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Quantum limit and reentrant superconducting phases in the Q1D conductor Li0.9Mo6O17
Physica B 460 (2015) 231–235
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Quantum limit and reentrant superconducting phases in the Q1D conductor Li0.9Mo6O17 O. Sepper a, A.G. Lebed a,b,n a b
Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA L.D. Landau Institute for Theoretical Physics, 2 Kosygina Street, 117334 Moscow, Russia
art ic l e i nf o
a b s t r a c t
Available online 24 November 2014
We solve the theoretical problem of restoration of superconductivity in a triplet quasi-one-dimensional, layered superconductor in an ultra-high magnetic field. With the field perpendicular to the conducting chains as well as having a component normal to the layers, we suggest a new quantum limit superconducting phase and derive an analytical expression for the transition temperature as a function of magnetic field, T ⁎ (H) . Using our theoretical results along with the known band and superconducting parameters of the presumably triplet superconductor Li0.9Mo6O17, we determine the orientation of H that maximizes T ⁎ (H) for a given value of the field. Subsequently, we show that reentrant superconductivity in this compound is attainable with currently available non-destructive pulsed magnetic fields of order H ≃ 100 T , when such fields are perpendicular to conducting chains and parallel to the layers. For its possible experimental discovery, we give a detailed specification on how small angular inclinations of the magnetic field from its best experimental geometry decrease the superconducting transition temperature of the reentrant phase. & 2014 Elsevier B.V. All rights reserved.
PACS: 74.20.Rp 74.70.Kn 74.25.Op Keywords: High magnetic field Triplet superconductor Unconventional superconductor
1. Introduction Theoretical studies of quasi-one-dimensional (Q1D) layered conductors under high magnetic fields have led to the description of a range of phenomena that result from a quasi-classical 3D → 2D field induced dimensional crossover [1–4] characterized by electron trajectories that extend through length scales much larger than the inter-plane distances in these layered compounds. These phenomena include the field induced spin- [1,5–7] and charge[1,2,8,9] density-wave transitions, Danner–Kang–Chaikin oscillations [1,10], Lebed magic angles [1,11,12], and Lee–Naughton–Lebed oscillations [1,13–15]. In contrast, a quantum 3D → 2D dimensional crossover [1,16] occurs when a magnetic field localizes electrons on Q2D layers, with typical sizes of electron orbits comparable or less than the inter-plane distance. Thus, superconductivity is restored [16] on the layers through suppression of orbital destructive effects. This can happen if the Pauli paramagnetic spin-splitting is absent – as would be the case for superconductors with equal spin-triplet pairing. In this regard, a strong possibility of triplet pairing has been recently proposed [17–19] in the Q1D layered transition metal oxide – Li0.9Mo6O17, n Corresponding author at: Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA. fax: þ 1 520 621 4721. E-mail address: [email protected] (A.G. Lebed).
http://dx.doi.org/10.1016/j.physb.2014.11.078 0921-4526/& 2014 Elsevier B.V. All rights reserved.
where experimental measurements of upper critical magnetic field, Hcx2, parallel to the most conducting axis yielded values 5 times the so-called Clogston–Chandrasekhar limit [20]. A quantitative theoretical analysis [18,19] has shown that, indeed, a triplet pairing scenario can account for such Hc2 values through pure orbital destructive effects, so long as the out-of-plane superconducting coherence length exceeds the inter-plane distance. In what follows, we solve a fully quantum mechanical problem of a Q1D layered conductor in an ultra-high magnetic field perpendicular to conducting chains. First, we examine the case when the field has a component normal to the x–y plane (i.e. subtends an angle α with respect to the layers), resulting in localization of electrons along the conducting chains. We show that this leads to restoration of superconductivity as a quantum limit phase through a 3D → 1D dimensional crossover [21], where the characteristic sizes of electron orbits becoming comparable to both inter-plane and inter-chain length scales. We demonstrate that it corresponds to experimentally available destructive pulsed magnetic fields of the order of H ≃ 500 − 700 T . Subsequently, we suggest experimental discovery of the reentrant superconducting phase in Li0.9Mo6O17 using the currently available pulsed (non-destructive) magnetic fields of order H ≃ 100 T . Our calculations show that such fields, parallel to the layers and perpendicular to the conducting chains lead to the appearance of the reentrant superconductivity phenomenon below a reentrant transition temperature of T ⁎ (H = 100T ) ≃ Tc /2 ≃ 1 K in the superconductor
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Li0.9Mo6O17. We calculate the angular dependence of T ⁎ (α , H) for small angles to account for inclination of the same magnetic field from the optimal geometry of the corresponding experiment. The latter will allow to conduct the experiments with necessary accuracy.
2. Quasi-classical electron orbits in high magnetic fields In this section we qualitatively consider electron trajectories in a high magnetic field to show how confinement in the y and z directions leads to the 3D → 1D dimensional crossover. To this end, let the magnetic field be perpendicular to the conducting chains, and subtend an angle α in the y–z plane, as shown in Fig. 1. The magnetic field and the vector potential corresponding to this configuration are
H = (0, cos α, sin α) H ,
A = (0, sin α, − cos α) Hx.
(1)
We use the tight-binding approximation to describe electron motion in the Q1D conductor, for which the anisotropic dispersion relation is
ϵ (p) = − 2t x cos(px a x ) − 2t y cos(py a y ) − 2t z cos(pz a z ),
(2)
where ti are the electron hopping integrals along different crystallographic directions, and ai are the crystal lattice constants. The quasi-classical electron orbits in a magnetic field are governed by the Lorentz force,
dp ⎛ e ⎞ = ⎜ ⎟ v × H, ⎝c ⎠ dt
(3)
where electrons moving on Q1D Fermi surface have the Fermi velocity vx ≈ vF = const. and
vi =
∂ϵ (p) = 2ti ai sin(pi ai ), ∂pi
i = y , z.
