Upper critical field H_c2 in Bechgaard salts (TMTSF)₂PF₆

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)₂PF₆. 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 PF₆ 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)₂PF₆ and (TMTSF)₂ClO₄ are clearly beyond the Pauli limit [4–7], suggesting triplet pairing. Recent NMR data [8, 9] from (TMTSF)₂PF₆ 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

In the following we shall examine the upper critical field of these superconductors following the standard method initiated by Gor'kov [10] and extended by Luk'yanchuk and Mineev [11] for unconventional superconductors. Also we take the quasiparticle energy in the normal state as in the standard model for Bechgaard salts [2]

$$\xi (k)=v(|k_a|-k_F)-2t_b\cos (\overrightarrow{b}\overrightarrow{k})-2t_c\cos (\overrightarrow{c}\overrightarrow{k})$$

(1)

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)₂PF₆ and (TMTSF)₂ClO₄ 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_c2s. 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 $\Delta (\overrightarrow{k})~\left(\frac{1}{\sqrt{2}}sgn(k_a)+i\sin {\chi }_2\right)\cos {\chi }_2$, looks most promising, where χ₁ = $\overrightarrow{b}\overrightarrow{k}$ and χ₂ = $\overrightarrow{c}\overrightarrow{k}$ 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.

Before proceeding, we show |(Δ($\overrightarrow{k}$)| of two chiral f-wave superconductors in Fig. 1a) and 1b).

Figure 1

Sketch of the order parameters. |Δ($\overrightarrow{k}$)| of chiral f-wave and chiral f'-wave SC are sketched in a) and b) respectively, where $|\Delta (k)|~{\left[(1+\cos 2{\chi }_1)(1-\frac{1}{2}\cos 2{\chi }_2)\right]}^{\frac{1}{2}}$ and $|\Delta (k)|~{\left[(1+\cos 2{\chi }_2)(1-\frac{1}{2}\cos 2{\chi }_2)\right]}^{\frac{1}{2}}$ for chiral f and chiral f'.

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 Sr₂RuO₄ [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 UPt₃ and Sr₂RuO₄ 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: $\overrightarrow{\Delta }(k)~sgn(k_a)$

Following [17, 18] the upper critical field is determined by

$$-\ln t={\displaystyle {\int }_0^{\infty }\frac{du}{\sinh u}(1-K_1)}$$

(2)

$$-C\ln t={\displaystyle {\int }_0^{\infty }\frac{du}{\sinh u}(C-K_2)}$$

(3)

where

$$K_1=〈e^{-\rho u^2|s{|}^2}\left(1+2C{\rho }^2{u}^4{s}^4\right)〉$$

(4)

$$K_2=〈e^{-\rho u^2|s{|}^2}\left(\frac{1}{6}{\rho }^2{u}^4{s}^{\ast 4}+C\left(1-8\rho u^2|s{|}^2+12{\rho }^2{u}^q|s{|}^4-\frac{16}{3}{\rho }^3{u}^6|s{|}^6+\frac{2}{3}{\rho }^4{u}^8|s{|}^8\right)\right)〉$$

(5)

and $t=\frac{T}{T_c},\rho =\frac{v_a{v}_{c}e{H}_{c2}(T)}{2{(2\pi T)}^2},s=\left(\frac{1}{\sqrt{2}}sgn(k_a)+i\sin {\chi }_2\right),{\chi }_2=\overrightarrow{c}\overrightarrow{k}$, and ⟨...⟩ means average over χ₂. Here v _a , v _c are the Fermi velocities parallel to the a axis and the c axis respectively.

Here we assumed that Δ($\overrightarrow{r}$) is given by [17, 18]:

$$\Delta (\overrightarrow{r})~\left(1+C{(a^{+})}^4\right)〉$$

(6)

where $〉={\displaystyle \sum C_n{e}^{-eB{x}^2-nk(x+iz)-\frac{{(nk)}^2}{4eB}}}$ is the Abrikosov state [19], C _n the occupancy of the n^thLandau 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 $\rho =\frac{2}{7\zeta (3)}(-\ln t)=0.237697(-\ln t)$ and $C=\frac{93\zeta (5)}{647\zeta (3)}\rho$.

For t → 0 on the other hand we obtain

$${\rho }_0={\rho }_0(0)=\underset{t\to 0}{\lim }\rho t^2=\frac{v_a{v}_{c}e{H}_{c2}(0)}{2{(2\pi T_c)}^2}=0.1583$$

(7)

and C = -0.031. From these we obtain

$$h(0)=\frac{H_{c2}(0)}{\frac{\partial H_{c2}(t)}{\partial t}{|}_{t=1}}=0.6659$$

(8)

Both ρ₀(t) and C(t) are evaluated numerically and shown in Fig. 2a) and b) respectively. Here ρ₀(t) = t²ρ(t) = v _a v _c eH_c2(t)/2(2πT _c )².

