Open Access
Add to BibSonomy
Abstract
We report a study of surface reconstruction effects in yttrium iron garnets using density functional theory. This work responds to the need to explain the physical basis of recent experimental results showing a significant enhancement in Faraday rotation in iron garnets. These materials are extensively used in the telecom industry for nonreciprocal device applications. Understanding the physical basis of the heightened Faraday response at the surface is important for the development of ultrathin nonreciprocal devices. Our results show that the bandgap near the surface is significantly reduced compared to that of the bulk, and that spin-orbit coupling effects become more important near the surface. We find that the decrease in band gap results in an enhancement and change in direction in the Faraday rotation in the visible range all the way to the deep red. Electronic transition matrix elements for surface-sensitive ultra-thin layers are calculated and compared to those of bulk samples, leading to the conclusion that the octahedrally-oxygen-coordinated iron sublattice in these ferrimagnetic materials is more strongly affected by surface reconstruction than the other, anti-ferromagnetically coupled, tetrahedral sublattice. We explain how these changes contribute to the enhancement in magneto-optic response near the surface as compared to the bulk.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Yttrium iron garnet ($Y_{3}Fe_{5}O_{12}$, YIG) is a ferrimagnetic insulator used extensively in radars, attenuators, optical isolators and circulators, filters, and switches [1–12]. Recent studies in iron garnet films have shown significant enhancement in their near-surface magneto-optic response compared to the bulk [13–18]. They provide experimental analytical evidence relating surface reconstruction to changes in specific Faraday rotation within a few nm of the surface.
The magneto-optic Faraday Effect (MOFE) produces a nonreciprocal optical response for light propagating in iron garnets. There have been many experiments performed to study MOFE properties, but very few first-principles calculations relating band structure changes to the magneto-optic (MO) response. There has also been no study of their surface properties in iron garnets. This paper reports density functional theory (DFT) calculations using the Vienna Ab-Initio Simulation Package (VASP) software to study the magneto-optical properties of the top surface slab with respect to the bulk.
The present work analyzes surface reconstruction in yttrium iron garnet for a (100) surface and compares the changes in the density of states, optical absorption, and Faraday rotation to those of the bulk material. Surface reconstruction refers to the formation of a different atomic structure at the surface from that of the bulk of the material. This restructuring is a result of the introduction of a discontinuity in the periodic structure of the crystal by the surface. In our case, it means a change in atomic locations at the surface compared to the bulk in the in-plane direction. Surface ionic relaxation is also considered here, where the change in the atomic location from bulk in terms of the bulk coordinate system is considered. The response of the top 1 nm-thick surface layer is compared to that of the bulk. We find a significant narrowing in the optical bandgap at the Fermi level as well as spin-orbit coupling effects on the surface-sensitive model. Its effects on the Faraday rotation response at the surface compared to the bulk are studied.
Recent experimental works have studied changes brought about in the Faraday response in ultra-thin films and by directly probing surface effects in thicker films. These studies have sought to investigate surface-sensitive properties in these materials [13–18]. The present study looks into the physical origin of these differences, as well as possible areas of miniaturization for these materials. We predict a multi-fold increase in specific Faraday rotation at the surface over the bulk in the green to deep red spectral region, and even higher at shorter wavelengths in the visible regime. This is discussed in detail in Section 3.4, below.
This paper will describe the first-principles density functional theory (DFT) methodology utilized to investigate the magneto-optic (MO) properties. The results of this study will be outlined in Section 3, and a summary of the findings will be provided in Section 4.
Faraday rotation for surface reconstructed YIG materials is compared in the present work to that of bulk configuration. Linearly polarized light entering a magneto-optic material in the magnetization direction suffers a rotation in the plane of polarization [19].
This rotation for a given magneto-optic material probed at frequency $\omega$ below the bandgap is given by
where $\Theta _{F}$ is the specific Faraday rotation, $\eta _{+}$ and $\eta _{-}$ are the refractive indices for the right circularly polarized (RCP) light and the left circularly polarized (LCP) light, respectively. $\eta _{+}$ and $\eta _{-}$ refractive indices arise because of the difference in virtual transition probabilities for spin-up and spin-down electrons in the system. These indices are related to the dielectric permittivity as shown in Eq. (2):The difference in refractive indices for RCP and LCP photons is parameterized by the off-diagonal components in the dielectric permittivity tensor. These components, given in Eq. (3), relate to the gyrotropic efficiency of the Faraday rotator.
We examine the case of materials magnetized in the 001 direction for the bulk. The corresponding dielectric permittivity tensor is shown in Eq. (3) (Dionne) [19]. This orientation has a four-fold symmetry. The Faraday rotation is calculated in Section 3.4 by computing the difference in refractive index for opposite optical helicities. Calculation of the dielectric polarizability (which gives rise to the gyrotropy) for surface and bulk is used, based on the virtual electronic transitions.
Virtual transitions in YIG operate through an inter-sublattice covalent model because the spin is conserved in these processes, as described in Dionne [19], Fig.7.14. Here, as proposed by Clogston [20], the transition follows the selection rules $\Delta$S = 0 and $\Delta$L = $\pm$1, by acquiring orbital angular momentum from hybridization with the excited-state terms. So, the ground state (GS) term of octahedrally-oxygen-coordinated $Fe^{3+}$ ions hybridize with the neighboring lowest excited-state (ES) term of tetrahedrally-oxygen-coordinated $Fe^{3+}$ ions and vice-versa.