(4)
By substituting the component of the magnetic field, Eqs. (3) and (4) can be integrated with respect to time to find the form of quasi-classical orbits:
y (t) =
2t y ω y (α )
⎡ ⎤ a y cos ⎢ω y (α) t ⎥, ⎢⎣ ⎥⎦
z (t) =
⎡ ⎤ 2t z a z cos ⎢ω z (α) t ⎥, ⎢⎣ ⎥⎦ ω z (α )
where the angle dependent oscillation frequencies are
ω y (α ) =
evF a y H c
sin α,
ω z (α ) =
evF a z H cos α. c
(6)
These results directly demonstrate that for high enough magnetic field the amplitudes of the quasi-classical periodic electron orbits (inversely proportional to H) become smaller than the inter-chain and inter-plane distances – effectively localizing electrons on the conducting chains. This occurs for a magnetic field
⎧ 2t y c 2t z c ⎫ ⎬. H > H⁎ = max ⎨ , ⎩ evF sin α evF cos α ⎭
(7)
This is the qualitative basis for the quantum limit superconductivity phenomenon. In the next section we present a quantitative solution to this problem and show that 2ti/ωi are important quantum parameters.
3. Electron wave functions and the spin-triplet superconducting order parameter In this section we derive an integral gap equation to describe the restoration of superconductivity in ultra-high magnetic field. We use Gor'kov's equation for non-uniform superconductivity under the assumption of equal spin-triplet pairing and derive an analytical solution under the conditions that give rise to the quantum limit superconducting phase. To this end, consider the Q1D Fermi surface consisting of two slightly warped sheets, shown in Fig. 2. The anisotropic electron spectrum with t z < < t y < < t x in Eq. (2) can be linearized on the right ( þ) and left ( ) sheets of the Fermi surface:
ϵ± (p) = ± vF (px − pF ) − 2t y cos(py a y ) − 2t z cos(pz a z ).
(8)
The Hamiltonian is obtained from Eq. (8) by the method of Peierls' substitution: px → − i∂ x , py → py − (e/c) A y , pz → pz − (e/c) A z :
(5)
Fig. 1. The magnetic field makes an angle α with respect to the y-axis, perpendicular to the conducting chains.
Fig. 2. Q1D Fermi surface consists of two slightly warped sheets centered at px = ± pF and extending in the z direction. The triplet order parameter changes its sign on the right ( þ ) and left ( ) sheets.
O. Sepper, A.G. Lebed / Physica B 460 (2015) 231–235
⎡ ω y (α ) ⎤ d ^± / = ∓ ivF x⎥ − 2t y cos ⎢py a y − dx vF ⎦ ⎣
4. Quantum limit reentrant superconducting phase in Li0.9Mo6O17
⎡ ω z (α ) ⎤ x⎥. − 2t z cos ⎢pz a z + vF ⎦ ⎣
(9)
The spin interaction is ignored in the Hamiltonian, since we are considering equal spin-triplet pairing, and the Schrödinger-like ^± equation H Ψϵ± (x; py , pz ) = ϵΨϵ± (x; py , pz ) can be solved for the exact eigenfunctions:
In this section we derive an analytical expression for the critical temperature as a function of magnetic field, T ⁎ (H), at which superconductivity restores as the quantum limit phase. We note that as H → ∞, the arguments of the Bessel functions in Eq. (14) tend to zero, with J0 (0) = 1, giving us an equation that defines the critical temperature Tc at zero field:
1 = g
1 e ±iϵx / v F 2π v F
Ψϵ± (x; py , pz ) = ×
⎧ ⎛ ω y (α ) ⎞ ⎪ 2t y ⎡ ⎢ sin ⎜py a y − exp ⎨ x⎟ ∓i ⎪ vF ⎠ ⎝ ⎩ ω y (α) ⎢⎣
×
−
The finite temperature Green's functions can be obtained from the electron wavefunctions according to the standard procedure [22]:
Gi±ωn (x, x′; py , pz ) =
∫
Ψϵ± (x; py , pz ) Ψϵ± (x; py , pz )⁎
iωn − ϵ
dϵ ,
(11)
where ωn = πT (2n + 1) are the fermion Matsubara frequencies. Having obtained the Green's functions, we use Gor'kov's equations to derive the gap integral equation for the superconducting nucleus. We consider the simplest triplet order parameter that changes its sign on the sheets of the Fermi surface, and is para^ metrized according to Δ^ (x; px ) = ^I sgn(px ) Δ (x), where I is a unit matrix in spin space, sgn ( ± pF ) = ± 1, and
∫ K (x, x′) Δ (x′),
(12)
where g is the effective electron coupling constant, and the kernel is
K (x, x′) = T ∑ Gi+ωn (x, x′; py , pz ) G−−iωn (x, x′; − py , − pz ) ωn
p y , pz
, (13)
After straightforward calculations, it is possible to show that the gap integral equation for triplet superconducting order parameter is
∫|x − x |> d ′
2πTc dz . ⎛ 2πTc z ⎞ vF sinh ⎜ ⎟ ⎝ vF ⎠
(15)
⎛ ω (α ) ⎞ ⎤ 4t y2 ⎡ ⎢1 − cos ⎜ y z⎟ ⎥ 2 ω y (α) ⎢⎣ ⎝ vF ⎠ ⎥⎦
⎡ ⎛ ω z (α ) ⎞ ⎤ 4t z2 ⎢ z⎟ ⎥, 1 cos − ⎜ ⎝ vF ⎠ ⎥⎦ ω z2 (α) ⎢⎣
(16)
(10)
.
