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# Ticket #8771 (closed enhancement: fixed)

Opened 2 years ago

## Only show Wysiwyg editor for wiki pages

Reported by: Assigned to: dreas@spamexperts.com jun66j5 low TracWysiwygPlugin minor rjollos 0.12

### Description

We would prefer to only see the Wysiwyg editor when editing wiki pages. Ideally it does not show for Trac tickets. Could that be included as an optional feature?

## Change History

### 01/18/12 03:35:54 changed by rjollos

• cc set to rjollos.

### 01/18/12 19:43:10 changed by jun66j5

• status changed from new to closed.
• resolution set to fixed.

(In [11158]) fixed #8771, Adds [tracwysiwyg] templates option. It's able to show WYSIWYG editors for pages that the user wants.

### 01/18/12 19:52:21 changed by jun66j5

• type changed from defect to enhancement.

If you want show the WYSIWYG editor for wiki pages;

[tracwysiwyg]
templates = wiki_*.html

If you don't want show for ticket pages;

[tracwysiwyg]
templates = !ticket*.html,*.html

### 01/30/12 04:36:54 changed by rjollos

I noticed you don't maintain a description of the configuration options on the project wiki page, which I think is a good idea because they can become out of sync so easily with the source code and can differ depending on the Trac release. However, I added a note about how to find the configuration options because I've seen the question raised fairly regularly for various plugins. I just wanted to mention this since I made an edit to your project wiki page and I think it is always a good idea to keep the author in the loop of any changes that are made.

### Add/Change #8771 (Only show Wysiwyg editor for wiki pages)

Change Properties


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### A Sameness Surely

Triangle ABC has a right angle at C. ACRS and CBPQ are squares. ST and PU are perpendicular to AB produced. Show that ST + PU = AB

### Three Balls

A circle has centre O and angle POR = angle QOR. Construct tangents at P and Q meeting at T. Draw a circle with diameter OT. Do P and Q lie inside, or on, or outside this circle?

### Angle Trisection

It is impossible to trisect an angle using only ruler and compasses but it can be done using a carpenter's square.

# Encircling

##### Stage: 4 Challenge Level:

Congratulations James Page of Hethersett High School, Norwich on your solution to this question. Again there are many possible methods of showing that the radius of the circle is equal to the side of the square.

The blue line $OD$ is a diameter of the circle and a line of symmetry.

$\angle EDC = 150^{\circ}$ and $\triangle ECD$ isosceles so $\angle CED = \angle CDE =15^{\circ}$.

Now $\angle OED = 30^{\circ}$ so $\angle OEC = \angle OED - \angle CED = 15^{\circ}$ and hence the red line $EC$ is a line of symmetry for the quadrilateral $OEDC$ proving that the radius of the circle, $OC$ is equal in length to the side of the square.

format: plain


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Letter | Published:

# Ferromagnetic exchange, spin–orbit coupling and spiral magnetism at the LaAlO3/SrTiO3 interface

## Abstract

The electronic properties of the polar interface between insulating oxides is a subject of great interest1,2,3. An exciting development is the observation of robust magnetism4,5,6,7,8 at the interface of two non-magnetic materials, LaAlO3 (LAO) and SrTiO3 (STO). Here we present a microscopic theory for the formation and interaction of local moments that depends on essential features of the LAO/STO interface. We show that correlation-induced moments arise owing to interfacial splitting of orbital degeneracy. We find that conduction electrons with a gate-tunable Rashba spin–orbit coupling mediate ferromagnetic exchange with a twist. We predict that the zero-field ground state is a long-wavelength spiral. Its evolution in an external field accounts semi-quantitatively for torque magnetometry data5 and describes qualitative aspects of the scanning superconducting quantum interference device measurements6. We make several testable predictions for future experiments.

## Main

Recent experiments on the LAO/STO interface have seen tantalizing magnetic signals4,5,6,7,8, often persisting up to high temperatures 100 K. A large magnetization of 0.3– 0.4μB per interface Ti was observed by torque measurements5 in an external field. In contrast, scanning superconducting quantum interference device (SQUID) experiments6 found an inhomogeneous state with a dense set of local moments with no net magnetization, only isolated micron-scale ferromagnetic patches. Our goal is to reconcile these seemingly contradictory observations and to gain insight into the itinerant versus local moment nature of the magnetism, the exchange mechanism and the ordered state.

