Chiral Triplet Quasi-Two-Dimensional Superconductor in a Parallel Magnetic Field

We calculate the parallel upper critical magnetic field ${{H}_{\parallel }}(0)$ for an in-plane isotropic quasi-two-dimensional chiral triplet superconductor at zero temperature, $T = 0$. In particular, the ratio ${{H}_{\parallel }}(0){\text{/}}\left( {{\text{|}}dH_{\parallel }^{{GL}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{c}}}}}{{T}_{c}}} \right)$ = 0.815 is defined, where ${\text{|}}dH_{\parallel }^{{GL}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{c}}}}}$ is the so-called Ginzburg–Landau slope of the upper critical magnetic field, T_c is a superconducting transition temperature at $H = 0$. We show that the theoretically obtained above mentioned value strongly contradicts to the experimentally measured ones in a candidate for a chiral triplet superconductivity Sr₂RuO₄, which provides one more argument against the chiral triplet scenario of superconductivity in this compound. Our results may be useful for establishing chiral triplet superconductivity in other quasi-two-dimensional candidates for this phenomenon.

Since discovery of superconductivity in the quasi-two-dimensional (Q2D) conductor Sr₂RuO₄ [1], it has been intensively investigated for more than 25 years (for reviews, see [2, 3]). Some analogy of this Q2D superconductor with the superfluid ³He was recognized from the beginning and the existence of a chiral triplet superconducting phase in the Sr₂RuO₄ was suggested [4]. This scenario of superconducting pairing was supported by the observations of no change of the Knight shift between normal and superconducting phases [5, 6] and breaking of the time reversal symmetry in the superconducting phase [7, 8]. On the other hand, there were arguments against the chiral triplet superconductivity scenario, which were almost ignored that time by scientific community. One of the first argument was the paramagnetic limitation of the parallel upper critical magnetic field in Sr₂RuO₄ [9, 10]. In addition, the predicted in the chiral triplet scenario edge currents were not found in the Sr₂RuO₄ [11, 12] but were found zeros of superconducting gap on Q2D Fermi surface (FS) [13, 14], which is against the fully gaped chiral triplet scenario [4]. Recently, the strongest experimental argument against the triplet scenario of superconductivity in Sr₂RuO₄ was published [15], where strong drop of the Knight shift in superconducting state of the above-mentioned material was experimentally discovered.

As seen from the above discussion, the situation with the chiral triplet scenario of superconductivity in  Sr₂RuO₄ is still rather controversial. The goal of this work is two-fold. First, we improve and make our pioneering argument [9] in favor of singlet superconductivity in Sr₂RuO₄ to be firm. The point is that  in [9] (see also recent work [16]) we calculated the ratio ${{H}_{\parallel }}(0){\text{/}}\left( {{\text{|}}dH_{\parallel }^{{{\text{GL}}}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{{\text{c}}}}}}}{{T}_{{\text{c}}}}} \right) = 0.75$, where ${\text{|}}dH_{\parallel }^{{{\text{GL}}}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{{\text{c}}}}}}}$ is the so-called Ginzburg–Landau (GL) slope, for s-wave Q2D superconductivity and compared it with the experimental one, 0.45–0.5 [17–19]. To make our argument against the chiral triplet scenario to be firm, below we calculate the above-mentioned ratio exactly for the in-plane isotropic chiral triplet superconductor with d vector order parameter [4, 20],

$$\mathbf{d} = \mathbf{z}{\kern 1pt} {{\Delta }_{0}}({{k}_{x}} \pm i{{k}_{y}}),$$

(1)

and obtain even stronger inconsistency,

$${{H}_{\parallel }}(0) = 0.815{\text{|}}dH_{\parallel }^{{{\text{GL}}}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{{\text{c}}}}}}}{{T}_{{\text{c}}}},$$

(2)

with the experimental values [17–19], where, to the best of our knowledge, Eq. (2) is the first time obtained in this work. The second our goal is to suggest one more test for chiral triplet superconductivity, which may already exist in the slightly in-plane anisotropic Q2D triplet superconductor UTe₂ [21] and, as we hope, will be discovered in some other Q2D compounds in the future.