g 2
∞
J0 (···) J0 (···) ≈ 1 −
exp ⎧ ⎤⎫ ⎛ ⎪ ⎪ ω (α ) ⎞ 2t z ⎡ ⎢ sin ⎜pz a z + z ⎨ ±i x⎟ − sin(pz a z ) ⎥ ⎬ ⎪ ⎪ vF ⎠ ⎝ ⎦⎭ ⎩ ω z (α ) ⎣
Δ (x) =
∫d
The conditions under which the quantum limit phase is achieved allow us to use the special solution, Δ (x) = const ., and expand the product of the two Bessel functions in Eq. (14) in powers of the small amplitudes, 2ti/ωi < < 1:
⎤⎫ ⎪ − sin(py a y ) ⎥ ⎬ ⎪ ⎥⎦ ⎭
Δ (x) = g
233
2πTΔ (x′) dx′ ⎛ 2π T ⎞ |x − x′|⎟ vF sinh ⎜ ⎝ vF ⎠
where d ∼ vF /Ω is the cutoff distance.
∫0
∞
1 − cos(βx) dx ≈ ln(2βγ), sinh x
(17)
where γ ≈ 1.781 is the value of the exponential of the Euler constant, and Eq. (15) we arrive at an implicit analytical expression for the quantum limit transition temperature, T ⁎ (α , H), at which superconductivity is restored:
⎤ ⎡ ⎡ γω (α) ⎤ ⎛T ⎞ 2 4t y2 y c ⎥ + 4t z ln ⎢ γω z (α) ⎥. ln ⎜⎜ ⁎ ⎟⎟ = 2 ln ⎢ ⁎ ⁎ 2 ⎣⎢ πT ⎥⎦ ⎣⎢ πT ⎥⎦ ω z (α) ⎝ T ⎠ ω y (α )
(18)
Numerical results for Eq. (18) can be obtained for Li0.9Mo6O17, for which the band and superconducting parameters have been calculated in Ref. [18]. We plot the superconducting transition temperature as a function of the angle, α, for given values of magnetic field, H = 700 T, 500 T . The results are shown in Fig. 3. Note that the zero field critical temperature for Li0.9Mo6O17 is Tc ≈ 2.2 K . A broad maximum is achieved in both cases at α ≈ 58°, with T ⁎ = 1.7 K and 1.4 K . Although a particular geometry optimizes T ⁎ (H), the values of transition temperatures near their maximums are accessible for a wide range of angles.
5. Experimental detection of reentrant superconductivity in Li0.9Mo6O17
⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ 8t y ⎪ ω y (α ) ω y (α ) × J0 ⎨ sin ⎢ (x − x′) ⎥ sin ⎢ (x + x′) ⎥ ⎬ ⎢⎣ 2vF ⎪ ω y (α ) ⎣⎢ 2vF ⎦⎥ ⎦⎥ ⎪ ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ 8t z ⎪ ω z (α ) ω z (α ) × J0 ⎨ (x + x′) ⎥ ⎬ sin ⎢ (x − x′) ⎥ sin ⎢ ⎣ 2v F ⎦ ⎣ 2v F ⎦⎪ ⎪ ω z (α ) ⎪ ⎪ ⎩ ⎭
where the change of variables, z = x′ − x , along with averaging over x + x′ has been performed. Using the approximation
(14)
In this section we suggest possible experimental observation of reentrant superconducting phase [16] in the Q1D conductor Li0.9Mo6O17, using the currently experimentally available magnetic fields of order 100 T. To this end, we consider a magnetic field perpendicular to the conducting chains and parallel to the layers, as shown in Fig. 4. Using the steps similar to the ones that led to Eq. (14), we obtain the gap integral equation corresponding to the above configuration [16]:
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Fig. 3. Quantum limit superconducting transition temperate as a function of the angle α plotted for magnetic fields H = 500 T and H = 700 T . Both curves show a broad maximum at α ≈ 58°, with transition temperatures of T ⁎ = 1.4 K and T ⁎ = 1.7 K respectively.
Fig. 5. Reentrant superconducting transition temperature (in degrees Kelvin) for Li0.9Mo6O17 as a function of magnetic field oriented perpendicular to conducting chains and parallel to the layers, as depicted in Fig. 4.
t z = 14 K and ω z (H) = evF a z H /c = 0.58 H (K /T ) are obtained using the parameters for Li0.9Mo6O17 calculated in Ref. [18]. Thus, Eq. (21) can be numerically solved to obtain the field dependence of the reentrance transition temperature, shown in Fig. 5. Based on the above curve, we can see that the reentrant superconducting phase in Li0.9Mo6O17 can be obtained at the experimentally available non-destructive pulsed magnetic field and temperature of
T⁎ ≈ 1 K
at H ≈ 100 Tesla.
Our next step is to theoretically explore the variation in T ⁎ (α , H) due to angular deviation in the optimal experimental geometry. Using the general gap integral equation, Eq. (14), in the high-field regime, we introduce a normalized Ginzburg–Landau like trial solution with a variational parameter κ: 2
Δ (x) = (2κ /π)1/4 e−κx .
Fig. 4. Magnetic field perpendicular to the conducting chains and parallel to the layers of a Q1D layered conductor.