LAO and STO are both band insulators, but the TiO2 layers at the interface are n-doped when LAO is terminated by a LaO+ layer. The polar catastrophe1 arising from a stack of charged LaO+ and AlO2 layers grown on STO is averted by the transfer of 0.5 electrons per interface Ti. Oxygen vacancies are also known to provide additional electrons at the interface1.

What is the fate of these electrons at the interface? Transport data suggest that only a small fraction of the electrons (5–10% of the 0.5 electrons per Ti) are mobile2,9,10,11. Interestingly, magneto-transport studies show a large, gate-tunable Rashba spin–orbit coupling (SOC) for these conduction electrons, arising from broken inversion at the interface12. Most of the electrons (comparable to 0.5 electrons per Ti) seem to behave like local moments in the magnetic measurements5,6 discussed above.

We propose a microscopic model of electrons in Ti t2g states at the LAO/STO interface that leads to the following results: local moments form in the top TiO2 layer owing to correlations, with interfacial splitting of t2g degeneracy playing a critical role; conduction electrons mediate ferromagnetic double-exchange interactions between the moments; Rashba SOC for the conduction electrons leads to a Dzyaloshinskii–Moriya interaction and a compass anisotropy term with a definite ratio of their strengths; the zero-field ground state is a long-wavelength spiral with a SOC-dependent pitch; the spiral transforms into a ferromagnetic state in an external field H.

We provide a semi-quantitative understanding of the torque results for the magnetization M(H), and reconcile some of the key differences between the torque and SQUID measurements. We also point to a new magnetoelastic coupling effect that can be important for polar interfaces. Our model naturally explains the coexistence of magnetism and superconductivity at very low temperatures5,6,8. Finally, we make a number of specific predictions that can be tested experimentally.

## Symmetry-based considerations

Many of our conclusions regarding magnetism can be understood qualitatively on the basis of symmetry alone. Although symmetry dictates the form of the interactions, the microscopic analysis shows how local moments Sr arise on a square lattice at the interface, and provides quantitative insights into the parameter dependence of exchange couplings. The isotropic Heisenberg exchange is , with and lattice spacing b. Its sign is not determined by symmetry, but our microscopic analysis leads to a ferromagnetic J>0. Inversion symmetry breaking at the interface implies two SOC terms: the Dzyaloshinskii–Moriya interaction with , and the compass anisotropy .

One can see, quite generally, that the ground state of such a model can be a long-wavelength spiral that looks locally ferromagnetic, thus minimizing the J term, with pitch determined by the Dzyaloshinskii–Moriya term. Our microscopic results for D,A,A′ unequivocally predict such a spiral state for H = 0, which turns into an in-plane ferromagnet for H1 T. These observations permit us to understand the torque data in a field5 semi-quantitatively, and to see why over most of the sample the H = 0 scanning SQUID measurements find no net moment6.

## Electronic structure

Both density functional theory13,14 (DFT) and spectroscopy15,16,17, show that, in the top TiO2 layer dxy states have lower energy than dxz,dyz (Fig. 1a). In addition to z confinement raising the xz,yz levels, the mismatch of in-plane LAO and STO lattice parameters leads to an out-of-plane distortion that lowers dxy. The dxy orbitals delocalize primarily in the xy plane, and similarly for dxz and dyz. DFT (refs 18, 19) and photoemission20 find the in-plane hopping between dxy orbitals t0.3 eV, whereas the out-of-plane tt/30. The resulting xz (yz) bands are quasi-one-dimensional (1D), dispersing primarily along x (y), and confined along z.

## Local moments

Coulomb interactions, disorder and coupling to phonons can all play a role in localizing charge. Nevertheless, interactions are essential for forming local moments. Experiments11,21 show that the quasi-1D xz,yz bands, which should be the most sensitive to disorder, contribute to transport, ruling out Anderson localization. We thus make a simple model where we treat correlations first and can then add the effects of phonons and disorder.