The difference in the virtual transition probabilities from $^{6}S^{O}$ to $^{4}T_{1g}^{T}$(spin down transition) levels and from $^{6}S^{T}$ to $^{4}T_{1g}^{O}$ (spin up transition) levels gives rise to the Faraday rotation. Here the superscripts O and T stand for octahedrally-coordinated, and tetrahedrally-coordinated, respectively. The spin-up and spin-down transition energies decrease with a decrease in the bandgap, thus yielding an increase in the refractive index [21]. The difference in virtual transition energies and in electronic transition probability amplitudes for RCP and LCP photon-induced excitations, translates into a Faraday rotation. The presence of incident RCP or LCP probe photons with energies below the band gap, disturbs the stationary electronic states of the system. Ground and excited states are thus coupled, even though the probe photon does not have enough energy to induce real optical absorption. This produces new stationary states (eigenstates) in the form of quantum superpositions of two previously unperturbed states: 1. probe-photon and ground-state electron and 2. excited electron without probe photon. These new superposition states disturb the phase acquired by the probe photon traversing the magneto-optic sample as opposed to vacuum. Given that RCP- and LCP- induced virtual transitions are different because of selection rules, RCP and LCP photons acquire different phases. That phase difference produces a Faraday rotation, since the latter is, in fact, a rotation in linear polarization. This is the result of the phase difference between its circular polarization components. The change in band structure at the surface changes the rate of virtual transitions for RCP and LCP photons as well. The transition analysis for bulk and surface is carried out in detail in Section 3.2.1. Further, we study of the change in virtual transitions in Section 3.4, where we show that the change in band structure at the surface favors enhancement in Faraday rotation.
In the bulk, the spin-orbit coupling does not play much of a role in the Faraday rotation for unsubstituted YIG. But the Y orbitals play a significant role in spin-orbit coupling for surface-sensitive electrons. Electronic transition analysis also shows a further splitting of the $d_{eg}$ and $d_{t2g}$ levels. There is a transition from the high spin-orbital GS term to low spin-orbital term for the Fe ions. This results from high crystal field splitting energy compared to the spin-polarized virtual transition energy gap (Chapter 2 , Dionne [19]). The result of this low spin state on the surface is the combination of the Jahn-Teller (J-T) effect with S-O coupling.
Our findings suggest a correlation between a decrease in film thickness and an increase in Faraday rotation, accompanied by a change in direction after reaching the point of minimum FR. In Section 3.4, we provide a comprehensive electronic transition analysis for spin up and spin down, both for bulk and surface (100) models.
2. Material system: structure and computational methods
Many experimental data and first-principles analyses have been conducted by other researchers for bulk YIG. We use the parameters given in this section to validate and achieve convergence for our bulk YIG model with the experimental and theoretical experiments conducted in the past. Using the bulk parameters as our initial data set, we conduct DFT studies for the surface slab model to study the importance of surface reconstruction on the band gap, hence the Faraday rotation.
Bulk YIG is of Ia3d space group cubic structure [22,23] as shown in Fig. 1(a). The Fe ions are located in tetrahedrally- ($Fe^{T}$) and octahedrally-oxygen-coordinated ($Fe^{O}$) sub-lattices, and the yttrium atoms are in a dodecahedrally-oxygen-coordinated sub-lattice [24]. Each unit cell contains 16 Fe atoms in the octahedral positions (16a), 24 Fe atoms in the tetrahedral positions (24d), 96 oxygen atoms at 96h positions, and 24 yttrium atoms in 24c positions in the conventional unit cell. Thus, there are two octahedrally-coordinated $Fe^{3+}$ ions and three tetrahedrally-coordinated $Fe^{3+}$ ions per formula unit (f.u). Experimentally, the lattice constant a = 12.37 Å [22].
Fig. 1. (a) $\frac{1}{8}$ of a YIG conventional unit cell. The oxygen atoms are shown as red, yttrium atoms are shown as silver spheres, $Fe^{T}$ are shown as green spheres, and $Fe^{O}$ atoms are shown as brown spheres. The three-dimensional configuration of the primitive elements in each sub-lattice is depicted in Fig. 1(a) (b) Slab model of the (100) YIG conventional unit cell with 20 Å vacuum. The slab consists of 10 layers. The rectangle denotes the first layer of the slab. The square on the right shows the (100) slab surface. Six atoms were picked and labeled (1) and (2) (two for each coordination type). Their relative positions are studied for surface reconstruction in Table 2.
We construct the surface slab model of 10 layers for a (100) surface with 20 Å as a vacuum between two super-cell layers shown in Fig. 1(b) using vesta and the atomic simulation environment [25]. We use the Wyckoff atomic positions and the experimental lattice parameter for YIG (Table 1) for all the bulk and surface slab YIG calculations. In our earlier experimental results of ultra-thin iron garnets for near-surface studies, we get an enhanced magneto-optic response in (100) surface [16]. This motivates our first-principle calculations on its parent compound YIG for the same orientation. We focused on YIG, the parent compound to Bi-iron garnets, as a step towards a broader understanding of the behavior of Bi-substituted compounds.
Table 1. Listed below are the structural parameters of $Y_{3}Fe_{5}O_{12}$ that were used for the DFT calculations. The experimental lattice constant for YIG is a = 12.37 Å [22].
The (100) surface was terminated at $Fe^{T}$. Due to symmetry, the termination at $Fe^{T}$ and Y has the same result. $Fe^{O}$ termination has higher surface energy. The result below was calculated for the (100) surface with $Fe^{T}$ termination.
In order to gain convergence for the bulk, we compare the ground state energy and the lattice constants of the material with the experimental value. We also cross-check our initial simulation results for the density of states and bandgap with the materials project database to gain convergence [23].