Δ (x) =
g 2
∫|x − x |> d ′
2πTΔ (x′) dx′ vF sinh
(
2πT |x vF
− x′|
)
⎧ ⎫ ⎡ ωz ⎤ ⎡ ωz ⎤⎪ ⎪ 8t z × J0 ⎨ sin ⎢ (x − x′) ⎥ sin ⎢ (x + x′) ⎥ ⎬ , ⎣ 2v F ⎦ ⎣ 2v F ⎦⎪ ⎪ ωz ⎩ ⎭
The trial solution in Eq. (22) along with the expansion in Eq. (16) is substituted into Eq. (14) to evaluate in integrals and optimize the resulting expression with respect to κ, giving the maximum transition temperature as a function of the angle: T ⁎ (α , H). As a result, for small angles, α, we obtain an expression for the Ginzburg–Landau like dependence of T ⁎ (α , H), in the high magnetic field regime corresponding to the reentrant superconducting phase:
⎛ ⎞ H T ⁎ (α, H) = T ⁎ (H) ⎜1 − sin α⎟, HGL ⎝ ⎠ (19)
evF a z H . c
HGL =
4 2 π 2cTc2 . 7ζ (3) t y evF a y
(24)
(20)
Using the expansions and approximation similar to the ones in Eqs. (15)–(17), we arrive at the implicit expression for the reentrance transition temperature, T ⁎ (H):
⎡ T ⎤ ⎡ γω (H) ⎤ 4t z2 c z ⎥= ⎥. ln ⎢ ⁎ ln ⎢ ⁎ 2 ⎣ T (H ) ⎦ ω z (H ) ⎣ π T (H ) ⎦
(23)
where
where the oscillation frequency is
ωz =
(22)
(21)
For the solution (21) of the above equation, the numerical values
[Note that the obtained angular dependence (23) and (24) has a meaning different from the standard Ginzburg–Landau one, since in our case it is valid only at high magnetic fields.] These results, along with the value of T ⁎ (α = 0) ≃ 1 K at H ≃ 100 T can be used to calculate the angular dependence of the reentrant transition temperature, plotted in Fig. 6. We note that the sharp drop observed for angles near α = 0 shows that one needs a careful alignment of the magnetic field during the
O. Sepper, A.G. Lebed / Physica B 460 (2015) 231–235
235
This information is important for accurate alignment of a sample in magnetic field. If confirmed experimentally, the reentrant superconductive phase in Li0.9Mo6O17 would be the first example of survival of superconductivity in ultra-high magnetic fields and would in addition unequivocally confirm spin-triplet pairing nature in this compound.
References
Fig. 6. Angular variation of reentrant superconducting transition temperature, T ⁎ (α) , at H = 100 T for small inclination angles, α (in degrees) from the optimal experimental geometry.
corresponding experiment. As follows from Fig. 6, the accuracy of the alignment has to be better than δα = 0.2° in y–z plane (see Fig. 1). On the other hand, it is known that magnetic fields of order of H = 15 T can destroy superconductivity in Li0.9Mo6O17 when the field is aligned parallel to the conducting chains (see Ref. [17]). Therefore, the component of the experimental magnetic field parallel to the conducting chains has to be less than δH∥ ≃ 5 T . Thus, accurate angular orientation is important for detection of the reentrant superconducting phase at fields of order of H ≃ 100 T , where small inclination of the field (in particular, towards the z -axis) can destroy superconductivity.
6. Conclusion In this paper we have presented theoretical studies of quantum limit and reentrant superconductivity phenomena in the layered Q1D conductor Li0.9Mo6O17. Our results show that superconductivity can be restored in this compound with a reentrant transition temperature T ⁎ (H = 100 T) ≃ 1 K when a field of the order of H ≃ 100 Tesla is aligned parallel to the layers and perpendicular to the conducting chains. We noted that such magnetic fields are currently experimentally available as pulsed non-destructive fields. Furthermore, we have specified how the reentrance transition temperature, T ⁎ (α , H), depends on arbitrary, as well as small deviations from the optimal experimental geometry.
[1] A.G. Lebed (Ed.), The Physics of Organic Superconductors and Conductors, Springer, Berlin, 2008. [2] L.P. Gor'kov, A.G. Lebed', On the stability of the quasi-one-dimensional metallic phase in magnetic fields against the spin density wave formation, J. Phys. Lett. 45 (9) (1984) 433–440. [3] M. Héritier, G. Montambaux, P. Lederer, Stability of the spin density wave phases in (TMTSF)2ClO4: quantized nesting effect, J. Phys. Lett. 45 (19) (1984) 943–952. [4] A.G. Lebed, Phase diagram of layered quasi-one-dimensional conductors in a magnetic field, Sov. Phys. JETP 62 (3) (1985) 595. [5] P.M. Chaikin, M.-Y. Choi, J.F. Kwak, J.S. Brooks, K.P. Martin, M.J. Naughton, E. M. Engler, R.L. Greene, Tetramethyltetraselenafulvalenium perchlorate, (TMTSF)2ClO4, in high magnetic fields, Phys. Rev. Lett. 51 (1983) 2333–2336. [6] M. Ribault, D. Jérome, J. Tuchendler, C. Weyl, K. Bechgaard, Low-field and anomalous high-field hall effect in (TMTSF)2ClO4, J. Phys. Lett. 44 (23) (1983) 953–961. [7] T. Ishiguro, K. Yamaji, G. Saito, Organic Superconductors, Springer, Berlin, 1998. [8] D. Zanchi, A. Bjeliš, G. Montambaux, Phase diagram for charge-density waves in a magnetic field, Phys. Rev. B 53 (1996) 1240–1250. [9] A.G. Lebed, Universal field-induced charge-density-wave phase diagram: theory versus experiment, Phys. Rev. Lett. 103 (2009) 046401. [10] G.M. Danner, W. Kang, P.M. Chaikin, Measuring the fermi surface of quasi-onedimensional metals, Phys. Rev. Lett. 72 (1994) 