Proximity effect in superconductor/conical magnet heterostructures

The results presented below utilise self-consistent solutions of the microscopic BdG equations in the clean limit. The method employed is similar to our previous work on Ho containing heterostructures [12], so only the most relevant formulas are repeated here. Generally, the spin-dependent BdG equations can be written as [12–15]

$$\begin{eqnarray}\left(\begin{array}{c} {{H}_{0}}-{{h}_{z}} & -{{h}_{x}}+\text{i}{{h}_{y}} & {{\Delta}_{\uparrow \uparrow}} & {{\Delta}_{\uparrow \downarrow}} \\ -{{h}_{x}}-\text{i}{{h}_{y}} & {{H}_{0}}+{{h}_{z}} & {{\Delta}_{\downarrow \uparrow}} & {{\Delta}_{\downarrow \downarrow}} \\ \Delta_{\uparrow \uparrow}^{*} & \Delta_{\downarrow \uparrow}^{*} & -{{H}_{0}}+{{h}_{z}} & {{h}_{x}}+\text{i}{{h}_{y}} \\ \Delta_{\uparrow \downarrow}^{*} & \Delta_{\downarrow \downarrow}^{*} & {{h}_{x}}-\text{i}{{h}_{y}} & -{{H}_{0}}-{{h}_{z}} \\ \end{array}\right)\left(\begin{array}{c} {{u}_{n\uparrow}} \\ {{u}_{n\downarrow}} \\ {{v}_{n\uparrow}} \\ {{v}_{n\downarrow}} \\ \end{array}\right)={{\varepsilon}_{n}}\left(\begin{array}{c} {{u}_{n\uparrow}} \\ {{u}_{n\downarrow}} \\ {{v}_{n\uparrow}} \\ {{v}_{n\downarrow}} \\ \end{array}\right).\end{eqnarray} \tag{ 1 }$$

Here, ε_n denotes the eigenvalues, and u_nσ and v_nσ are quasiparticle and quasihole amplitudes for spin σ, respectively. After some simplifications [12, 13, 16] the tight-binding Hamiltonian ${{\mathcal{H}}_{0}}$ reads

$$\begin{eqnarray}&&{{\mathcal{H}}_{0}}=-t\underset{n}{\sum} \left(c_{n}^{\dagger}{{c}_{n+1}}+c_{n+1}^{\dagger}{{c}_{n}}\right)+\underset{n}{\sum} \left({{\varepsilon}_{n}}-\mu \right)c_{n}^{\dagger}{{c}_{n}},\end{eqnarray} \tag{ 2 }$$

with $c_{n}^{\dagger}$ and c_n being electronic creation and destruction operators at multilayer index n, respectively. Choosing the next-nearest neighbour hopping parameter t = 1 and the chemical potential (Fermi energy) μ = 0 sets the energy scales.

The vector components of the conical exchange field are written as [10–12]

$$\begin{eqnarray}&&\mathbf{h}={{h}_{0}}\left\{\text{cos} \alpha \mathbf{y}\text{+sin} \alpha \left[\text{sin} \left(\frac{\beta \text{y}}{\text{a}}\right)\mathbf{x}\text{+cos} \left(\frac{\beta \text{y}}{\text{a}}\right)\mathbf{z}\right]\right\},\end{eqnarray} \tag{ 3 }$$

with h₀ = 0.1 being the strength of the conical magnet's exchange field and a = 1 being the lattice constant.

The general form of the pairing matrix in (1) can be rewritten according to Balian and Werthamer [17, 18]

$$\begin{eqnarray}\left(\begin{array}{c} {{\Delta}_{\uparrow \uparrow}} & {{\Delta}_{\uparrow \downarrow}} \\ {{\Delta}_{\downarrow \uparrow}} & {{\Delta}_{\downarrow \downarrow}} \\ \end{array}\right)=\left(\Delta+\widehat{\sigma}\cdot \mathbf{d}\right)\text{i}{{\hat{\sigma}}_{2}}=\left(\begin{array}{c} -{{d}_{x}}+\text{i}{{d}_{y}} & \Delta+{{d}_{z}} \\ -\Delta+{{d}_{z}} & {{d}_{x}}+\text{i}{{d}_{y}} \\ \end{array}\right),\end{eqnarray} \tag{ 4 }$$

with $.\hat{.}.$ indicating a 2  ×  2 matrix. Utilising the Pauli matrices $\widehat{\sigma}$, the superconducting order parameter is described by a singlet (scalar) part Δ and a triplet (vector) part d, respectively.

