Pis’ma v ZhETF, vol. 110, iss. 12, pp. 812 - 813

Optical properties of p_x + ip_y superconductors with strong impurities

P. A. Ioselevich^+∗1), P. M. Ostrovsky^∗×

⁺National Research University Higher School of Economics, 101000 Moscow, Russia

^∗L. D. Landau Institute for Theoretical Physics, 119334 Moscow, Russia

^×Max-Planck-Institute for Solid State Research, 70569 Stuttgart, Germany

Submitted 15 November 2019

Resubmitted 27 November 2019

Accepted 27

November 2019

DOI: 10.1134/S0370274X19240081

Observation of the polar Kerr effect in Sr₂RuO₄

impurity band consisting of Yu-Shiba-Rusinov (YSR)

[1] has lead to theoretical studies of the Hall response

type bound states hosted by the impurities [10-12]. We

σ_xy(ω) in p_x + ip_y-wave superconductors [2-9]. A non-

derive a semi-circle-shaped density of states centered√

zero σ_xy implied by the Kerr effect requires either a

at E₀ with the bandwidth w =

8n_impΔ| sin³ δ|/πν.

multi-band structure or impurities in the superconduc-

This is valid in the limit of a well-resolved band, w ≪

tor. So far only weak impurities have been considered.

≪E₀, Δ - E₀.

We generalize existing theory to strong impurities, con-

Equations (2), (3) also describe smoothing of the

sidering their effect on the spectrum of the chiral su-

Bardeen-Cooper-Schrieffer cusp in the density of states

perconductor and on its Hall response in the limit of

at energies close to Δ. This happens on the energy scale

strong, point-like impurities of low concentration n_imp.

n^2imp/Δν² which is parametrically smaller than the YSR

Point-like impurities are parametrized by their scat-

bandwidth w.

tering phase δ in the s-channel. It enters the T matrix,

The density of states resulting from Eqs. (2), (3) is

which describes scattering off a single impurity in a clean

illustrated by Fig. 1.

metal to all orders in its potential,

T₀ =

(1)

πν(isignImE - τ₃ cotδ)

where E is energy, ν is metallic density of state and τ₃

acts in Nambu space. To describe a disordered super-

conducting state, we write down the Dyson equation

G^-1 = G^-1S - n_imp

(2)

where G_S = (E - τ₃ξ(p) -

Δ)^-1 is the clean supercon-

ducting Green’s function with

Δ=Δ(τ1p_x - τ₂p_y)/|p|

being the p_x + ip_y order parameter.

T is a T matrix

describing scattering in the disordered superconducting

Fig. 1. (Color online) Density of states of the supercon-

state. It is related to T₀ via

ductor with strong point-like impurities at cos δ = 0.7,

nimp = 0.06Δν. The semi-circle shape of the YSR impu-

T -T₀ =

T( G - G₀)T₀,

(3)

rity band is distorted due to vicinity to the continuum

where G₀

= (E - τ₃ξ)^-1 is the clean normal-state

Having established the effects of strong impurities

Green’s function. Together, Eqs. (2), (3) form a self-

on the spectrum, we turn to the calculation of σ_xy(ω).

consistent equation set on the disorder-averaged ob-

The leading contribution in small n_imp is given by the

jects

G,T. Solving these, we find that

G(E) exhibits

diagram on the inset of Fig. 2. It contains all orders of

a short cut at energies close to E0 = Δcosδ, where

scattering off a single impurity, with the shaded trian-

Δ is the superconducting gap. This indicates a sub-gap

gles representing T matrices. Calculating the diagram,

¹⁾e-maol: pioselevich@itp.ac.ru

we arrive at

812

Письма в ЖЭТФ том 110 вып. 11 - 12

2019

Optical properties of p_x + ip_y superconductors with strong impurities

813

n_impe²v²Δsin³ δ

R(x ± i0) = (2Δ² - E₀x - x²) ×

σ_xy(ω) =



4ω⁴

arccos^xΔ

[

]



√

|x| < Δ,

× R(E₀ + ω + i0) + R(E₀ - ω - i0) - 2R(E₀) ,

(4)



Δ² - x²



arccosh

√

x > Δ,

(5)

