Self-Consistent Consideration of Fluctuations in Singlet Superconducting Phases with s and d Symmetry

Abstract

We analyze the behavior of thermal fluctuations of the superconducting order parameter with extended s-wave and ${{d}_{{{{x}^{2}} - {{y}^{2}}}}}$-wave symmetry. For this purpose, we develop a method of self-consistent consideration of the order parameter fluctuations and charge carrier scatterers by fluctuations of coupled electron pairs using the theory of functional integration. The study is performed based on the quasi-two-dimensional one-band model with attraction between electrons located at neighboring sites. We obtain the distribution functions of the phase fluctuation probabilities depending on temperature, charge carrier concentrations, and model parameters. It is shown that the phase of the order parameter in the superconducting region is coherent, and the density of states has a dip at the Fermi level. In approaching the incoherent region of the phase diagram, the dip in the density of states disappears simultaneously with the loss of phase coherence. At the same time, the order parameter amplitude averaged over fluctuations remains finite at any temperature and concentration of charge carriers. Our results show that the pseudogap state cannot be explained in the frames of this scenario.

Green’s Functions in TCPA

As noted above, Hamiltonian ${{\mathcal{H}}_{{HF}}}$ (4) of the model considered here in the HF approximation describes the properties of the system disregarding the superconducting OP fluctuations. The consideration fluctuations requires the introduction of fluctuating Δ$\mathcal{U}$ potential (4) and leads to the problem of nondiagonal disorder in disordered systems. This problem can be solved in the given case using the two-site coherent potential approximation (TCPA) [32–34]. In the case considered here, this approximation implies that the scattering electron pair with fluctuating potential Δ$\mathcal{U}$ (4) is inbuilt into the effective medium consisting of electron pairs with effective parameters determined by the self-energy part $\hat {\Sigma }$(iω_n),

$$\begin{gathered} \hat {\Sigma }(i{{\omega }_{n}}) = \sum\limits_{j,\delta }^{} {\hat {c}_{{j\delta }}^{\dag }{{\Sigma }_{{j\delta }}}(i{{\omega }_{n}}){{{\hat {c}}}_{{j\delta }}},} \\ {{\Sigma }_{{j\delta }}}(i{{\omega }_{n}}) = \left[ {\begin{array}{*{20}{c}} {\Sigma _{{j,j}}^{ \uparrow }(i{{\omega }_{n}})}&{\Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(i{{\omega }_{n}})} \\ {\Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(i{{\omega }_{n}})}&{\Sigma _{{j + \delta ,j{\text{ + }}\delta }}^{ \downarrow }(i{{\omega }_{n}})} \end{array}} \right], \\ \end{gathered} $$

$$\begin{gathered} \Sigma _{{j,j}}^{ \uparrow }(i{{\omega }_{n}}) = {{\Sigma }^{ \uparrow }}(i{{\omega }_{n}}){\text{/}}4, \\ \Sigma _{{j + \delta ,j + \delta }}^{ \downarrow }(i{{\omega }_{n}}) = {{\Sigma }^{ \downarrow }}(i{{\omega }_{n}}){\text{/}}4, \\ \end{gathered} $$

(A.1)

$$\begin{gathered} \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(i{{\omega }_{n}}) = {{\Sigma }^{{ \uparrow \downarrow }}}(i{{\omega }_{n}})\exp (i{{\alpha }_{\delta }}), \\ \Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(i{{\omega }_{n}}) = {{\Sigma }^{{ \uparrow \downarrow }}}(i{{\omega }_{n}})\exp ( - i{{\alpha }_{\delta }}), \\ \end{gathered} $$

which has the same functional form as ${{\hat {\mathcal{H}}}_{{HF}}}$ (4) and preserves complete symmetry of the system under investigation. Therefore, temperature Green function G_jδ(iω_n, λ) (10) in this approximation is described by the Dyson equation with fluctuating potential Δ$\mathcal{U}$ (4):

$$\begin{gathered} {{G}_{{j\delta }}}(i{{\omega }_{n}},\delta ) = {{F}_{{j\delta }}}(i{{\omega }_{n}}) \\ \, + {{F}_{{j\delta }}}(i{{\omega }_{n}})[\lambda \Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(i{{\omega }_{n}})]{{G}_{{j\delta }}}(i{{\omega }_{n}},\lambda ) \\ = {{F}_{{j\delta }}}(i{{\omega }_{n}}) + {{F}_{{j\delta }}}(i{{\omega }_{n}}){{T}_{{j\delta }}}(i{{\omega }_{n}}){{F}_{{j\delta }}}(i{{\omega }_{n}}), \\ \end{gathered} $$

