Pis’ma v ZhETF, vol. 111, iss. 4, pp. 249 - 250

February 25

Four-fold anisotropy of the parallel upper critical magnetic field

in a pure layered d-wave superconductor at T = 0

A. G. Lebed^+∗1), O. Sepper⁺

⁺Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA

^∗L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences, 117334 Moscow, Russia

Submitted 23 December 2019

Resubmitted 14 January 2020

Accepted 16

January 2020

DOI: 10.31857/S0370274X20040098

Since the discovery of unconventional d-wave super-

Below, we consider a layered superconductor with

conductivity in high-temperature superconductors [1],

the following Q2D electron spectrum, which is an

physical consequences of d-wave electron pairing have

isotropic within the conducting plane:

been intensively investigated. One of such physical prop-

ǫ(p) = ǫ(p_x, p_y) - 2t_⊥ cos(p_zc^∗), t_⊥ ≪ ǫ_F ,

(1)

erties is a four-fold symmetry of the parallel upper criti-

cal magnetic field in these quasi-two-dimensional (Q2D)

where

superconductors [2-5]. From the beginning, it was rec-

(p^2x + p^2y)

p^2F

ognized that the four-fold anisotropy of the parallel up-

ǫ(p_x, p_y) =

ǫ_F =

(2)

per critical magnetic field disappears in the Ginzburg-

[Here, m is the effective in-plane electron mass, t_⊥ is

Landau (GL) region (see, for example, book [6]) and

the integral of overlapping of electron wave functions

has to be calculated as a non-local correction to the

in a perpendicular to the conducting planes direction;

GL results [3, 4]. Another approach was calculation of

ǫ_F and p_F are the Fermi energy and Fermi momentum,

the parallel upper critical magnetic field at low temper-

respectively; ℏ ≡ 1.] At the beginning, the parallel mag-

atures and even at T = 0 [2,7-9] using approximate

netic field is assumed to be applied along x axis,

method [10], which was elaborated for unconventional

superconductors with closed electron orbits in an ex-

H = (H,0,0),

(3)

ternal magnetic field. Note that Q2D conductors in a

parallel magnetic field are characterized by open elec-

where vector potential of the field is convenient to

tron orbits, which makes the calculations [2, 7-9] to be

choose in the form:

unappropriate.

The goal of our article is to suggest an appropri-

A = (0,0,Hy).

(4)

ate method to calculate the parallel upper critical mag-

Electron motion within the conducting plane is sup-

netic field in a Q2D d-wave superconductor. For this

posed to be free (2), therefore, we can make the following

purpose, we explicitly take into account almost cylindri-

substitutions in the electron energy (1) and (2):

cal shape of its Fermi surface (FS) and the existence of

(

)

(

)

open electron orbits in a parallel magnetic field. We use

∂

the Green’s functions formalism to obtain the Gor’kov’s

p_x → -i

, p_y → -i

(5)

∂x

∂y

gap equation in the field. As an important example, we

numerically solve this integral equation to obtain the

whereas, for the perpendicular electron motion, we can

four-fold anisotropy of the parallel upper critical mag-

perform the so-called Peierls substitution:

netic field in a d_x2_-y2 -wave Q2D superconductor with

)

(ω_c

ev_F c^∗H

isotropic in-plane FS. In particular, we demonstrate

p_zc^∗ → p_zc^∗ -

y, ω_c =

(6)

v_F

that the so-called supercondcting nuclei at T = 0 oscil-

late in space in contrast to the previous results [2, 7-9].

As a result, the electron Hamiltonian in the magnetic

field (3) can be represented as:

(

)

(

)

∂²

-2t_⊥ cos p_zc^∗ -

ω_c y

(7)

¹⁾e-mail: lebed@email.arizona.edu

= -2m ∂x2

∂y²

v_F

Письма в ЖЭТФ том 111 вып. 3 - 4

2020

249

250

A. G. Lebed, O. Sepper

By means of the quasi-classical approach [11] to the

Hamiltonian (7), it is possible to find electron Green’s

functions in a magnetic field. Then, using linearized

Gor’kov’s gap equation for unconventional non-uniform

superconductivity [12, 13], it is possible to obtain for

d_x2-y2 electron pairing,

U (φ, φ₁) = g cos(2φ) cos(2φ₁),

(8)

the following gap equation at T = 0:

{

}

∫ ∞

2t_⊥ω_c

Δ_α(y) = g

J₀

[z(2y + z sin φ₁)]

v²

× [1 + cos(4α) cos(4φ₁)]Δ_α(y + z sin φ₁)

(9)

φ1

Fig. 1. (Color online) Angular dependence of the ratio

hc2(α) = Hc2(α)/

(0), where Hc2(α) is the direction

where now angle α is the angle between the magnetic

dependent parallel upper critical magnetic field at T = 0

field and x axes in (x, y)-plane; < ... >φ1 means average

and H^GLc2(0) is the Ginzburg-Landau parallel upper criti-

over angle φ₁. For further solution of Eq. (9), it is useful

cal magnetic field slope at T = 0

to introduce new convenient variables:

than that reported before [2]. In addition, from Fig.1, it

√2t_⊥ω_c

√2t_⊥ω_c y.