Figure 2

Upper critical field for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime }$Normalised H_c2(t) and C(t) for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime }$ are shown in a) and b) respectively. Here black, red and blue lines are chiral f'-wave, chiral p-wave and simple p-wave respectively. Chiral f-wave has the same H_c2(t) as chiral p-wave.

Chiral p-wave SC: $\overrightarrow{\Delta }(k)=1/\sqrt{2}sgn(k_a)+i\sin ({\chi }_2)$

Here $\frac{1}{\sqrt{2}}sgn(k_a)+i\sin ({\chi }_2)$ is the analogue of e^ιϕif in the 3D systems in the quasi 1D system.

For a chiral state the Abrikosov function is written as [20]:

$$\Delta (\overrightarrow{r},\overrightarrow{k})~(s+C{s}^{\ast }{(a^{†})}^2)〉$$

(9)

where $s=\frac{1}{\sqrt{2}}sgn(k_a)+i\sin ({\chi }_2)$. Then we obtain eq. 2–3 with

$$K_1=〈e^{-\rho u^2|s{|}^2}\left(|s{|}^2-2C{s}^4\right)〉$$

(10)

$$K_2=〈e^{-\rho u^2|s{|}^2}\left(-s^4+C|s{|}^2\left(1-4\rho u^2|s{|}^2+2{\rho }^2{u}^4|s{|}^4\right)\right)〉$$

(11)

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 ρ₀ = 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: $\widehat{\Delta }(k)~\widehat{d}s\cos {\chi }_1$

H_c2(t) is determined from eq. 2–3 where now:

$$K_1=〈(1+\cos 2{\chi }_1)e^{-\rho u^2|s{|}^2}\left(|s{|}^2-2\rho u^2{s}^4\right)〉$$

(12)

$$K_2=〈(1+\cos 2{\chi }_1)e^{-\rho u^2|s{|}^2}\left(-\rho u^2{s}^4+C|s{|}^2\left(1-4\rho u^2|s{|}^2+2{\rho }^2{u}^4|s{|}^4\right)\right)〉$$

(13)

Here now $〈\mathrm{...}〉$ means the average over both χ₁ and χ₂. As in previous sections, $s=\frac{1}{\sqrt{2}}sgn(k_a)+i\sin ({\chi }_2)$ (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 χ₁ is readily integrated out.

Chiral f'-wave SC: $\widehat{\Delta }(k)~\widehat{d}s\cos {\chi }_2$

Now we have a set of equations similar to the chiral f-wave except (1 + cos 2χ₁) in both eqs. 13 has to be replaced by $\frac{4}{3}$(1 + cos 2χ₁). 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 ρ₀ = 0.3734.

We show ρ₀ 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: $\Delta (\overrightarrow{k})=sgn(k_a)$

The equation for H_c2(t) is given by [17, 18] and can be written as in eq. 2–3 with:

$$K_1=〈e^{-\rho u^2|s{|}^2}\left(1+2C{\rho }^2{u}^4{s}^4\right)〉$$

(14)

$$K_2=〈e^{-\rho u^2|s{|}^2}\left({\rho }^2{u}^2{s}^4+C\left(1-8\rho u^2|s{|}^2+12{\rho }^2{u}^4|s{|}^4-\frac{16}{3}{\rho }^3{u}^6|s{|}^6+\frac{2}{3}{\rho }^4{u}^8|s{|}^8\right)\right)〉$$

(15)

where $t=\frac{T}{T_c},\rho =\frac{v_b{v}_{c}e{H}_{c2}(t)}{2{(2\pi T)}^2}$ and s = (sin χ₁ + ι sin χ₂) with χ₁ = $\overrightarrow{b}\overrightarrow{k}$ and χ₂ = $\overrightarrow{c}\overrightarrow{k}$.

Then for t → 1, we find $C=-\frac{93\zeta (5)}{508\zeta (3)}\rho$ and $\rho =\frac{2}{7\zeta (3)}(-\ln t)=0.2377(-\ln t)$. 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 α₀ = -⟨ln|s|²⟩ = 0.220051 and ${\beta }_0=-〈\frac{s^4}{|s{|}^4}〉=\frac{4}{\pi }-1=0.0170$. From these we obtain h(0) = 0.73673.

Both h(t) and C(t) are evaluated numerically and we show them in Fig. 3a) and 3b) respectively.