In our model the pairing interaction is assumed to apply only for s-wave singlet Cooper pairs, and so the pairing potential is restricted to a scalar quantity Δ. This fulfils the self-consistency condition

$$\begin{eqnarray}&&\Delta\left(\mathbf{r}\right)=\frac{g\left(\mathbf{r}\right)}{2}\underset{n}{\sum} \left({{u}_{n\uparrow}}\left(\mathbf{r}\right)v_{n\downarrow}^{*}\left(\mathbf{r}\right)\left[1-f\left({{\varepsilon}_{n}}\right)\right] +{{u}_{n\downarrow}}\left(\mathbf{r}\right)v_{n\uparrow}^{*}\left(\mathbf{r}\right)f\left({{\varepsilon}_{n}}\right)\right),\end{eqnarray} \tag{ 5 }$$

where we are summing only over positive eigenvalues ε_n, and where f(ε_n) denotes the Fermi distribution function evaluated as a step function for zero temperature. Setting up the heterostructure as shown in figure 1 the effective superconducting coupling parameter g(r) equals 1 in the n_SC = 250 layers of spin-singlet s-wave superconductor to the left of the interface and vanishes elsewhere. The conical exchange field, defined according to (3), is added to the n_CM = 500 layers of conical magnet to the right of the interface.

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The superconducting pairing correlation between spins α and β can generally be evaluated as on-site interaction for times t = τ and t' = 0 as

$$\begin{eqnarray}&&{{f}_{\alpha \beta}}\left(\mathbf{r},\tau ,0\right)=\frac{1}{2}\bigg\langle {{\hat{\Psi}}_{\alpha}}\left(\mathbf{r},\tau \right){{{\hat \Psi}}_{\beta}}\left(\mathbf{r},0\right)\bigg\rangle \end{eqnarray} \tag{ 6 }$$

with ${{{\hat \Psi}}_{\sigma}}\left(\mathbf{r},\tau \right)$ being the many-body field operator for spin σ at time τ. The time-dependence is introduced through the Heisenberg equation of motion. Being local in space the pairing correlation evaluated using (6) leads to vanishing triplet contribution for τ = 0 in accordance with the Pauli principle [19]. However for finite times τ nonvanishing contributions emerge, an example of odd-frequency triplet pairing [2]. Substituting the field operators valid for our setup and phase convention the spin-dependent triplet pairing correlations are given by

$$\begin{eqnarray}&&\begin{array}{*{20}{c}} {{f_0}(y,\tau ) = \frac{1}{2}({f_{ \uparrow \downarrow }}(y,\tau ) + {f_{ \downarrow \uparrow }}(y,\tau ))} \\ \nonumber \end{array}\, = \frac{1}{2}\mathop \sum \limits_n ({u_{n \uparrow }}(y)v_{n \downarrow }^*(y) + {u_{n \downarrow }}(y)v_{n \uparrow }^*(y)){\zeta _n}(\tau ) \\ &&\nonumber \ \begin{array}{*{20}{c}} {{f_1}(y,\tau ) = \frac{1}{2}({f_{ \uparrow \uparrow }}(y,\tau ) - {f_{ \downarrow \downarrow }}(y,\tau ))}\end{array}\, \\ &&\nonumber = \frac{1}{2}\mathop \sum \limits_n ({u_{n \uparrow }}(y)v_{n \uparrow }^*(y) - {u_{n \downarrow }}(y)v_{n \downarrow }^*(y)){\zeta _n}(\tau )\end{eqnarray} \tag{ 7 }$$

depending on position y and time parameter τ (set to τ = 10 in the present numerical work), and with ζ_n(τ) given by

$$\begin{eqnarray}&&{{\zeta}_{n}}\left(\tau \right)=\text{cos} \left({{\varepsilon}_{n}}\tau \right)-\text{i sin} \left({{\varepsilon}_{n}}\tau \right)\left(1-2f\left({{\varepsilon}_{n}}\right)\right).\end{eqnarray} \tag{ 8 }$$