Note that superconducting phase (1) is not destructed by the Pauli paramagnetic spin-splitting effects in a parallel magnetic field [9, 20, 22]. Moreover, in accordance with [23–28], it has also to restore superconductivity in very high magnetic fields due to quantum effects of the Bragg reflections from boundaries of the Brillouin zone. In this work, we consider the so-called quasiclassical parallel orbital upper critical magnetic field [16, 29, 30], ${{H}_{\parallel }}(0)$, which destroys superconductivity in Q2D superconductors at low temperatures in an intermediate region of the fields.

Let us consider a layered superconductor with the following in-plane isotropic Q2D electron spectrum:

$$\epsilon (\mathbf{p}) = \epsilon ({{p}_{x}},{{p}_{y}}) - 2{{t}_{ \bot }}\cos ({{p}_{z}}c\text{*}),\quad {{t}_{ \bot }} \ll {{\epsilon }_{{\text{F}}}},$$

(3)

where

$$\epsilon ({{p}_{x}},{{p}_{y}}) = \frac{{(p_{x}^{2} + p_{y}^{2})}}{{2m}},\quad {{\epsilon }_{{\text{F}}}} = \frac{{p_{{\text{F}}}^{2}}}{{2m}}.$$

(4)

[In Eqs. (3) and (4), ${{t}_{ \bot }}$ is the overlapping integral of electron wavefunctions in a perpendicular to the conducting planes direction, m is the in-plane electron mass, ${{\epsilon }_{{\text{F}}}}$ and p_F are the Fermi energy and momentum, respectively; $\hbar \equiv 1$.] In a parallel magnetic field, which is applied along x axis,

$$\mathbf{H} = (H,0,0),$$

(5)

it is convenient to choose the vector potential of the field in the form

$$\mathbf{A} = (0,0,Hy).$$

(6)

For electron motion within the conducting planes and between the planes, we make use of the so-called Peierls substitution method [22] in the electron energy spectrum (3) and (4):

$${{p}_{x}} \to - i\left( {\frac{\partial }{{\partial x}}} \right),\quad {{p}_{y}} \to - i\left( {\frac{\partial }{{\partial y}}} \right),$$

(7)

and

$${{p}_{z}}c\text{*} \to {{p}_{z}}c\text{*} - \left( {\frac{{{{\omega }_{c}}}}{{{{v}_{{\text{F}}}}}}} \right)y,\quad {{\omega }_{{\text{c}}}} = \frac{{e{{v}_{{\text{F}}}}c{\text{*}}H}}{c},$$

(8)

where e is the elementary charge and c is the speed of light.

After the Peierls substitutions in a magnetic field (7) and (8), the electron Hamiltonian (3) and (4) becomes:

$$\hat {H} = - \frac{1}{{2m}}\left( {\frac{{{{\partial }^{2}}}}{{\partial {{x}^{2}}}} + \frac{{{{\partial }^{2}}}}{{\partial {{y}^{2}}}}} \right) - 2{{t}_{ \bot }}\cos \left( {{{p}_{z}}c{\text{*}} - \frac{{{{\omega }_{c}}}}{{{{v}_{F}}}}y} \right).$$

(9)

If we take into account that ω_c ≪ ${{\epsilon }_{F}}$, we can define the  following electron wavefunctions in the mixed, $({{p}_{x}},y,{{p}_{z}})$, representation:

$$\begin{gathered} \Psi _{\epsilon }^{ \pm }(x,y,z) = \exp (i{{p}_{x}}x)\exp [ \pm ip_{y}^{0}({{p}_{x}})y]\exp (i{{p}_{z}}z) \\ \times \;\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}),\quad p_{y}^{0}({{p}_{x}}) = \sqrt {p_{F}^{2} - p_{x}^{2}} , \\ \end{gathered} $$

(10)

where $\epsilon$ is the energy measured from the Fermi energy ${{\epsilon }_{F}}$. Note that Eq. (9) is very general. For instance, it takes into account the quantum effects for open electron orbits in a magnetic field—the Bragg reflections from boundaries of the Brillouin zones (see [23–28]). As we already mentioned, below we calculate the so-called quasiclassical upper critical magnetic field [16, 29, 30] and, thus, do not take into account possible stabilization of superconductors in very high magnetic fields [23–28], which is possibly observed in the Q2D superconductor UTe₂.