(A.2)

where F_jδ(iω_n) is the Fourier transform of the effective temperature Green function and T_jδ(iω_n) is the Fourier transform of the electron pair scattering matrix with fluctuating potential (4). Then self-energy part Σ_jδ(iω_n) can be determined self-consistently from the requirement of the absence of scattering by this electron pair on the average, 〈T_jδ(iω_n, λ)〉 = 0 or 〈G_jδ(iω_n, λ)〉 = F_jδ(iω_n).

Explicit expressions for the matrix elements of Green’s functions in the representation of the Nambu matrices can be obtained using the Fourier transformation of the creation (annihilation) operators $\hat {c}_{{js}}^{\dag }$(${{\hat {c}}_{{js}}}$),

$$\hat {c}_{{js}}^{\dag }({{\hat {c}}_{{js}}}) = \frac{1}{N}\sum\limits_k^{} {\exp [ \mp ik{{R}_{j}}]\hat {c}_{{ks}}^{\dag }({{{\hat {c}}}_{{ks}}}),} $$

(A.3)

where N is the number of sites in the system, R_j are the square lattice vectors, $\hat {c}_{{ks}}^{\dag }$(${{\hat {c}}_{{ks}}}$) are the creation (annihilation) operators for electron with momentum k and spin projection s. As a result of this transformation, Hamiltonian ${{\hat {\mathcal{H}}}_{{HF}}}$ (4) of the given system in the HF approximation can be written in the representation of Nambu matrices

$${{\hat {c}}_{k}} = \left[ \begin{gathered} {{{\hat {c}}}_{{k \uparrow }}} \\ \hat {c}_{{ - k \downarrow }}^{\dag } \\ \end{gathered} \right],\quad \hat {c}_{k}^{\dag } = [c_{{k \uparrow }}^{\dag }\,\,{{\hat {c}}_{{ - k \downarrow }}}]$$

(A.4)

in form

$$\begin{gathered} {{{\hat {\mathcal{H}}}}_{{HF}}}(\bar {\Delta },\alpha ) = \frac{1}{N}\sum\limits_k^{} {\hat {c}_{k}^{\dag }{{\mathcal{H}}_{{HF}}}(k){{{\hat {c}}}_{k}},} \\ {{\mathcal{H}}_{{HF}}}(k) = \left[ {\begin{array}{*{20}{c}} {\mathcal{H}_{{HF}}^{{ \uparrow \uparrow }}(k)}&{\mathcal{H}_{{HF}}^{{ \uparrow \downarrow }}(k)} \\ {\mathcal{H}_{{HF}}^{{ \downarrow \uparrow }}(k)}&{\mathcal{H}_{{HF}}^{{ \downarrow \downarrow }}(k)} \end{array}} \right], \\ \end{gathered} $$

$$\begin{gathered} \mathcal{H}_{{HF}}^{{ \uparrow \uparrow }}(k) = {{\varepsilon }_{k}} - \mu ,\quad \mathcal{H}_{{HF}}^{{ \downarrow \downarrow }}(k) = - {{\varepsilon }_{k}} + \mu , \\ \mathcal{H}_{{HF}}^{{ \uparrow \downarrow }}(k) = - 2V\bar {\Delta }{{V}_{k}}(\alpha ), \\ \mathcal{H}_{{HF}}^{{ \downarrow \uparrow }}(k) = (\mathcal{H}_{{HF}}^{{ \uparrow \downarrow }}(k)){\text{*}}, \\ \end{gathered} $$