(10)

is clear that the calculated by us anisotropic term is not

v_F

of a pure cos(4α) form as was stated in the all previous

[Here, at T = 0 we define the upper critical magnetic

calculations [2,7-9].

field in the framework of the Landau theory of the sec-

The author is thankful to N. N. Bagmet (Lebed) for

ond order phase transitions, as it is done, for example,

useful discussions.

in [12, 13]. Therefore, we disregard the possible appear-

Full text of the paper is published in JETP Letters

ance of the so-called quantum phase transitions.] In new

journal. DOI: 10.1134/S0021364020040037

variables Eq. (9) can be written as follows

1. C. C. Tsuei, J. R. Kirtley, C. C. Chi, Lock See Yu-Jahnes,

∫ ∞

A. Gupta, T. Shaw, J. Z. Sun, and M. B. Ketchen, Phys.

Δ_α(y) = g

J₀[z(2y + zsinφ₁)] ×

Rev. Lett. 73, 593 (1994).

2. H. Won and K. Maki, Physica B 199-200, 353 (1994).

× [1 + cos(4α) cos(4φ₁)]Δ_α(y + zsin φ₁)

(11)

3. K. Takanaka and K. Kuboya, Phys. Rev. Lett. 75, 323

φ1

(1995).

Below, we solve integral Eq.(11) numerically. The

4. T. Sugiyama and M. Ichioka, J. Phys. Soc. Jpn. 67, 1738

typical example of its solution for the superconduct-

(1998).

5. Y. Koyama, T. Sasagawa, Y. Togawa, K. Otzschi, J. Shi-

ing nucleus, Δ_α(y), is shown in Fig. 1 of [11], where it

moyama, K. Kitazawa, and K. Kishio, J. Low Temp.

oscillates and changes its sign with the changing co-

Phys. 117, 551 (1999).

ordinate y. These oscillations are consequences of the

6. A. A. Abrikosov, Fundamentals of Theory of Metals, El-

open nature of electron trajectories in a parallel mag-

sevier Science, Amsterdam (1988).

netic field for the Q2D electron spectrum (1), (2). As we

7. G. Wang and K. Maki, Phys. Rev. B 58, 6493 (1998).

have shown, they result in the appearance of the oscil-

8. F. Weickert, P. Gegenwart, H. Won, D. Parker, and

lating Bessel function in the gap Eq. (11). This typical

K. Maki, Phys. Rev. B 74, 134511 (2006).

behavior of the superconducting nuclei is in a sharp con-

9. H. A. Viera, N. Oeschler, S. Seiro, H. S. Jeevan,

trast with the calculations of the four-fold anisotropy

C. Geibel, D. Parker, and F. Steglich, Phys. Rev. Lett.

at T = 0, which were done before [2,7-9]. The rea-

106, 207001 (2011).

son for that is the fact that the previous calculations

10. I. A. Luk’yanchuk and V. P. Mineev, Sov. Phys. JETP

didn’t take into account open nature of electron orbits

66, 1168 (1987).

11. A. G. Lebed, JETP Lett. 110, 173 (2019) [Pis’ma v

in Q2D conductors in a parallel magnetic field. Numeri-

ZhETF 110, 163 (2019)].

cally calculated from Eq.(11) angular dependence of the

12. V. P. Mineev and K. V. Samokhin, Introduction to Un-

parallel upper critical magnetic field of a Q2D d_x2_-y2 su-

conventional Superconductivity, Gordon and Breach Sci-

perconductor is shown in Fig.1, where there is a sharp

ence Publisher, Australia (1999).

peak at α = 0 and a shallow minima at α = ±45⁰.

13. A. G. Lebed and K. Yamaji, Phys. Rev. Lett. 80, 2697

The calculated magnitude of the four-fold anisotropy is

(1998).

[H_c2(0⁰)- H_c2(45⁰)]/H_c2(22.5⁰) = 0.13, which is higher

Письма в ЖЭТФ том 111 вып. 3 - 4

2020