Superconductivity and spin-density waves in multiband metals

We present a detailed description of two-band quasi-two-dimensional metals with $s$-wave superconducting (SC) and antiferromagnetic spin-density-wave (SDW) correlations. We present a general approach and use it to investigate the influence of the difference between the shapes and the areas of the two Fermi surfaces on the phase diagram. In particular, we determine the conditions for the coexistence of SC and SDW orders at different temperatures and dopings. We argue that a conventional $s$-wave SC order coexists with SDW order only at very low $T$ and in a very tiny range of parameters. An extended $s$-wave superconductivity, for which SC gap changes sign between the two bands, coexists with antiferromagnetic SDW over a much wider range of parameters and temperatures but even for this SC order the regions of SDW and SC can still be separated by a first-order transition. We show that the coexistence range becomes larger if SDW order is incommensurate. We apply our results to iron-based pnictide materials, in some of which coexistence of SDW and SC orders has been detected.

Figure 1

(Color online) Left: electronic structure of the two-band model considered in this paper, in the unfolded Brillouin zone. The hole FS is in the center, with SC order parameter ${\Delta }_c$, and the electron FSs are at $\left(0,\pi \right)$ and $\left(\pi ,0\right)$, with SC order parameter ${\Delta }_f$. The magnetic order with momentum ${\mathbf{Q}}_0=\left(0,\pi \right)$ hybridize hole and electron FSs separated by ${\mathbf{Q}}_0$ but leaves FSs at $\left(\pm \pi ,0\right)$ intact. Right: by doping or pressure one may adjust the size and shape of hole and electron bands, and also SDW order parameter can be incommensurate, with momentum ${\mathbf{Q}}_0+\mathbf{q}$. These effects are described by FS detuning parameter, ${\delta }_{\mathbf{k}\mathbf{q}}=\left[{\xi }_f\left(\mathbf{k}+\mathbf{q}\right)+{\xi }_c\left(\mathbf{k}\right)\right]/2$.

Figure 2

(Color online) The appearance of gapless excitations in the presence of SDW order. The dotted lines indicate FSs for electrons, ${\xi }_f=0$, and holes, ${\xi }_c=0$. The dashed curve is an “effective” FS, ${\xi }_{\mathbf{k}\mathbf{q}}=0$. (a) When $q=0$ and $m$ is large compared to FS mismatch, $m>{\delta }_{\hat{\mathbf{k}}0}$, all excitations are gapped; (b) when $m$ is small, gapless excitations are preserved along the two modified FSs at ${\xi }_{\mathbf{k},0}=\pm {\left({\delta }_{\hat{\mathbf{k}},0}^2-m^2\right)}^{1/2}$ ($\left|{\delta }_{\hat{\mathbf{k}},0}\right|>m$ in the shaded region). Such gapless state, however, only exists at high temperatures while at low $T$ it is pre-emptied by a first-order transition to the normal state (Ref. 42); (c) to prevent the first-order transition, magnetic order is formed at an incommensurate vector $\mathbf{Q}={\mathbf{Q}}_0+\mathbf{q}$. This improves electron-hole nesting on some part of the FS but allows for gapless excitations at the opposite side; (d) when the two FSs are of different shapes, the nested parts become gapped due to SDW order and on the rest of the FSs the excitations are little affected by SDW order. The density of states for these cases is shown in Fig. 3.

Figure 3

(Color online) The FS averaged DOS for pure SDW state and FS mismatch ${\delta }_{\hat{\mathbf{k}}\mathbf{q}}={\delta }_0+{\delta }_2\text{cos}2\varphi$. We use dimensionless variables denoted by bars, ${\overline{\delta }}_0={\delta }_0/2\pi T_s=0.13$, zero-temperature SDW gap ${\overline{m}}_0=m\left(T=0\right)/2\pi T_s=0.28$ and vary ${\overline{\delta }}_2={\delta }_2/2\pi T_s$. For ${\overline{\delta }}_2=0$, $N\left(ϵ\right)$ vanishes below $\overline{ϵ}=\overline{m}-{\overline{\delta }}_0$ and has two sharp BCS peaks at $\overline{ϵ}=\overline{m}\pm {\overline{\delta }}_0$. At finite ${\overline{\delta }}_2$, each of the two peaks splits into a “band” bounded by two weaker nonanalyticities separated by $2{\overline{\delta }}_2$. The gap in the DOS behaves as $\overline{m}-{\overline{\delta }}_0-{\overline{\delta }}_2$ and closes when ${\overline{\delta }}_0+{\overline{\delta }}_2\ge \overline{m}$, metallic states forms. The DOS remains the same if we replace the ellipticity parameter ${\overline{\delta }}_2$ by the incommensurability parameter $\overline{q}$.