It is easy to prove that, in this case, we can represent the electron Hamiltonian (9) for the wavefunctions $\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}})$ in Eq. (10) as

$$\begin{gathered} \left\{ {\frac{1}{{2m}}\left[ {p_{{\text{F}}}^{2} \pm 2ip_{y}^{0}({{p}_{x}})\frac{d}{{dy}}} \right] - 2{{t}_{ \bot }}\cos \left( {{{p}_{z}}c{\text{*}} - \frac{{{{\omega }_{{\text{c}}}}}}{{{{v}_{{\text{F}}}}}}y} \right)} \right\} \\ \times \;\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}) = (\epsilon + {{\epsilon }_{{\text{F}}}})\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}). \\ \end{gathered} $$

(11)

Note that it is easy to represent Eq. (11) in the following more convenient way:

$$\begin{gathered} \left[ { \pm iv_{y}^{0}({{p}_{x}})\frac{d}{{dy}} - 2{{t}_{ \bot }}\cos \left( {{{p}_{z}}c{\text{*}} - \frac{{{{\omega }_{{\text{c}}}}}}{{{{v}_{{\text{F}}}}}}y} \right)} \right]\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}) \\ = \epsilon \psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}),\quad v_{y}^{0}({{p}_{x}}) = p_{y}^{0}({{p}_{x}}){\text{/}}m. \\ \end{gathered} $$

(12)

We point out that Eq. (12) is still too general. For instance, for a pure case, it contains the above discussed quantum effects of an electron motion in a magnetic field. As mentioned before, below, we study the quasiclassical case, where, as shown in [16] and [30], it is possible to take into account only linear term with respect to magnetic field. As a result, we obtain:

$$\begin{gathered} \left[ { \pm iv_{y}^{0}({{p}_{x}})\frac{d}{{dy}} - 2{{t}_{ \bot }}\cos ({{p}_{z}}c\text{*}) - \left( {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}y}}{{{{v}_{{\text{F}}}}}}} \right)\sin ({{p}_{z}}c\text{*})} \right] \\ \times \;\psi _{\epsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}) = \epsilon \psi _{\varepsilon }^{ \pm }({{p}_{x}},y,{{p}_{z}}). \\ \end{gathered} $$

(13)

In this work, we use the so-called Matsubara Green’s functions ${{G}^{ \pm }}(i{{\omega }_{n}};{{p}_{x}};y,{{y}_{1}};{{p}_{z}})$ [31] for interacting electrons in the mixed $({{p}_{x}},y,{{p}_{z}})$ representation. For noninteracting electrons in the magnetic field, equation for Matsubara Green’s function can be written as [32]

$$\left[ {i{{\omega }_{n}} \mp iv_{y}^{0}({{p}_{x}})\frac{d}{{dy}}} \right. + 2{{t}_{ \bot }}\cos ({{p}_{z}}c\text{*})$$

$$ + \;\left. {\left( {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}y}}{{{{v}_{{\text{F}}}}}}} \right)\sin ({{p}_{z}}c\text{*})} \right]$$

(14)

$$ \times \;{{G}^{ \pm }}(i{{\omega }_{n}};{{p}_{x}};y,{{y}_{1}};{{p}_{z}}) = \delta (y - {{y}_{1}}),$$

where ${{\omega }_{n}}$ is the so-called Matsubara frequency [31]. It is important that Eq. (14) can be solved analytically [16, 25]. As a result, we obtain:

$${{G}^{ \pm }}(i{{\omega }_{n}};{{p}_{x}};y,{{y}_{1}};{{p}_{z}}) = - i\frac{{{\text{sgn}}({{\omega }_{n}})}}{{v_{y}^{0}({{p}_{x}})}}\exp \left[ { \pm \frac{{{{\omega }_{n}}(y - {{y}_{1}})}}{{v_{y}^{0}({{p}_{x}})}}} \right]$$

$$ \times \;\exp \left[ { \mp i\frac{{2{{t}_{ \bot }}\cos ({{p}_{z}}c\text{*})(y - {{y}_{1}})}}{{v_{y}^{0}({{p}_{x}})}}} \right]$$

