dependent impurity potential.We show that the anisotropy of the scattering potential can alter signi?cantly the spatial dependence of the quasiparticle density of states in the vicinity of the impurity.
A search for the symmetry of the cuprate superconduct-ing state involves among others theoretical studies of the disordered systems [1,2,3].The most direct probe of a simple defect impact on superconductivity is provided by the scanning tunneling microscopy (STM)measure-ment of the position-dependent quasiparticle density of states.The STM images of Zn and Ni substitutions at the planar Cu sites in Bi 2Sr 2CaCu 2O 8+δreveal a dis-tinct four-fold symmetry of the local density of states (LDOS)[4,5]predicted for the d-wave superconductor response to disorder [6,7].In addition to providing detailed information on the superconducting state,this kind of experiment may shed light on the nature of the quasiparticle scattering centers.Although the main fea-ture of the four-fold symmetry of the tunneling currents is captured by the model of isotropic impurity scatter-ing the observed spatial dependence of the quasiparticle density of states is far more complex,and may originate from a non trivial structure of the impurity potential.This aspect of the quasiparticle scattering process has been raised in the discussion of macroscopic measure-ments in these compounds [8,9,10,11].It has been shown that the unexpected for the d-wave superconduc-tivity weak impurity-induced suppression of the critical temperature can be understood within a scenario of a momentum-dependent (anisotropic)impurity scattering potential [12,13,14,15].In the present paper we dis-cuss the e?ect of anisotropic impurity potential on the local density of states in the vicinity of a single impurity and verify to what extend this scenario can reproduce the STM maps of Bi 2Sr 2CaCu 2O 8+δ.There are studies of the momentum-dependent impurity scattering which indicate the existence of the impurity-bound states even for the potential strength far from resonant [17].There-fore,we focus here on the symmetry of the LDOS and its spatial dependence.For this purpose and for analytical simplicity we consider scattering in the Born approxi-mation.Hereafter we show that the anisotropy of the scattering potential alters the local density of states in the d-wave superconductor but preserves its tetragonal https://www.sodocs.net/doc/4316056989.html,pared to the e?ect of the isotropic im-purity potential it enhances the spatial variation of the quasiparticle density of states and can lead to a long-range spatial modulation of the LDOS spectra.We note that the anisotropy of the impurity potential,if the same as the anisotropy of the superconducting order parame-ter,enhances the relative magnitude of oscillations in the
LDOS for the direction of a gap maximum,especially in the vicinity of the impurity,whereas the impurity po-tential with maxima in the nodal region inverts this ten-dency and leads to a larger quasiparticle density of states along the gap node direction.
We consider the momentum-dependent impurity po-tential [12,13,14,15,16]
?v (k ,k ′)=[v i +v a f (k )f (k ′)]?τ3
(1)
where v i ,v a are isotropic and anisotropic scattering am-plitudes,respectively,f (k )is the anisotropy function,
and ?τi (i =1,2,3)are Pauli matrices.We assume f (k )=±1such as its Fermi surface (FS)average van-ishes,i.e., f =
F S
dS k n (k )f (k )=0,where n (k )
is the normalized angle resolved FS density of states,
F S dS k n (k )= 1.Potential determined by f (k )=sgn k 2
x
?k 2y =sgn (cos 2φ),which is in phase with the d-wave superconducting order parameter leads to a par-ticularly moderate suppression of the critical tempera-ture [12,15,16].In the above φis a polar angle.
We study the e?ect of the impurity potential (1)on the quasiparticle states using the one particle Green’s function of the d x 2?y 2-wave superconducting state in the particle-hole notation
?G 0(k ,ωn )=[iωn ?τ0?ξk ?τ3??(k )?τ1]?1(2)
where ?(k )=? k 2
x ?k 2
y
=?cos 2φrepresents the su-perconducting order parameter,ωn is the Matsubara fre-quency,ξk =εk ?εF is the quasiparticle energy in the normal state,εF is the Fermi energy,and ?τ0is the iden-tity matrix.For simplicity we assume a parabolic dis-persion and a two-dimensional (planar)superconductiv-ity.Neglecting the anisotropy of the Fermi surface we concentrate on the feature of coupled anisotropies of the order parameter and the impurity potential.The impu-rity e?ect is studied at zero temperature by the analytic continuation of (2)
?G 0(k ,ω)=?G 0(k ,ωn )iωn =ω+i 0+
(3)
In order to discuss the real space distribution of the quasi-particle states we take a two-dimensional Fourier trans-form of the Green’s function (3),which for the parabolic
band εk =k 2/2m and for positive ωsmaller than the
Fermi energy reads
?G
0(r ,ω)=∞
?∞
d 2
k
2(2π)2
2π 0
dφk ω?τ0+?(k )?τ1ω2??2(k )
e i k +r
+e ?i k ?r +?τ3 e i k +r ?e ?i k ?r (4)
where similarly
to
the
three-dimensional
case
[18]
k ±=
√ω2??2(k )
1/2
and
Im
π
Im {δG 11(r ,ω)}
(10)
We have evaluated the LDOS for equal impurity scatter-ing strength in isotropic and anisotropic channel πN 0v i =πN 0v a =0.1,where N 0is the density of states per spin at the Fermi level in the normal state,and for the co-herence length ξ0=12πk ?1
F .The distance scale in the
?gures is set by k ?1