The pairing amplitudes in (7) are defined relative to a single spin-quantisation direction, here z. In the SC/FM proximity effect this has a natural choice as the FM spin direction. However in the present calculations for a SC/CM interface there is no such fixed reference axis, since the rotation of the CM moment leads to a spatially varying local magnetisation direction. Therefore it is helpful to represent the triplet pairing correlations in a form which is independent of the spin-quantisation axis chosen. Recalling (4) we define ${\hat \Delta}$ as the triplet pairing matrix for an ordinary spin-triplet superconductor

$$\begin{eqnarray}{\hat \Delta}=\left(\widehat{\sigma}\cdot \mathbf{d}\right)\text{i}{{\hat{\sigma}}_{2}}=\left(\begin{array}{c} -{{d}_{x}}+\text{i}{{d}_{y}} & {{d}_{z}} \\ {{d}_{z}} & {{d}_{x}}+\text{i}{{d}_{y}} \\ \end{array}\right),\end{eqnarray} \tag{ 9 }$$

which obeys the following identity [18]

$$\begin{eqnarray}&&{\hat \Delta}{{{\hat \Delta}}^{\dagger}}=\bigg|\mathbf{d}{{\bigg|}^{2}}{{\hat{\sigma}}_{0}}+\text{i}\left(\mathbf{d}\times {{\mathbf{d}}^{*}}\right)\cdot {\hat \sigma},\end{eqnarray} \tag{ 10 }$$

where ${{\hat{\sigma}}_{0}}$ denotes the identity matrix. For bulk triplet superconductors unitary pairing states are defined by the condition id × d^* = 0 and for these states 2|d| is the quasiparticle energy gap. For non-unitary triplet states id × d^*≠ 0 and is a (real) vector in the direction of the net Cooper pair spin, i.e., it is a measure of the pair spin magnetic moment [10, 18]. The quantities |d| and id × d^* are obviously also gauge invariant, unlike the complex vector d itself.

In the present work there is no pairing interaction in the triplet channel, (5), and so d = 0. Therefore, instead of this we focus on the triplet pair correlation function matrix which can be written similarly to the matrix (9) as [10, 20]

$$\begin{eqnarray}\hat{f}=\left(\widehat{\sigma}\cdot \mathbf{f}\right)\text{i}{{\hat{\sigma}}_{2}}=\left(\begin{array}{c} -{{f}_{x}}+\text{i}{{f}_{y}} & {{f}_{z}} \\ {{f}_{z}} & {{f}_{x}}+\text{i}{{f}_{y}} \\ \end{array}\right).\end{eqnarray} \tag{ 11 }$$

(Note the additional factor i in this definition compared to [20] to be consistent with the definition of $\text{\hat \Delta}$ in (4).) Now the analogue to (10) reads

$$\begin{eqnarray}&&\hat{f}{{\hat{f}}^{\dagger}}=\bigg|\mathbf{f}{{\bigg|}^{2}}{{\hat{\sigma}}_{0}}+\text{i}\left(\mathbf{f}\times {{\mathbf{f}}^{*}}\right)\cdot \mathbf{\hat{\sigma}}.\end{eqnarray} \tag{ 12 }$$

The two analogues to |d| and d × d^*, namely |f| and f × f^*, can be expressed conveniently in terms of the f-vector components, depending on the spin-dependent triplet pairing correlations as

$$\begin{eqnarray}&&\begin{array}{*{35}{l}} {{f}_{x}}=\frac{1}{2}\left(-{{f}_{\uparrow \uparrow}}+{{f}_{\downarrow \downarrow}}\right) \\ {{f}_{y}}=-\frac{i}{2}\left({{f}_{\uparrow \uparrow}}+{{f}_{\downarrow \downarrow}}\right) \\ {{f}_{z}}=\frac{1}{2}\left({{f}_{\uparrow \downarrow}}+{{f}_{\downarrow \uparrow}}\right). \\ \end{array}\end{eqnarray} \tag{ 13 }$$

The coordinate axis independent quantity, |f|², can be interpreted as the mean density of spin-triplet Cooper pairs, and the quantity if × f^* can be interpreted as a vector whose magnitude and direction denote the net spin moment arising from the non-unitary (equal-spin) part of the triplet Cooper pair density. Of course the quantities |f| and if × f^* are also gauge invariant under overall phase changes of the superconducting order parameter.

The results presented in this section are for pairing correlations at a SC/CM interface, obtained via a parameter scan of opening angle α and turning (or pitch) angle β in the ranges from 0° to 180°, calculated in steps of 5°. According to the definition (3) of α, conical magnetic structures with α < 90° (α > 90°) have magnetic moment orientations pointing away (towards) the superconducting side of the interface. α = 90° denotes the special case of a helical magnet with β, now known as the pitch angle. For the limiting cases with β = 0° (β = 180°) the magnetic moments between adjacent layers are ordered ferromagnetically (antiferromagnetically), respectively.

