Zero temperature phase diagram of a d-wave superconductor with Anderson impurities
ABSTRACT We study the model of a d-wave superconductor interacting with finite concentration of Anderson impurities at zero temperature. The interaction between impurity and conduction electrons is taken into account within the large-$N$ approximation. We discuss the obtained phase diagram and its dependence on the main energy scales. Comment: 5 pages, 3 figures, Physics of Magnetism 2008 proceedings
arXiv:0808.3576v1 [cond-mat.supr-con] 26 Aug 2008
Vol. 97 (2000)
ACTA PHYSICA POLONICA A
Proceedings of the European Conference “Physics of Magnetism 08”, Pozna´ n 2008
Zero temperature phase diagram of a d-wave
superconductor with Anderson impurities
L. S. Borkowski
Quantum Physics Division, Faculty of Physics,
A. Mickiewicz University, Umultowska 85, Pozna´ n, Poland
(Version date: June 10, 2008)
We study the model of a d-wave superconductor interacting with finite
concentration of Anderson impurities at zero temperature. The interaction
between impurity and conduction electrons is taken into account within the
large-N approximation.We discuss the obtained phase diagram and its
dependence on the main energy scales.
PACS numbers: 74.81.-g,74.25.Dw
Magnetic and nonmagnetic impurities in correlated electron systems are
an important probe of the properties of the host material. There is large and
growing body of research on impurities in high-temperature superconductors.[1, 2]
The refinement of experimental techniques probing local properties stimulated the
theoretical work on defects in those systems. The investigation of the effects of Zn,
Ni and other dopants in YBCO and BSCCO answered some important questions
and raised new ones.
Measurements on some compounds show that the superconducting order
parameter is not uniform over the entire sample.- Scanning tunnelling spec-
troscopy measurement showed that modulation of the structure of Bi2Sr2CaCu2O8+x
is correlated locally with the magnitude of the energy gap. The spatial varia-
tion of ∆0(r) may result e.g. from the structural supermodulation affecting the
strength of local pairing interaction.
The intrinsic spatial variation of the superconducting gap raises the possi-
bility of observing impurity states on both sides of the quantum phase transition
in the same sample. Theoretical work on magnetic impurities in systems with
reduced density of states near the Fermi surface- showed that the resonant
impurity states may be viewed as a sensitive probe of the superconducting state.
If the coupling to the magnetic impurity is small compared to the energy scale
associated with the gap, the impurity is decoupled from conduction electrons.
This impurity quantum phase transition occurs at finite coupling, provided the
particle-hole symmetry is broken, and may be studied by STM techniques. The
low-energy behavior of the model depends on the exponent r in the conduction
electron density of states, N(ǫ) ∼ |ǫ|r, where the Fermi level is fixed at ǫ = 0.
The interaction of impurities with the conduction electron band may be
studied in the Anderson model with a BCS-type pairing interaction,
k,mfmb + h.c.]
This model allows studying also the mixed valence regime where the impu-
rity occupation number is less than 1 and charge fluctuations are dominant. We
assume a two-dimensional d-wave order parameter of the form ∆(k) = ∆0cos(2φ),
where φ is the angle in the kx− kyplane. The constraint of single occupancy of
the impurity site by adding a term λ(nf− 1) to the Hamiltonian, where λ is the
Lagrange multiplier and taking λ → ∞. Minimizing the free energy with respect
to the resonant level energy ǫf and z =< b†>=< b > and taking the mean field
approximation we obtain
dωf(ω)Gf(ω + i0+),(2)
dωf(ω)G0(ω + i0+)Gf(ω + i0+)].(3)
Equations equations (2) and (3) are solved self-consistently with the gap
The self-energy in the full conduction electron Green’s function G is av-
eraged over impurity positions. We solve these equations self-consistently and
obtain the phase diagram as a function of the parameters of the model.
The obtained phase diagram is shown in Fig. 1. The approximate slope
of the impurity quantum phase transition line, −0.26, agrees with results of the
numerical renormalization group (NRG) method and large-N one-impurity
When the bare impurity level E0 lies closer to the Fermi energy the self-
consistent treatment of finite concentration of impurities leads to reentrant behavior.
In that part of the phase diagram the pair-breaking is weaker and is spread over
wide energy range.
Fig. 2 shows the impurity occupation number nfat the impurity decoupling
transition. It reaches 1 at the transition. For Γ0≫ ∆0, nf(E0) slowly approaches
1, |dnf/dE0|Γ0=const≪ 1, as E0/Γ0→ (E0/Γ0)critical. However for small Γ0the
dependence of nfon E0/Γ0becomes singular and the slope |dnf/dE0|Γ0=const≫
-1-0.8-0.6-0.4-0.2 0 0.2
Fig. 1. The phase diagram of the d-wave superconductor with nondegerate Anderson
impurities for several impurity concentrations. The lines are guide to the eye. The
order parameter amplitude is ∆0 = 0.01D, where D is half of the conduction electron
bandwidth. All energies are scaled in units of D. The dotted line indicates the impurity
quantum phase transition.
Fig. 2. The impurity occupation number as a function of the ratio E0/Γ0 for several
values of Γ0. At the impurity transition nf → 1.
-1-0.8 -0.6-0.4 -0.2 0
Fig. 3. The location of the critical point (dotted line) and the impurity decoupling
transition (broken line).