For the surface-slab model the position for the first layer $Fe^{T}$, $Fe^{O}$, and Y atoms with respect to the pristine structure positions change by 2.43 %, 1.59 % and 3.04 %, respectively. The $Fe^{T}$, $Fe^{O}$, and Y bonds with oxygen distort at the surface as a result of the structural discontinuity introduced by the break in periodicity. These changes minimize the Free energy at the surface-vacuum interface. The relaxed layers reach convergence after 4 layers of asymmetric relaxation from the surface. In addition, the surface was also allowed to reconstruct in the x-y direction. We do not perform a long-range reconstruction here. The x-y reconstruction was allowed in the 1*1 conventional super-cell. The surface energy was also considered to achieve convergence in this process. As a result, we observe an increase in p-orbital charge density in the $Fe^{O}$ ions. Concomitantly, there is also an increase in d-orbital charge density in the $Fe^{T}$ ions.
The initial structural parameters and the subsequent calculations including the bulk and surface calculations were performed using VASP. For all the calculations we use the generalized gradient approximation (GGA) with Projector Augmented Wave pseudopotential (PAW) [26,27]. This was combined with the exchange-correlation functional (DFT + U). The U values are introduced to take care of the spurious self-interaction energy by GGA [28] for the exchange-correlation functional.
For the density of states calculations for the bulk structure, we use U-J = 3 eV , which gives the closest bandgap to the experimental results i.e. 2 eV for spin up and 1.6 eV for spin down (experimental bandgap = 2.4 eV [7]). The k-mesh chosen for the bulk simulations were 3x3x3 and for the surface simulations were 3x3x1. A finer mesh was used for the density of states plots (8x8x8 for bulk and 8x8x1 for the slab model). The plane wave energy cut-off was chosen at 400 eV for the bulk and 500 eV for the surface. For the surface slab model, the spin-orbit coupling was considered.
DFT+U calculations were performed for all elements for the surface density of states calculations. Magnetic moments and the lattice constants remained constant for various U parameters; whereas the density of states and the band structure changed significantly. For the results below corresponding to the slab model at the surface, we chose U-J = 5 eV, 3 eV, and 2 eV for Fe, O, and Y, respectively for the d , p and d orbitals respectively. These effective U values were considered for their lowest ground state energy and to maintain the semiconductor character for the slab model. There is a possibility that a YIG (100) surface could have a metallic bandgap. But, with no current experimental evidence, we perform our calculations for a small bandgap. The d orbitals of Y enhance the spin-orbit coupling in Fe d electrons; thus an effective U for Y d orbitals was investigated for the surface model. The spin-orbit coupling yielded an energy minimum along the (100) axis. The magnetic field for the surface slab was turned on for the (100) axis i.e., parallel to the spin-orbit coupling results. Similarly to our experimental results from S-TEM for bismuth substituted iron garnets [16], we don’t see any change in the lattice constant for the surface slab model when compared with the bulk model, although the bond lengths for the surface atoms get distorted. (theoretical lattice constant for YIG, a = 12.35 Å). Thus, we see significant surface reconstruction effects for YIG for a 1 nm depth from the surface with 0.5 nm layers relaxed. In Table 2, we show the change in distance and angles for two sets of $Fe^{T}$, $Fe^{O}$, and Y atoms from the first relaxed and reconstructed layer. We see that the surface reconstruction mostly affects the $Fe^{O}$ atoms. From our earlier S-TEM HAADF images for bismuth-substituted iron garnets, we see the distance between two atoms for bulk to surface change from 3.08 Å to 2.94 Å or 2.21 Å [16]. Our present YIG results have reconstruction in the surface to a similar order.
Table 2. Distances and angles for two sets of $Fe^{T}$, $Fe^{O}$ and Y atoms. The signifiers (1) and (2) stand for the positions of two atoms in the bulk and the top relaxed surface layer (shown as (1) and (2) in Fig. 1(b))
3. Results
3.1 Density of states
Figures 2(a) and 2(b) show the density of states calculated for bulk and surface models, respectively. The splitting of energy levels in the YIG bulk density of states is less impacted by spin-orbit coupling than the YIG surface.
Fig. 2. (a) Density of States of YIG Bulk model with VBM as 0 eV. (b) Density of States of YIG Surface slab (100) model with VBM as 0 eV.
The occupation number difference between the two energy-split levels caused by the hybridization of the GS and ES is the major cause of the magneto-optical properties in the YIG bulk at finite temperatures. But, at 0 K the difference in virtual transitions between spin-up and spin-down is the main source of the difference in refractive index responsible for the Faraday effect. No appreciable spin-orbit coupling contributions impact the density of states for the bulk in the simulations [29].
The density of states for the bulk structure was also calculated, and the plots were deconvoluted to spin projection, orbital projection of the bands of the elements, and atomic DOS. The effect of Y electrons is most prominent in the conduction band (d electrons) ($\approx$4.5 eV and above), which is far from the conduction orbitals of Fe ($\approx$2 eV - 3.5 eV) and O ($\approx$2.1 eV - 3.7 eV). This also shows that the Y conduction band is not involved with the Fe and O conduction bands. The spin-orbit coupling effect arises from coupling the ground state of $Fe^{T}$ with the excited state of $Fe^{O}$ atoms and vice-versa. The $Fe^{T}$ atoms have spin-up as the majority spin orientation in the valence band and spin-down as the majority spin in the conduction band. It is also found that the valence band near the Fermi energy is mostly $d_{t2g}$, and the conduction band near Fermi energy is $d_{eg}$. $Fe^{O}$ has spin-down as the majority spin in the valence band and spin-up as the majority spin in the conduction band. The splitting found in octahedrally-coordinated iron at 2.5 eV corresponds to the crystal field splitting and is about $\approx$0.5 eV for YIG bulk (Fig. 2(d)).