3714–3717. [11] A.G. Lebed, Anisotropy of an instability for a spin density wave induced by a magnetic field in a Q1D conductor, JETP Lett. 43 (1986) 174. [12] A.G. Lebed, P. Bak, Theory of unusual anisotropy of magnetoresistance in organic superconductors, Phys. Rev. Lett. 63 (1989) 1315–1317. [13] M.J. Naughton, I.J. Lee, P.M. Chaikin, G.M. Danner, Critical fields and magnetoresistance in the molecular superconductors (TMTSF)2X, Synth. Met. 85 (1997) 1481–1485. [14] I.J. Lee, M.J. Naughton, Effective electrons and angular oscillations in quasione-dimensional conductors, Phys. Rev. B 57 (1998) 7423–7426. [15] A.G. Lebed, M.J. Naughton, Interference commensurate oscillations in quasione-dimensional conductors, Phys. Rev. Lett. 91 (2003) 187003. [16] A.G. Lebed, Reversible nature of the orbital mechanism for the suppression of superconductivity, JETP Lett. 44 (1986) 114. [17] J.-F. Mercure, A.F. Bangura, X. Xu, N. Wakeham, A. Carrington, P. Walmsley, M. Greenblatt, N.E. Hussey, Upper critical magnetic field far above the paramagnetic pair-breaking limit of superconducting one-dimensional Li0.9Mo6O17 single crystals, Phys. Rev. Lett. 108 (2012) 187003. [18] A.G. Lebed, O. Sepper, Possible triplet superconductivity in the quasi-one-dimensional conductor Li0.9Mo6O17, Phys. Rev. B 87 (2013) 100511. [19] O. Sepper, A.G. Lebed, Nodeless versus nodal scenarios of possible triplet superconductivity in the quasi-one-dimensional layered conductor li0.9mo6o17, Phys. Rev. B 88 (2013) 094520. [20] A.M. Clogston, Upper limit for the critical field in hard superconductors, Phys. Rev. Lett. 9 (1962) 266–267. [21] M. Rasolt, Z. Tešanović, Theoretical aspects of superconductivity in very high magnetic fields, Rev. Mod. Phys. 64 (1992) 709–754. [22] A.A. Abrikosov, L.P. Gorkov, I.E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics, Revised Edition, Dover, New York, 1975.
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Quantum limit and reentrant superconducting phases in the Q1D conductor Li0.9Mo6O17 O. Sepper a, A.G. Lebed a,b,n a b
Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA L.D. Landau Institute for Theoretical Physics, 2 Kosygina Street, 117334 Moscow, Russia
art ic l e i nf o
a b s t r a c t
Available online 24 November 2014
We solve the theoretical problem of restoration of superconductivity in a triplet quasi-one-dimensional, layered superconductor in an ultra-high magnetic field. With the field perpendicular to the conducting chains as well as having a component normal to the layers, we suggest a new quantum limit superconducting phase and derive an analytical expression for the transition temperature as a function of magnetic field, T ⁎ (H) . Using our theoretical results along with the known band and superconducting parameters of the presumably triplet superconductor Li0.9Mo6O17, we determine the orientation of H that maximizes T ⁎ (H) for a given value of the field. Subsequently, we show that reentrant superconductivity in this compound is attainable with currently available non-destructive pulsed magnetic fields of order H ≃ 100 T , when such fields are perpendicular to conducting chains and parallel to the layers. For its possible experimental discovery, we give a detailed specification on how small angular inclinations of the magnetic field from its best experimental geometry decrease the superconducting transition temperature of the reentrant phase. & 2014 Elsevier B.V. All rights reserved.
PACS: 74.20.Rp 74.70.Kn 74.25.Op Keywords: High magnetic field Triplet superconductor Unconventional superconductor
1. Introduction Theoretical studies of quasi-one-dimensional (Q1D) layered conductors under high magnetic fields have led to the description of a range of phenomena that result from a quasi-classical 3D → 2D field induced dimensional crossover [1–4] characterized by electron trajectories that extend through length scales much larger than the inter-plane distances in these layered compounds. These phenomena include the field induced spin- [1,5–7] and charge[1,2,8,9] density-wave transitions, Danner–Kang–Chaikin oscillations [1,10], Lebed magic angles [1,11,12], and Lee–Naughton–Lebed oscillations [1,13–15]. In contrast, a quantum 3D → 2D dimensional crossover [1,16] occurs when a magnetic field localizes electrons on Q2D layers, with typical sizes of electron orbits comparable or less than the inter-plane distance. Thus, superconductivity is restored [16] on the layers through suppression of orbital destructive effects. This can happen if the Pauli paramagnetic spin-splitting is absent – as would be the case for superconductors with equal spin-triplet pairing. In this regard, a strong possibility of triplet pairing has been recently proposed [17–19] in the Q1D layered transition metal oxide – Li0.9Mo6O17, n Corresponding author at: Department of Physics, University of Arizona, 1118 E. 4-th Street, Tucson, AZ 85721, USA. fax: þ 1 520 621 4721. E-mail address: [email protected] (A.G. Lebed).
http://dx.doi.org/10.1016/j.physb.2014.11.078 0921-4526/& 2014 Elsevier B.V. All rights reserved.