(15)

$$ \times \;\exp \left[ { \mp i\frac{{{{t}_{ \bot }}{{\omega }_{{\text{c}}}}({{y}^{2}} - y_{1}^{2})}}{{v_{y}^{0}({{p}_{x}}){{v}_{{\text{F}}}}}}\sin ({{p}_{z}}c\text{*})} \right].$$

Let us consider the so-called Gor’kov’s gap equation for the two-component order parameter (see, e.g., [33]),

$$\Delta (\mathbf{k},\mathbf{q}) = {{\Delta }_{x}}(\mathbf{q}){{\delta }_{x}}(\mathbf{k}) + {{\Delta }_{y}}(\mathbf{q}){{\delta }_{y}}(\mathbf{k}),$$

(16)

where $[{{\phi }_{x}}(\mathbf{k}),{{\phi }_{y}}(\mathbf{k})]$ are basis functions of two-dimensional representation of tetragonal group transforming as $({{k}_{x}},{{k}_{y}})$. The Gor’kov’s equation in this case can be written as [33]:

$$\begin{gathered} \Delta (\mathbf{k},\mathbf{q}) = T\sum\limits_n \int \frac{{{{d}^{3}}{{\mathbf{k}}_{1}}}}{{{{{(2\pi )}}^{3}}}}V\sum\limits_{i = x,y} {{\delta }_{i}}(\mathbf{k}){{\delta }_{i}}({{\mathbf{k}}_{1}}) \\ \times \;{{G}^{ \pm }}(i{{\omega }_{n}};{{\mathbf{k}}_{1}})\;{{G}^{ \mp }}( - i{{\omega }_{n}}; - {{\mathbf{k}}_{1}} + \mathbf{q})\Delta ({{\mathbf{k}}_{1}},\mathbf{q}). \\ \end{gathered} $$

(17)

Using Fourier transformations of the known Green’s functions (15), it is possible to derive the so-called gap equation in the mixed representation, determining the upper critical magnetic field for the two-component order parameter (16). Below, we use the Q2D Fermi surface (3) with isotropic in-plane electron 2D spectrum (4) and the following isotropic chiral triplet electron–electron superconducting interactions:

$$\begin{gathered} V\sum\limits_{i = x,y} {{\delta }_{i}}(\mathbf{k}){{\delta }_{i}}({{\mathbf{k}}_{1}}) \\ = 2g[\cos \phi \cos {{\phi }_{1}} + \sin \phi \sin {{\phi }_{1}}] = 2g\cos (\phi - {{\phi }_{1}}). \\ \end{gathered} $$

(18)

As a result of lengthy but straightforward calculations, we obtain the Gor’kov’s gap equation in the form

$$\Delta (\phi ,y) = \int\limits_0^{2\pi } {\frac{{d{{\phi }_{1}}}}{{2\pi }}} g\cos (\phi - {{\phi }_{1}})\int\limits_{|y - {{y}_{1}}| > d|\sin {{\phi }_{1}}|}^\infty {} $$

$$ \times \;\frac{{2\pi Td{{y}_{1}}}}{{{{v}_{F}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}\sinh \left( {\frac{{2\pi T{\text{|}}y - {{y}_{1}}{\text{|}}}}{{{{v}_{{\text{F}}}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}} \right)}}$$

(19)

$$ \times \;{{J}_{0}}\left[ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}({{y}^{2}} - y_{1}^{2})} \right]\Delta ({{\phi }_{1}},{{y}_{1}}),$$

where g is the effective electron coupling constant, d is the cut-off distance, ${{J}_{0}}(...)$ is the zero order Bessel function. In Eq. (19) the superconducting gap $\Delta (\phi ,y)$ depends on a center of mass of the BCS pair, y, and on the position on the cylindrical FS (4), where ϕ and ${{\phi }_{1}}$ are the polar angles measured from the x axis.

