Non-Collinear magnetic order in the double perovskite

1Introduction¶

In this chapter we discuss magnetism in the metallic double perovskites A$_2$BB$'$O$_6$. These involve a transition metal ion B, with a large magnetic moment, and a non magnetic ion B$'$. While many double perovskites (DP) in this category are ferromagnetic there are hints of antiferromagnetic phases at higher electron doping. These are driven by electron delocalisation, instead of the short range superexchange seen in magnetic insulators.

We will present a comprehensive study of the magnetic ground state and $T_c$ scales of the minimal double perovskite model in three dimensions, using a combination of spin-fermion Monte Carlo and variational calculations. The model is defined on the three dimensional cubic lattice, where B and B$'$ sites are in a rock salt pattern (Figure 1). The sublattice of the magnetic B atoms is face centered cubic (FCC) and geometrically disallows a Neel state. As a result, the antiferromagnetic tendency manifests itself as spiral order, or non-coplanar ‘flux’ like phases. We will map out the possible magnetic phases for varying electron density, the level separation between the B and B’ ions, and the crucial B$'$-B$'$ (next neighbour) hopping $t'$.

Previous study of double perovskites in two dimensions (Sanyal & Majumdar (2009)) revealed three collinear phases, namely ferromagnet (FM), a diagonal stripe phase (FM lines coupled antiferromagnetically) and a ‘G type’ phase (up spin surrounded by down and vice versa). In 2D the B sublattice is square and bipartite, so there is no frustration. If we had a 3D simple cubic B lattice the counterparts of the 2D phases would be FM, A type (planar), C type (line like) and G type (Neel). The B ion lattice in 3D is FCC, so, while the FM and planar (A type) phases can exist, the C type phase is modified and the G type phase is disallowed.

Figure 1b indicates why it is impossible to have an ‘up’ ($\uparrow$) B ion to be surrounded by only ‘down’ ($\downarrow$) B ions, i.e., the G type arrangement. Two B neighbours of a B ion are also neighbours of each other, frustrating G type order. The suppression of the G type phase, which occupies a wide window in 2D, requires us to move beyond collinear phases in constructing the phase diagram for 3D. We will discuss the variational family in Section Section 1.1.

Using a combination of Monte Carlo and variational minimization, our main results are the following:

• We map out the magnetic ground state at large Hund’s coupling for varying electron density and B-B$'$ level separation. In addition to FM, and collinear A and C type order, the phase diagram includes large regions of non-collinear ‘flux’ and spiral phases and windows of phase separation.

• Modest B$'$B$'$ hopping leads to significant shift in the phase boundaries, and “particle-hole asymmetry”.

• We provide an estimate of the $T_c$ of these phases from the Monte Carlo where possible, or make a rough estimate based an energy difference calculation.

This chapter is organized as follows. In section Section 1.1 we define the effective single band model for the metallic double perovskites and describe the methods we use. Section Section 2 discusses our results on the particle-hole symmetric model, and section Section 3 describes the effect of a introducing electron hopping $t'$ between the non-magnetic sites. Section Section 4 discusses some issues of modeling the real double perovskites. We summarize and conclude the chapter in section Paragraph.

1.1Effective single band model¶

The alternating arrangement of B and B$'$ ions in the ordered cubic double perovskites is shown in Figure 1. We use the following one band model on that structure:

The $f^{\dagger}$ correspond to the B ions and the $m^{\dagger}$ to the B$'$. $\epsilon_{B}$ and $\epsilon_{B'}$ are ‘on-site’ energy on the B and B$'$ sites respectively, e.g., the $t_{2g}$ level energy of Fe and Mo in SFMO. $\mu$ is the chemical potential and ${\hat N} = \sum_{i \sigma} (f^{\dagger}_{i \sigma}f_{i \sigma} + m^{\dagger}_{i \sigma}m_{i \sigma})$ is the total electron number operator. $t$ is the hopping amplitude between nearest neighbour B and B$'$ ions. We augment this model later to study first neighbour B$'-$B$'$ hopping $t'$ as well ($-t'\sum_{\langle ij\rangle \in B'}[m^{\dagger}_{i \sigma} m_{j \sigma} +\textrm{hc}]$). $J$ arises from the Hund’s coupling on the $d$ shell. The Hund’s coupling itself keeps the 5 core electrons polarised into a $S=5/2$ state, and the exclusion principle forces the conduction electron spin to be antiparallel to this core spin. We will use $\vert {\bf S}_i \vert =1$, and absorb the magnitude of ${\bf S}$ in $J$. ${\sigma}^{\mu}_{\alpha\beta}$ are the Pauli matrices.