where experimental measurements of upper critical magnetic field, Hcx2, parallel to the most conducting axis yielded values 5 times the so-called Clogston–Chandrasekhar limit [20]. A quantitative theoretical analysis [18,19] has shown that, indeed, a triplet pairing scenario can account for such Hc2 values through pure orbital destructive effects, so long as the out-of-plane superconducting coherence length exceeds the inter-plane distance. In what follows, we solve a fully quantum mechanical problem of a Q1D layered conductor in an ultra-high magnetic field perpendicular to conducting chains. First, we examine the case when the field has a component normal to the x–y plane (i.e. subtends an angle α with respect to the layers), resulting in localization of electrons along the conducting chains. We show that this leads to restoration of superconductivity as a quantum limit phase through a 3D → 1D dimensional crossover [21], where the characteristic sizes of electron orbits becoming comparable to both inter-plane and inter-chain length scales. We demonstrate that it corresponds to experimentally available destructive pulsed magnetic fields of the order of H ≃ 500 − 700 T . Subsequently, we suggest experimental discovery of the reentrant superconducting phase in Li0.9Mo6O17 using the currently available pulsed (non-destructive) magnetic fields of order H ≃ 100 T . Our calculations show that such fields, parallel to the layers and perpendicular to the conducting chains lead to the appearance of the reentrant superconductivity phenomenon below a reentrant transition temperature of T ⁎ (H = 100T ) ≃ Tc /2 ≃ 1 K in the superconductor
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Li0.9Mo6O17. We calculate the angular dependence of T ⁎ (α , H) for small angles to account for inclination of the same magnetic field from the optimal geometry of the corresponding experiment. The latter will allow to conduct the experiments with necessary accuracy.
2. Quasi-classical electron orbits in high magnetic fields In this section we qualitatively consider electron trajectories in a high magnetic field to show how confinement in the y and z directions leads to the 3D → 1D dimensional crossover. To this end, let the magnetic field be perpendicular to the conducting chains, and subtend an angle α in the y–z plane, as shown in Fig. 1. The magnetic field and the vector potential corresponding to this configuration are
H = (0, cos α, sin α) H ,
A = (0, sin α, − cos α) Hx.
(1)
We use the tight-binding approximation to describe electron motion in the Q1D conductor, for which the anisotropic dispersion relation is
ϵ (p) = − 2t x cos(px a x ) − 2t y cos(py a y ) − 2t z cos(pz a z ),
(2)
where ti are the electron hopping integrals along different crystallographic directions, and ai are the crystal lattice constants. The quasi-classical electron orbits in a magnetic field are governed by the Lorentz force,
dp ⎛ e ⎞ = ⎜ ⎟ v × H, ⎝c ⎠ dt
(3)
where electrons moving on Q1D Fermi surface have the Fermi velocity vx ≈ vF = const. and
vi =
∂ϵ (p) = 2ti ai sin(pi ai ), ∂pi
i = y , z.
(4)
By substituting the component of the magnetic field, Eqs. (3) and (4) can be integrated with respect to time to find the form of quasi-classical orbits:
y (t) =
2t y ω y (α )
⎡ ⎤ a y cos ⎢ω y (α) t ⎥, ⎢⎣ ⎥⎦
z (t) =
⎡ ⎤ 2t z a z cos ⎢ω z (α) t ⎥, ⎢⎣ ⎥⎦ ω z (α )
where the angle dependent oscillation frequencies are
ω y (α ) =
evF a y H c
sin α,
ω z (α ) =
evF a z H cos α. c
(6)
These results directly demonstrate that for high enough magnetic field the amplitudes of the quasi-classical periodic electron orbits (inversely proportional to H) become smaller than the inter-chain and inter-plane distances – effectively localizing electrons on the conducting chains. This occurs for a magnetic field
⎧ 2t y c 2t z c ⎫ ⎬. H > H⁎ = max ⎨ , ⎩ evF sin α evF cos α ⎭
(7)
This is the qualitative basis for the quantum limit superconductivity phenomenon. In the next section we present a quantitative solution to this problem and show that 2ti/ωi are important quantum parameters.
3. Electron wave functions and the spin-triplet superconducting order parameter In this section we derive an integral gap equation to describe the restoration of superconductivity in ultra-high magnetic field. We use Gor'kov's equation for non-uniform superconductivity under the assumption of equal spin-triplet pairing and derive an analytical solution under the conditions that give rise to the quantum limit superconducting phase. To this end, consider the Q1D Fermi surface consisting of two slightly warped sheets, shown in Fig. 2. The anisotropic electron spectrum with t z < < t y < < t x in Eq. (2) can be linearized on the right ( þ) and left ( ) sheets of the Fermi surface:
ϵ± (p) = ± vF (px − pF ) − 2t y cos(py a y ) − 2t z cos(pz a z ).
(8)
The Hamiltonian is obtained from Eq. (8) by the method of Peierls' substitution: px → − i∂ x , py → py − (e/c) A y , pz → pz − (e/c) A z :
(5)
Fig. 1. The magnetic field makes an angle α with respect to the y-axis, perpendicular to the conducting chains.
Fig. 2. Q1D Fermi surface consists of two slightly warped sheets centered at px = ± pF and extending in the z direction. The triplet order parameter changes its sign on the right ( þ ) and left ( ) sheets.