The key to understanding correlation effects is the splitting of t2g degeneracy at the interface that leads to a quarter-filled xy band (0.5 electrons per Ti) in the top TiO2 layer. A modest on-site Hubbard U and next-neighbour Coulomb V then lead to a charge-ordered insulator (COI; Fig. 1b). We obtain a simple analytical result for the metal–insulator transition using a slave-rotor approach22. Coupling to the breathing-mode phonon further stabilizes the COI; (see Methods). DFT calculations need a rather large U8 eV to stabilize a COI (ref. 23). We see from Fig. 1 that, for realistic values24 of U = 4 eV (U/t13) and V 0.5–1 eV (V/t1.5–3), we are deep in the chequerboard COI state.

## Exchange

Consider then the problem of local moments in dxy orbitals on a chequerboard lattice (Fig. 2a) interacting through Hund’s coupling JH with a small density nc of conduction electrons in xz, yz bands. Given JH1 eV, it is much more reasonable to work in the non-perturbative double-exchange limit JHt than in the perturbative RKKY limit JHt. We treat the S = 1/2 moments classically. This is justified a posteriori because of the small quantum fluctuations in the ferromagnetic and long-wavelength spiral states.

We find with where r denotes positions of occupied sites in the COI (see Methods). is obtained by interchanging and replacing D→−D in . The term couples nearest neighbours along the diagonal (Fig. 2a). The spins have normalization Sr2 = 1. The form of and , as well as JJ′, arises from the two-sublattice structure of the COI and the quasi-1D nature of xz, yz bands. We find Jnct/4 and Jnct′/4 for low carrier density nc (the number of electrons per Ti in each band).

The new aspect here is the Rashba SOC for the conduction electrons, where k is their momentum, are Pauli matrices and a is the Ti–Ti distance. The SOC strength λ is determined by the electric field at the interface. Experiments12 find a gate-voltage-tunable λ1.3–6.3 meV (λ/t0.004–0.022). SOC then gives rise to the Dzyaloshinskii–Moriya and compass terms in through the double-exchange mechanism. For λ/t1, we find Dncλ, A2ncλ2/t and A′ = 0. Thus, D/Jλ/t and A/J(λ/t)2 with AJ/D2 = 1/2 independent of parameters.

## Spiral ground state

We have examined the ground-state properties of in an external field H using a variety of techniques. Using a variational approach we find that, for AJ/D2<1, the H = 0 ground state is a coplanar spiral with spins lying in a plane perpendicular to the interface. Here ; see Methods. For AJ/D2>1, the ground state is an in-plane ferromagnet with . For all AJ/D2, the ground state is infinitely degenerate (arbitrary 0≤φ<2π), even though has symmetry. This peculiar degeneracy of compass anisotropy25 is expected to be broken by order-by-disorder.

The microscopic value AJ/D2 = 1/2 implies a zero-field spiral variational ground state. We have verified the stability of the spiral at H = 0 using two independent, unbiased calculations (see Supplementary Information): finite-temperature Monte Carlo and T = 0 conjugate gradient energy minimization. In Fig. 2b we show an example of spiral ordering in real space with a wavelength of a few lattice spacings. For realistic values of λ/t0.02, our variational results give a pitch of (2π/Q0)600 Å.

In Fig. 2c–h, we show the evolution of the spiral state as a function of H applied in the xz plane at an angle θH = 15° with the z axis, (the geometry used in the torque experiments5). In the variational calculation (Fig. 2c) the spiral state () develops into a fully magnetized state above a field Hpeak, which corresponds to a maximum in the in-plane component Mx of M = 〈Sr〉. We plot in Fig. 2d,e Mx and Mz as functions of H, where we also see excellent agreement between variational and Monte Carlo results. For the Monte Carlo simulations we used larger SOC λ/t = π/25 that led to smaller pitch spirals. Below we use realistic SOC λ values to study long-wavelength spirals within a variational approach.

The evolution from spiral to ferromagnet as a function of field is best seen in the spin structure factor Iq;|〈Sq〉|2 shown in Fig. 2f–h, where Sq is the Fourier transform of Sr. The spiral at H = 0 shows two peaks in Iq at (Fig. 2f), which lie on the white circle, the locus of Q0 corresponding to the degenerate states in equation (2). At intermediate fields, a uniform component develops along with peaks (Fig. 2g), whereas for H>Hpeak one obtains a ferromagnet (Fig. 2h).