The model has parameters $J$, $\epsilon_B$, $\epsilon_{B'}$, and $\mu$ (or $n$). Since only the level difference matters, we set $\epsilon_{B'} =0$. We have set $t=1$, and use $J/t \gg 1$ so that the conduction electron spin at the B site is slaved to the core spin orientation. The B site levels are shifted to $\epsilon_B \pm J/2$ due to the strong on site coupling. To keep the difference between the lower B level and the B$'$ level finite we set the parameter $\epsilon_B = \Delta + J/2$, where $\Delta$ remains finite even as $J/t \rightarrow \infty$. We explore the phases as a function of $n$ and $\Delta/t$. We will present results for

• ($a$) the nearest neighbour hopping model ($t'/t=0$) and

• ($b$) when a non-zero B$'$-B$'$ hopping $t'/t=\pm 0.3$ is introduced.

A schematic for the on-site levels, and how they hybridize, is shown in Figure 2. The structural unit cell of the system has 2 atoms (one B, one B$'$), which amounts to 4 atomic levels (2 up spin, 2 down spin). The two spin levels at the B site are separated by $JS$ and overlap with 2 spin degenerate levels of the B$'$ site at $\epsilon_{B'}=0$. We take $\Delta$ in the range (0-10). One B band become centered at $\Delta$ and second goes to $JS + \Delta$. In this situation the down spin B and two B$'$ bands overlap while up spin B band is always empty. The accessible electron density window includes the lowest three bands, so our electron density will be in the range $[0,3]$ per formula unit.

To get a general feel of the band structure of the particle hole symmetric case, we notice that we have three levels (excluding the highest $f_\uparrow$ level at $JS +\Delta$ which remains empty and is redundant for our purpose) in atomic limit. These include one spin slaved $f_\downarrow$ level at $\Delta$, and the two $m_\uparrow$, $m_\downarrow$, levels which overlap with the $f_\downarrow$ levels depending on the spin configurations. This overlap leads to electron delocalisation and band formation.

In the ferromagnetic case, Figure 2(a), only one spin channel (say $m_\downarrow$) gets to delocalize through $f$ sites and forms two bands, separated by a band gap of $\Delta$, while other spin state (say $m_\uparrow$) is localized at 0.

For collinear AF configurations, the rough band scheme is as shown in Figure 2(b). The conduction path gets divided into two sublattices, such that each spin channel gets to delocalize in one sublattice (in which all the core spins point in same direction, making the sublattice ferromagnetic.) See Figure 3(a)-(b), and Figure 3(c)-(d) for the details of the conduction path. In one such sublattice, only one of the $\uparrow$ or $\downarrow$ delocalized, the other remains localized. The roles of $\uparrow$ and $\downarrow$ are reversed in going from one sublattice to other, as a result one gets spin-degenerate localized and dispersive bands for AF phases.

1.2Methods of solution¶

1.2.1Monte Carlo¶

The model involves spins and fermions, and if the spins are ‘large’, $2S \gg 1$, they can be approximated as classical. This should be reasonable in materials like SFMO where $S = 5/2$. Even in the classical limit these spins are annealed variables and their ground state or thermal fluctuations have to be accessed via iterative diagonalization of the electronic Hamiltonian. As mentioned in chapter Section 1, we use a cluster based Monte Carlo (MC) method where the cost of a spin update is estimated via a small cluster Hamiltonian instead of diagonalizing the whole system (Kumar & Majumdar (2006)). We typically use a $12 \times 12 \times 12$ system with the energy cost of a move estimated via a $4\times 4\times 4$ cluster built around the reference site.

We principally track the magnetic structure factor to explore the nature of magnetic order

where $\langle ... \rangle$ denote thermal average. Although the magnetic lattice is FCC, the electrons delocalize on the combined B-B$'$ system which is a cubic lattice. Hence we define our wavenumbers ${\bf q}$ with respect to the full B-B$'$ lattice. As a result even a simple state like the ferromagnet corresponds to peaks at ${\bf q} =(0,0,0)$ and ${\bf q} =(\pi,\pi,\pi)$ and not just ${\bf q} =(0,0,0)$. This is because the spin field is also defined on B$'$ sites and it has to have zeros value on these sites.

The possibility of complex order in the ground state the Monte Carlo may require large annealing time, and involve noisy data. So, to complement the MC results we have also used a variational scheme discussed below.

1.2.2Variational scheme¶

We now move to the variational approach. The variational states that we consider here are more complicated than that discussed in the last chapter, and are ${\cal O}(N^2)$ in number, so we need some caution in implementing the scheme. We compute the energy for periodic configurations where,

with $\theta_{\bf r} = {\bf q}_{\theta}\cdot{\bf r}$ and $\phi_{\bf r} = {\bf q}_{\phi}\cdot{\bf r}$ with $p_{\bf r}= 1$ if ${\bf r} \in B$ and $p_{\bf r}= 0$ if ${\bf r} \in B^{\prime}$. ${\hat x}$, etc., are unit vectors in the corresponding directions.