In the limit of vanishing interaction Γ0→ 0, E0= 0 is the singular point of
the model. For E0> 0 the superconducting state survives even for large impurity
For any finite n there is a critical point (Γ0c,E0c). For Γ0 slightly larger
than Γ0c there is a superconductor-normal state transition at E01 < E0c and
another normal-superconductor transition at E02, where E01 < E02 < E0c. At
fixed n, E02− E01 ≃ α(n)(Γ0− Γ0c), where α(n) weakly depends on n. The
location of this critical point in the E0-Γ0plane is shown in Fig. 3.
It would be interesting to test this theoretical picture in experiment. Near
the impurity quantum phase transition the impurity state is very sensitive to
small changes of hybridization Γ0 or impurity level energy E0. In compounds
with spatially varying energy gap this could lead to impurities existing on the two
sides of the transition line in different parts of the sample.
The phase diagram calculated in this work might also be relevant in some
heavy-fermion compounds where similar competition occurs between energy scales
associated with the Kondo screening and the superconducting correlations. Stud-
ies of CeCu2(Si1−xGex)2under varying hydrostatic pressure reveal two supercon-
ducting domes in the phase diagram.[20, 21] The existing interpretation of this
dependence on pressure relies on additional valence-fluctuation mediated pairing
However our work suggests that the second superconducting dome in
CeCu2(Si1−xGex)2at high pressure may follow from weakened pair-breaking. The
change of pressure shifts the chemical potential and brings the system to the
mixed-valence regime when the bare f-level E0of Ce ions approaches EF. The
phase diagram in Fig. 1 shows that in this limit superconducting correlations are
less affected by pair-breaking.
The large-N method used in the present calculation gives qualitatively sim-
ilar results for larger N. Also the symmetry of the order parameter should not
introduce drastic changes to the phase diagram. The reentrant behavior as a func-
tion of E0 results from the competition between the formation of the impurity
resonance and superconducting correlations and depends mainly on the ratio of
the relevant energy scales.
Extension of the theory beyond the mean field should not change the phase
diagram qualitatively. A more detailed description of physics in the vicinity of
the impurity transition line requires careful treatment of low-energy scattering in
specific superconducting compounds.
 H. Alloul, J. Bobroff, M. Gabay, and P.J. Hirschfeld, Rev. Mod. Phys., to appear,
and references therein.
 A. V. Balatsky, I. Vekhter, and J.-X. Zhu, Rev. Mod. Phys. 78, 373 (2006), and
 T. Cren, D. Roditchev, W. Sacks, J. Klein, J.-B. Moussy, C. Deville-Cavellin, and
M. Lagues, Phys. Rev. Lett. 84, 147 (2000).
 S.-H. Pan, J. P. O’Neal, R. L. Badzey, C. Chamon, H. Ding, J. R. Engelbrecht,
Z. Wang, H. Eisaki, S. Uchida, A. K. Guptak, K. Ng, E. W. Hudson, K. M. Lang,
and J. C. Davis, Nature 413, 282 (2001).
 C. Howald, P. Fournier, and A. Kapitulnik, Phys. Rev. B 64, 100504(R) (2001).
 K. M. Lang, V. Madhavan, J. E. Hoffman, E. W. Hudson, H. Eisaki, S. Uchida,
and J. C. Davis, Nature 415, 412 (2002).
 K. McElroy, J. Lee, J. A. Slezak, D.-H. Lee, H. Eisaki, S. Uchida, and J. C. Davis,
Science 309, 1048 (2005).
 J. A. Slezak, Ph.D. thesis, Cornell University 2007.
 Y. He, S. Graser, P. J. Hirschfeld, and H.-P. Cheng, Phys. Rev. B, to appear.
 H. Q. Yuan, F. M. Grosche, M. Deppe, C. Geibel, G. Sparn, and F. Steglich,
Science 302, 2104 (2003).
 D. Withoff and E. Fradkin, Phys. Rev. Lett. 64, 1835 (1990).
 L. S. Borkowski and P.J. Hirschfeld, Phys. Rev. B 46, 9274, 1992.
 L. S. Borkowski and P. J. Hirschfeld, J. Low Temp. Phys. 96, 185 (1994).
 C. Gonzalex-Buxton and K. Ingersent, Phys. Rev. 54, R15614 (1996).
 C. Gonzalez-Buxton and K. Ingersent, Phys. Rev. B 57, 14254 (1998).
 J.-X. Zhu and C. S. Ting, Phys. Rev. B 63, 020506(R) (2000).
 A. Polkovnikov, S. Sachdev, and M. Vojta, Phys. Rev. Lett 86, 296 (2001).
 G.-M. Zhang, H.Hu, and Lu Yu, Phys. Rev. Lett. 86, 704 (2001).
 L. S. Borkowski, Phys. Rev. B 78, 020507 (2008).
 H. Q. Yuan, F. M. Grosche, M. Deppe, C. Geibel, G. Sparn, and F. Steglich, New
J. Phys. 6, 132 (2004).
 P. Gegenwart, Q. Si, F. Steglich, Nature Physics, 4, 186 (2008).
 K. Miyake and H. Maebashi, J. Phys. Soc. Japan 71, 1007 (2002).