Effect of a magnetic field on tunneling conductance in normal metal d-wave superconductor interfaces

Abstract

Through the analytic solutions of the Bogoliubov de Gennes (BdG) equations the effect of a static and homogeneous magnetic field applied parallel to the interface of an NIS (N: Normal metal, S: superconductor and I: Insulator) junction on the differential conductance is calculated. For a d xy - symmetry we obtain zero bias conductance peak that can be split by a magnetic field. The shift of the zero bias conductance peak depends on the spread (beta) of the tunneling electrons in k space, on the magnitude of the applied field H and on the ratio between the Fermi energy of the superconductor and the normal region, E FS/E FN. Finally we estimate the minimum value of the magnetic field, Hmin, that splits the zero bias conductance peak. In general Hmin depends on beta, E FS/E FN, the strength of the insulating barrier Z and on the temperature T.

Zero bias conductance peak; d-wave superconductivity; Magnetic field

SURFACES, INTERFACES, AND THIN FILMS

Effect of a magnetic field on tunneling conductance in normal metal d-wave superconductor interfaces

William J. Herrera; J. Virgilio Niño

Departamento de Física, Universidad Nacional de Colombia, Bogotá- Colombia

ABSTRACT

Through the analytic solutions of the Bogoliubov de Gennes (BdG) equations the effect of a static and homogeneous magnetic field applied parallel to the interface of an NIS (N: Normal metal, S: superconductor and I: Insulator) junction on the differential conductance is calculated. For a d_xy - symmetry we obtain zero bias conductance peak that can be split by a magnetic field. The shift of the zero bias conductance peak depends on the spread (b) of the tunneling electrons in k space, on the magnitude of the applied field H and on the ratio between the Fermi energy of the superconductor and the normal region, E_FS/E_FN. Finally we estimate the minimum value of the magnetic field, H_min, that splits the zero bias conductance peak. In general H_min depends on b, E_FS/E_FN, the strength of the insulating barrier Z and on the temperature T.

Keywords: Zero bias conductance peak; d-wave superconductivity; Magnetic field

I. INTRODUCTION

In high temperature superconductivity different experiments have been interpreted assuming a d-wave symmetry of the pair potential, see for example [1]. A feature of d symmetry is the zero-bias conductance peak (ZBCP) observed in tunneling conductance in NIS junctions [1]-[6]. The ZBCP appears when the angle between the a axis of the superconductor and the vector normal to the interface is nonzero; it is a maximum when this angle is ±p/4, (110) orientation. This peak is due to the formation of the zero energy states (ZES) that are originated by the Andreev reflection at the interface; it undergoes a difference in phase of p due to the anisotropy of the pair potential [7]-[10]. Different experiments have shown that the ZBCP can be split due to application of a magnetic field [2]-[5]. From numerical solutions of the Eilenberger's equations, it has been shown that the effect of the magnetic field is to produce a shift of the ZES proportional to the applied magnetic field [11]-[12], other experiments show [13]-[14] that the effect of the magnetic field is to decrease of the heigth of the ZBCP. In this work we solve the BdG equations for this system and show that the splitting depends on Z, b,c and T, The above mentioned experimental characteristics are explained in this work.

II. THEORY

The quasiparticles in a superconductor are described by the BdG equations. For steady states and anisotropic superconductors these equations are [15]

where H_e(r₁) = ( -iÑ-eA(r₁))²/2m+V( r₁) -µ is an electronic hamiltonian, with A(r₁) the vector potential associated with the magnetic fields present in the system, V(r₁) the scalar potential and m the chemical potential, (r₁,r₂) is the pair potential, u(r₁)and v(r₁) are the wave function for the electron- and hole-like components of a quasiparticle. The insulating barrier of height V₀ and thickness d is located in x > 0, this barrier can be modeled by a delta function, V(x) = U₀d(x), where U₀ = V₀d. We concentrate on cuprate superconductor junctions. It is supposed that the quasiparticle moves on the CuO₂ plane with the a and b axes in the x-y plane, the interfaces are normal to the x-axis, see Fig. 1. The Fourier transform of the pair potential is modeled by

where R = r₁-r₂,r = (r₁+r₂)/2 , Q(x) the Heaviside function and D() is the pair potential that undergoes a quasiparticle with momentum k, for s-symmetry D() = D₀ and for d_x-y-symmetry D() = D₀sin2q_s,where q_s is the quasiparticle angle in the superconductor region q_s = sin^-1(k_y/| k| ). The magnetic field H, is applied parallel to the z-axis, therefore the vector potential can be written as A(r) = A_y(x). We consider the situation of a high Tc superconductor where the coherence length of the pair potential x is much smaller than the penetration length l of the magnetic field and approximate A_y = -Hl. As the potentials depends only on x, the solutions of the BdG equations can be written as Y(r) = .

Considering an incoming electron from the normal region, the wave functions (x) and (x) are given taking into account the in the Andreev approximation [15] by

where

The wavenumbers k_FN and k_FS are determined by the Fermi energy in the normal and superconducting regions respectively, = E_FS/E_FN = c². The quasiparticles with and wavenumber move in the pair potential D₊ and D_- respectively

The effect of the magnetic field is an energy shift that depends on k_y and H. One finds a, b, C and D using the boundary conditions in x = 0. The electron-electron and electron-hole reflection coefficients are respectively, R_e = |b| ², R_h = |a| ².