The vector field ${\bf S}_{\bf r}$ is characterized by the two wavevectors ${\bf q}_{\theta}$ and ${\bf q}_{\phi}$. For a periodic configuration, these should be ${\bf q}_{\theta} = \frac{2\pi}{L}(q_1,q_2,q_3)$ and ${\bf q}_{\phi} = \frac{2\pi}{L}(p_1,p_2,p_3)$, where $q_i$’s and $p_i$’s are integers, each of which take $L$ values in $\{0,1,2,3,...L-1\}$. There are $\sim L^{6}$ ordered magnetic configurations possible, within this family, on a simple cubic lattice of linear dimension $L$.

The use of symmetries, e.g., permuting components of $q_{\theta}$, etc., reduces the number of candidates somewhat, but they still scale as $\sim L^{6}$. For a general combination of ${\bf q}_{\theta}, {\bf q}_{\phi}$ the eigenvalues of H cannot be analytically obtained because of the non trivial mixing of electronic momentum states. We have to resort to a real space diagonalization. The Hamiltonian matrix size is $2 N \times 2N$ (where $N=2L^{3}$) and the diagonalization cost is $\sim N^3$. So, an exhaustive comparison of energies based on real space diagonalization costs $\sim N^5$, possible only for $N \le 8^3$.

We have adopted two strategies:

i. we have pushed this `${\bf q}_{\theta}, {\bf q}_{\phi}$’ scheme to large sizes via a selection process described below, and

ii. for a few collinear configurations, where Fourier transformation leads to a small matrix, we have compared energies on sizes $\sim 100^3$.

First, scheme (i). For $L=8$ we compare the energies of all possible phases, to locate the optimal pair $\{ {\bf q}_{\theta}, {\bf q}_{\phi} \}_{min}$ for each $\mu$. We then consider a larger system with a set of states in the neighbourhood of $\{ {\bf q}_{\theta}, {\bf q}_{\phi} \}_{min}$. If we consider $\pm \pi/L$ variation about each component of ${\bf q}_{\theta,min}$, {\it etc}, that involves 3⁶ states. The shortcoming of this method is that it explores only a restricted neighbourhood, dictated by the small size result. We have used $L = 12,16,20$ within this scheme.

The phases that emerge as a result of the above process are (i) FM, (ii) A-type, (iii) C-type, (iv) ‘flux’, and (v) three spirals SP$_1$, SP$_2$, SP$_3$. A-type consists of ($1,1,1$) FM planes with alternate planes having opposite spin orientation (see Figure 3a). If we convert each of these planes to alternating FM lines, so that the overall spin texture is alternating FM lines in all directions, we get C-type phase (see Figure 3c).

The ‘flux’ phase is different from the spiral families described using period vectors ${\bf q}_{\theta},{\bf q}_{\phi}$. It is the augmented version of ‘flux’ phase used in cubic lattice double exchange model by Alonso et al (Table-I of ref. Alonso et al. (2001)). It has spin-ice like structure, and is described by

The spiral SP$_n$ phases are characterized by commensurate values of ${\bf q}_{\theta},{\bf q}_{\phi}$ (See Table 1 for details of periods and the S(q) peaks).

The simplest, SP$_1$ can be viewed as $\frac{\pi}{2}$-angle pitch in the $(110)$,$(101)$ and $(011)$ directions. The other two spirals SP$_2$ and SP$_3$ are respectively C-type and A-type modulations upon SP$_1$. Just as flipping alternate $1,1,1$ planes in a FM leads to the A type phase, flipping the spins in the $(111)$ planes alternatively in SP$_1$, leads to SP$_3$. Analogously, flipping FM lines in a FM and leads to C-type order - and a similar exercise on SP$_1$ leads to SP$_2$. This modulation is also seen in the $S({\bf q})$ peaks of SP$_2$ and SP$_3$. See the Table 1, where all the three spirals have 4 S({\bf q}) peaks common, and SP$_2$ and SP$_3$ possess extra S({\bf q}) peaks of the A-type and C-type correlations.

In scheme (ii) we take collinear phases that occur in the phase diagram and compare their energy on very large lattices. This does not require real space diagonalization. The simple periodicity of these phases leads to coupling between only a few $\vert {\bf k} \rangle$ states. The resulting small matrix can be diagonalized and the eigenvalues summed numerically. We also included the ‘flux’ phase in this comparison. The details of this calculation, and the the magnetic phase diagram from comparison of FM, A, C and flux phases on large lattice size is discussed in Appendix Section 1.1.

