Upper critical field H_{c2 }in Bechgaard salts (TMTSF)_{2}PF_{6}
- Ana D Folgueras^{1, 2}Email author and
- Kazumi Maki^{3}
Received: 30 November 2007
Accepted: 09 December 2008
Published: 09 December 2008
Abstract
The symmetry of the superconductivity in Bechgaard salts is still unknown, though the triplet pairing has been established by H_{c2 }and NMR for (TMTSF)_{2}PF_{6}. The large upper critical field at T = 0K (H_{c2 }~ 5 Tesla) both for $\overrightarrow{H}\left|\right|\overrightarrow{a}$ and $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$ also indicates strongly the triplet pairing.
Here we start with a low energy effective Hamiltonian and study the temperature dependence of the corresponding H_{c2}(T)'s.
The present analysis suggests that one chiral f-wave superconductor should be the most likely candidate near the upper critical field.
PACS Codes: 74.70.Kn ; 74.20.Rp; 74.25.Op.
Introduction
The Bechgaard salt (TMTSF)_{2} PF_{6} is the first organic superconductor discovered in 1980 [1]. Until very recently the superconductivity was believed to be conventional s-wave [2]. More recently the symmetry of the superconductivity has become one of the central issues [3]. The upper critical field at T = 0K in (TMTSF)_{2}PF_{6} and (TMTSF)_{2}ClO_{4} are clearly beyond the Pauli limit [4–7], suggesting triplet pairing. Recent NMR data [8, 9] from (TMTSF)_{2}PF_{6} supports triplet superconductivity.
Here we shall first derive H_{c2}(T) for a variety of p-wave and f-wave superconductors [8]. Later, we will discuss the relation between the nuclear spin relaxation rate and the nodal lines.
Theoretical model
with v : v_{ b }: v_{ c }~ 1 : 1/10 : 1/300 and v = v_{ a }, v_{ b }= $\sqrt{2}{t}_{b}b$ and v_{ c }= $\sqrt{2}{t}_{c}c$; for example, P. M. Grant [12] gives v_{ c }~ 1 meV, t_{ b }~ 26.2 meV and t_{ a }~ 365 meV.
There are earlier analysis of H_{c2 }of Bechgaard salts starting from the one dimensional models [13, 14]. However, those models predict diverging H_{c2}(T) for T → 0K or the reentrance behaviour, which have not been observed in the experiments [4, 5]. The one dimensional model, like the one proposed by Lebed [13, 15] is valid only when 2t_{ c }< 2.14T_{ c }~ 3 K, in Bechgaard salts it is believed that the transfer integral in the c direction is 2t_{ c }~ 10 – 30 K while the superconducting transition temperature is T_{ c }~ 1.2 K, so the 1D model is unrealistic. Also, the quasilinear T dependence of H_{c2}(T) in both (TMTSF)_{2}PF_{6} and (TMTSF)_{2}ClO_{4} is very unusual.
We consider a 3D model, though strongly anisotropic. We start with a continuum model, where the cristal anisotropy is incorporated only through the great anisotropies of the Fermi velocities. We have considered chiral superconductors because these symmetries have been shown to lead to higher H_{c2}s. In the absence of an applied magnetic field, we could obtain one of those chiral states as a combination of two different order parameters (with two different transition temperatures), but the external magnetic field breaks the time reversal symmetry, allowing the formation of a chiral state in the superconducting phase (see [16]).
Among the symmetries we have considered, the chiral f'-wave superconductor with $\Delta (\overrightarrow{k})~\left(\frac{1}{\sqrt{2}}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}\right)\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}$, looks most promising, where χ_{1} = $\overrightarrow{b}\overrightarrow{k}$ and χ_{2} = $\overrightarrow{c}\overrightarrow{k}$ are $\overrightarrow{b}$ and $\overrightarrow{c}$ the cristal vectors.