Using the general Eq. (16), we need to seek two solutions of Eq. (19) in the form

$$\Delta (\phi ,y) = {{\Delta }_{x}}(y)\cos (\phi ) + {{\Delta }_{y}}(y)\sin (\phi ),$$

(20)

for given direction of a magnetic field. Since we consider the isotropic 2D FS (4) and the isotropic chiral triplet electron-electron interactions (18), it is possible to make sure that the in-plane upper critical magnetic field does not depend on direction of the field. Therefore, we can consider magnetic field applied along x axis (5), as we suggested before. For such magnetic field, it is easy to show that there are the following two solutions:

$${{\Delta }_{1}}(\phi ,y) = {{\Delta }_{x}}(y)\cos (\phi ),$$

(21)

and

$${{\Delta }_{2}}(\phi ,y) = {{\Delta }_{y}}(y)\sin (\phi ),$$

(22)

which obey the following integral equations:

$${{\Delta }_{x}}(y) = \int\limits_0^{2\pi } {\frac{{d{{\phi }_{1}}}}{{2\pi }}} g{{\cos }^{2}}({{\phi }_{1}})\int\limits_{|y - {{y}_{1}}| > d|\sin {{\phi }_{1}}|}^\infty {} $$

$$ \times \;\frac{{2\pi Td{{y}_{1}}}}{{{{v}_{{\text{F}}}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}\sinh \left( {\frac{{2\pi T{\text{|}}y - {{y}_{1}}{\text{|}}}}{{{{v}_{{\text{F}}}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}} \right)}}$$

(23)

$$ \times \;{{J}_{0}}\left[ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}({{y}^{2}} - y_{1}^{2})} \right]{{\Delta }_{x}}({{y}_{1}}),$$

and

$${{\Delta }_{y}}(y) = \int\limits_0^{2\pi } {\frac{{d{{\phi }_{1}}}}{{2\pi }}} g\mathop {\sin }\nolimits^2 ({{\phi }_{1}})\int\limits_{|y - {{y}_{1}}| > d|\sin {{\phi }_{1}}|}^\infty {} $$

$$ \times \;\frac{{2\pi Td{{y}_{1}}}}{{{{v}_{{\text{F}}}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}\sinh \left( {\frac{{2\pi T{\text{|}}y - {{y}_{1}}{\text{|}}}}{{{{v}_{{\text{F}}}}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}} \right)}}$$

(24)

$$ \times \;{{J}_{0}}\left[ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}{\text{|}}\sin {{\phi }_{1}}{\text{|}}}}({{y}^{2}} - y_{1}^{2})} \right]{{\Delta }_{y}}({{y}_{1}}),$$

respectively. Note that the solutions (21) and (22) degenerate in the absence of a magnetic field. In the presence of the magnetic field (5) with the vector potential (6), Eqs. (23) and (24) are not degenerated anymore and correspond to different solutions, ${{\Delta }_{x}}(y) \ne {{\Delta }_{y}}(y)$, with different upper critical magnetic fields, $H_{\parallel }^{x} \ne H_{\parallel }^{y}$. The actual upper critical field will be the maximum value of them. Therefore, below, we need to solve and study the both integral equations, (23) and (24).

Here, we introduce more convenient variable of the integrations,

$${{y}_{1}} - y = z{\text{|}}\sin {{\phi }_{1}}{\text{|}},$$

(25)

and rewrite the integral equations (23) and (24) in the following ways:

$${{\Delta }_{x}}(y) = \int\limits_0^{2\pi } {\frac{{d{{\phi }_{1}}}}{{2\pi }}} g\mathop {\cos }\nolimits^2 ({{\phi }_{1}})\int\limits_{|z| > d}^\infty {\frac{{2\pi Tdz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi T{\text{|}}z{\text{|}}}}{{{{v}_{{\text{F}}}}}}} \right)}}} $$

$$ \times \;{{J}_{0}}\left[ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}}}[z(2y + z{\text{|}}\sin {{\phi }_{1}}{\text{|}})]} \right]$$