The fundamental pairing amplitudes which we calculate are shown in figure 2, which shows results for a SC/CM interface containing Ho as the conical magnet, i.e. fixing α = 80° and β = 30°. The upper (lower) left panels show the real (green) and imaginary (orange) parts of the equal-spin (unequal-spin) spin-triplet correlations f₀ (f₁) as defined according to (7). The interface is indicated by the dashed line at layer n = 0, with the superconductor on the left (layer n < 0) and the CM on the right (n ⩾ 0). The upper (lower) middle panels depict the real (green) and imaginary (orange) parts of f_↑↑ (f_↓↓) for the same system. According to (7), these pairing correlations are the two contributions to f₁. Finally, the upper and lower right panels of figure 2 depict the magnitude of the f-vector and the magnitude of f × f^* as introduced in (12), respectively.

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Starting with f₀ and f₁, shown in the left panels of figure 2, one notices only slight oscillations of the real parts around the interface layer, whereas the oscillations of the imaginary parts proceed well into the conical magnet layer. It has to be noted that the faster oscillations visible in the imaginary part are related to the chosen turning angle β, i.e., the layer index difference between adjacent maxima correspond to one full turn of the conical magnetic structure. This is superimposed on a slower oscillation responsible for some pronounced intensity decay in the imaginary parts of f₀ and f₁ as one goes further into the conical magnet side of the interface.

Looking now at the middle panels of figure 2, depicting the real and imaginary parts of f_↑↑ and f_↓↓, one notices a different behaviour for real and imaginary parts. The real parts of these quantities differ only in the interface region, and are of equal sign and size in parts of the conical magnet away form the interface. Evaluating f₁ according to (7) leads to the vanishing contributions away from the interface. The very fast oscillations shown in the real parts away from the interface are an artefact of the simple model used here and should vanish once the full k dependence (parallel to the interface layer) and/or impurity scattering effects are taken into account, thereby allowing for deviations from the clean limit. The imaginary parts of f_↑↑ and f_↓↓, however, are of equal size but opposite sign leading to the nonvanishing contributions to f₁ shown in the lower left panel of figure 2.

The upper right panel of figure 2 depicts the corresponding magnitude of the f-vector, as introduced in (12). The slightly out-of-phase relation between f₀ and f₁ leads to a faster oscillation in |f| compared to the oscillations in f₀ and f₁ alone. The very fast oscillations right of the interface stem from the contributions f_↑↑ and f_↓↓ as shown in the middle panels.

Finally the lower right panel in figure 2 depicts the magnitude of f × f^*, giving insight in the spin magnetic properties of the Cooper pairs. It can be seen that the magnetic properties of the pairing decay very rapidly on both sides of the interface, with stronger intensities to be seen in the conical magnet side. The pairing spin magnetic moments are only visible for approximately one turn of the conical magnet, while the magnetic moments in the superconductor side of the interface describe the so-called inverse proximity effect.

The influence of the turning angle β on the spin-triplet pairing correlations f₀ and f₁ is shown in the upper and lower panels of figure 3, respectively. Keeping the opening angle α fixed to 80° and allowing the turning angle β to vary between 0° and 180° yields the full data set depicted in the middle panels of figure 3 (logarithmic scale). The left (linear scale) and right panels (logarithmic scale) of figure 3 show the yz and top view of the data shown in the middle panels, respectively.

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In figure 3 the maximum values for any β clearly stem from the interface region, with a rapidly decaying oscillation in the conical magnet region. The influence of β on the pairing correlation f₀ is largest for β = 0° and β = 180°, which is best seen from the upper left panel of figure 3. Going away from those extremal β values, the pairing correlation decays very quickly to approximately one third of its maximum value for β = 90°.

Looking at the f₁ pairing correlations in the lower panels of figure 3 the maxima are slightly shifted away from the extremal β values but show again a decrease towards β = 90° and fall again down to roughly one third of the maximum f₁. In contrast to f₀ in the upper panels with maximal contributions from β = 0° and β = 180°, the respective β contributions to f₁ vanish. The right panels show a top view of the data shown in the middle panels to depict the slight oscillations with changing β. These patterns are similar for f₀ and f₁.

Keeping in mind that the data displayed in figure 3 corresponds to α = 80°, the middle and right panels allow an estimate of the turning angle's influence on the decay length of f₀ and f₁. Starting from the nonvanishing contributions to f₀ for β = 0° (β = 180°) in the interface region, the decay length is largest (smallest) for the data shown here. For the other values of β the decay length has the same order of magnitude, being influenced by some oscillations due to the multilayer index n. Looking now at the f₁ values close to β = 0° (β = 180°) in the interface region the decay length shows a similar behaviour to f₀, being largest (smallest) for β = 0° (β = 180°) values. Again, for the other values of β we observe decay lengths of the same order of magnitude and similar to the ones seen for f₀.