×

x² - Δ²

where R(x) is a cumbersome integral (see full text) de-









iπ - arccosh

pending on the ratios of x and the three energy pa-

 ∓

√

x < -Δ.

rameters E₀, Δ, and temperature T. The function R(x)

x² - Δ²

is real for |x| < Δ but acquires an imaginary part at

Only the threshold process at ω = Δ + E₀ survives at

|x| > Δ. This leads to threshold behavior in σ_xy(ω) as

T = 0, as seen on Fig.2.

seen on Fig.2. As ω is increased, an imaginary part in

Experiment [1] was limited to ω ∼ 1 eV. In this limit,

σ_xy first appears at ω = Δ - E₀. This threshold corre-

ω ≫ Δ,T we obtain

sponds to processes where occupied YSR states are ion-

n_impe²v²Δsin³ δ

σ_xy(ω) = -

ized into the continuum, and hence this process is sup-

[

4ω³

]

E₀

4E₀

pressed as ∼ exp(-E₀/T ) at low temperatures T ≪ E₀.

× itanh

(6)

πω

max{Δ, T }

The second threshold appears at ω = Δ + E₀ and cor-

responds to Cooper pairs being optically excited into a

In the low-T , weak impurity limit, T ≪ Δ, E0 → Δ,

pair of YSR quasiparticle and continuum quasiparticle.

this reproduces [7] up to parameters used to describe

This process is not thermally activated and therefore

the impurities.

survives at T → 0.

In conclusion, we have considered strong point-like

impurities in p_x + ip_y superconductors. We have shown

such impurites to produce a YSR-type impurity band at

E₀ = Δcosδ. We have calculated the anomalous Hall re-

sponse σ_xy as a function of temperature and frequency.

At high frequencies ω ≫ Δ, T its behavior is similar

to that of weak impurities, while behavior at ω ∼ Δ

is much richer, exhibiting thresholds at ω = Δ - E₀

and ω = Δ + E₀. The first corresponds to the onset

of ionization of YSR states into the continuum and is

thermally activated while the second involves creation

of continuum + YSR quasiparticle pairs and survives at

T = 0.

This work was supported by the Basic Research Pro-

gram of Higher School of Economics.

Fig. 2. (Color online) Imσxy(ω) (solid lines) and Reσxy(ω)

Full text of the paper is published in JETP Letters

(dashed lines) in a.u. at Δ = 1, E0 = 0.6. Black lines are

journal. DOI: 10.1134/S0021364019240019

at T = 0 while red lines are at T = 0.1. Inset: diagram

representing σxy(ω) in the linear order in nimp

1. J. Xia, Y. Maeno, P. T. Beyersdorf, M. M. Fejer, and

A. Kapitulnik, Phys. Rev. Lett. 97, 167002 (2006).

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In the next order in n_imp a signal should also ap-

3. V. P. Mineev, Phys. Rev. B 76, 212501 (2007).

pear at 2E₀ - w < ω < 2E₀ + w corresponding to the

4. V. P. Mineev, Phys. Rev. B 77, 180512 (2008).

excitation of a pair of YSR quasiparticles. Furthermore

5. R. Roy and C. Kallin, Phys. Rev. B 77, 174513 (2008).

we note that our calculation breaks down at ω < w

6. J. Goryo, Phys. Rev. B 78, 060501(R) (2008).

7. R. M. Lutchyn, P. Nagornykh, and V. M. Yakovenko,

where transitions within the YSR band become impor-

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tant and the whole ladder diagram must be taken into

8. S. Li, A. V. Andreev, and B. Z. Spivak, Phys. Rev. B 92,

account. However, the latter process is thermally acti-

100506 (2015).

vated, therefore at low temperatures our result is rele-

9. E. J. König and A. Levchenko, Phys. Rev. Lett. 118,

vant up to exponentially small ω ∼ exp(-#E₀/T ). Note

027001 (2017).

that σ_xy(ω) shows no features at ω = 2Δ indicating

10. L. Yu, Acta Phys. Sin. 21, 75 (1965).

11. H. Shiba, Progress of Theoretical Physics 40,

435

that processes where two continuum quasiparticles are

(1968).

created are irrelevant to the Hall response.

12. A. Rusinov, Sov. Phys. JETP 29, 1101 (1969).

At zero temperature, the function R(x) simplifies to

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2019