## Torque and magnetization

The torque is non-zero only in the presence of spin-space anisotropy, otherwise M would simply align with H. In Fig. 3a, we compare our variational results with the magnetization Mττ/H versus  H derived from torque magnetometry5, where Mτ is the component of M perpendicular to H. (The experimental data have been shown after background subtraction; see Supplementary Information.)

The M(H) curve shows no hysteresis5, because the spiral state has no spontaneous magnetization at H = 0. The in-plane component of the field () induces a uniform magnetization in the spiral state (Fig. 2). This leads to an increasing Mτ for H<Hpeak in Fig. 3a, where Hpeak, at which Mτ is a maximum, depends on θH. For H>Hpeak, the ground state is a ferromagnet (Fig. 2) with the compass term giving rise to a large easy-plane anisotropy A0.15–0.3 T (with λ0.016–0.022 and nc = 0.05) that tries to keep M in the plane. For large enough H the field dominates over the anisotropy, pulls M out of the plane and Mτ decreases. These two regimes are shown schematically in Fig. 3a inset.

In Fig. 3a we show two M(H) curves that differ in their choice of AJ/D2. The red curve with AJ/D2 = 1/2 has too rapid a high-field drop in Mτ(H). (Here A = 2ncλ2/t = 0.3 is tuned to match Hpeak and λ/t = 0.022, nc = 0.05.) The blue curve with AJ/D2 = 0.8 (with λ/t = 0.028) leads to better agreement with experiment. Such a phenomenological choice of AJ/D2 amounts to changing the anisotropy while keeping Dzyaloshinskii–Moriya fixed. Although A and D were both determined by Rashba SOC in our microscopic theory, dipolar interactions and atomic SOC also contribute to A (see Supplementary Information).

The peak position Hpeak;ncλ2/t in our microscopic model and Mτ(H) for different λ collapse onto a single curve when plotted versus H/Hpeak (see Fig. 3b,c). In practice, other sources of anisotropy might modify this scaling. Nevertheless, both nc and λ are gate tunable and we expect Hpeak to vary substantially with bias.

## Scanning SQUID experiments

We now discuss how we can reconcile the torque5 and scanning SQUID (ref. 6) results. At H = 0 most of the sample has spiral order and hence no net magnetization, consistent with ref. 6. A detailed modelling of the observed inhomogeneity is beyond the scope of this paper, nevertheless we can offer a plausible picture based on our theory. The energy gain of spiral over ferromagnet is quite small: Δε(D2/JA)/2 = ncλ2/t0.1–0.2 K. Therefore, small terms ignored up to now might well upset the balance in favour of the ferromagnet. We find that magnetoelastic coupling to polar distortions, a new aspect of these interfaces, can reverse the balance between spiral and ferromagnetic states and stabilize ferromagnetic patches (see Methods). These ferromagnetic patches can then point in any in-plane direction, consistent with compass anisotropy.

## Discussion

We comment briefly on how our theory, which builds on insights from DFT (refs 13, 18, 19, 23), differs from other approaches. It is hard to obtain the large exchange or net moment seen in LAO/STO from itinerant models26, which may be relevant for GdTiO3/SrTiO3. On symmetry grounds, these itinerant systems will also have a SOC-induced spiral ground state similar to the one discussed here. Our model differs from ref. 27 in several respects. We are in the non-perturbative double-exchange limit and not the RKKY regime. In our model, xz,yz carriers mediate exchange. The xy electrons in lower layers have essentially no interaction with moments in the xy orbitals of the top TiO2 layer. This results from the small overlap t′ between xy orbitals along as well as the level mismatch due to confinement. Conventional superconductivity arises in xy states in lower layers and is decoupled from the local moments.

## Conclusions

Our theory has several testable predictions. Local moment formation is associated with (π,π) charge order at the interface, although disorder will probably disrupt the order. The zero-field state is predicted to be a coplanar spiral state with several striking properties, including a wave vector that scales linearly with Rashba SOC and hence is gate tunable. The evolution from a spiral to a ferromagnetic state in an external field has characteristic signatures in the spin structure factor. The easy-plane anisotropy strength is also dominated by SOC and exhibits substantial gate voltage dependence that can be tested in torque magnetometry experiments. The ferromagnetic exchange J = nct/440 K (for nc = 0.05 and t = 0.3 eV) should also be tunable by changing nc, the density of carriers. A theoretical analysis of the finite temperature properties of our model is left for future investigations.