(26)

$$ \times \;{{\Delta }_{x}}(y + z{\text{|}}\sin {{\phi }_{1}}{\text{|}}),$$

and

$${{\Delta }_{y}}(y) = \int\limits_0^{2\pi } {\frac{{d{{\phi }_{1}}}}{{2\pi }}} g{{\sin }^{2}}({{\phi }_{1}})\int\limits_{|z| > d}^\infty {\frac{{2\pi Tdz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi T{\text{|}}z{\text{|}}}}{{{{v}_{{\text{F}}}}}}} \right)}}} $$

$$ \times \;{{J}_{0}}\left[ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}}}[z(2y + z{\text{|}}\sin {{\phi }_{1}}{\text{|}})]} \right]$$

(27)

$$ \times \;{{\Delta }_{y}}(y + z{\text{|}}\sin {{\phi }_{1}}{\text{|}}).$$

Let us first study Eq. (26). Here, we derive the GL equation for (26) to determine the so-called GL slope of the upper critical magnetic field, ${\text{|}}dH_{\parallel }^{{x({\text{GL}})}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{{\text{c}}}}}}}$. In order to derive the GL equation, we expand the superconducting gap and the Bessel function in Eq. (26) with respect to small parameter, ${\text{|}}z{\text{|}} \ll {{v}_{{\text{F}}}}{\text{/}}(\pi {{T}_{{\text{c}}}})$:

$$\begin{gathered} {{\Delta }_{x}}(y + z{\text{|}}\sin {{\phi }_{1}}{\text{|}}) \approx {{\Delta }_{x}}(y) + \frac{1}{2}{{z}^{2}}{{\sin }^{2}}{{\phi }_{1}}\left[ {\frac{{{{d}^{2}}{{\Delta }_{x}}(y)}}{{d{{y}^{2}}}}} \right], \\ {{J}_{0}}\left\{ {\frac{{2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}}}{{v_{{\text{F}}}^{2}}}[z(z{\text{|}}\sin {{\phi }_{1}}{\text{|}} + 2y)]} \right\} \approx 1 - \frac{{4t_{ \bot }^{2}\omega _{{\text{c}}}^{2}}}{{v_{{\text{F}}}^{4}}}{{z}^{2}}{{y}^{2}}. \\ \end{gathered} $$

(28)

Next step is to substitute the obtained expansions (28) into the integral gap equation (26) and average over the polar angle ${{\phi }_{1}}$:

$$ - \frac{1}{8}\left[ {\frac{{{{d}^{2}}{{\Delta }_{x}}(y)}}{{d{{y}^{2}}}}} \right]\int\limits_0^\infty {\frac{{2\pi {{T}_{{\text{c}}}}{{z}^{2}}dz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi {{T}_{{\text{c}}}}z}}{{{{v}_{{\text{F}}}}}}} \right)}}} $$

$$ + \;{{y}^{2}}{{\Delta }_{x}}(y)\frac{{4t_{ \bot }^{2}\omega _{{\text{c}}}^{2}}}{{v_{{\text{F}}}^{4}}}\int\limits_0^\infty {\frac{{2\pi {{T}_{{\text{c}}}}{{z}^{2}}dz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi {{T}_{{\text{c}}}}z}}{{{{v}_{{\text{F}}}}}}} \right)}}} $$

(29)

$$ + \;{{\Delta }_{x}}(y)\left[ {\frac{1}{g} - \int\limits_d^\infty {\frac{{2\pi Tdz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi Tz}}{{{{v}_{{\text{F}}}}}}} \right)}}} } \right] = 0.$$

Note that in the absence of a magnetic field the superconducting transition temperature T_c obeys the equation

$$\frac{1}{g} = \int\limits_d^\infty {\frac{{2\pi {{T}_{{\text{c}}}}dz}}{{{{v}_{{\text{F}}}}\sinh \left( {\frac{{2\pi {{T}_{{\text{c}}}}z}}{{{{v}_{{\text{F}}}}}}} \right)}}} .$$