Figure 3

Upper critical field for $\overrightarrow{H}\left|\right|\overrightarrow{a}$ Normalised H_c2(t) and C(t) for $\overrightarrow{H}\left|\right|\overrightarrow{a}$ are shown in a) and b) respectively. Here black, red, blue and green lines are chiral f'-wave, chiral f-wave, chiral p-wave and simple p-wave respectively.

Chiral p-wave SC: $\Delta (k)~\left(\frac{1}{\sqrt{2}}sgn(k_a)+i\sin {\chi }_2\right)$

Now H_c2(t) is determined by a similar set of equations as Ec. 10–11. Now, s = (sin χ₁ + ι sin χ₂). In particular we find for t → 1 C = -0.027735 and ρ = 0.212598(ln t) while for t → 0 C = -0.067684 and ρ₀ = 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: $\widehat{\Delta }(k)~\widehat{d}s\cos {\chi }_1$

Again we use a similar set of equations as Ec. 12–13, with s = (sin χ₁ + ι sin χ₂), we find for t → 1 C = -0.0356236 and ρ = 0.2744495(ln t) while for t → 0 C = 0.066 and ρ₀ = 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: $\widehat{\Delta }(k)~\widehat{d}s\cos {\chi }_2$

Now we find for t → 1 C = -0.05 and ρ = -0.2910(ln t), while for t → 0 C = -0.1019 and ρ₀ = 0.2090.

We have shown again h(t) and C(t) in Fig. 3a) and 3b) respectively.

Comparing these results with H_c2(T) from (TMTSF)₂PF₆ and (TMTSF)₂ClO₄ [4, 5], we can conclude that for both $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime }$ and $\overrightarrow{H}\left|\right|\overrightarrow{a}$, the chiral f'-wave SC is most consistent with experimental data. In particular these states have relatively large h(0) (see Table 1). On the other hand almost the same H_c2(0) for $\overrightarrow{H}\left|\right|{\overrightarrow{b}}^{\prime }$ and $\overrightarrow{H}\left|\right|\overrightarrow{a}$ has to be still accounted.

Table 1 Summary of results. Here ${\rho }_0(0)=\frac{{\widehat{v}}^{2}e{H}_{c2}(0)}{2{(2\pi T_c)}^2}$ and $h(0)=\frac{H_{c2}(0)}{\frac{\partial H_{c2}(t)}{\partial t}{|}_t=1}$

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 χ₁ - χ₂ plane), the chiral f-wave SC at χ₁ = $\pm \frac{\pi }{2}$, while chiral f'-wave SC at χ₂ = $\pm \frac{\pi }{2}$.

These nodal lines may be detected if the nuclear spin relaxation rate is measured in a magnetic field rotated within the b' - c* plane.

Following the standard procedure given in [21], the quasiparticle density of states in the vortex state for T <<T _c and E = 0 is given by

$$N\left(0,\overrightarrow{H}\right)=\frac{2}{{\pi }^2}{v}^2\sqrt{eH}{\left(1+\cos {\theta }^2\sin {\chi }_{10}^2\right)}^{\frac{1}{2}}$$

(16)

where χ₁₀ is the position of the nodal line (i.e. the angle that defines the line on which Δ(k) = 0). So for the chiral f-wave SC we find χ₁₀ = $\frac{\pi }{2}$ and N (0, $\overrightarrow{H}$) exhibits the simple angular dependence. On the other hand when nodal lines are on the χ₂ axis, the θ dependence will be too small to see. Finally this gives

$$T_1^{-1}(\overrightarrow{H})/T_{1N}^{-1}={\left(\frac{2}{{\pi }^2}\right)}^2{\overrightarrow{v}}^2(eH)\left(1+\cos {\theta }^2\right)$$

(17)

for the chiral f-wave SC.

We show the θ dependence of $T_1^{-1}$ in Fig. 4 for a few candidates. The chiral f-wave SC has the strongest θ dependence (solid line) while the chiral h-wave SC (dashed line) and the chiral p-wave SC (dotted line) have a similar θ dependence.

Figure 4

Nuclear spin relaxation rate. The angle dependent nuclear spin relaxation rate for a few nodal superconductors is shown. Chiral f-wave, chiral h-wave and chiral p-wave are represented in red, blue and black lines respectively. If the nodes lie parallel to χ₁, then it is invisible to NMR, so chiral f'-wave gives the same results as chiral f-wave.

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 $(\overrightarrow{\Delta }(k)~\left(\frac{1}{\sqrt{2}}sgn(K_a)+i\sin {\chi }_2\right)\cos {\chi }_2)$ 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 v² = $v^2=\sqrt{v_a{v}_c}~{1.510}^4$ ~ 1.510⁴ 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.