Moreover, if the superconductor belongs to one of the nodal superconductors [17, 18] and if nodes lay parallel to ${\overrightarrow{k}}_{c}$ within the two sheets of the Fermi surface, the angle dependent nuclear spin relaxation rate ${T}_{1}^{-1}$ in a magnetic field rotated within the b' - c* plane will tell the nodal directions.
1 Results and discussion
Upper critical field for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$
In the following we neglect the spin component of $\overrightarrow{\Delta}(\overrightarrow{k})$. Most likely the equal spin pairing is realised in Bechgaard salts as in Sr_{2}RuO_{4} [3]. In this case the spin component is characterized by a unit vector $\widehat{d}$. Also $\widehat{d}$ is most likely oriented parallel to ${\overrightarrow{c}}^{\ast}$. Let's assume $\widehat{d}\left|\right|{\overrightarrow{c}}^{\ast}$, though H_{c2}(T) is independent of $\widehat{d}$ as long as the spin orbit interaction is negligible. Experimental data from both UPt_{3} and Sr_{2}RuO_{4} indicate that the spin-orbit interactions in these systems are not negligible but extremely small [3]. We consider a variety of triplet superconductors (see some of them in Fig. 1), most of them chiral variants, as we find in general that the chiral variant has larger H_{c2 }than the non-chiral one:
Simple p-wave SC: $\overrightarrow{\Delta}(k)~sgn({k}_{a})$
and $t=\frac{T}{{T}_{c}},\rho =\frac{{v}_{a}{v}_{c}e{H}_{c2}(T)}{2{(2\pi T)}^{2}},s=\left(\frac{1}{\sqrt{2}}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}\right),{\chi}_{2}=\overrightarrow{c}\overrightarrow{k}$, and ⟨...⟩ means average over χ_{2}. Here v_{ a }, v_{ c }are the Fermi velocities parallel to the a axis and the c axis respectively.
where $\u3009={\displaystyle \sum {C}_{n}{e}^{-eB{x}^{2}-nk(x+iz)-\frac{{(nk)}^{2}}{4eB}}}$ is the Abrikosov state [19], C_{ n }the occupancy of the n^{ th }Landau level (we assume there is only one occupied Landau level) and ${a}^{+}=\frac{1}{\sqrt{2eB}}\left(-i{\partial}_{z}-{\partial}_{x}+2ieHz\right)$ is the raising operator.
Then in the vicinity of t → 1 we find $\rho =\frac{2}{7\zeta (3)}(-\mathrm{ln}\phantom{\rule{0.5em}{0ex}}t)=0.237697(-\mathrm{ln}\phantom{\rule{0.5em}{0ex}}t)$ and $C=\frac{93\zeta (5)}{647\zeta (3)}\rho $.
Chiral p-wave SC: $\overrightarrow{\Delta}(k)=1/\sqrt{2}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}({\chi}_{2})$
Here $\frac{1}{\sqrt{2}}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}({\chi}_{2})$ is the analogue of e^{ ιϕ }if in the 3D systems in the quasi 1D system.
and the same expressions for t, ρ,...
For t → 1 we find $C=1-\sqrt{1.5}=-0.2247$ and ρ = 0.3838(-ln t).
On the other hand, for t → 0 we obtain C = -0.3660 and ρ_{0} = 0.27343.
From these we obtain h(0) = 0.71324. We obtain ρ(t) and C(t) numerically. They are shown in Fig. 2a) and 2b) respectively.
Chiral f-wave SC: $\widehat{\Delta}(k)~\widehat{d}s\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{1}$
Here now $\u3008\mathrm{...}\u3009$ means the average over both χ_{1} and χ_{2}. As in previous sections, $s=\frac{1}{\sqrt{2}}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}({\chi}_{2})$ (s depends on the direction of the magnetic field). Then it is easy to see that the chiral f-wave SC has the same H_{c2}(t) and C(t) as the chiral p-wave SC, since the variable χ_{1} is readily integrated out.