O. Sepper, A.G. Lebed / Physica B 460 (2015) 231–235
⎡ ω y (α ) ⎤ d ^± / = ∓ ivF x⎥ − 2t y cos ⎢py a y − dx vF ⎦ ⎣
4. Quantum limit reentrant superconducting phase in Li0.9Mo6O17
⎡ ω z (α ) ⎤ x⎥. − 2t z cos ⎢pz a z + vF ⎦ ⎣
(9)
The spin interaction is ignored in the Hamiltonian, since we are considering equal spin-triplet pairing, and the Schrödinger-like ^± equation H Ψϵ± (x; py , pz ) = ϵΨϵ± (x; py , pz ) can be solved for the exact eigenfunctions:
In this section we derive an analytical expression for the critical temperature as a function of magnetic field, T ⁎ (H), at which superconductivity restores as the quantum limit phase. We note that as H → ∞, the arguments of the Bessel functions in Eq. (14) tend to zero, with J0 (0) = 1, giving us an equation that defines the critical temperature Tc at zero field:
1 = g
1 e ±iϵx / v F 2π v F
Ψϵ± (x; py , pz ) = ×
⎧ ⎛ ω y (α ) ⎞ ⎪ 2t y ⎡ ⎢ sin ⎜py a y − exp ⎨ x⎟ ∓i ⎪ vF ⎠ ⎝ ⎩ ω y (α) ⎢⎣
×
−
The finite temperature Green's functions can be obtained from the electron wavefunctions according to the standard procedure [22]:
Gi±ωn (x, x′; py , pz ) =
∫
Ψϵ± (x; py , pz ) Ψϵ± (x; py , pz )⁎
iωn − ϵ
dϵ ,
(11)
where ωn = πT (2n + 1) are the fermion Matsubara frequencies. Having obtained the Green's functions, we use Gor'kov's equations to derive the gap integral equation for the superconducting nucleus. We consider the simplest triplet order parameter that changes its sign on the sheets of the Fermi surface, and is para^ metrized according to Δ^ (x; px ) = ^I sgn(px ) Δ (x), where I is a unit matrix in spin space, sgn ( ± pF ) = ± 1, and
∫ K (x, x′) Δ (x′),
(12)
where g is the effective electron coupling constant, and the kernel is
K (x, x′) = T ∑ Gi+ωn (x, x′; py , pz ) G−−iωn (x, x′; − py , − pz ) ωn
p y , pz
, (13)
After straightforward calculations, it is possible to show that the gap integral equation for triplet superconducting order parameter is
∫|x − x |> d ′
2πTc dz . ⎛ 2πTc z ⎞ vF sinh ⎜ ⎟ ⎝ vF ⎠
(15)
⎛ ω (α ) ⎞ ⎤ 4t y2 ⎡ ⎢1 − cos ⎜ y z⎟ ⎥ 2 ω y (α) ⎢⎣ ⎝ vF ⎠ ⎥⎦
⎡ ⎛ ω z (α ) ⎞ ⎤ 4t z2 ⎢ z⎟ ⎥, 1 cos − ⎜ ⎝ vF ⎠ ⎥⎦ ω z2 (α) ⎢⎣
(16)
(10)
.
g 2
∞
J0 (···) J0 (···) ≈ 1 −
exp ⎧ ⎤⎫ ⎛ ⎪ ⎪ ω (α ) ⎞ 2t z ⎡ ⎢ sin ⎜pz a z + z ⎨ ±i x⎟ − sin(pz a z ) ⎥ ⎬ ⎪ ⎪ vF ⎠ ⎝ ⎦⎭ ⎩ ω z (α ) ⎣
Δ (x) =
∫d
The conditions under which the quantum limit phase is achieved allow us to use the special solution, Δ (x) = const ., and expand the product of the two Bessel functions in Eq. (14) in powers of the small amplitudes, 2ti/ωi < < 1:
⎤⎫ ⎪ − sin(py a y ) ⎥ ⎬ ⎪ ⎥⎦ ⎭
Δ (x) = g
233
2πTΔ (x′) dx′ ⎛ 2π T ⎞ |x − x′|⎟ vF sinh ⎜ ⎝ vF ⎠
where d ∼ vF /Ω is the cutoff distance.
∫0
∞
1 − cos(βx) dx ≈ ln(2βγ), sinh x
(17)
where γ ≈ 1.781 is the value of the exponential of the Euler constant, and Eq. (15) we arrive at an implicit analytical expression for the quantum limit transition temperature, T ⁎ (α , H), at which superconductivity is restored:
⎤ ⎡ ⎡ γω (α) ⎤ ⎛T ⎞ 2 4t y2 y c ⎥ + 4t z ln ⎢ γω z (α) ⎥. ln ⎜⎜ ⁎ ⎟⎟ = 2 ln ⎢ ⁎ ⁎ 2 ⎣⎢ πT ⎥⎦ ⎣⎢ πT ⎥⎦ ω z (α) ⎝ T ⎠ ω y (α )
(18)
Numerical results for Eq. (18) can be obtained for Li0.9Mo6O17, for which the band and superconducting parameters have been calculated in Ref. [18]. We plot the superconducting transition temperature as a function of the angle, α, for given values of magnetic field, H = 700 T, 500 T . The results are shown in Fig. 3. Note that the zero field critical temperature for Li0.9Mo6O17 is Tc ≈ 2.2 K . A broad maximum is achieved in both cases at α ≈ 58°, with T ⁎ = 1.7 K and 1.4 K . Although a particular geometry optimizes T ⁎ (H), the values of transition temperatures near their maximums are accessible for a wide range of angles.
5. Experimental detection of reentrant superconductivity in Li0.9Mo6O17
⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ 8t y ⎪ ω y (α ) ω y (α ) × J0 ⎨ sin ⎢ (x − x′) ⎥ sin ⎢ (x + x′) ⎥ ⎬ ⎢⎣ 2vF ⎪ ω y (α ) ⎣⎢ 2vF ⎦⎥ ⎦⎥ ⎪ ⎪ ⎪ ⎩ ⎭ ⎧ ⎫ ⎪ ⎪ ⎡ ⎤ ⎡ ⎤ ⎪ 8t z ⎪ ω z (α ) ω z (α ) × J0 ⎨ (x + x′) ⎥ ⎬ sin ⎢ (x − x′) ⎥ sin ⎢ ⎣ 2v F ⎦ ⎣ 2v F ⎦⎪ ⎪ ω z (α ) ⎪ ⎪ ⎩ ⎭
where the change of variables, z = x′ − x , along with averaging over x + x′ has been performed. Using the approximation
(14)
In this section we suggest possible experimental observation of reentrant superconducting phase [16] in the Q1D conductor Li0.9Mo6O17, using the currently experimentally available magnetic fields of order 100 T. To this end, we consider a magnetic field perpendicular to the conducting chains and parallel to the layers, as shown in Fig. 4. Using the steps similar to the ones that led to Eq. (14), we obtain the gap integral equation corresponding to the above configuration [16]:
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Fig. 3. Quantum limit superconducting transition temperate as a function of the angle α plotted for magnetic fields H = 500 T and H = 700 T . Both curves show a broad maximum at α ≈ 58°, with transition temperatures of T ⁎ = 1.4 K and T ⁎ = 1.7 K respectively.