Figure 4 now shows the influence of β on the magnitudes of the f-vector and f × f^*, as introduced in (12), respectively. The upper panels present the magnitude of f and the lower panels the magnitude of f × f^*, each shown in the same parameter ranges and views as figure 3. Not surprisingly, the magnitude of the f-vector resembles the appearance of f₀ and f₁ from figure 3. The maxima occur for β = 0° and β = 180°, as for f₀, whereas the decaying patterns are similar to the ones already familiar from figure 3. Additionally, the most prominent contributions to the triplet pairing correlations again originate from the interface region. Only in the top view figure (upper right panel) can be seen a somewhat blurred superposition of the right panels of figure 3. This can be attributed to the slight out-of-phase relation of f₀ and f₁, as already seen in figure 2 and discussed in section 3.1.

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However, the magnitude of f × f^* describing the triplet spin magnetic properties behaves surprisingly differently. As seen in the lower panels of figure 4, it is zero for β = 0°, then increases up to a first maximum around β = 45°, and subsequently decreases again to form a small plateau around β = 90°. After that the magnitude of f × f^* has another sharp increase to its maximum value, before vanishing again for β = 180°. The fact that this vanishes for β = 0° and β = 180° is to be expected since these are collinear magnetic structures (ferro- or antiferromagnetic, respectively) and so there are no spin-flip processes at the SC/CM interface.

We now consider variations in the full range of both conical magnet angles α and varying β. As has been discussed in section 3.2 the most prominent contributions to either f₀ and f₁ or |f| and |f × f^*| originate from the interface region, as can be seen clearly from figures 3 and 4. The height profile of the respective quantities for a fixed α and varying β are shown in the left panels of those figures, displaying the yz views. Since the maxima of f₀ and f₁ are always occurring in the interface region this gives us the opportunity to have a look at the data for all angles α and β. This is displayed in the plots shown in figure 5 with the left and right panels depicting the yz views of f₀ and f₁ now shown with varying opening angle α. As can be seen, for α = 0° and α = 180° the contributions to f₀ and f₁ vanish. With increasing α the triplet pairing correlations f₀ and f₁ increase up to a maximum around α = 90° to then decay to zero again afterwards. In figure 5 the appearance along the β coordinate is similar for all α values and has already been shown, as for example in figure 3.

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The full angle dependence on the magnitudes of the f-vector and × f^* is shown in the left and right panels of figure 6, respectively. Focusing for the moment on the left panel showing the yz views of the magnitude of the f-vector one notices that for α = 0° and α = 180° it has a nonvanishing contribution, in contrast to the vanishing contributions of f₀ and f₁ depicted in figure 5. This can be understood knowing that vanishing contributions to f₁ are due to equal size but opposite sign contributions from f_↑↑ and f_↓↓ generating f₁. Looking additionally at the separate contributions f_↑↑ and f_↓↓ can provide a deeper understanding of spin-triplet generation at specific interfaces. Coming back to the data at hand, maxima in the magnitude of the f-vector appear for α = 90° and β either 0° or 180°, i.e., a complete ferromagnet or antiferromagnet with the magnetic moments oriented along the z-axis. All deviations from this ferromagnetic-like arrangement of magnetic moments between adjacent layers, either changing just β (corresponding to a helical magnet), or changing both α and β (a CM) leads to a decrease in the magnitude of the f-vector associated with the unitary triplet pairing correlations in the SC/CM interface.

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The right panel of figure 6 now shows the magnitude of f × f^* associated with the non-unitary triplet pairing, or spin magnetic moment of the Cooper pair. The pattern of this quantity with respect to varying β are the same as already discussed in section 3.2 and shown in figure 4, namely vanishing contributions for β = 0°, increase up to β = 45°, again decreasing to the plateau around β = 90°, and then the sharp increase to the maximum value. The maximum values of the non-unitary pairing correlations clearly occur for values of α near to 90° and β = 180°, although the correlation becomes zero exactly at the point α = 90°, β = 180°, since this is a antiferromagnetic structure.

One can see in these plots that the CM angle values of α = 80° and β = 30°, corresponding to Ho as used in the experiments, are fairly typical in their behaviour. In fact in figure 6 there is a broad maximum in the non-unitary triplet pairing function f × f^* for values similar to those of Ho.