Electronic structure of vortex state in a FFLO superconductor

Physica C 445–448 (2006) 186–189 www.elsevier.com/locate/physc

Electronic structure of vortex state in a FFLO superconductor Masanori Ichioka *, Hiroto Adachi, Takeshi Mizushima, Kazushige Machida Department of Physics, Okayama University, 3-1-1, Tsushima, Okayama 700-8530, Japan Available online 3 May 2006

Abstract Vortex states in a Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) superconductor are microscopically investigated by the quasi-classical Eilenberger theory. We calculate the spatial structure of the pair potential, paramagnetic magnetization (i.e., Knight shift in a magnetic field) and electronic states, including paramagnetic depairing effect in addition to the orbital depairing effect. Topologies of 2p phase winding at the vortex line and p phase shift at the FFLO nodal plane affect the distribution of paramagnetic magnetization and low energy electronic states. We also discuss the NMR resonance line shape observed in the high field phase of CeCoIn5. Ó 2006 Elsevier B.V. All rights reserved. PACS: 74.25.Op; 74.25.Jb; 74.25.Ha Keywords: FFLO superconductor; Vortex; Quasi-classical theory; CeCoIn5

1. Introduction The Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state [1,2] was proposed in superconductors with large paramagnetic depairing effect. Since the Fermi surfaces for up- and down-spin electrons split due to the Zeeman effect under magnetic fields, Cooper pairs of up- and down-spin electrons are likely to have non-zero momentum for the center of mass coordinate of the pair, inducing spatial modulations of the pair potential [3–9]. The possible FFLO state is widely discussed in various research fields, ranging from superconductors in condensed matter, neutral Fermion superfluids in an atomic cloud [10], to color superconductivity in high energy physics [11]. One of the candidates for the FFLO state is a new superconducting phase at high fields in a quasi-two-dimensional (Q2D) heavy fermion superconductor CeCoIn5 [12–15]. So far, many calculations for the FFLO state have been done by neglecting vortex structure. However, we have to take account of vortices, since the FFLO state appears at

*

Corresponding author. Tel.: +81 86 251 7806; fax: +81 86 251 7830. E-mail address: [email protected] (M. Ichioka).

0921-4534/$ - see front matter Ó 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physc.2006.03.111

high fields in the mixed states. As for the FFLO modulation in the vortex state, there are two possible configurations, i.e., the modulation vector of the FFLO state is parallel or perpendicular to the applied magnetic field. In our study, the former case is investigated by the quasi-classical theory [4,6,16,17]. In a previous quasi-classical study by Tachiki et al. [4] for this case, the FFLO modulation is analyzed after reducing the FFLO vortex states to a problem of the one-dimensional system along the magnetic field direction. In our study, three-dimensional structures of the vortex and the FFLO modulation are determined by the self-consistent calculation based on the quasi-classical theory. On the other hand, the vortex and nodal plane structures in the FFLO state were studied by the Bogoliubov de-Gennes (BdG) theory for a single vortex in a cylindrical superconductor [18]. This study showed that the topologies of pair potential’s phase structure affect the distribution of paramagnetic moment and low energy electronic states. The purpose of this paper is to investigate these characters of the vortex and the FFLO nodal plane structure by the quasi-classical theory. After explaining formulation of our calculations in Section 2, we discuss the spatial structure of the pair potential, paramagnetic magnetization

M. Ichioka et al. / Physica C 445–448 (2006) 186–189

(i.e., Knight shift in a magnetic field) and electronic states under given period of the FFLO modulation in Section 3. We also discuss the NMR resonance line shape in the FFLO state. The last section is devoted to summary and discussions. 2. Quasi-classical theory with paramagnetic effect Under magnetic fields, the superconductivity is suppressed by the paramagnetic depairing effect due to the Zeeman splitting term lBB in addition to the orbital depairing effect by the vector potential A. Quasi-classical Green’s functions gðxl þ i~ lB; k; rÞ, f ðxl þ i~ lB; k; rÞ, f þ ðxl þ i~ lB; k; rÞ are calculated by Eilenberger equations [4,6,16,17] fxl þ i~ lB þ v  ðr þ iAÞgf ¼ Dðr; kÞg; fxl þ i~ lB  v  ðr  iAÞgf þ ¼ D ðr; kÞg; v  rg ¼ D ðr; kÞf  Dðr; kÞf þ ;