Fig. 5. Reentrant superconducting transition temperature (in degrees Kelvin) for Li0.9Mo6O17 as a function of magnetic field oriented perpendicular to conducting chains and parallel to the layers, as depicted in Fig. 4.
t z = 14 K and ω z (H) = evF a z H /c = 0.58 H (K /T ) are obtained using the parameters for Li0.9Mo6O17 calculated in Ref. [18]. Thus, Eq. (21) can be numerically solved to obtain the field dependence of the reentrance transition temperature, shown in Fig. 5. Based on the above curve, we can see that the reentrant superconducting phase in Li0.9Mo6O17 can be obtained at the experimentally available non-destructive pulsed magnetic field and temperature of
T⁎ ≈ 1 K
at H ≈ 100 Tesla.
Our next step is to theoretically explore the variation in T ⁎ (α , H) due to angular deviation in the optimal experimental geometry. Using the general gap integral equation, Eq. (14), in the high-field regime, we introduce a normalized Ginzburg–Landau like trial solution with a variational parameter κ: 2
Δ (x) = (2κ /π)1/4 e−κx .
Fig. 4. Magnetic field perpendicular to the conducting chains and parallel to the layers of a Q1D layered conductor.
Δ (x) =
g 2
∫|x − x |> d ′
2πTΔ (x′) dx′ vF sinh
(
2πT |x vF
− x′|
)
⎧ ⎫ ⎡ ωz ⎤ ⎡ ωz ⎤⎪ ⎪ 8t z × J0 ⎨ sin ⎢ (x − x′) ⎥ sin ⎢ (x + x′) ⎥ ⎬ , ⎣ 2v F ⎦ ⎣ 2v F ⎦⎪ ⎪ ωz ⎩ ⎭
The trial solution in Eq. (22) along with the expansion in Eq. (16) is substituted into Eq. (14) to evaluate in integrals and optimize the resulting expression with respect to κ, giving the maximum transition temperature as a function of the angle: T ⁎ (α , H). As a result, for small angles, α, we obtain an expression for the Ginzburg–Landau like dependence of T ⁎ (α , H), in the high magnetic field regime corresponding to the reentrant superconducting phase:
⎛ ⎞ H T ⁎ (α, H) = T ⁎ (H) ⎜1 − sin α⎟, HGL ⎝ ⎠ (19)
evF a z H . c
HGL =
4 2 π 2cTc2 . 7ζ (3) t y evF a y
(24)
(20)
Using the expansions and approximation similar to the ones in Eqs. (15)–(17), we arrive at the implicit expression for the reentrance transition temperature, T ⁎ (H):
⎡ T ⎤ ⎡ γω (H) ⎤ 4t z2 c z ⎥= ⎥. ln ⎢ ⁎ ln ⎢ ⁎ 2 ⎣ T (H ) ⎦ ω z (H ) ⎣ π T (H ) ⎦
(23)
where
where the oscillation frequency is
ωz =
(22)
(21)
For the solution (21) of the above equation, the numerical values
[Note that the obtained angular dependence (23) and (24) has a meaning different from the standard Ginzburg–Landau one, since in our case it is valid only at high magnetic fields.] These results, along with the value of T ⁎ (α = 0) ≃ 1 K at H ≃ 100 T can be used to calculate the angular dependence of the reentrant transition temperature, plotted in Fig. 6. We note that the sharp drop observed for angles near α = 0 shows that one needs a careful alignment of the magnetic field during the
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235
This information is important for accurate alignment of a sample in magnetic field. If confirmed experimentally, the reentrant superconductive phase in Li0.9Mo6O17 would be the first example of survival of superconductivity in ultra-high magnetic fields and would in addition unequivocally confirm spin-triplet pairing nature in this compound.
References
Fig. 6. Angular variation of reentrant superconducting transition temperature, T ⁎ (α) , at H = 100 T for small inclination angles, α (in degrees) from the optimal experimental geometry.
corresponding experiment. As follows from Fig. 6, the accuracy of the alignment has to be better than δα = 0.2° in y–z plane (see Fig. 1). On the other hand, it is known that magnetic fields of order of H = 15 T can destroy superconductivity in Li0.9Mo6O17 when the field is aligned parallel to the conducting chains (see Ref. [17]). Therefore, the component of the experimental magnetic field parallel to the conducting chains has to be less than δH∥ ≃ 5 T . Thus, accurate angular orientation is important for detection of the reentrant superconducting phase at fields of order of H ≃ 100 T , where small inclination of the field (in particular, towards the z -axis) can destroy superconductivity.
6. Conclusion In this paper we have presented theoretical studies of quantum limit and reentrant superconductivity phenomena in the layered Q1D conductor Li0.9Mo6O17. Our results show that superconductivity can be restored in this compound with a reentrant transition temperature T ⁎ (H = 100 T) ≃ 1 K when a field of the order of H ≃ 100 Tesla is aligned parallel to the layers and perpendicular to the conducting chains. We noted that such magnetic fields are currently experimentally available as pulsed non-destructive fields. Furthermore, we have specified how the reentrance transition temperature, T ⁎ (α , H), depends on arbitrary, as well as small deviations from the optimal experimental geometry.
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