(30)

After substitution of Eq. (30) into Eq. (29), we obtain the differential equation

$$ - \xi _{\parallel }^{2}\left[ {\frac{{{{d}^{2}}{{\Delta }_{x}}(y)}}{{d{{y}^{2}}}}} \right] + {{\left( {\frac{{2\pi H}}{{{{\phi }_{0}}}}} \right)}^{2}}\xi _{ \bot }^{2}{{y}^{2}}{{\Delta }_{x}}(y) - \tau {{\Delta }_{x}}(y) = 0,$$

(31)

which is called the GL one. During the derivation of the GL equation, determining the upper critical magnetic field slope near T_c, $\tau = ({{T}_{{\text{c}}}} - T){\text{/}}{{T}_{{\text{c}}}} \ll 1$, we introduce the so-called parallel and perpendicular GL coherent lengths:

$${{\xi }_{\parallel }} = \frac{{\sqrt {7\zeta (3)} {{v}_{{\text{F}}}}}}{{8\pi {{T}_{{\text{c}}}}}}, \quad {{\xi }_{ \bot }} = \frac{{\sqrt {7\zeta (3)} {{t}_{ \bot }}c{\text{*}}}}{{2\sqrt 2 \pi {{T}_{{\text{c}}}}}}.$$

(32)

Note that $\zeta (x)$ is the so-called Riemann function [34],

$$\int\limits_0^\infty {\frac{{{{z}^{2}}dz}}{{\sinh (z)}}} = \frac{7}{3}\zeta (3),$$

(33)

whereas ${{\phi }_{0}} = \frac{{\pi c}}{e}$ is the magnetic flux quantum. It is important that ${{\xi }_{\parallel }}$ in Eqs. (31) and (32) is different from that in [16, 30], since in this work we consider two-component superconducting gap. To obtain the parallel upper critical field for Eq. (26) near T_c, it is necessary to solve the Schrödinger-like equation (31) and to find its lowest energy level. Since Eq. (31) is mathematically equivalent to the Schrödinger equation for a harmonic oscillator, we obtain the following equation for the parallel upper critical field in the GL region, $\tau \ll 1$:

$$H_{\parallel }^{{x({\text{GL}})}}(T) = \left( {\frac{{{{\phi }_{0}}}}{{2\pi {{\xi }_{\parallel }}{{\xi }_{ \bot }}}}} \right)\tau = \left[ {\frac{{8\sqrt 2 {{\pi }^{2}}cT_{{\text{c}}}^{2}}}{{7\zeta (3)e{{v}_{{\text{F}}}}{{t}_{ \bot }}c{\text{*}}}}} \right]\tau ,$$

(34)

which, as expected, differs from that in [16, 30].

More complicated problem is to solve Eq. (26) at $T = 0$ and, thus, to find the upper critical magnetic field at zero temperature, $H_{\parallel }^{x}(0)$. This is possible to do only by means of numerical calculations. To this end, let us introduce more convenient variables,

$$\tilde {z} = \frac{{\sqrt {2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}} }}{{{{v}_{{\text{F}}}}}}z, \quad \tilde {y} = \frac{{\sqrt {2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}} }}{{{{v}_{{\text{F}}}}}}y,$$

(35)

and rewrite the gap Eq. (26) for the case $T = 0$:

$$\begin{gathered} {{{\tilde {\Delta }}}_{x}}(\tilde {y}) = g\int\limits_{ - \pi /2}^{\pi /2} {\frac{{d{{\phi }_{1}}}}{\pi }} \int\limits_d^\infty {\frac{{d\tilde {z}}}{{\tilde {z}}}} {{J}_{0}}[\tilde {z}(2\tilde {y} + \tilde {z}\sin {{\phi }_{1}})] \\ \times \;2{{\cos }^{2}}({{\phi }_{1}}){{{\tilde {\Delta }}}_{x}}(\tilde {y} + \tilde {z}\sin {{\phi }_{1}}), \\ \end{gathered} $$