Chiral f'-wave SC: $\widehat{\Delta}(k)~\widehat{d}s\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}$
Now we have a set of equations similar to the chiral f-wave except (1 + cos 2χ_{1}) in both eqs. 13 has to be replaced by $\frac{4}{3}$(1 + cos 2χ_{1}). We obtain, for t → 1, C = -0.2247 and ρ = 0.5181(-ln t). On the other hand, for t → 0 we find C = -0.3660 and ρ_{0} = 0.3734.
We show ρ_{0} and C(t) of the chiral f'-wave in Fig. 2a) and 2b) respectively.
Note that C(t) is the same for three chiral states (chiral p-wave, chiral f-wave and chiral f'-wave) as well as chiral p-wave studied in [20].
Therefore for the magnetic field $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$, the chiral f'-wave have the largest H_{c2}(t) if we assume T_{ c }and v, v_{ c }are the same. Also H_{c2}(t) of these states are closest to the observation.
Upper critical field for $\overrightarrow{H}\left|\right|\overrightarrow{a}$
In this section, we assume the applied magnetic field runs parallel to the direction defined by $\overrightarrow{a}$. We calculate the upper critical field in these circunstances for different symmetries of the order parameter, following the same procedure as the one we used in previous section.
Simple p-wave SC: $\Delta (\overrightarrow{k})=sgn({k}_{a})$
where $t=\frac{T}{{T}_{c}},\rho =\frac{{v}_{b}{v}_{c}e{H}_{c2}(t)}{2{(2\pi T)}^{2}}$ and s = (sin χ_{1} + ι sin χ_{2}) with χ_{1} = $\overrightarrow{b}\overrightarrow{k}$ and χ_{2} = $\overrightarrow{c}\overrightarrow{k}$.
Then for t → 1, we find $C=-\frac{93\zeta (5)}{508\zeta (3)}\rho $ and $\rho =\frac{2}{7\zeta (3)}(-\mathrm{ln}\phantom{\rule{0.5em}{0ex}}t)=0.2377(-\mathrm{ln}\phantom{\rule{0.5em}{0ex}}t)$. While for t → 0 $C=\frac{3}{2{\beta}_{0}}-\sqrt{{\left(\frac{3}{2{\beta}_{0}}\right)}^{2}+\frac{1}{12}}=-0.0170129$ and ${\rho}_{0}=\frac{{v}_{b}{v}_{c}e{H}_{c2}(0)}{2{(2\pi {T}_{c})}^{2}}=\frac{1}{4\gamma}$, where α_{0} = -⟨ln|s|^{2}⟩ = 0.220051 and ${\beta}_{0}=-\u3008\frac{{s}^{4}}{|s{|}^{4}}\u3009=\frac{4}{\pi}-1=0.0170$. From these we obtain h(0) = 0.73673.
Chiral p-wave SC: $\Delta (k)~\left(\frac{1}{\sqrt{2}}sgn({k}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}\right)$
Now H_{c2}(t) is determined by a similar set of equations as Ec. 10–11. Now, s = (sin χ_{1} + ι sin χ_{2}). In particular we find for t → 1 C = -0.027735 and ρ = 0.212598(ln t) while for t → 0 C = -0.067684 and ρ_{0} = 0.139672. We obtain h(0) = 0.6566. We show h(t) and C(t) in Fig. 3a) and 3b) respectively.
Chiral f-wave SC: $\widehat{\Delta}(k)~\widehat{d}s\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{1}$
Again we use a similar set of equations as Ec. 12–13, with s = (sin χ_{1} + ι sin χ_{2}), we find for t → 1 C = -0.0356236 and ρ = 0.2744495(ln t) while for t → 0 C = 0.066 and ρ_{0} = 0.1920 and h(0) = 0.6997. Both h(t) and C(t) are evaluated numerically and shown in Fig. 3a) and 3b).
Chiral f'-wave SC: $\widehat{\Delta}(k)~\widehat{d}s\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}$
Now we find for t → 1 C = -0.05 and ρ = -0.2910(ln t), while for t → 0 C = -0.1019 and ρ_{0} = 0.2090.
We have shown again h(t) and C(t) in Fig. 3a) and 3b) respectively.