Particle hole symmetry in the model: The electrons move on the cubic lattice divided into two FCC sublattices each of consists of only B or B$'$ sites. For each of these sublattice, one can define particle-hole transformation (Fazekas (1999)) as $f_i\rightarrow f^{\dagger}_i$ and $m_{i\sigma} \rightarrow-m^{\dagger}_{i\sigma}$. This transforms the Hamiltonian as

When $t'=0$, this simplifies to

This symmetry reflects in the phase diagram as the repetition of the phases after half-filling. Introducing the $t'$ hopping destroys this symmetry, but a reduced symmetry still remains, which relates

This is reflected in the phase diagrams of particle-hole asymmetric case.

2Results: nearest neighbour model¶

We now discuss the results in the particle-hole symmetric case, i.e, $t'=0$. The $t' \neq 0$ situation is discussed in the next section. For each of these cases we first discuss the ground state phase diagram, then the nature of magnetic correlations at finite temperature, and finally the estimate of $T_c$.

Although the MC approach is ‘unbiased’, the results are are affected by finite size and the use of single spin based update. We use MC results to establish the dominant magnetic correlations in a parameter window and then explore these more carefully using the variational scheme. The MC based $T_c$ estimate is also checked against a rough energy difference calculation that be implemented on the variational ground state.

2.1Monte Carlo¶

We studied a $N=12\times 12\times 12$ system using the cluster based update scheme. We used a large but finite $J$ to avoid explicitly projecting out any electronic states (undefined (n.d.)), since that complicates the Hamiltonian matrix but allows only a small increase in system size. The magnetic phases were explored for $\Delta = 0,~4$ and 10.

In Figure 4, the magnetic ground state is shown, obtained via a combination of Monte Carlo and variational calculations. We see that the phase diagram is symmetric in density. For small $\Delta$, in the range $0-4$, we have FM, followed by A-type, C-type, and ‘flux’ phase. The order reverses as we go in the other half of the density window. The G-type phase which was largest stable phase in 2D (Fig.2 and Fig.5 in Ref. (Sanyal & Majumdar (2009))) is almost taken over by the ‘flux’ phase. The stability of the ‘flux’ phase decreases with $\Delta$ and it does not show up for $\Delta >4$. The phases that dominate the phase diagram are listed in table Table 1.

Two complications arise in the Monte Carlo approach in the present model.

i. Additional degeneracy of some AF phases: In addition to the usual spin rotation symmetry, the A-type and C-type phases have four fold degeneracy. The A-type phase for example can arise from planes that are normal to any of the four body diagonals (not just (111)). This leads to growth of domains with different orientation as the system is cooled and multiple peaks in the structure, Table 2.

ii. Geometrical frustration leads to the presence of multiple minima with similar energy and increases the equilibriation time.

(a)	(b)	(c)

An illustrative plot of peak features in $S({\bf q})$ as function of temperature $T$, is shown in Figure 5 for some typical densities. For FM, S(${\bf q}_{FM}$) shows monotonic decrease of T$_c$ with increasing $\Delta$. For A-type and ‘flux’ phase, the S(q) data shows a number of sub-dominant q peaks whose number keeps increasing as we move to more complicated phases with increasing density. These sub-dominant peaks arise due to a combination of $(a)$ geometrical frustration and $(b)$ our cluster based update.

In Figure 6 we show the full structure factor in the the three phases, namely FM, A-type and flux like. The data is visualized in following scheme. At every ${\bf q}$, a sphere is drawn, whose size, and colour scales with the value of $S({\bf q})$. The colour scheme has white colour for the low values of $S({\bf q})$, so that in the present scheme, only the most prominent ${\bf q}$ are displayed, and the smaller ones are essentially invisible.

Low T	High T

For FM (Figure 6 (a)-(b)), we see two bright spots at $(0,0,0)$ and $(\pi,\pi,\pi)$ at low temperature, which correspond to the ${\bf q}$ values for ideal ferromagnet (see Table 1). At higher temperature $T=0.06t$, its weight goes down, but other ${\bf q}$ are still negligible, as the temperature is lower than $T_c$.

The A-type phase should ideally have peaks at $(\frac{\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$, and $(\frac{3\pi}{2},\frac{3\pi}{2},\frac{3\pi}{2})$, or one of its symmetry related ${\bf q}$s (Table 2). However, at low temperature, as in Figure 6(c) it has comparable weights corresponding to A$_1$ and A$_4$, and few subdominant neighbouring ${\bf q}$s. These weight quickly diminish with increasing temperature (Figure 6(d)).

The Figure 6(e) shows the structure factor at density where we expect the flux phase. Here the maximum weight lies close to ${\bf q}$ values corresponding to flux phase (Table 1), but the weight is more dispersed, and there are a lot of sub-dominant ${\bf q}$s. With increasing temperature, they diminish quickly.