## Methods

### Charge-ordering Mott transition.

We consider the quarter-filled, extended Hubbard model on a 2D square lattice with onsite U and nearest-neighbour Coulomb V. Using slave-rotor22 mean field theory for U and Hartree–Fock for V, we obtain an analytical result for the transition from metal to charge ordered insulator (COI). The chequerboard COI is stable for U>8〈t〉 when U<4V, and for (U−2V)V/U>〈t〉 when U>4V. Here 〈t0.66t is the kinetic energy of occupied states. Slave-rotor mean field theory has been found to be in excellent semi-quantitative agreement with dynamical mean field theory for several problems. In fact, our 2D square lattice results are quite similar to dynamical mean field theory of the Wigner–Mott transition on a Bethe lattice28.

The COI is stabilized by coupling to the breathing-mode phonon, which has the same (π,π) wave vector as the charge order (see Supplementary Information). The only change above is that , where Eph is the energy gained though lattice distortion. Typical values of Eph0.05–0.10 eV (ref. 29).

### Double exchange.

The local moments Sr on the sites of a 2D chequerboard lattice are described by their orientation (θr,φr). In the large JH limit, a conduction electron on the local-moment sublattice (a) has its spin parallel to Sr. Thus, we write the spin-full fermion operators , with orbital index α = (xz),(yz), in terms of spinless fermions arα, through . Here Fr = cos(θr/2) and $Fr↓ = sin (θr/2)e−iϕr$.

Both spin projections σ = ↑,↓ are allowed on empty sublattice (b) sites, for which we use a common (global) quantization axis. The kinetic energy is then given by where and .

The Rashba terms and lead to the SOC Hamiltonian We can rewrite this in terms of the spinless a fermions as where and we define λ(xz)↑ = −λ(xz)↓ = λ and λ(yz)↑ = λ(yz)↓ = iλ.

To obtain the parameters J, J′, D and A of of equation (1) starting from the microscopic , we match the energies of several low-lying configurations computed for both and . See Supplementary Information for details.

### Variational calculation.

We study the ground-state properties of using a variational ansatz Sr = M+R(Q)cos(Q·r)+I(Q)sin(Q·r) with . The normalization Sr2 = 1 is satisfied by the constraints M·R = M·I = R·I = 0; R2 = I2 and M2+R2 = 1. We numerically minimize the energy per spin ε to obtain optimal values for M, R, I and Q as a function of H, and calculate the torque.

At zero field, we find a spiral ground state of equation (2) with Q = Q0 and M = 0 for AJ/D2<1 and an in-plane ferromagnet for AJ/D2>1. We can see this analytically using the ansatz Sr = Rcos(Q·r)+Isin(Q·r). We write and with θ[0,π] and φ[0,2π). We first minimize with respect to Q (for Qa1), find the optimal and the energy ε(Q)−2(J+J′)−A+(AD2/J)(cos2θ)/2, which is independent of φ. For AJ/D2<1, the optimal choice of cosθ = 1 leads to the spiral of equation (2) with Q = Q0 and an energy gain Δε = (D2/JA)/2 relative to the ferromagnetic state. For our microscopic model Δε = ncλ2/t.

### Polar distortions.

Displacements of oxygen ions that bridge Ti–O–Ti bonds30 at the interface affect the local electric field and hence the Rashba SOC λ(u) = λλ1u. Here λ1>0 because the local electric field decreases if negatively charged oxygen ions move up (u>0). The resulting coupling to magnetism can be modelled by AA(u) = 2ncλ2(u)/t and DD(u) = ncλ(u) in the energy ε(Q), in addition to the cost of distortion. The A and D terms in gain energy by increasing λ(u) with u<0. However, this costs an electrostatic energy −qEu and an elastic energy Ku2/2, where is the electric field acting on oxygen ions at the interface and K is the elastic constant. We minimize the total energy with respect to u to find that energy gain of the spiral is reduced: Δε(ncλ2/t)[1−2qEλ1/Kλ]. With suitable choice of parameters, one can make Δε<0 and stabilize the ferromagnetic state over the spiral.