Summary of results. Here ${\rho}_{0}(0)=\frac{{\widehat{v}}^{2}e{H}_{c2}(0)}{2{(2\pi {T}_{c})}^{2}}$ and $h(0)=\frac{{H}_{c2}(0)}{\frac{\partial {H}_{c2}(t)}{\partial t}{|}_{t}=1}$
symmetry | C(0) | C(1) | $-\frac{\partial \rho}{\partial t}{|}_{t}=1$ | ρ_{0}(0) | h(0) | |
---|---|---|---|---|---|---|
$\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$ | p-wave | -0.031 | 0 | 0.2377 | 0.1583 | 0.6659 |
chiral p-wave | -0.2247 | -0.3660 | 0.3838 | 0.2734 | 0.71324 | |
chiral f'-wave | -0.2247 | -0.3660 | 0.5181 | 0.3734 | 0.72073 | |
$\overrightarrow{H}\left|\right|\overrightarrow{a}$ | p-wave | -0.017 | 0 | 0.2377 | 0.1751 | 0,7366 |
chiral p-wave | -0.066 | -0.028 | 0.2126 | 0.1396 | 0,6566 | |
chiral f-wave | -0.066 | -0.035 | 0.2744 | 0.1920 | 0.6997 | |
chiral f'-wave | -0.1019 | -0.05 | 0.2910 | 0.2090 | 0,7182 |
Nodal lines in Δ($\overrightarrow{k}$)
We have seen that from the temperature dependence of H_{c2}(T), we can deduce the chiral f-wave and chiral f'-wave superconductors are the most favourable candidates. They have nodal lines on the Fermi surface (i.e. the χ_{1} - χ_{2} plane), the chiral f-wave SC at χ_{1} = $\pm \frac{\pi}{2}$, while chiral f'-wave SC at χ_{2} = $\pm \frac{\pi}{2}$.
These nodal lines may be detected if the nuclear spin relaxation rate is measured in a magnetic field rotated within the b' - c* plane.
for the chiral f-wave SC.
Conclusion
We have computed the upper critical field of Bechgaard salts for a variety of nodal superconductors with the standard microscopic theory. The results are shown in Fig. 2 and 3. We find:
a) Assuming all these superconductors have the same T_{ c }, the chiral f'-wave SC $(\overrightarrow{\Delta}(k)~\left(\frac{1}{\sqrt{2}}sgn({K}_{a})+i\mathrm{sin}\phantom{\rule{0.5em}{0ex}}{\chi}_{2}\right)\mathrm{cos}\phantom{\rule{0.5em}{0ex}}{\chi}_{2})$ appears to be the most favourable with largest H_{c2}'s for both $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$ and $\overrightarrow{H}\left|\right|\overrightarrow{a}$.
b) However, non of these states exhibit the quasilinear temperature dependence of H_{c2}(T) as observed in [3].
c) Also the present theory predicts H_{c2}(0) ~ (v_{ a }v_{ c })^{-1} and (v_{ b }v_{ c })^{-1} for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$ and $\overrightarrow{H}\left|\right|\overrightarrow{a}$ respectively. This means H_{c2}(0) for $\overrightarrow{H}\left|\right|\overrightarrow{a}$ is about 5 time larger than the one for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime}$ contrary to observation.
d) From H_{c2}(0) ~ 5T and T_{ c }= 1.5 K we can extract v^{2} = ${v}^{2}=\sqrt{{v}_{a}{v}_{c}}~{1.510}^{4}$ ~ 1.510^{4} cm s^{-1}, consistent with the known values of v_{ a }, v_{ c }.
We have also shown that the nodal lines should be visible through the angle dependent ${T}_{1}^{-1}$ in NMR with the magnetic field rotating in the c*-b' plane.
Declarations
Acknowledgements
We thank S. Brown, P. Chaikin, S. Haas and H. Won for useful discussion. ADF also acknowledges gratefully the discussion with J. Ferrer and F. Guinea. The authors would also like to aknowledge the useful comments of the reviewers during the correction process.
Authors’ Affiliations
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