Using the structure factor data, we establish the $n-T_c$ phase diagram for $\Delta=0,4,10$ that is plotted in Figure 7. The Monte Carlo captures mainly three collinear phases, namely FM, A-type, and a $\uparrow\uparrow\downarrow\downarrow$ phase. The $\uparrow\uparrow\downarrow\downarrow$ phase corresponds to two FM up planes followed by two FM down planes and so forth. As the carrier density is increased by increasing $\mu$, we find a FM phase followed by the A-type AF. A $\uparrow\uparrow\downarrow\downarrow$ phase appears in a thin window surrounded by FM itself. We suspected that this as a finite size effect, and a comparison with the energy of the FM on larger lattices (20³), shows that the FM is indeed the ground state in the thermodynamic limit, and so we consider FM and $\uparrow\uparrow\downarrow\downarrow$ collectively as FM only, and presence of $\uparrow\uparrow\downarrow\downarrow$ is not indicated in the phase diagram.

The FM is stable at the ends of the density window, and its region of occurrence is slowly enhanced as we increase $\Delta$, see Figure 4 as well. The $T_c$ however decreases with increasing $\Delta$ since the degree of B-B$^{\prime}$ mixing (and kinetic energy) decreases.

With increase in $n$ the 2D system is known to make a transition to a line-like phase, and then a ‘G type’ phase (up spin surrounded by down, etc). The equivalent in 3D would be a progression from FM to a ‘planar’ (A type) phase, then a ‘line like’ (C type) phase and finally to a G type phase if possible. Numerical complexity and the geometric constraint modify the picture as below.

We do access the A type phase with some difficulty but our Monte Carlo cannot access the long range ordered C type phase. However, we see clear evidence of C type correlations in the structure factor. Comparing the energy of the ideal C type phase with the short range correlated phase that emerges from the MC we infer that C type order is indeed preferred over a parameter window. However, we cannot estimate a reliable $T_c$ scale. In the next section we will see the variational results on the C type phase and get a rough estimate of the $T_c$.

The G type phase is geometrically disallowed on the B sublattice due to its FCC structure. An examination of the structure factor in the density window $n =[1,2]$ suggests ‘flux’ like correlations at small $\Delta$ which evolves into a spiral at larger $\Delta$. The frustration reduces the $T_c$ of the phases in this density window compared to that of the FM. We can calculate the energy difference between a fully random spin configuration and the variational ground state. This $\delta E$ serves as a crude measure of the $T_c$ of the phase.

When $t'=0$, the electron delocalisation happens through B-B$'$-B paths only (see the conduction paths, for example of collinear phases A and C in Figure 3(b) and Figure 3(d) respectively). In this case all the phases have an atomic level located at $\epsilon_{B'}(=0)$ in the limit $J\rightarrow \infty$. This is directly seen in the density of states (DOS) of these phase. In Figure 8 we show the DOS for the F, A, C, ‘flux’ and paramagnet phases. This dispersionless level gives constant $T_c$ in density region $n=[1,2]$. This feature, and several others, are modified by finite B$'$-B$'$ hopping, which leads to broadening of this level. It makes the DOS of the various magnetic phases asymmetric (in energy) and also destroys the particle-hole symmetry in the phase diagram.

2.2Variational scheme¶

Using the variational scheme discussed earlier one can establish a ground state phase diagram. This is consistent, overall, with correlations that we see in the Monte Carlo, and now allows us to compare candidate phases on large sizes. This establishes the window of existence of the collinear phases, FM, A and C.

For the middle part of the density window no simple phase is suggested by the Monte Carlo. The variational scheme suggests the phases to be SP$_1$, SP$_2$, SP$_3$, and ‘flux’. See the configurations in Figure 3 (e)-(f) and $S({\bf q} )$ details from Table 1.

The variational scheme also allows a rough estimate of the $T_c$ scale of the ordered state. Ideally, one should compute the energy of ‘spin wave’ configurations about the non trivial ground state, fit these to an effective Heisenberg model, and then work out the $T_c$ of that model. Here we simply compute the ‘binding energy’ of each ordered phase, i.e, its energy gain with respect to the paramagnetic phase at the same electron density, and use the size normalised value as an estimate of $T_c$.

where $E_{VC}$ corresponds to the variational minimum and $E_{para}$ to the paramagnet.

This calculation is no substitute for the full MC, and is only meant to supplement the MC based $T_c$ information and provide a rough estimate where MC is noise limited.

3Results: next neighbour hopping¶

The model with only ‘nearest neighbour’ (BB$'$) hopping has a rich phase diagram. However, this has the artificial feature of a non dispersive level. In reality all materials have some degree of B$'$B$'$ (Sanyal et al. (2009)) hopping due to larger size of B$'$ ion, and we wish to illustrate the qualitative difference that results from this hopping. We explored two cases, $t'=0.3$ and $t'=-0.3$ for these particle-hole asymmetric cases.