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## Acknowledgements

We acknowledge stimulating conversations with L. Li, L. Balents, W. Cole, J. Mannhart, K. Moler, W. Pickett, S. Satpathy and N. Trivedi, and the support of DOE-BES DE-SC0005035 (S.B.), NSF-DMR-1006532 (O.E.) and NSF MRSEC DMR-0820414 (M.R.).

## Affiliations

1. ### Department of Physics, The Ohio State University, Columbus, Ohio 43210, USA

• Sumilan Banerjee
• , Onur Erten
•  & Mohit Randeria

## Authors

### Contributions

S.B., O.E. and M.R. contributed to the theoretical research described in the paper and the writing of the manuscript.

### Competing interests

The authors declare no competing financial interests.

## Corresponding author

Correspondence to Mohit Randeria.

## PDF files

1. 1.

### Supplementary Information

Supplementary Information

### DOI

https://doi.org/10.1038/nphys2702

• ### Interface-driven chiral magnetism and current-driven domain walls in insulating magnetic garnets

• Can Onur Avci
• , Ethan Rosenberg
• , Lucas Caretta
• , Felix Büttner
• , Maxwell Mann
• , Colin Marcus
• , David Bono
• , Caroline A. Ross
•  & Geoffrey S. D. Beach

Nature Nanotechnology (2019)

• ### Strain-tunable magnetism at oxide domain walls

• D. V. Christensen
• , Y. Frenkel
• , Y. Z. Chen
• , Y. W. Xie
• , Z. Y. Chen
• , Y. Hikita
• , A. Smith
• , L. Klein
• , H. Y. Hwang
• , N. Pryds
•  & B. Kalisky

Nature Physics (2019)

• ### Study of Rashba Spin–Orbit Field at LaAlO3/SrTiO3 Heterointerfaces

• Mi-Jin Jin
• , Shin-Ik Kim
• , Seon Young Moon
• , Daeseong Choe
• , Jungmin Park
• , Vijayakumar Modepalli
• , Junhyeon Jo
• , Inseon Oh
• , Seung-Hyub Baek
•  & Jung-Woo Yoo

Journal of Electronic Materials (2019)

• ### Influence of In-Gap States on the Formation of Two-Dimensional Election Gas at ABO3/SrTiO3 Interfaces

• Cheng-Jian Li
• , Hong-Xia Xue
• , Guo-Liang Qu
• , Sheng-Chun Shen
• , Yan-Peng Hong
• , Xin-Xin Wang
• , Ming-rui Liu
• , Wei-min Jiang
• , Lin He
• , Rui-Fen Dou
• , Chang-Min Xiong
• , Wei-ming Lü
•  & Jia-Cai Nie

Scientific Reports (2018)

• ### Formation of Two-dimensional Electron Gas at Amorphous/Crystalline Oxide Interfaces

• ChengJian Li
• , YanPeng Hong
• , HongXia Xue
• , XinXin Wang
• , Yongchun Li
• , Kejian Liu
• , Weimin Jiang
• , Mingrui Liu
• , Lin He
• , RuiFen Dou
• , ChangMin Xiong
•  & JiaCai Nie

Scientific Reports (2018)


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# Given oracle for Max-3SAT compute clauses that cannot be satisfied

We know that Max-3SAT is NP-hard to compute exactly (and also hard to approximate to a particular constant multiplicative factor). However, suppose you are given an oracle for Max-3SAT that tells you the whether the maximum number of clauses you can satisfy when an equation is the input is $k$. Can you use a polynomial number of calls to the oracle to figure out the clauses that are not satisfied when provided an optimal assignment of the variables?

A simple statement that could help with this is that suppose you know the clauses that are not satisfied by an optimal assignment of variable values, then removing any subset of these clauses will not result in more clauses that can be satisfied. Is this statement true?