ð1Þ

~ ¼ lB B0 =pk B T c . In where g = (1  ff+)1/2, Re g > 0 and l Eq. (1), pair potential D(r, k) = D(r)/(k), where r is the center of mass coordinate of the Cooper pair, and k is a relative momentum of the Cooper pair. Since we report the s-wave pairing case in this paper, the pairing function /(k) = 1. Throughout this paper, length, temperature, Fermi velocity, magnetic field and vector potentials are scaled by R0, Tc, vF , B0 and B0R0, respectively. Here, 1=2 R0 ¼  hvF =2pk B T c , B0 ¼  hc=2jejR20 , and vF ¼ hv2 ik is an averaged Fermi velocity on the Fermi surface. h  ik indicates the Fermi surface average. Energy, D and Matsubara frequency xl are scaled by pkBTc. The pair potential is selfconsistently calculated by X DðrÞ ¼ g0 N 0 T ð2Þ h/ ðkÞff þ f þ gik ; 0
P with ðg0 N 0 Þ ¼ ln T þ 2T 0
where the second term in right hand side is the diamagnetic contribution by supercurrent, and the first term is the paramagnetic contribution by the paramagnetic moment ! BðrÞ 2T X M para ðrÞ ¼ M 0  hIm gik ; ð4Þ ~B 0
187

2 1=2 anisotropy ratio c ¼ nc =nb  hv2c i1=2 k =hvb ik  0:5. Since a magnetic field is applied along a-axis direction in our calculation, (x, y, z) of the coordinate for vortices corresponds to (b, c, a) of the crystal coordinate. We solve Eq. (1) and Eqs. (2)–(4) alternately, to obtain self-consistent solution, by fixing a unit cell of the vortex lattice and a period L of the FFLO modulation [16,17]. The unit cell of the vortex lattice is given by r = w1(u1  u2) + w2u2 + w3u3 with 0.5 6 wi 6 0.5 (i = 1, 2, 3), u1 = (a, 0, 0), u2 = (a/2, ay, 0) and u3 = (0,p0, ffiffiffi L). Reflecting anisotropy ratio c, we set ay =a ¼ 3c=2 with c = 0.5. For the FFLO modulation, we assume D(x, y, z) = D(x, y, z + L) and D(x, y, z) = D(x, y, z). Then, D(r) = 0 at the FFLO nodal plane z = 0, and ±0.5L, as shown in Fig. 1. In the calculation for the electronic states, we solve Eq. (1) with ixl ! E + ig to obtain the local density of states (LDOS) by D n oE N r ðr; EÞ ¼ Re gðxl þ ir~ lB; k; rÞjixl !Eþig ð5Þ

for each spin component. We typically use g = 0.01. 3. FFLO vortex structure The pair potential has 2p-phase winding around a vortex line, and p-phase shift at the nodal plane of the FFLO modulation. Therefore, when we see the phase structure in the xz-plane (y = 0) as schematically shown in Fig. 1, the phase changes p at the vortex line along trajectories across a vortex line. Similarly, along trajectories across a FFLO nodal plane, p-phase change occurs at the nodal plane. However, along trajectories through the intersection point of a vortex line and a FFLO nodal plane, the phase shift is 2p, combining p due to a vortex and p due to a nodal plane. In the following we discuss how these topologies of the phase structure affect the distribution of paramagnetic moments and low energy electronic states. Fig. 2(a) shows the amplitude of the order parameter within a unit cell in xz-plane at B ¼ 0:18B0 (a = 9.0R0) and T = 0.2Tc, when we set L = 50R0. There we see that jD(r)j is suppressed near the vortex line at x = y = 0 and

Fig. 1. Configurations of vortex lines and FFLO nodal planes are presented schematically in the xz-plane (y = 0). Intervortex distance is a in the x-direction. The distance between FFLO nodal planes is L/2. Hatched region shows a unit cell, where + and  indicate signs of the pair potential. 0, p, and 2p indicate the phase shift along trajectories presented by arrows.

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0.5 0.6 0.3 0.0

0

1 0.0

z/L

-0.5 (a)

0.5

M

0

z/L

-0.5 0 x/a

-0.5 0.5

0 x/a

(b)

-0.5 0.5

Fig. 2. Amplitude of pair potential (a) and paramagnetic moment (b) at B ¼ 0:18B0 , T = 0.2Tc and L = 50R0. We, respectively, show jD(r)j and Mpara(r)/M0 within a unit cell in the xz-plane (y = 0).

the FFLO nodal plane at z = 0, ± 0.5L, where the sign of the pair potential changes. Local paramagnetic moment Mpara(r)/M0 is presented in Fig. 2(b), where we see that Mpara(r) is enhanced at a vortex line and FFLO nodal planes, but it is not enhanced at the intersection point of a vortex and a nodal plane. These structures of Mpara(r) are related to the local spectra of the low energy electrons around the vortex and the nodal plane. The LDOS for up and down spin electrons is presented in Fig. 3. In the quasi-classical theory, N"(E, r) and N#(E, r) are symmetric by E M E. The spectra of the DOS for up- (down-) spin electrons is shifted to positive ~B due to the Zeeman shift. At the (negative) energy by l vortex center far from the nodal plane (Fig. 3(a)), N"(E, r) μ+

μ− (a)

N

8

1 0

(b)

N1 0

(c)