For a surface YIG model, the story changes. The Y conduction band is close to the Fe and O conduction bands. The contribution to the density of states from the spin-orbit coupling increases. There is an increase in correlation for Fe, O, and Y as well. The crystal field splitting broadens in the slab model, but we still see a $\approx$0.2 eV gap (denoted with double-ended arrow in Fig. 2). The octahedrally-coordinated Fe $d_{t2g}$ have $d_{yz}$ states as the Fermi-level active components for spin-up electrons, indicating the out-of surface $d_{yz}$ (z-direction) being more sensitive for the surface structure. The bandgap for spin up (spin down) density of states, decreases to 0.1 eV (0.8 eV). This has a huge impact on the refractive index as well as on the Faraday rotation; as shown in Section 3.2. For the unoccupied levels, we see spin up $d_{t2g}$ in $Fe^{O}$ and spin down for both $d_{eg}$ and $d_{t2g}$ in $Fe^{T}$. The change in occupation results in a change in electronic transition amplitude when the material is probed. (shown in Section 3.4)
3.2 Band structure and electronic structure
The band structure for the bulk and surface YIG were calculated along the Kpath. The Kpath was generated for high symmetry points of our system using the vaspkit python module [30,31] (Fig. 3).
Fig. 3. (a) Brillouin zone of YIG bulk with the high symmetry lines [30]. (b) Brillouin zone for YIG surface with high symmetry lines. Along the k-path , the band structure with respect to energy was plotted. (c) Band structure of bulk YIG deconvoluted to spin-up (a) and down (b) states along the k-path ‘$\Gamma$HN$\Gamma$PH’. 0 eV corresponds to the VBM [31]. (d) Band structure of slab YIG deconvoluted to spin-up (a) and down (b) states along the k-path ‘$\Gamma$XSY$\Gamma$’. 0 eV corresponds to the Valence Band Maximum (VBM).
For YIG bulk, the band structure plotted in Fig. 3(c) shows that the conduction band minimum (CBM) and the valence band maximum (VBM) are both located at $\Gamma$ point. This makes the bulk structure a direct band-gap semiconductor with a bandgap of $\approx$ 2 eV($\approx$1.6 eV) for spin-up (spin-down) transitions. The result underestimates the experimental value $\approx$2.4 eV (Fig. 3(c)). The VBM (CBM) for spin-up corresponds to -0.03 eV (1.9416 eV) and VBM (CBM) for spin-down corresponds to -0.1412 eV (1.3441 eV). Thus, VBM is a spin-up band and CBM is a spin-down band. The spin-up transition takes place from $Fe^{T}$ GS to $Fe^{O}$ ES and the spin-down transition takes place from $Fe^{O}$ GS to $Fe^{T}$ ES orbital, as shown in Section 3.2.1.
The band structure (including spin-orbit coupling) has interesting properties for the YIG surface. First, what is noticeable and significant is that the band structure for the spin up and spin down reverses in the slab model. In bulk, the band gap of spin-up is greater than spin-down, whereas, in the surface spin-down band gap is greater than spin-up. The second point to notice is the reduced bandgap for the surface-sensitive structure compared to the bulk structure. We use these two salient features to analyze the Faraday rotation dependence on the surface structure.
In the band structure and density of states analysis, the PAW-GGA+U predictions systematically underestimate the bandgaps. The actual band gap for thin samples is expected to lie within the systematic error gap of PAW-GGA+U for the predicted value (as in the case of the bulk bandgap). Nevertheless, the trend of a band gap reduction at the surface compared to the bulk, and the findings shown in Section 3.4 behind the trend both hold true. The decrease in bandgap for thinner YIG samples has also been shown experimentally, [32] where the bandgap for YIG was found to be lower by 1 eV from that of bulk YIG for a 19-nm-thick film.
Another significant point to be considered in the band structure analysis is the indirect bandgap in the surface model compared to the direct bandgap in the bulk model. An indirect bandgap leads to poor optical conductivity [33]. Most of the bands for spin-up in the conduction band near the Fermi level come from the dangling oxygen atoms. A comparison, in Fig. 2(b), of the total density of states (a) and those for oxygen (b), shows that most of the near-Fermi level states comprise oxygen atoms at the surface. As oxygen atoms are not involved in the Faraday rotation effect, the enhancement in Faraday rotation should be inhibited by it. We performed an electronic-transition matrix analysis to study the effect of the indirect bandgap in the optical transitions and the presence of dangling oxygen atoms at and hence the Faraday rotation (Details in Section 3.4).
3.2.1 Electronic transition analysis
In this section, we examine in more detail the transition mechanisms in YIG. Figure 4 shows the Fe band structure deconvoluted to the projected d-orbitals.The bubble size in the figure adds another dimension to the number of the occupied or the available bands at the particular point in energy and the k-path, and represents the density of data points. As the quantum origin of the Faraday rotation stems from the difference in transition rates for spin-up and spin-down electrons in the Fe ions, the projected orbital band structure configuration allows one to examine the occupied d-orbitals below Fermi energy for bulk and surface YIG. It also displays the unoccupied orbitals in the conduction band (CB) in Fe where the transition is more probable. The plots are deconvoluted to spin up and spin down Fe orbitals near the Fermi level. Further, each plot is deconvoluted into $d_{eg}$, and $d_{t2g}$ orbitals.
Figure 4(a) and Fig. 4(b) show the bulk occupied and unoccupied bands and Fig. 4(c) and Fig. 4(d), the surface occupied and unoccupied bands at the Fermi level.