(A.5)

$$\begin{gathered} {{\varepsilon }_{k}} = - 2t(\cos {{k}_{x}} + \cos {{k}_{y}}) + 4t'\cos {{k}_{x}}\cos {{k}_{y}}, \\ {{V}_{k}}(\alpha ) = \cos \alpha (\cos {{k}_{x}} + \cos {{k}_{y}}) \\ \, + i\sin \alpha (\cos {{k}_{x}} - \cos {{k}_{y}}), \\ \end{gathered} $$

where ε_k is the dispersion law for the energy of electrons on a square lattice with hops to the nearest and next-to-nearest sites and V_k(α) is the dispersion law for the superconducting OP with the symmetry specified by the value of phase α. Fluctuating potential (4) in this representation can be written as

$$\begin{gathered} \Delta \hat {\mathcal{U}}(\Delta ,\phi ,\tau ) = \frac{1}{N}\sum\limits_k^{} {\hat {c}_{k}^{\dag }(\tau )} \Delta \mathcal{U}(k){{{\hat {c}}}_{k}}(\tau ), \\ \Delta \mathcal{U}(k) = \left[ {\begin{array}{*{20}{c}} 0&{\Delta {{\mathcal{U}}^{{ \uparrow \downarrow }}}(k)} \\ {(\Delta {{\mathcal{U}}^{{ \uparrow \downarrow }}}(k)){\text{*}}}&0 \end{array}} \right], \\ \Delta {{\mathcal{U}}^{{ \uparrow \downarrow }}}(k) = 2V[\bar {\Delta }{{V}_{k}}(\alpha ) - \Delta (\phi ){{V}_{k}}(\phi )]. \\ \end{gathered} $$

(A.6)

The effective medium is described by Hamiltonian

$$\begin{gathered} {{{\hat {\mathcal{H}}}}_{{{\text{eff}}}}}(E) = {{{\hat {\mathcal{H}}}}_{{HF}}} + \hat {\Sigma }(E), \\ \hat {\Sigma }(E) = \frac{1}{N}\sum\limits_k^{} {\hat {c}_{k}^{\dag }{{\Sigma }_{k}}(E){{{\hat {c}}}_{k}},} \\ \end{gathered} $$

$${{\Sigma }_{k}}(E) = \left[ {\begin{array}{*{20}{c}} {{{\Sigma }^{ \uparrow }}(E)}&{\Sigma _{k}^{{ \uparrow \downarrow }}(E)} \\ {\Sigma _{k}^{{ \downarrow \uparrow }}(E)}&{{{\Sigma }^{ \downarrow }}(E)} \end{array}} \right],$$

(A.7)

$$\begin{gathered} \Sigma _{k}^{{ \uparrow \downarrow }}(E) = 2{{\Sigma }^{{ \uparrow \downarrow }}}(E){{V}_{k}}(\alpha ), \\ \Sigma _{k}^{{ \downarrow \uparrow }}(E) = 2{{\Sigma }^{{ \uparrow \downarrow }}}(E)V_{k}^{*}(\alpha ), \\ \end{gathered} $$

where Σ_k(E) is the self-energy part (A.1) in the representation of Nambu matrices (A.4). For considering the OP with the symmetry types under investigation, it is necessary that coefficient Σ^↑↓(E) in nondiagonal matrix elements of self-energy part (A.7) be a real-valued function of energy, while diagonal elements Σ^↑(E) and Σ^↓(E) are generally complex-valued functions. Effective Green function (A.2) in the representation of Nambu matrices (A.4) can be expressed in terms of effective medium Hamiltonian ${{\hat {\mathcal{H}}}_{{{\text{eff}}}}}$(E) (A.7):

$$\begin{gathered} {{F}_{k}}(E) = {{[E - {{\mathcal{H}}_{{{\text{eff}}}}}(k)]}^{{ - 1}}} = \left[ {\begin{array}{*{20}{c}} {F_{k}^{ \uparrow }(E)}&{F_{k}^{{ \uparrow \downarrow }}(E)} \\ {F_{k}^{{ \downarrow \uparrow }}(E)}&{F_{k}^{ \downarrow }(E)} \end{array}} \right], \\ F_{k}^{{ \uparrow ( \downarrow )}}(E) = \frac{{E \pm {{\varepsilon }_{k}} \mp \mu - {{\Sigma }^{{ \downarrow ( \uparrow )}}}(E)}}{{(E - E_{k}^{ + })(E - E_{k}^{ - })}}, \\ \end{gathered} $$

$$F_{k}^{{ \uparrow \downarrow ( \downarrow \uparrow )}}(E) = \frac{{\Sigma _{k}^{{ \uparrow \downarrow ( \downarrow \uparrow )}}(E) + \mathcal{H}_{{HF}}^{{ \uparrow \downarrow ( \downarrow \uparrow )}}(k)}}{{(E - E_{k}^{ + })(E - E_{k}^{ - })}},$$