Figure 4

(Color online) The SDW-N transition for commensurate SDW order. The parameters ${\overline{\delta }}_0$ and ${\overline{\delta }}_2$ describe the difference between the area of hole and electron pockets and the ellipticity of the electron pocket, respectively. Here and in all subsequent figures dotted lines mark first-order transitions, solid and dashed lines mark second-order transitions. Panel (a): variation in the transition temperature with ${\delta }_0$ for fixed ${\delta }_2$. The transition is second order at small ${\delta }_0$ but becomes first order at larger ${\delta }_0$. At ${\delta }_2=0$, the transition becomes first order at $T_s^{\ast }\approx 0.56T_s$. Panel (b): variation in the transition temperature with ${\delta }_2$ for fixed ${\delta }_0$. $T_s\left({\delta }_0,{\delta }_2\right)$ monotonically decreases with increasing ${\delta }_2$ and vanishes at the same value ${\overline{\delta }}_2\approx 0.28073$, independent of ${\delta }_0$.

Figure 5

Same as in Fig. 4 but when the system is allowed to choose between commensurate and incommensurate SDW orders. Solid lines are second-order transition lines into a state with a commensurate SDW order and dashed lines are second-order transition lines into an SDW state with an incommensurate SDW order (the magnetic analog of FFLO state). For all ${\delta }_2>0$, incommensuration occurs before the commensurate transition becomes first order (the onsets of incommensuration and first-order transition coincide for ${\delta }_2=0$). Observe that incommensuration develops at progressively smaller ${\delta }_0$ as ${\delta }_2$ increases and $T_s\left({\delta }_0\right)$ decreases but the range of ${\delta }_0$ over which incommensurate SDW order exists actually increases with increasing ${\delta }_2$.

Figure 6

(Color online) The phase diagram of SDW and SC $s^{+-}$ states when only a commensurate SDW order is allowed $\left(q=0\right)$. We set $T_s/T_c=3$ and varied either (a) the relative radius of circular hole and electron pockets or (b) the form of one of the pockets. The pure SC $s^{+-}$ and SDW states are separated by first-order transition, and there is no co-existence region (Refs. 41, 42).

Figure 7

(Color online) Appearance of coexistence when both ${\delta }_0$ and ${\delta }_2$ are finite. We set $T_s/T_c=2$ and $q=0$. Panels (a1)–(a3)—phase diagrams in variables $T,{\delta }_0$ at fixed ${\delta }_2$ and panels (b1)–(b3)—phase diagrams in variables $T,{\delta }_2$ at fixed ${\delta }_0$. Panels (a1) and (b1)—there appears a region near $T=0$, where SDW and SC $s^{+-}$ orders coexist. Panels (a2) and (b2)—the coexistence region broadens and reaches $T=T_c$. Panels (a3) and (b3)—the transition at low $T$ becomes first order between pure SDW and SC states but narrow coexistence region is still present near $T_c$. A complimented zero-temperature phase diagram is presented in Fig. 10.

Figure 8

(Color online) (a) Same as in Fig. 7b but for $T_s/T_c=5$. [(b) and (c)] SDW and SC gaps, in units $\overline{m}$ and $\overline{\Delta }=\Delta /2\pi T_c$, and the free energy (b) as functions of ${\overline{\delta }}_0$ along the line $T/T_c=0.2$, and (c) as functions of temperature for ${\overline{\delta }}_0=0.17$.

Figure 9

(Color online) FS averaged DOS as a function of energy at $T/T_c=0.1$ for different values of ${\delta }_0$ (i.e., different dopings). We set ${\overline{\delta }}_2=0.2$ and $T_s/T_c=5$. The characteristic values of the SDW order parameter for this range of parameters is $\overline{m}\approx 0.28$ (zero-$T$ limit). Upper panel: DOS for small ${\overline{\delta }}_0$, when the system remains in the pure SDW state. This figure is similar to Fig. 3. The DOS vanishes below $ϵ=m-{\delta }_0-{\delta }_2$ and has logarithmic nonanalytic behavior at $ϵ=m\pm {\delta }_0+{\delta }_2$, and sudden drops at $ϵ=m\pm {\delta }_0-{\delta }_2$. Lower panel: DOS for larger ${\delta }_0$, when SDW and SC orders coexist. Sharp peaks at small $ϵ$ are due to opening of the superconducting gap $\Delta$. Once SDW order disappears at ${\overline{\delta }}_0\approx 0.21$, the DOS acquires BCS form with the maximal gap ${\Delta }_{max}/2\pi T_c\approx 0.28$.

Figure 10

(Color online) The zero-temperature phase diagram: SDW, SC states and their coexistence region for various ${\delta }_2$ and ${\delta }_0$ and two different $T_s/T_c=2$ (left) and $T_s/T_c=5$ (right). We only allow the system to develop a commensurate SDW order $\left(q=0\right)$. The dashed line denotes first appearance of gapless excitations in the pure SDW state [$m={\delta }_0+{\delta }_2$, Eq. (2.45)]. The coexistence region at $T=0$ (shaded area) extends down to ${\delta }_0=0$ but is not present at small ${\delta }_2$. The width of the coexistence region increases with the relative strength of SDW interaction, as determined by ratio $T_s/T_c$. The squares mark the location of the crossing point between $T_s\left({\delta }_0,{\delta }_2\right)$ and $T_c$. Open squares indicate that the SDW-SC transition near $T_c$ is first order while filled squares signal the presence of the coexistence phase near $T_c$. Note that the regions where coexistence phase is present at $T=0$ and near $T_c$ are not identical.