(36)

where

$${{\tilde {\Delta }}_{x}}(\tilde {y}) = \Delta \left( {\frac{{{{v}_{{\text{F}}}}\tilde {y}}}{{\sqrt {2{{t}_{ \bot }}{{\omega }_{{\text{c}}}}} }}} \right).$$

(37)

Here, we summarize procedure of the numerical solution of the integral Eq. (36) and obtain the following new result:

$$H_{\parallel }^{x}(0) = 10.78\frac{{cT_{{\text{c}}}^{2}}}{{e{{v}_{{\text{F}}}}{{t}_{ \bot }}c{\text{*}}}}.$$

(38)

Note that solution for the superconducting gap, ${{\tilde {\Delta }}_{x}}(\tilde {y})$ of Eq. (36) is not of an exponential shape and changes its sign several times in space, in contrast to the 3D case [29, 35]. Using Eqs. (34) and (38), we have

$$H_{\parallel }^{x}(0) = 0.815{\text{|}}dH_{\parallel }^{{x({\text{GL}})}}{\text{/}}dT{{{\text{|}}}_{{T = {{T}_{{\text{c}}}}}}}{{T}_{{\text{c}}}}.$$

(39)

Now, let us summarize the investigation of Eq. (24), which is done in the same way as our study of Eq. (26). In particular, it is possible to show that the GL field near transition temperature is equal to

$$H_{\parallel }^{{y({\text{GL}})}}(T) = \left[ {\frac{{8\sqrt 2 {{\pi }^{2}}cT_{{\text{c}}}^{2}}}{{7\zeta (3)\sqrt 3 e{{v}_{{\text{F}}}}{{t}_{ \bot }}c{\text{*}}}}} \right]\tau ,$$

(40)

which is $\sqrt 3$ times smaller than in Eq. (34). Solution of the Eq. (27) at $T = 0$ also gives value of the upper critical field, which is approximately two times less than the upper critical field (38), obtained for Eq. (36):

$$H_{\parallel }^{y}(0) = 0.5 \times H_{\parallel }^{x}(0).$$

(41)

On the above-mentioned grounds, we can conclude that the solutions (21) of Eq. (23) are essential for our calculations of the parallel upper critical magnetic field in the chiral triplet in-plane isotropic Q2D superconductor. Therefore, experimentally measured ratio has to be close to the value (2), (39). It is important that the theory developed by us above cannot be obtained in the framework of the Gor’kov [29] and Werthamer–Helfand–Hohenberg [35] approaches. There are two reasons for that. First, electrons in a Q2D superconductor in a parallel magnetic field move along open trajectories in p space, in contrast to closed electron orbits of [29, 35]. Second, we have two-component order parameter, instead of one-component [29, 35].

As we already mentioned, in the candidate for the chiral triplet in-plane isotropic superconductivity, Sr₂RuO₄, the corresponding experimental coefficients [17–19] are almost two times smaller than the calculated in this work (2), which is a strong argument against the chiral triplet scenario. In this context, it is important that the parallel upper critical magnetic field in the above-mentioned compound is a good measurable quantity, unlike the upper critical magnetic fields in some other Q2D superconductors. In our opinion, the experimental value of the ${{H}_{\parallel }}(0)$ at low temperatures are restricted by the paramagnetic spin-splitting effects, which are always present in the singlet and some triplet superconducting phases. This point of view is also supported by the first order nature of the phase transition in parallel magnetic fields at low temperatures [17–19]. The recent Knight shift experiments [15] also strongly demonstrate the existence of the paramagnetic spin-splitting effects in superconducting phase of the Sr₂RuO₄. In conclusion, let us discuss the physical model we have used for the calculations of the parallel upper critical magnetic field. It is known [36] that in the Sr$_{2}$RuO$_{4}$ perpendicular to the conducting layers coherence length (32) is much larger than the interplane distance. Therefore, we have used Eq. (36), instead of the so-called Lawrence–Doniach model [37, 38]. We have also make use of the fact that metallic phase of the Sr₂RuO₄ is a good Fermi liquid [2, 3].