• To clarify, are you looking for the clauses that aren't satisfied in any optimal assignment, or the clauses that aren't satisfied in some optimal assignment? (Also, yes, your statement is true.) – Tom Tseng Jul 22 '16 at 21:35
• Clauses that aren't satisfied in some optimal assignment. – user1246462 Jul 23 '16 at 15:34
• In that case, do you mean [the set S given by [C is in S if and only if there is an optimal assignment that does not satisfy C]] or ["the" set S such that there exists an optimal assignment A such that S is given by [C is in S if and only if A does not satisfy C]]?   – user6973 Jul 23 '16 at 20:11
• The scare quotes are because there can easily be more than one "set S such that ... does not satisfy C]" - Consider the 3-variable 8-clause instances in which each variable occurs exactly once in each clause and the 8 sets of literals are all distinct.  If that's what you mean, then how should S be chosen?  (For example, are you really just after a "set S such that ... does not satisfy C]", which the algorithm can choose, rather than the "set S such that ... does not satisfy C]", specified in some unknown-to-me way?)     – user6973 Jul 23 '16 at 20:11

Given an instance of 3SAT with $m$ clauses, you can find the set of clauses that are not satisfied in some optimal assignment with $O(m)$ calls to the oracle.
The algorithm: Call the oracle on the original input to find $k^*$, the maximum number of clauses that can be satisfied. For each clause, try removing the clause and then call the oracle. If the maximum number of clauses that can be satisfied is still $k^*$, then leave the clause removed. Otherwise, put the clause back in.
Observation 1: Throughout the algorithm, the non-removed clauses are always a superset of the satisfied clauses of some optimal assignment. (To see this, proceed by contrapositive. Suppose that the non-removed clauses are not a superset of the satisfied clauses of any optimal assignment. Then we cannot satisfy $k^*$ of the non-removed clauses simultaneously, or else those $k^*$ clauses would be the clauses of an optimal assignment for which the non-removed clauses are a superset.)
Observation 2: The algorithm will end with $k^*$ non-removed clauses. (Suppose that we have strictly more than $k^*$ clauses. By Observation 1, there is some optimal assignment whose satisfied clauses are a subset of the non-removed clauses. Then we can remove the clauses that aren't satisfied by this optimal assignment and still have $k^*$ satisfied clauses, and these clauses would have been removed when the algorithm looped over the clauses.)


text:
# Derivative of a integral function?

How does one work the following?

$$\frac{d}{dx}\int^x_y f(x,u)du$$

I know that (given certain assumptions about the function f)

$$\frac{d}{dx}\int^x_y f(w,u)du=f(w,x)$$

and

$$\frac{d}{dx}\int^c_y f(x,u)du=\int^c_y \frac{df}{dx}(x,u)du$$

but how do we put them together?

$$\frac{d}{dx}\int^x_y f(x,u)du = \int^x_y \frac{d}{dx}f(x,u)du+f(x,x).$$

Thanks, Pere.

HallsofIvy
$$\frac{d}{dx}\int_{\alpha(x)}^{\beta(x)} f(x,t)dt= f(x,\beta(x))\frac{d\beta}{dx}- f(x, \alpha(x))\frac{d\alpha}{dx}+ \int_{\alpha(x)}^{\beta(x)} \frac{\partial f}{\partial x} dt$$


text:
## 5. Extended example: the style guide

This section reproduces Dr. Steve Simpson's style guide using the nmtsrsab2012.sty style file.

• See the PDF rendering of this file. It is a guide to writing the content of your paper, as well as being an example of the use of the nmtsrsab2012.sty style.

• example.bib is the BibTeX file used in this example.

example.tex
\documentclass[12pt]{article}
%================================================================
% Preamble
%----------------------------------------------------------------
%
% Packages used
%
\usepackage{graphicx}
\usepackage[round]{natbib}
\usepackage{nmtsrsab2012}
%
% Definitions
%
\registrationNumber{1889}
\date{April 13, 2012}
%================================================================
\begin{document}
\title%
{%
SRS extended abstract author guidelines: Replace this text
with your own descriptive title in sentence case
}%
\correspondingAuthor{Author Name 1}{Department Name}{email1@nmt.edu}
\author{Author Name 2}{Department Name}{email2@nmt.edu}
\author[Other Institution]{Author Name 3}{Department Name}{email2@nmt.edu}
\authorsEnd
%================================================================
\section{Introduction}

The introduction to your extended abstract should clearly explain
the nature and motivation of your study.  State the purpose of
your study in simple sentences and in language that can be
understood by a scientifically-literate person from any
department or discipline.  Also be sure to state clearly the
importance of your study in a way that someone outside your
particular field can appreciate and understand.  For example,
while people in your field might understand the significance of
determining a tropical storm's vorticity, a person outside your
field might understand better if you explain that your research
helps understand how tropical storms develop.  If you need
to use discipline-specific terminology, provide a definition or
explanation.  Define abbreviations and acronyms the first time
that they are used.