The $n-\Delta$ ground state phase diagram for $t'=\pm 0.3$, obtained by a combination of MC, and variational calculation, is shown in Figure 10 . Turning on $t'$ has a significant effect on the phase diagram, compare with the $t'=0$ case, Figure 4. The particle hole symmetry $(n \rightarrow 3-n)$ is destroyed but a reduced symmetry $(n,~\Delta,~t')\rightarrow (3-n,-\Delta,-t')$ still holds. The phase diagram is richer in the middle of the density window where crossing among various phases occurs at different densities. Due to the symmetry mentioned above it is enough to discuss the $\Delta > 0$ case with $t'=\pm 0.3$.

The A-type phase becomes very thin in the left, but unaffected by $\Delta$, while in right side it widens up in the low $\Delta$ and gets replaced by the spiral quickly as we go up in $\Delta$. ‘flux’ and C-type both become stable for high $\Delta$ with a gradual shift in the high density. For $t'=-0.3$, at very small $\Delta$ in the left and the middle part A-type and the spiral are major candidates with small window for C-AF and ‘flux’. The behaviour in this part is not very sensitive to sign of $t'$.

Focusing on $t'=-0.3$, as go up from $\Delta=0$ to $\Delta\sim 5$ the AF phases become less and less stable and are almost wiped out from the left part of the density, and FM becomes stable there. The largest stability window of FM occurs roughly near $\Delta \sim 5$, where its stable up-to $n \sim 1.8$. Going further with higher $\Delta$, FM looses its stability, from C type, ‘flux’ and spirals. However, there is very thin strip of stability of the FM in the band edge in the left part, and towards the middle density, there is re-entrance of the FM phase. In the right part of the density window we have FM, A type and spiral. Increasing $\Delta$ reduces the stability of A type to FM, making it vanish near $\Delta\sim 7$, while FM window keeps increasing with $\Delta$.

Overall, $t'\Delta > 0$, shows a rich set of antiferromagnetic phases, while $t'\Delta < 0$, gives a ferromagnetic window, suppressing antiferromagnetic phases.

3.1Monte Carlo¶

Low T	High T

In Figure 11(a),(b) we show the structure factor data at two densities, for (a) A type and (b) C type phases, to demonstrate one remarkable difference from the particle-hole symmetric case. Figure 12 shows the full structure factor at low (T=0) and some high temperature ($T=0.05t$) at the same densities for A and C type phases.

As we saw earlier in Figure 5 (temperature dependence) and Figure 6 (full $S({\bf q})$) for $t'=0$ the structure factor data were very noisy for AF phases, with many sub-dominant q peaks around the central peak. The saturation value for the A-type peak in the symmetric case was $\sim 10^{-2}$, while now it is $\sim 0.2$, close to the ideal value of 0.25! This is also evident for the full structure factor. Figure 12(a) shows clean peak at $(\frac{3\pi}{2},\frac{\pi}{2},\frac{\pi}{2})$ and $(\frac{\pi}{2},\frac{3\pi}{2},\frac{3\pi}{2})$, with other ${\bf q}$s essentially zero. The largest weights of C-type (Figure 12(c)) are close to $(\pi,0,0)$ and $(0,\pi,\pi)$, with very few subdominent ${\bf q}$s (recall table Table 1 for ${\bf q}$ values of clean phases).

The sharp change in the structure factor makes the identification of the $T_c$ scale more reliable. Although inclusion of $t'$ does not remove the noise completely, it is reduced over a reasonable part of the phase diagram.

Figure 13(a) presents the $n-T_c$ phase diagram for $t'=0.3$ and $\Delta=0$ established from Monte Carlo, along with the $\delta E$ from the variational approach (Figure 13(b)). In this case, the phases that appear as a function of density $n$ are FM, A type, spiral, C type, A type and FM again. For FM, the window of stability gets reduced in the left (low density) part but enhanced in the right (high density) part. The $\uparrow \uparrow \downarrow \downarrow$ phase appears again, but being a finite size artifact, is absorbed in the FM (and not shown).

The $T_c$ is usually reduced, from the symmetric ($t'=0$) case, as $BB'$ hopping provides conduction paths that are non-magnetic. There is a wider space with moderate $T_c$ for A type phase, located asymmetrically in density. It is more stable, in the right window, than left window. The correlations of spiral and C type phases are also captured with relatively less noise, see Figure 11(b) for example of C-type correlation. The $n-T$ phase diagram for $t'=-0.3$ and $\Delta=0$, can be obtained from transformation $n\longrightarrow 3-n$, i.e., reversing the density axis of Figure 13(a),(b).