III. DIFFERENTIAL CONDUCTANCE

Using the model developed by Blonder et al.[16] the differential conductance for an angle q_S and for T = 0K is calculated from R_e and R_h coefficients as

with

s_N is the differential conductance when D = 0 (NIN junction), Z is the strength of the barrier and q_N is the quasiparticle angle in the normal region and is determined by the momentum conservation condition in y direction

Firstly the case of c = 1 (q_N = q_Sº q) is analyzed. The relative total differential conductance is found by integration in the k-space as

where

and b is related to the spread of tunneling electron in k- space, it is given by

It is important to note that the insulating barrier is characterized by the parameter Z and b. If in the equation (6) H = 0, our results agree with [7]-[8]. When H ¹ 0 the differential conductance s_S(eV,q) has a shift given by V_A(q). The Gaussian distribution, , diminishes 99% for a angle given by q_c = . The maximum peak energy shift is determined by V_A(q_max), where q_max is the maximum angle for which an electron tunnels the insulating barrier. This angle is p/2-e (e® 0) if 0 < b < 4.2 or approximately q_c if b > 4.2.

The maximum value of V_A is max{D(q)} = D₀, therefore the maximum magnetic field is

with H_c the bulk critic magnetic field.

The average shift is dV = á| V_A|ñ = elH á| sinq|ñ /m. For b >> 1,dV is approximately

The average shift dV is strongly affected by b, this behavior is due to the fact that as b increases, the tunneling cone diminishes and therefore the shift energy decreases. Fig. 2 shows s_R at different values of b, the splitting of ZBCP increases as b decreases. In the inset of the Fig. 2 it is contrated the average shift obtained in Eq. (14) with the numerically value. Fig. 3 shows s_R for different values of H with Z =3, ZBCP is split and the shift depends on the magnetic field and b, as shown the inset of Fig. 3. Fig. 4(a) shows s_R for different values of H with Z =0.5. Is observed that beyond some value the magnetic field ( H_min ) the ZBCP is split. This value depends on Z and b as shown the figures 4(a) and 4(b). For an estimation of H_min one can compare dV with the width G₀ of the ZBCP. For an angle q, the width is given by

Averaging over q the width is approximately

As b increases , decreases, this is because the tunneling cone decreases and therefore average gap diminishes. The splitting appears when dV ~ /2, from this relation we obtained H_min

with p ~ 1. H_min is independent of b , this is due to the fact that as b increases, the width of ZBCP decreases but also dV decreases and these effects compensate each other. It is important to stand out that although in this case H_min is independent of b, H_max depends on b. If H > H_min the magnetic field induces a splitting of the ZBCP, but if H_min > H_max it is not possible to observe this splitting, as is shown in figures 4(a) and 4(b) for b = 5 and b = 20 respectively, and H_min@ 1.4H_c with p = 0.8. If b = 5, H_max = 1.2H_c < H_min and the splitting does not take place. If b = 20, H_max@ 2.2H_c and ZBCP is split. In both cases if H < H_min the effect of the magnetic field is to decrease the height of the ZBCP.

Now we study the effect of the Fermi energy difference between the normal and superconductor regions. This difference is quantified by the parameter c = k_FS/k_FN, the average angle in Eq. (10) is modified by

where q_m is the maximum angle in the superconductor, forc < 1 , q_m = p/2 and for c > 1, q_m = sin^-1(1/c) . If bc² >> 1 from Eq's. (10), (14), (15), dV and ₀ are

In this case H_min and H_max are

In Figs. 5(a) and 5 (b) s_R is plotted against V for c = 0.5 and c = 1.5 respectively. From Eq. (22) H_min = 0.8H_c and H_min = 1.6H_c in agreement with the numerical results shown in the figures.

Finally we study the differential conductance at finite temperature, in this case s_R is calculated from

with f(E) the Fermi-Dirac distribution function at temperature T: Fig. 6 shows s_R for T = 0.1T_c and for different values of Z. The main effect of T is to increase the width of the peak to H = 0 and therefore to increase H_min. In order to analyze the width of the ZBCP we considerer first the case k_BT << ₀, we find that width of the ZBCP is

and H_min is given by

Therefore the value H_min at finite temperature is larger than H_min for T = 0. The relative differenceDH/H_min(0) = (H_min(T)-H_min(0))/H_min(0) is proportional to .gif" align="absmiddle">, and therefore in this case the relative increase of H_min is always less than one, as shown in Fig, 6(a) where H_min@ 1.6H_c ; from Eq's. (17) and (22) H_min(0) = 1.4H_c and H_min(T) = 1.57H_c as shown in the figures. For k_BT >> ₀ (T)

In this case the width is determined main by the temperature ~ K_BT, H_min(T) is proportional to T and b^1/2, and can be larger than H_min(0), in contrast to the previous case , see Fig. 6(b).

IV. CONCLUSIONS

Starting from the solutions of the Bogoliubov of Gennes equations in an NIS junctions we have determined the effect of a magnetic field on the differential conductance. For d_xy symmetries the differential conductance presents a ZBCP. The magnetic field induces a splitting of the ZBCP when H > H_min, where H_min depends on Z, c,and T, in this case the shift of the peak is proportional to H/(b^1/2c). When H < H_min the effect of the magnetic field is to decrease of the height of the ZBCP. If H_min > H_max, the splitting does not take place when the magnetic field is applied; the value of H_max depends on c,and b. This results can be used for the interpretation of the tunneling characteristic in high Tc superconducting junctions in applied magnetic fields.

Acknowledgments

The authors have received support from División de Investigaciones de la Universidad Nacional de Colombia sede Bogotá.

Received on 8 December, 2005

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Publication Dates

• Publication in this collection
  04 Dec 2006
• Date of issue
  Sept 2006

History

• Received
  08 Dec 2005