N1 0 -1

-0.5

0

E

0.5

1

Fig. 3. Local density of states for up-spin electrons N"(E,r) (solid lines) and down-spin electrons N#(E,r) (dashed lines) at three points r = (x, y, z). (a) At the vortex center far from the FFLO nodal plane, x = y = 0 and z = 0.25L. (b) At vortex center in the FFLO nodal plane, x = y = z = 0. (c) Far from the vortex and the FFLO nodal plane, x = 0.5a, y = 0 and z = 0.25L.

and N#(E, r), respectively, have a ‘‘zero energy peak’’ at ~B and E ¼ l  ~ E ¼ lþ  l lB. The zero energy peak comes from the bound states due to the sign change of the pair potential by the p-phase shift at the vortex line. Also at the FFLO nodal plane far from the vortex line, there appear similar zero-energy peaks due to the p-phase shift (not presented in the figure). Since peak states of up- (down-) spin electrons at E > 0 (E < 0) is empty (filled) states, the imbalance in the occupation of spins induces paramagnetic moments given by Z 0 M para ðrÞ ¼ lB ð6Þ fN " ðE; rÞ  N # ðE; rÞgdE: 1

This is a reason why the paramagnetic moment is enhanced at the vortex core and the FFLO nodal plane. However, along trajectories through the intersection point of a vortex and a nodal plane, D(r) does not change the sign because of 2p phase shift, as mentioned before. Thus, the zero-energy peak does not appear as shown in Fig. 3(b). Instead, N"(E, r) has two broad peaks at finite energies shifted upper or lower from l+. In this spectrum for N"(E, r) (N#(E, r)), lower (higher) energy peak is partially occupied for E < 0 and the other peak is empty (occupied). Then, since the imbalance of up- and down-spin electron density is smaller compared with the spectrum in Fig. 3(a), Mpara(r) is suppressed at the vortex center in the FFLO nodal plane. These features of the spatial structure of the paramagnetic moment and the LDOS are qualitatively consistent with those obtained by the BdG theory [18]. We also show the LDOS spectrum far from the vortex and nodal plane in Fig. 3(c), where we see the superconducting gap structure as in the spectrum at a zero field. The modulation of the spectrum near gap edge is due to the vortex lattice effect at finite fields. The paramagnetic moment Mpara is suppressed in the superconducting state. This Knight shift is observed in the NMR experiment [19,20]. Outside the vortex core, Mpara(r) decreases from M0 on lowering T in the singlet pairing case. However, Mpara(r) does not decrease at the vortex core or the FFLO nodal plane. The effective field for a nuclear spin is given by Beff(r) = B(r) + AhfMpara(r) with the hyperfine coupling constant Ahf. When the contribution of the hyperfine coupling is dominant, the NMR signal can detect the Knight shift of the paramagnetic moments. From the spatial structure of Mpara(r), we calculate the distribution function P(M) = òd(M  Mpara(r))dr, which corresponds to the resonance line shape in NMR experiments when contribution of hyperfine coupling is dominant. If we calculate the distribution function P(B) from the internal field B(r), instead of Mpara(r), we obtain the Redfield pattern of the resonance line shape in the case when Ahf is negligible. The distribution function P(M) in the conventional vortex lattice state without FFLO modulation is shown by a dashed-line in Fig. 4. The peak position M corresponds to the Knight shift outside of the vortex core. In the presence of the FFLO modulation, as presented by a solid line

M. Ichioka et al. / Physica C 445–448 (2006) 186–189

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P(M)

be a clue to identify the FFLO state in the experiment. The double peak structure in the resonance line shape of NMR, observed in a high field phase in CeCoIn5 [15], may be considered as a feature of the FFLO vortex states. References

0

1

M / M0 Fig. 4. Calculated line shape P(M) for the Knight shift as a function of Mpara/M0 without (a dashed line) and with (a solid line) the FFLO modulation. B ¼ 0:18B0 , T = 0.2Tc and L = 50R0.

in Fig. 4, the line shape P(M) becomes double peak structure. In addition to the peak seen in a dashed line, a new peak due to the FFLO nodal planes appears near Mpara(r)  M0. These features of P(M) are basically consistent with the observed NMR resonance line shape in the FFLO phase in CeCoIn5 [15]. The volume contributions by the two-dimensional sheet structure of the FFLO nodal plane are large, compared with those by the one-dimensional structure of the vortex line.

[1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

4. Summary and discussions In order to investigate the structure of the vortex and the FFLO nodal plane in the FFLO state, we calculate the spatial structure of the pair potential, paramagnetic moment and the electronic states under given period of the FFLO modulation by the quasi-classical theory. The topologies of the phase structure in the pair potential affect the distribution of paramagnetic moment, and low energy electronic states. These structures are consistent with those obtained by the BdG theory [18]. We hope that these features will

[15] [16] [17] [18] [19] [20]

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