3.2Variational scheme¶

We also used the variational scheme to get an independent feel for the ground state and the $\delta E$ for $t'=\pm0.3$. For $t'=0.3$ the trends from MC are well reproduced by the variational scheme on large $(20^3)$ systems at \D=0. We observe reduced stability of FM at low density and enhancement at high density.

Note that the overall correspondence between the Monte Carlo and the variational approach is much better here than in the $t'=0$ case, Figure 7 and the Figure 9.

In Figure 14 we have shown the $\delta E(n)$ calculated for 20$^3$ size, for $t'=\pm0.3$. For $t'=0.3$ and $\Delta=0,4,10$ (Figure 14(a)-(c)) shows asymmetric $\delta E$ (or $T_c$), for ferromagnet. The stability windows of the A type is enhanced, the large window of flux phase that was for $t'=0$, is reduced in competition to spiral and C type. For the $t'=-0.3$, $\Delta=0$ (Figure 14(d)) is just density reversed due to symmetry. However, the intermediate $\Delta=4$ (Figure 14(e)) has a large stability window of ferromagnet. Here, though the $\Delta$ and $t'$ are non-zero, due to unusually large stability window, the $\delta E$(or $T_c$) is large. For larger $\Delta=10$ (Figure 14(f)), the ferromagnet appears in the middle and right with moderately reduced $\delta E$ (or $T_c$), while the same is heavily suppressed for antiferromagnetic phases.

To summarize, from the MC and variational data we learn that, apart from asymmetry in the phase diagram, collinear FM and A type phases become stable in wide density window. Their $T_c$ however is slightly reduced than the symmetric case. The S(q) data showing less noise for A, C type and spirals indicates that the energy landscape become ‘smoother’ by $t'$ so that annealing process becomes easier to get to the ground state. The energy differences $\delta E$ as well as MC estimated $T_c$s show overall decrease with $t'$. This is understandable as, by introducing $t'$ we allow electrons to more on the ‘non-magnetic’ sublattice B$'$. Now the energy of any phase, depends on the energy gain via the hopping process. From the nearest $f-m$ hopping, this gain scales as $\frac{t^2}{\Delta}$ subject to spin configurations, while the from the next nearest $m-m$ hopping, this gain simply scales as $t'$, and doesn’t care upon spin configurations. So more we increase $t'$ and $\Delta$, the more we are making the energy of the system insensitive to spin-configurations. The asymptotic limit of this is $\frac{t^2}{\Delta}\rightarrow 0$ when every phase has same energy as paramagnet.

In the couple of paragraphs below we try to create an understanding of how the phase diagram is affected by $t'$. Unfortunately we do not understand the effects over the entire density window yet.

For $t'=-0.3$, the FM loses its stability to AF phases even at low $n$. That is puzzling since one would expect the FM phase to have the largest bandwidth. We recall that in the $t'=0$ case, there is a localized band coming from B$'$ level for all the phases. The dispersion of this previously localized level causes the $m$ and $f$ to have a ${\bf k}$ dependent separation, which was $\Delta$ for all ${\bf k}$ in the symmetric case. The separation for these levels in the asymmetric case is $\Delta_{\bf k}=\Delta - \epsilon'_{\bf k}$, which varies from $\Delta - 12|t'|$, to $\Delta + 12|t'|$ in 3D. In 2D it varies from from $\Delta - 4|t'|$, to $\Delta + 4|t'|$. Figure 15 shows the lowest eigenvalues of F,A,C and flux phases with $\Delta$ for $t'=\pm0.3$ and we see how a crossing occurs at $t'=-0.3$.

We make a general observation about the change in the ‘energy landscape’ of the model, in the space of spin configurations, with changing $t'$. If $t' \gg t$ then the electrons could delocalize on the wide $t'$ based band populating the non magnetic sites only. Magnetic order would make little difference to electronic energies and the energy landscape would be featureless. There are no ordered states as minimum in this landscape. Contrast to $t'=0$, where delocalisation takes place necessarily through the magnetic sites and deep minima in in the landscape represent ordered states in real space. There are presumably shallow metastable states possible too.

At intermediate $t'$ the ordered states are shallower, since the gain from magnetic ordering is lower, but the metastable seem to be affected even more, if the relative ease of Monte Carlo annealing is any indicator.

4Discussion¶

The real double perovskites are multi-band materials, involving additional interaction effects and antisite disorder beyond what we have considered here. Nevertheless, we feel it is necessary to understand in detail the phase diagram of the `simple’ model we have studied, and then move to more realistic situations. In what follows, we first provide a qualitative comparison of the trends we observe with experimental data, and then move to a discussion of issues that are ignored in the present model.

