Phase diagram and spectroscopy of Fulde-Ferrell-Larkin-Ovchinnikov states of two-dimensional $d$-wave superconductors

Experimental evidence suggests that the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state may be realized in the unconventional, heavy-fermion superconductor ${\mathrm{CeCoIn}}_5$. We present a self-consistent calculation of the field versus temperature phase diagram and order parameter structures for the FFLO states of quasi-two-dimensional $d$-wave superconductors. We calculate the spatially nonuniform order parameter, free energy density, and local density of states for magnetic fields parallel to the superconducting planes. We predict that the lower critical magnetic field transition between the spatially uniform and nonuniform FFLO state is second order. We discuss the signatures of the nonuniform FFLO state which should be observable in scanning tunneling microscopy measurements of the local density of states.

Figure 1

(Color online) The phase diagram of the FFLO state of a two-dimensional $d$-wave superconductor. The Larkin-Ovchinnikov (LO) state is stabilized in the high-$B$, low-$T$ region of the phase diagram. The solid lines are second order phase transition lines that determine the upper critical field, $B_{c2}$, and separate the normal and FFLO states below $T_{\mathrm{FFLO}}\approx 0.56T_c$. Below $T\lesssim 0.06T_c$ a first order transition (long-dashed line) occurs between order parameter modulations along $⟨110⟩$ and $⟨100⟩$ directions. At the lower critical field, $B_{c1}$, a second order transition (circles-solid) occurs between the uniform and nonuniform $⟨110⟩$-oriented LO phase. The unphysical transition line from the uniform state into the $⟨100⟩$-oriented nonuniform state is shown for comparison (short-dashed). The Chandrasekhar-Clogston phase transition line between the uniform superconducting and normal state would be of first order below $T_{\mathrm{FFLO}}$ (dot-dashed), but is unphysical, while it is of second order and physical above $T_{\mathrm{FFLO}}$ (thick dot-dashed). The corresponding order parameter modulations for a field scan between points (a) and (d) are shown in Fig. 2.

Figure 2

(Color online) Order parameter modulations at $T=0.15T_c$ for different points in the LO state as indicated by (a)–(d) in Fig. 1. The period of the order parameter decreases with increasing field $B$. At the lower magnetic field, $B_{c1}$, a transition from the uniform to nonuniform superconducting state occurs at position (a), which is signaled by a single domain wall (kink). We define the superconducting coherence length ${\xi }_0=v_f∕\left(2\pi T_c\right)$ and dimensionless magnetic field $b=\mu B∕\left(2\pi T_c\right)$, with the zero-temperature order parameter ${\Delta }_0∕\left(2\pi T_c\right)=0.241$.

Figure 3

(Color online) Second order instability lines between normal and FFLO state as function of relative orientation ${\varphi }_0$ of one-dimensional stripes, i.e., order parameter modulations. For temperatures above $T^{*}\approx 0.06T_c$, the modulation along $⟨110⟩$ has the highest instability field, while below this temperature the orientation $⟨100⟩$ is energetically favored. The thin-dotted line is the unphysical critical field, or “supercooled transition,” if one assumes a second order transition between the normal and uniform $(\mathbf{q}=0)$ superconducting state.

Figure 4

(Color online) Dimensionless wave vector $Q=q{\xi }_0$ of order parameter modulation along the instability lines in Fig. 3. Left panel: $T$ dependence of $Q$ for different orientations, with ${\varphi }_0=0°\leftrightarrow ⟨100⟩$ and ${\varphi }_0=45°\leftrightarrow ⟨110⟩$. Right panel: Angular dependence of $Q$ (dots) and upper critical field $B_{c2}$ (solid line) at $T=0$.

Figure 5

(Color online) Angle-averaged, spin-dependent local density of states (LDOS) for a single domain wall solution. (Top) Profile of order parameter at $T∕T_c=0.15$ and $b=0.175$. The panels below show LDOS at locations (a) and (b). (Center) LDOS at position (a), where order parameter is zero, i.e., at the domain wall. The Andreev bound states are shifted from zero energy by $+b$ for spin-up and $-b$ for spin-down electrons. (Bottom) LDOS at position (b), far from the domain wall, i.e., in bulk of $d$-wave superconductor. The Andreev states are bound to the region where the order parameter is suppressed and are very weak far from the domain wall.

Figure 6

(Color online) Angle-averaged LDOS for a periodic solution inside the LO state. (Top) Profile of order parameter at $T∕T_c=0.15$ and $b=0.200$. The panels below show LDOS at locations (a) and (b). (Center) LDOS at position (a), where order parameter vanishes. The broadened Andreev bound states, quasiparticle resonances, are shifted from zero energy by $+b$ for spin-up and $-b$ for spin-down electrons. (Bottom) LDOS at position (b) at maximum of order parameter. The Andreev bound states decayed and broadened even more.

Figure 7

(Color online) Spin-up/down DOS averaged over a single period of the order parameter for three different $b$ fields in the LO state. A broad bound state at $\epsilon ∕\left(2\pi T_c\right)=b$ is seen at lower fields, but continues to broaden as $b$ increases. Finally, the DOS becomes flat and “normal”-like. Here $T∕T_c=0.15$ and the corresponding upper critical field is $b_{c2}=0.25$.

Figure 8

(Color online) An example of a spatially averaged DOS for $⟨100⟩$ modulated order parameter at $T=0.15T_c$ and $b=0.200$. Note that the physically stable solution has $⟨110⟩$ modulation. (Upper left) Profile of periodic order parameter with three marked positions. (Lower left) DOS at the domain wall $(x=0)$, the Andreev bound state is clearly seen at $\epsilon ={ϵ}_{\mathrm{B}}$. (Lower right) DOS at distance $x=\lambda ∕8$ from domain wall, the Andreev bound state is slightly suppressed. (Upper right) DOS at the maximum of order parameter, the Andreev bound state is completely suppressed. In addition to the broadened Andreev bound state at $\epsilon ={ϵ}_{\mathrm{B}}$ there are singularities due to multiple Andreev reflections.

Figure 9

(Color online) (Upper left) Profile of order parameter for a single domain wall along $⟨110⟩$. (Lower left) Profile of free energy density $\Delta f$ for single domain wall. (Upper right) Profile of order parameter for periodic domain walls along $⟨110⟩$. (Lower right) Profile of free energy density $\Delta f$ for periodic domain walls.

Figure 10

(Color online) (Upper left) Profile of order parameter for a single domain wall along $⟨100⟩$. (Lower left) Profile of free energy density $\Delta f$ for single domain wall. (Upper right) Profile of order parameter for periodic domain walls along $⟨100⟩$. (Lower right) Profile of free energy density $\Delta f$ for periodic domain walls.