Fig. 4. High resolution Fe band structure for bulk (top) and surface models (bottom) deconvoluted to spin $\uparrow$ (left) and spin $\downarrow$. Each figure is further deconvoluted to its orbitals $d_{eg}$ (cyan) and $d_{t2g}$ (blue)
In the bulk structure, all 3d level orbitals have single electron occupancy and there are degenerate bands at 0K. The Jahn-Teller theorem for a partially filled degenerate set of orbitals says that the degeneracy will be lifted because the system will go through a distortion that lowers its electronic energy [34]. So, from ligand symmetry we have two different distorted 3d orbital for octahedrally-coordinated Fe ions and tetrahedrally-coordinated Fe ions. The 3d orbitals split into $d_{t2g}$ and $d_{eg}$ levels. For the octahedrally-coordinated system, $d_{t2g}$ has lower energy than $d_{eg}$ and for the tetrahedrally-coordinated system, $d_{e}$ is lower in energy than $d_{t2}$ [19].
In Fig. 4 we show the occupied and unoccupied states for Fe ions near the Fermi gap. From the selection rules of transition from the GS to the ES we have $\Delta S = 0$ and $\Delta L = \pm 1$ [35]. Our analysis of the electronic transitions responsible for the Faraday effect in YIG follows the treatment given in [19]. The process involves inter-sublattice Fe-electron transitions from hybridized ${^6}S$ ground state levels to ${^4}T$ excited states. This phenomenon is evident in the projected band structure of Fe shown in Fig. 3 and Fig. 4. We see that at the $\Gamma$ point, for the spin-up transition (Fig. 4(a)) the VBM corresponds to $d_{t2}$ levels of $Fe^{T}$ sites and CBM corresponds to $d_{t2g}$ levels of $Fe^{O}$ sites. For the spin-down transition (Fig. 4(b)) the VBM corresponds to $d_{eg}$ levels of the $Fe^{O}$ sites and the CBM corresponds to $d_{e}$ levels of $Fe^{T}$ sites. This shows the inter-sub-lattice transition, which upholds the selection rules. The band structure analysis corresponds and holds true with the molecular orbital theory presented by [19].
In the surface model for the high-resolution Fe band structure, we see that the gap gets significantly reduced, and the band structure broadens along the energy axis near the Fermi level. At the $\Gamma$ point, the spin-up transition from the VBM to the CBM consists of $d_{t2}$ ($Fe^{T}$) to $d_{t2g}$($Fe^{O}$), and the spin-down transition from VBM to CBM consists of $d_{eg}$ ($Fe^{O}$) to $d_{e}$ ($Fe^{T}$) transition. The key feature here is that as a result of spin-orbit coupling, the 3d states are split into non-degenerate $d_{t2g}$/$d_{t2}$ and $d_{eg}$/$d_{e}$ levels at the surface (whereas in the bulk, they are degenerate).
This drastically changes the electronic transition energies as compared to the bulk, where the levels are degenerate. The Faraday rotation for surface-sensitive models, resulting in an enhanced effect, is thus given by splitting the 3d orbitals to $d_{t2g}$/$d_{t2}$ and $d_{eg}$/$d_{e}$ non-degenerate orbitals. There is an inter-sub-lattice transition for the spin up and spin down electrons from $Fe^{T}$ to $Fe^{O}$ site. However, the rates of these transitions are different for the surface compared to the bulk. There is a low probability for spin down $d_{t2g}$ occupied states near the Fermi level compared to almost no probability in the bulk region. The probability of unoccupied spin-up $d_{eg}$ states decreases at the surface compared to the bulk. This is evident in Fig. 4(c) and Fig. 4(d).
We use these results to find a relationship for the band structure with the refractive indices for RCP, and LCP photons [36]. We also use the transition amplitudes to analyze the region of probe wavelengths in depth for enhanced Faraday rotation at the surface compared to the bulk.
3.3 Magnetic moments
Table 3 displays the magnetic moment for YIG. Fe ions of the same type couple ferromagnetically and Fe ions of different types do so anti-ferromagnetically [29]. There are two types of Fe ions in every unit cell i.e., $Fe^{T}$ tetrahedrally coordinated with oxygen ions, and $Fe^{O}$ octahedrally coordinated with oxygen ions,in a ratio of 3:2, aligned antiferromagnetically. The unit cell has $\approx$5.0 $\mu$B total magnetic moment per formula unit (f.u.). In the bulk, the Fe ions were calculated at $\approx$4.0 $\mu$B each. There is a small but not insignificant magnetic moment contribution from the oxygen atoms. So, the oxygen magnetic moment together with the net magnetic moment from the Fe ions gives a total magnetization of $\approx$5.0 $\mu$B p.f.u. The calculated values agree well with measured values [37]. These values are in very good agreement with experimental magnetic moment values. For the surface, with 4 relaxed layers, the Fe ion magnetic moments were calculated to be $\approx$3.9 $\mu$B, for a total magnetization of $\approx$4.0 $\mu$B f.u. indicating a decrease in magnetization.
Table 3. Magnetic moment - spin and orbital with average values of magnetic moment for each type of atoms of bulk and surface YIG with the calculated band gap of bulk YIG (and experimental) and slab YIG.
The surface displays an overall decrease in magnetic moment. The average magnetic moment of $Fe^{T}$ increases from 3.94 $\mu$B/atom to 3.95 $\mu$B/atom and $Fe^{O}$ decreases from 4.03 $\mu$B/atom to 3.9 $\mu$B/atom. The values of the magnetic moments do not remain constant across the lattice, as happens in the bulk. Rather, they vary around a mean value that is displayed in Table 3. The mean value of the magnetic moments in the table is calculated by taking the mean of the magnetic moment for the given atoms in the system. The magnetic moment for each atom was calculated using the VASP’s non-collinear executive. In Table 2, we see that $Fe^{O}$, being most confined, gets affected the most by surface reconstruction. This results in a larger modification in its magnetic moment. These bonds for Fe at the surface are not purely tetrahedral or octahedral. The bonds get distorted, thus having a lower magnetic moment at the surface.