F .We observe a pronounced spa-tial variation of the quasiparticle density of states for anisotropic impurity potential.This feature is in agree-ment with the result of an enhanced LDOS due to a ?nite range of the scattering potential [19].In Figs.1a-c we show the distance dependence of the impurity-induced change to the LDOS at the quasiparticle energy ω=1.1?along the a-axis (φ=0),i.e.,gap maximum,and nodal (φ=π/4)direction for isotropic scattering (Fig.1a)and two representative anisotropic impurity potentials (1):f (k )=sgn k 2x ?k 2
y =sgn (cos 2φ)which is in phase with the d-wave order parameter (Fig.1b),and f (k )=sgn (k x k y )=sgn (sin 2φ)(Fig.1c)scattering out of the order parameter phase.We note that the im-purity potential in phase with the order parameter en-hances the magnitude of oscillations in the LDOS for the direction of a gap maximum,especially in the vicinity of the impurity,whereas the out of phase impurity potential inverts this tendency and leads to a larger quasiparticle density of states in the nodal regions.Signi?cant is also very weak anisotropy of the LDOS near the isotropic im-purity (Fig.1a)where discernible di?erences between φ=0and φ=π/4directions occur at a distance of
about 50k ?1
F from impurity [6].It shows that visible spa-tial distribution of intensity maxima and minima in the STM maps is induced by the anisotropy of defect poten-tial.Therefore,we suggest that the apparent anisotropy of the experimental images [4,5]may possibly result in part from the presence of the momentum-dependent scat-tering potential.We have also performed the LDOS cal-culation for quasiparticles below the gap threshold energy and displayed them for ω=0.1?in Figs.2a-c.Another interesting feature of the anisotropic scattering is a pres-ence of a long-range spatial modulation of the LDOS for the impurity potential in phase with the d-wave order pa-rameter,i.e.,f (k )=sgn k 2x ?k 2
y .Such a modulation is absent for isotropic or out of phase scattering (Fig.3).We summarize our discussion by showing in Figs.4-5
comprehensive pictures of the STM images at the fre-quenciesω=1.1?andω=0.1?around simple defects of isotropic and anisotropic Born potential in the ab plane of the d-wave superconductor.The potential strength, coherence length and distance units have been?xed as in Figs.1-3.The quasiparticle density of states is given in the units of the FS two-spin density of states N(0), N(0)=2N0,and varies from the lowest(black)to the highest(white)value according to the scale in each?gure. We have checked that similarly to the e?ect on the criti-cal temperature[13]any other scattering potential which is orthogonal to the order parameter(in the sense of the FS integral as the scalar product)gives qualitatively the same LDOS as the discussed above f(k)=sgn(sin2φ), i.e.,out of phase potential.The e?ect of potentials in phase with the order parameter like f(k)=cos2φagrees qualitatively with f(k)=sgn(cos2φ).Therefore,both studied anisotropies can be considered representative for the impurity potential(1).Although we have restricted our study to nonmagnetic impurity scattering,as a clos-ing remark we note that inclusion of spin S=1/2,1 or3/2scattering does not contribute any quantitative change to the results for the impurity potential obeying the Born approximation.
Concluding,we have shown that the anisotropy of the impurity potential enhances the quasiparticle LDOS in the vicinity of the impurity and leads to discernible dif-ferences of its intensity in horizontal and diagonal direc-tion.Depending on its symmetry the impurity potential can rotate the LDOS maxima and minima by45?de-grees and can induce a long-range spatial modulation of the quasiparticle density of states but cannot change the induced by the d-wave superconductivity four-fold sym-metry of the STM images.
Note added in proof:A similar modulation to the one in Fig.3(b)is seen along the gap nodal direction for the potential out of the order parameter phase.
We would like to acknowledge helpful discussions with Dr.L.Borkowski.The work was supported in part by KBN grant No.5P03B05820.
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δN (r )
N (0)
01020
304050
?0.10
?0.05
0.00
0.05
0.10
0.15
r k ?1
F
(b)
δN (r )
δN (r )
N (0)
01020
304050
?0.010
?0.005
0.000
0.005
0.010
0.015
r k ?1
F
(b)
δN (r )
δN (r )
N (0)
204060
80
100120140
?0.015
?0.010
?0.005
0.000
0.005
0.010
0.015
r
k ?1F
(b)
FIG.3:Long-range distance dependence of the impurity-induced change to the quasiparticle density of states along φ=0direction at the frequency ω=1.1?for a)isotropic impurity potential and the out of the order parameter phase potential,f (k )=sgn (k x k y );b)potential in phase with the
order parameter,f (k )=sgn (k 2x ?k 2
y ).
(a)
(b)
(c) FIG.4:LDOS-map at the frequencyω=1.1?around the Born impurity located at(0,0)in the ab plane of the d-wave superconductor for a)isotropic impurity potential;b)poten-tial in phase with the order parameter,f(k)=sgn(k2x?k2y);c) potential out of the order parameter phase,f(k)=sgn(k x k y). The density of states is given in the units of N(0)by the scale next to each map and the distance is measured in k?1
units.
F
(a)
(b)
(c) FIG.5:LDOS-map at the frequencyω=0.1?around the Born impurity located at(0,0)in the ab plane of the d-wave superconductor for a)isotropic impurity potential;b)poten-tial in phase with the order parameter,f(k)=sgn(k2x?k2y);c) potential out of the order parameter phase,f(k)=sgn(k x k y). Units as in Fig.4.