Figure 11

(Color online) The phase diagram for ${\delta }_2=0$, when the system can choose the value of $\mathbf{q}$. Incommensurate SDW order appears below $T_s^{\ast }=0.56T_s$ and leaves some parts of the FS ungapped, allowing for coexisting SC order. (a) $T_s/T_c=3$. The $\text{SDW}+\text{SC}$ phase appears only in a small region near $T_c$. At low $T$ the system still undergoes a first-order transition between commensurate SDW and SC states. (b) For larger $T_s/T_c=5$ (weaker SC interaction) the coexistence region widens and extends down to $T=0$. The $q=0$ SDW state has the lowest energy at $T=0$ for ${\overline{\delta }}_0\lesssim 0.195$.

Figure 12

(Color online) Same as in Fig. 8a but now we allow the system to choose the value of $q$. The phase diagram from Fig. 8 is shown by dashed lines. A finite $q$ emerges below a particular $T$ and moves the coexistence region to larger ${\delta }_0$, together with the SDW-N transition. This broadens the coexistence region and slightly changes the shape of $T_c\left({\delta }_0\right)$ inside the magnetic dome.

Figure 13

(Color online) The phase diagram for a conventional $s^{++}$ SC order parameter, at ${\delta }_2=0$, and varying ${\delta }_0$. We allow the system to choose $q$. The $\text{SDW}+\text{SC}$ state does not appear, even when SDW order becomes incommensurate.

Figure 14

(Color online) (a) Same as in Fig. 13 but for fixed ${\overline{\delta }}_2=0.2$ and $q=0$. For most of the phase diagram, the behavior is the same as for ${\delta }_2=0$ but there appears a very tiny range of $\text{SC}+\text{SDW}$ phase at the lowest $T$. The transition to purely SC state is first order at all $T$. (b) The zero-temperature phase diagram. SDW and SC $s^{++}$ states are separated by first-order transition virtually everywhere except a small region at finite ${\delta }_0$ and ${\delta }_2$, where $\text{SDW}+\text{SC}$ state emerges.

Figure 15

(Color online) The three-dimensional plot of the SDW-SC crossing surface, $T_s\left({\delta }_0,{\delta }_2\right)=T_c$. At each point on the surface we show the sign of $B$ and $\chi =AB-C^2$. In the region where $\chi >0$, the transition between SDW and SC states occurs via the coexistence region. For $\chi <0$, pure SDW and SC states are separated by the first-order transition. When $B<0$, the SDW-N conversion is of the first order and the present GL analysis is invalid.

Figure 16

(Color online) The case of $s^{++}$ SC. Panel (a): the behavior of $\chi \left({\delta }_0,{\delta }_2\right)=AB-C^2$ for different $T_s/T_c$ for three cases: $\chi \left(\delta ,0\right)$, $\chi \left(0,\delta \right)$, and $\chi \left(\delta ,\delta \right)$. For each case, $\delta$ is chosen to satisfy the condition $T_s\left({\delta }_0,{\delta }_2\right)=T_c$ for a given $T_s/T_c$. We see that in all three cases, $\chi <0$ no matter what the ration $T_s/T_c$ is. Panel (b): the coefficient $B\left({\delta }_0,{\delta }_2\right)$ along the line $T_s\left({\delta }_0,{\delta }_2\right)=T_c$. GL analysis is only valid when $B>0$.

Figure 17

(Color online) A partially gapped SDW state, where only a fraction $N_{SDW}$ of the electron and hole FSs is nested. On the remaining parts the dispersions ${\xi }_f$ and ${\xi }_c$ are very different. The SDW state appearing below $T_s$ gaps excitations only in the shaded/boxed areas [see Fig. 2d] while on the rest of the FSs the dispersions are close to the original ${\xi }_f$ and ${\xi }_c$. The SC state below $T_c$ does not compete with SDW in non-nested regions but competes with SDW state in the nested (boxed) regions.

Figure 18

(Color online) (a) The phase diagram for $s^{+-}$ superconductivity and SDW order parameter existing only in boxed regions of the FS in Fig. 17, with the relative width $N_{SDW}$. We manually set the doping dependence of $N_{SDW}$ to be $N_{SDW}=0.8-0.4\left({\overline{\delta }}_0/0.3\right)$ and neglected the effect of this variation in $N_{SDW}$ on $T_s$. Observe that SC and SDW orders coexist in a wide range of ${\delta }_0$ and $T$. [(b)and (c)] the order parameters, $\Delta$ and $m$, and the free energy, $F$, as functions of ${\delta }_0$ at a constant temperature, $T/T_c=0.2$ (b) and as functions of $T$ at a constant ${\overline{\delta }}_0=0.13$ (c). As a function of ${\delta }_0$, SDW order parameter starts decreasing when SC order appears and then jumps to zero and the system becomes a pure SC.