3.4 Magneto-optics properties
In this section, we calculate the Faraday rotation effect for surface and bulk-sensitive models based on bandgap and allowed transitions from occupied to unoccupied bands. This analysis considers the appearance of an indirect band gap in the surface models.
We calculate the specific Faraday rotation using the transition amplitudes for valence bands from -4 eV to 0 eV to the conduction bands using the transition analysis presented in Section 3.2.1. Equation (1) shows that $\Theta _{F}$ depends on the difference between $\eta _{+}$ and $\eta _{-}$. The refractive index depends on the scalar part of the dynamic polarizability tensor as shown in Eq. (4).
where $\rho$ is the atomic number density for YIG and $\alpha (\omega)$ is the scalar part of the dynamic polarizability tensor [38]. The initial and final states corresponding to $\eta _{+}$ and $\eta _{-}$ are the valence and conduction bands for spin-up and spin-down, respectively. The relation of $\alpha (\omega )$ with the transition matrix is given by Eq. (5) [38–40]:Here $A_{i\rightarrow f}$ gives the transition amplitude from the initial ${i}$ state to the final ${f}$ state. $E_{f}$ and $E_{i}$ correspond to the energy of the final and initial state (units eV), respectively. $\omega$ corresponds to the probe frequency, expressed in eV units. $\Gamma$ corresponds to the full effective natural linewidth that accounts for $\Gamma _{i}$ and $\Gamma _{f}$ and the broadening. For our calculations we take Fe $\Gamma = 4.7 \times 10^{-9} eV$ [41]. There has been evidence for lower linewidth in thinner structures. So, we have considered a 30% reduction in linewidth in the surface model, based on [13,42]. We use this relation to compare the Faraday rotation between bulk and surface models. Equations (1), (4) and (5) are used for the transition matrix Faraday rotation calculations.
The transition matrix amplitudes give us the probability of a particular transition from one band to another by following the selection rules that include parity forbidden transitions, and allowed transitions [31]. The parities have an impact on optical properties, and hence the Faraday rotation [33].
In our calculations, we only consider $\Gamma$ point transitions. The bulk structure has a direct band-gap at the $\Gamma$ point, whereas the surface sensitive structure has an indirect band-gap. Considering the transition probabilities is the best way to see how the indirect band gap affects the Faraday rotation at the $\Gamma$ point for surface models [33]. The optical absorption is dominated by the inter-band transitions at high symmetry points, i.e. the $\Gamma$ point [29].
We see from Fig. 5 the specific Faraday rotation plot for the probe energy (eV) and probe wavelength (nm). The transition data was calculated by sampling 1 out of 10 band numbers for the bulk and 1 out of 2 band numbers for the surface. The sample size is larger for the surface than bulk to consider the low energy gap near the Fermi-level and the band broadening at the Fermi level. The calculation for the bulk yields the accepted (measured) Faraday rotation and dispersion for YIG [29].
Fig. 5. Magneto-optic properties of bulk compared with the surface. Faraday rotation from Transition Matrix Analysis with respect to probe energy (left) and probe wavelength (right).
In Fig. 5 for the bulk analysis (blue line), we see a maximum Faraday rotation at $\approx$ 3.7 eV and the negative Faraday rotation is at its maximum at $\approx$ 4.5 eV. The theoretical calculations with an addition of a blue shift of $\approx$ 0.6 eV are in better agreement with the experimental data. This is because of the lower bandgap predictions in our methods compared to the experimental bandgaps. In the surface analysis (red line) we see a gradual increase in Faraday rotation from $\approx$ 2 eV ($\approx$715 nm). It has a broad probing region from 2 eV to 3.2 eV with higher values than the bulk. The results show a moving mean fit over the calculated $\Theta$.
In Fig. 5, we show the negative of the calculated surface Faraday rotation to highlight the enhancement compared to the bulk. There is a change in direction in Faraday rotation for surface-sensitive samples from the transition matrix calculations as well as predicted earlier in Section 3.2. In the surface band structure, we notice a presence of an indirect bandgap, shown in Fig. 3(d). It is evident from the transition matrix calculations that there is an increase in Faraday rotation for the surface-sensitive model. But, the optical activity reduces significantly due to the indirect bandgap and the presence of dangling oxygen bonds that do not contribute to the Faraday rotation. The use of stress and strain methods from lattice matching while growing YIG on different substrates can be investigated to tap into a very high Faraday rotation through direct bandgap.
Nevertheless, ultra-thin YIG models have high scope for higher Faraday rotation and hence a novel way to make on-chip optical isolators for lasers in the range of 500 nm to 729.7 nm (1.69 eV - $\sim$3 eV, visible range - deep red).
4. Summary
This paper analyzes the structural- and magneto-optic functionality changes between re-constructed surface (100) and bulk in yttrium iron garnets. The analysis uses VASP first-principle simulations to calculate the density of states and band structure reconfiguration between both cases. This study shows that $Fe^{O}$ surface atoms suffer are affected the most from the surface reconstruction, which can be noticed from their displacement and changes in the magnetic moment for the same.
This study predicts a significant enhancement in the role of spin-orbit coupling in ultrathin films (below 50 nm [15]) at the Fermi-level, resulting in a magnification of the Faraday rotation response. A crossover point in bandgap magnitude between spin-up and spin-down bandgaps is observed below a certain film thickness. This result is important as it leads to a reversal in the sense of Faraday rotation. Moreover, the strength of the specific Faraday rotation is generally found to increase with decreasing film thickness from this point on, as the band gap narrows. The strength of the Faraday rotation is parameterized through the off-diagonal components of the dielectric permittivity tensor, the gyrotropy parameter. So, the change in direction of Faraday rotation for the surface model suggests that with a decrease in the thickness of the sample, the gyrotropy reaches a global minimum and then keeps increasing as the thickness decreases. This correlates with a decrease in the bandgap at the surface (100) due to surface reconstruction.