4.1Comparison with experiments¶

There is limited experimental signature (Jana et al. (2012)) of metallic AF phases driven by the kind of mechanism that we have discussed. So, the comparison to experiments is, at the moment, confined to the $T_c$ scales (Navarro et al. (2001), Park et al. (2009)) etc, of the ferromagnetic DP’s.

From the ab initio calculations (Sanyal et al. (2009)) the sign $t'/t\sim 0.4 >0$ and $\Delta > 0$, so one can compare the right panel of Figure 4 qualitatively with experiments.

In a material like SFMO the electron density can be increased by doping La for Sr, i.e., compositions like Sr$_{2-x}$La$_x$FeMoO$_6$. This was tried (Navarro et al. (2001)) and the $T_c$ increased from 420K at $x=0$ to $\sim$ 490K at $x=1$. SFMO has threefold degeneracy of the active, $t_{2g}$, orbitals while we have considered a one band model. When we create a correspondence by dividing the electron count by the maximum possible per unit cell (3 in our case, 9 in the real material), in our units $x=0$ corresponds to $n=0.33$ and $x=1$ to $n=0.66$.

When $t'=0$, as a function of $n$ the $T_c$ peaks around $n=0.2$, Figure 7, quite far from the experimental value. However, in the presence of $t'=-0.3$ and $\Delta=4$, Figure 14(e), the peak occurs above $n=1$. So, modest $t'$ can generate the ferromagnetic window that is observed, and produce a $T_c \sim 0.1t$. For $t \sim 0.5$eV, this is in the right ballpark.

Ray et al experimentally estimate the onset of AF order upon La doping to be $x\sim 1.1-1.5$, which corresponds to $n\sim 2.1-2.5$. Our onset of AF order, for \D in [0-10], is $n\sim 0.5-0.75$, which for SFMO would be $n\sim 1.5-2.1$. This is quite close to experiments. However, inclusion of finite $t'/t>0$, pushes this to lower values.

4.2Additions to the model¶

(i)~Effect of orbital variables: Apart from renormalization of the electron count, multiple orbitals could, in principle, have a qualitative effect. If the local orbital degeneracy is lifted by Jahn-Teller effect then the resulting ‘orbital moment’ could order in some situations. This orbital ordered (OO) background can modify electron propagation and the magnetic state. This is known to happen at some dopings in the manganites. Even there, however, the broad sequence of magnetic phases is consistent with predictions from a one band model. The double perovskites do not seem to involve strong lattice effects, the orbital degeneracy survives, and there is no orbital order. This suggests that there is even better chance of a one band model being qualitatively correct here, compared to the manganites.

Had there been strong OO effects, the spin-spin coupling in that background may have picked up strong directionality, and the geometric frustration could have been irrelevant. This does not seem to happen in most double perovskites. Indeed, there are experiments on the insulating DP’s (Wiebe et al. (2002), Aharen & others (2010), Aharen & others (2009)) where the geometric frustration of the FCC lattice leads to a non trivial magnetic state, unrelieved by the presence of multiple orbitals.

We of course expect that the phase boundaries and $T_c$ scales that we calculate would be affected by the orbital degeneracy. However, the trends in the phase diagram with increasing density simply reflect a growing AF tendency and the non coplanar phases emerge due to the impossibility of Neel order.

(ii)~Antisite disorder: Attempts to increase $n$ via A site substitution also brings in greater antisite disorder (B-B$'$ interchange) and even the possibility of newer patterns of A site ordering (!) complicating the analysis. For example, one would try compositions of the form: A$_{2-x}$A$'_x$BB$'$O$_6$, where A and A$'$ have different valence in an attempt to change $n$. The assumption is that the A$'$ only changes $n$ without affecting other electronic parameters, i.e., A$'$ ions do not order and remain in an alloy pattern. This may not be true. In fact, at $x=1$, the material AA$'$BB$'$O$_6$ may have a specific A-A$'$-B-B$'$ ordering pattern that affects electronic parameters in a non trivial way and one cannot understand this material as a perturbation on A$_{2}$BB$'$O$_6$. In such a situation one needs guidance from experiments and ab initio theory to fix electronic parameters as $x$ is varied. All this before one even considers the inevitable antisite (B-B$'$) disorder and its impact on magnetism (Singh & Majumdar (2011)).

## Conclusions

We have studied a one band model of double perovskites in three dimensions in the limit of strong electron-spin coupling on the magnetic site. The magnetic lattice in the cubic double perovskites is FCC and increasing the electron density leads from the ferromagnet, through A and C type collinear antiferromagnets, to spiral or ‘flux’ phases close to half-filling. We estimate the $T_c$ of these phases, via Monte Carlo and variational calculation. The introduction of B$'$B$'$ hopping $t'/t \sim 0.3$ significantly alters the phase diagram and $T_c$ scales and creates a closer correspondence to the experimental situation on DP ferromagnets.