(A.8)

$$\begin{gathered} E_{k}^{ \pm } = \frac{{{{\Sigma }^{ \uparrow }}(E) + {{\Sigma }^{ \downarrow }}(E)}}{2} \pm \left[ {{{{\left( {{{\varepsilon }_{k}} - \mu + \frac{{{{\Sigma }^{ \uparrow }}(E) + {{\Sigma }^{ \downarrow }}(E)}}{2}} \right)}}^{2}}} \right. \\ {{\left. {_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{_{{}}}}}}}}}}}}}}}}}}}}}}}} + \,{{\Sigma }^{ \uparrow }}(E){{\Sigma }^{ \downarrow }}(E) + \,{\text{|}}\Sigma _{k}^{{ \uparrow \downarrow }}(E) + \mathcal{H}_{{HF}}^{{ \uparrow \downarrow }}(k){{{\text{|}}}^{2}}} \right]}^{{1/2}}}. \\ \end{gathered} $$

The explicit expressions for the matrix elements of the effective Green function in the representation of the Nambu matrices are obtained as a result of the Fourier transformation (A.3):

$$\begin{gathered} F_{{j,j}}^{{ \uparrow \uparrow }}(E) = \frac{1}{N}\sum\limits_k^{} {F_{k}^{ \uparrow }(E} ), \\ F_{{j + \delta ,j + \delta }}^{{ \downarrow \downarrow }}(E) = \frac{1}{N}\sum\limits_k^{} {F_{k}^{ \downarrow }(E} ), \\ \end{gathered} $$

$$\begin{gathered} F_{{j,j + \delta }}^{{ \uparrow \downarrow }}(E) = \frac{1}{N}\sum\limits_k^{} {\exp [ik{{R}_{\delta }}]F_{k}^{{ \uparrow \downarrow }}(E)} \\ = \frac{1}{N}\sum\limits_k^{} {\cos (k{{R}_{\delta }})F_{k}^{{ \uparrow \downarrow }}(E),} \\ \end{gathered} $$

(A.9)

$$\begin{gathered} F_{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E) = \frac{1}{N}\sum\limits_k^{} {\exp [ - ik{{R}_{\delta }}]F_{k}^{{ \downarrow \uparrow }}(E)} \\ = \frac{1}{N}\sum\limits_k^{} {\cos (k{{R}_{\delta }})F_{k}^{{ \downarrow \uparrow }}(E).} \\ \end{gathered} $$

For deriving this expressions, we have used properties F_–k(E) = F_k(E), which follows from the inverse symmetry of the dispersion law of the electron energy and superconducting OP (A.5). The nondiagonal matrix elements of effective Green function (A.8) in the representation of Nambu matrices (A.4) possess the property $F_{{{{k}_{y}},{{k}_{x}}}}^{{ \uparrow \downarrow ( \downarrow \uparrow )}}$(E) = $F_{{{{k}_{y}},{{k}_{x}}}}^{{ \downarrow \uparrow ( \uparrow \downarrow )}}$(E), which is associated with the symmetry of superconducting OP (A.5). Consequently, the nondiagonal matrix elements of the effective Green function in the representation of Nambu matrices have property $F_{{j,j \pm {{\delta }_{y}}}}^{{ \uparrow \downarrow }}$(E) = $F_{{j \pm {{\delta }_{x}},j}}^{{ \downarrow \uparrow }}$(E). The explicit expressions for the matrix elements of the Green function with fluctuating potential G_{j, δ}(E) can be derived from Dyson equation (A.2) as follows:

$$\begin{gathered} G_{{j,j}}^{{ \uparrow \uparrow }}(E) = \frac{{F_{{j,j}}^{{ \uparrow \uparrow }}(E) + {{\Sigma }^{ \downarrow }}(E)\det [{{F}_{{j\delta }}}(E)]}}{{\det [1 - {{F}_{{j\delta }}}(E)(\Delta {{{\hat {\mathcal{U}}}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E))]}}, \\ G_{{j + \delta ,j + \delta }}^{{ \downarrow \downarrow }}(E) = \frac{{F_{{j + \delta ,j + \delta }}^{{ \downarrow \downarrow }}(E) + {{\Sigma }^{ \uparrow }}(E)\det [{{F}_{{j\delta }}}(E)]}}{{\det [1 - {{F}_{{j\delta }}}(E)(\Delta {{{\hat {\mathcal{U}}}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E))]}}, \\ G_{{j,j + \delta }}^{{ \uparrow \downarrow }}(E) = \frac{{F_{{j,j + \delta }}^{{ \uparrow \downarrow }}(E) + \Delta \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(E)\det [{{F}_{{j\delta }}}(E)]}}{{\det [1 - {{F}_{{j\delta }}}(E)(\Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E))]}}, \\ G_{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E) = \frac{{F_{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E) + \Delta \Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E)\det [{{F}_{{j\delta }}}(E)]}}{{\det [1 - {{F}_{{j\delta }}}(E)(\Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E))]}}, \\ \end{gathered} $$

(A.10)

where

$$\begin{gathered} \Delta \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(E) = \Delta \mathcal{U}_{{j,j + \delta }}^{{ \uparrow \downarrow }} - \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(E), \\ \Delta \Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E) = \Delta \mathcal{U}_{{j + \delta ,j}}^{{ \downarrow \uparrow }} - \Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E), \\ \det [{{F}_{{j\delta }}}(E)] = F_{{j,j}}^{{ \uparrow \uparrow }}(E)F_{{j + \delta ,j + \delta }}^{{ \downarrow \downarrow }}(E) \\ \, - F_{{j,j + \delta }}^{{ \uparrow \downarrow }}(E)F_{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E); \\ \end{gathered} $$

(A.11)

$$\begin{gathered} \det [1 - {{F}_{{j\delta }}}(E)(\Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E))] \\ = 1 + F_{{j,j}}^{ \uparrow }(E)\Sigma _{{j,j}}^{ \uparrow }(E) + F_{{j + \delta ,j + \delta }}^{ \downarrow }(E)\Sigma _{{j + \delta ,j + \delta }}^{ \downarrow }(E) \\ \, - F_{{j,j + \delta }}^{{ \uparrow \downarrow }}(E)\Delta \Sigma _{{j,j + \delta }}^{{ \downarrow \uparrow }}(E) - F_{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E)\Delta \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(E) \\ + \det [{{F}_{{j\delta }}}(E)]\det [\Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E)], \\ \det [\Delta {{\mathcal{U}}_{{j\delta }}} - {{\Sigma }_{{j\delta }}}(E)] = \Sigma _{{j,j}}^{ \uparrow }(E)\Sigma _{{j + \delta ,j + \delta }}^{ \downarrow }(E) \\ \, - \Delta \Sigma _{{j,j + \delta }}^{{ \uparrow \downarrow }}(E)\Delta \Sigma _{{j + \delta ,j}}^{{ \downarrow \uparrow }}(E). \\ \end{gathered} $$

(A.12)

It can easily be verified that Δ$\mathcal{U}_{{j,j \pm {{\delta }_{y}}}}^{{ \uparrow \downarrow }}$ = Δ$\mathcal{U}_{{j \pm {{\delta }_{x}},j}}^{{ \downarrow \uparrow }}$ and $\Sigma _{{j,j \pm {{\delta }_{y}}}}^{{ \uparrow \downarrow }}$(E) = $\Sigma _{{j \pm {{\delta }_{x}},j}}^{{ \downarrow \uparrow }}$(E). As noted above, the nondiagonal matrix elements of the effective Green function possess the same property. Thus, it can be seen from expression (A.10) that the nondiagonal matrix elements of the Green function with the fluctuating potential also possess this property, and the determinants of matrices (A.12) are independent of indices j as well as of their nearest neighbors δ.