A splitting in $d_{eg}$ and $d_{t2g}$ orbitals increases in the surface-sensitive calculations. This suggests that the bandgap for virtual transition processes contributing to the Faraday rotation is lower in ultra-thin YIG films with respect to crystal field splitting. We notice the reduction in band gap in the surface models could lead to very high enhancement in Faraday rotation in ultra-thin samples.
Nevertheless, there is an increase in Faraday rotation, despite the indirect bandgap and presence of dangling oxygen atoms, in the visible-deep red region. One could exploit stress-strain and layering techniques to tune the bandgap to become direct and reduce the effect of the dangling oxygen bonds. This would open the possibility for higher Faraday rotation in this wavelength range. We, thus, ascribe the decrease in band gap and increase in Faraday rotation to surface reconstruction at the surface-vacuum interface for YIG (100) surface.
Funding
Photonica, Inc.
Acknowledgments
- • The VASP calculations were performed on the Superior HPC at Michigan Technological University and on the HPCs made available from the Extreme Science and Engineering Discovery Environment (XSEDE, now ACCESS), which is supported by National Science Foundation grant number ACI-1053575.
- • The authors thank R. Pandey, R. Pati, and G. Odegard. for valuable discussions and comments on the manuscript.
- • The authors would also like to thank the numerous colleagues who helped with DFT programming, XSEDE (ACCESS) proposal, plus discussions and comments on the manuscript.
- • S.S. Dash and M. Levy thankfully acknowledge support from Photonica, Inc.
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. N. K. Dissanayake, M. Levy, A. A. Jalali, and V. J. Fratello, “Gyrotropic band gap optical sensors,” Appl. Phys. Lett. 96(18), 181105 (2010). [CrossRef]
2. Z. Wu, M. Levy, V. J. Fratello, and A. M. Merzlikin, “Gyrotropic photonic crystal waveguide switches,” Appl. Phys. Lett. 96(5), 051125 (2010). [CrossRef]
3. Y. Shoji, T. Mizumoto, H. Yokoi, I.-W. Hsieh, and R. M. Osgood, “Magneto-optical isolator with silicon waveguides fabricated by direct bonding,” Appl. Phys. Lett. 92(7), 071117 (2008). [CrossRef]
4. F. Rachford, M. Levy, R. Osgood, A. Kumar, and H. Bakhru, “Magnetization and FMR studies of crystal-ion-sliced narrow linewidth gallium-doped yttrium iron garnet,” J. Appl. Phys. 87(9), 6253–6255 (2000). [CrossRef]
5. S. Geller, “Crystal and static magnetic properties of garnets,” Chemischer Informationsdienst. 10, no. 28 (1979).
6. A. Zvezdin and V. Kotov, Modern Magnetooptics and Magnetooptical Materials (1st ed.) (CRC Press, 1997).
7. S. Wittekoek, T. J. A. Popma, J. M. Robertson, and P. F. Bongers, “Magneto-optic spectra and the dielectric tensor elements of bismuth-substituted iron garnets at photon energies between 2.2-5.2 ev,” Phys. Rev. B 12(7), 2777–2788 (1975). [CrossRef]
8. S. Sung, X. Qi, and B. Stadler, “Integrating yttrium iron garnet onto nongarnet substrates with faster deposition rates and high reliability,” Appl. Phys. Lett. 87(12), 121111 (2005). [CrossRef]
9. K. Ando, N. Koshizuka, T. Okuda, and Y. Yokoyama, “Growth-induced optical birefringence in LPE-grown bi-based iron garnet films,” Jpn. J. Appl. Phys. 22(10A), L618–L620 (1983). [CrossRef]
10. H. A. Algra, J. M. Robertson, and H. Dötsch, “The incorporation of Ga in LPE-grown La, Ga: Yig films,” Appl. Phys. 22(2), 189–191 (1980). [CrossRef]
11. A. Eschenfelder, Magnetic Bubble Technology, Springer Series in Solid-State Sciences (Springer Berlin Heidelberg, 2012).
12. R. Wolfe, V. Fratello, and M. McGlashan-Powell, “Thin-film garnet materials with zero linear birefringence for magneto-optic waveguide devices (invited),” J. Appl. Phys. 63(8), 3099–3103 (1988). [CrossRef]
13. M. Levy, A. Chakravarty, H.-C. Huang, and R. M. Osgood, “Large magneto-optic enhancement in ultra-thin liquid-phase-epitaxy iron garnet films,” Appl. Phys. Lett. 107(1), 011104 (2015). [CrossRef]
14. M. Levy, O. V. Borovkova, C. Sheidler, B. Blasiola, D. Karki, F. Jomard, M. A. Kozhaev, E. Popova, N. Keller, and V. I. Belotelov, “Faraday rotation in iron garnet films beyond elemental substitutions,” Optica 6(5), 642–646 (2019). [CrossRef]
15. O. V. Borovkova, F. Jomard, S. A. Dagesyan, E. A. Popova, N. Keller, V. I. Belotelov, M. Levy, A. A. Voronov, M. A. Kozhaev, A. Chakravarty, H. Hashim, B. Blasiola, D. Karki, and C. Sheidler, “Enhanced magneto-optic response of the ultrathin iron-garnet films,” 2019 Thirteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials) pp. X–068–X–070 (2019).
16. S. S. Dash, P. Mukherjee, D. Haskel, R. A. Rosenberg, and M. Levy, “Boosting optical nonreciprocity: surface reconstruction in iron garnets,” Optica 7(9), 1038–1044 (2020). [CrossRef]
17. O. V. Borovkova, H. Hashim, M. A. Kozhaev, A. N. Kalish, S. A. Dagesyan, A. N. Shaposhnikov, V. Berzhansky, A. K. Zvezdin, L. V. Panina, and V. I. Belotelov, “Transverse magneto-optical effect in asymmetric plasmonic nanostructures,” 2020 Fourteenth International Congress on Artificial Materials for Novel Wave Phenomena (Metamaterials) pp. 330–332 (2020).
18. D. O. Ignatyeva, D. M. Krichevsky, V. I. Belotelov, F. Royer, S. Dash, and M. Levy, “All-dielectric magneto-photonic metasurfaces,” J. Appl. Phys. 132(10), 100902 (2022). [CrossRef]
19. G. F. Dionne, Magnetic Oxides (Springer, 2009).
20. A. Clogston, “Radium,” J. Phys. 20, 151 (1959).
21. N. Ravindra, P. Ganapathy, and J. Choi, “Energy gap-refractive index relations in semiconductors - an overview,” Infrared Phys. Technol. 50(1), 21–29 (2007). [CrossRef]
22. L.-S. Xie, G.-X. Jin, L. He, G. E. W. Bauer, J. Barker, and K. Xia, “First-principles study of exchange interactions of yttrium iron garnet,” Phys. Rev. B 95(1), 014423 (2017). [CrossRef]
23. A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards, S. Dacek, S. Cholia, D. Gunter, D. Skinner, G. Ceder, and K. A. Persson, “Commentary: The materials project: a materials genome approach to accelerating materials innovation,” APL Mater. 1(1), 011002 (2013). [CrossRef]
24. D. Rodić, M. Mitrić, R. Tellgren, and H. Rundlöf, “The cation distribution and magnetic structure of y3fe(5−x)alxo12,” J. Magn. Magn. Mater. 232(1–2), 1–8 (2001). [CrossRef]
25. A. H. Larsen, J. J. Mortensen, J. Blomqvist, et al., “The atomic simulation environment-a python library for working with atoms,” J. Phys.: Condens. Matter 29(27), 273002 (2017). [CrossRef]
26. B. Adolph, J. Furthmüller, and F. Bechstedt, “Optical properties of semiconductors using projector-augmented waves,” Phys. Rev. B 63(12), 125108 (2001). [CrossRef]
27. J. P. Perdew, K. Burke, and M. Ernzerhof, “Generalized gradient approximation made simple,” Phys. Rev. Lett. 77(18), 3865–3868 (1996). [CrossRef]
28. G. Kresse and D. Joubert, “From ultrasoft pseudopotentials to the projector augmented-wave method,” Phys. Rev. B 59(3), 1758–1775 (1999). [CrossRef]
29. W.-K. Li and G.-Y. Guo, “A first principle study on magneto-optical effects and magnetism in ferromagnetic semiconductors Y3Fe5O12 and Bi3Fe5O12,” arXiv, arXiv:2005.14133 (2020). [CrossRef]
30. Y. Hinuma, G. Pizzi, Y. Kumagai, F. Oba, and I. Tanaka, “Band structure diagram paths based on crystallography,” Comput. Mater. Sci. 128, 140–184 (2017). [CrossRef]
31. V. Wang, N. Xu, J.-C. Liu, G. Tang, and W.-T. Geng, “Vaspkit: A user-friendly interface facilitating high-throughput computing and analysis using VASP code,” Comput. Phys. Commun. 267, 108033 (2021). [CrossRef]
32. N. Thiery, V. Naletov, L. Vila, A. Marty, A. Brenac, J.-F. Jacquot, G. Loubens, M. Viret, A. Anane, V. Cros, J. Ben Youssef, N. Beaulieu, V. Demidov, B. Divinskiy, S. Demokritov, and O. Klein, “Electrical properties of epitaxial yttrium iron garnet ultrathin films at high temperatures,” Phys. Rev. B 97(6), 064422 (2018). [CrossRef]
33. W. Meng, X. Wang, Z. Xiao, J. Wang, D. B. Mitzi, and Y. Yan, “Parity-forbidden transitions and their impact on the optical absorption properties of lead-free metal halide perovskites and double perovskites,” J. Phys. Chem. Lett. 8(13), 2999–3007 (2017). [CrossRef]
34. “Jahn-Teller Distortions,” [Online; accessed 2022-11-24].
35. “Selection rules and transition moment integral,” [Online; accessed 2022-11-17].
36. L. Meier, “Manipulation of electron spins in quantum wells with magnetic and electric fields,” Doctoral Thesis, ETH (2022).
37. D. Rodic, M. Mitric, R. Tellgren, H. Rundlof, and A. Kremenovic, “True magnetic structure of the ferrimagnetic garnet Y3Fe5O12 and magnetic moments of iron ions,” J. Magn. Magn. Mater. 191(1-2), 137–145 (1999). [CrossRef]
38. D. V. Chubukov, L. V. Skripnikov, V. N. Kutuzov, S. D. Chekhovskoi, and L. N. Labzowsky, “Optical rotation approach to search for the electric dipole moment of the electron,” Atoms 7(2), 56 (2019). [CrossRef]
39. U. U. Peter, M. Oppeneer Summer School, “Theory of magneto-optics,” (2007).
40. P. Oppeneer, Handbook of Magnetic Materials (Vol. 13) (Elsevier, 2001).
41. D. Perepelitsa, “Mossbauer spectroscopy of 57Fe,” (2007).
42. C. C. S. Chan, B. P. L. Reid, R. A. Taylor, Y. Zhuang, P. A. Shields, D. W. E. Allsopp, and W. Jia, “Optical studies of the surface effects from the luminescence of single GaN/InGaN nanorod light emitting diodes fabricated on a wafer scale,” Appl. Phys. Lett. 102(11), 111906 (2013). [CrossRef]