End your introduction with a very clear \emph{forecasting
statement} describing specifically what you will discuss in this
abstract.  Begin the forecasting statement with a very clear
signal phrase, such as This presentation describes,'' or ''The
report examines,'' In the following sections,'' etc.

\section{Methods}

\subsection{Writing the methods section}

concisely.  Be thorough, but not excessive when describing your
methods.  You might need to customize the level-1 and level-2
you might add a Materials'' section or a description of the
Field Site.''

Your extended abstract should be double-spaced and use 12-point
the following minimum requirements:

\begin{itemize}
\item The abstract should be 5--6 pages long (approximately
1400--1700 words).

\item All references to secondary materials should be cited
using a recommended citation format in your field (e.g., APA,
IEEE, MLA, etc.). \citet*{schmo} says to please make sure that
all source material is properly quoted or paraphrased.  For
visit the Purdue Online Writing Lab (OWL)
\footnote{\texttt{http://owl.english.purdue.edu/}}.

\end{itemize}

Aside from these requirements, you may vary the format of your
report to account for the needs of your project and your field's
conventions.

\section{Results}

You will most likely use an array of tables and figures in your
Results'' section.  Be careful of plunking'' complex tables
and figures into your report without any explanation.  Every
table and figure should be numbered (tables are numbered on the
top, and figures are numbered on the bottom) and should be
referenced in the text.  For example, Figure \ref{plunk-fig} is a
cheesy bar graph showing the average number of times New Mexico
Tech students reported plunking'' figures into their results
sections.  What is significant in this figure is that students
tend to report fewer plunks'' as they progress through their
schooling, though they often end up right back where they started
in graduate school.  Also significant is that the figure was
mentioned above, and the figure's take away point'' was
explained in the text.

\begin{figure}
\includegraphics[scale=0.8]{plunks.pdf}
\caption%
{%
\bfseries
The mean number of times New Mexico Tech students report
plunking'' figures into their results section according
to students' years in school.
}%
\label{plunk-fig}
\end{figure}

While you may use any number of programs to create tables,
figures, and graphics, make sure that they are clear (both in
quality and image quality), that axes are properly labeled, and
that they include descriptive titles or captions.

\section{Conclusion}

In the conclusion section, avoid simply repeating your results.


text:
A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has
Question:

A group consists of 4 girls and 7 boys. In how many ways can a team of 5 members be selected if the team has (i) no girl? (ii) at least one boy and one girl? (iii) at least 3 girls?

Solution:

A group consists of 4 girls and 7 boys. Out of them, 5 are to be selected to form a team.

(i) If the team has no girls, then the number of ways of selecting 5 members $={ }^{7} C_{5}=\frac{7 !}{5 ! 2 !}=\frac{7 \times 6}{2}=21$

(ii) If the team has at least 1 boy and 1 girl, then the number of ways of selecting 5 members

$={ }^{4} C_{1} \times{ }^{7} C_{4}+{ }^{4} C_{2} \times{ }^{7} C_{3}+{ }^{4} C_{3} \times{ }^{7} C_{2}+{ }^{4} C_{4} \times{ }^{7} C_{1}$

$=140+210+84+7$

$=441$

(iii) If the team has at least 3 girls, then the number of ways of selecting 5 members

${ }^{4}{ }^{4} C_{3} \times{ }^{7} C_{2}+{ }^{4} C_{4} \times{ }^{7} C_{1}=84+7=91$


text:
I need to show that if $\bf{M}$ is an $m \times n$ matrix, then $||\bf{M}(\bf{h})||\leq \alpha ||\bf{h}||$ for some $\alpha$ by estimating the entries of $\bf{M}$. The hint I am given is that $\bf{M}(\bf{h}) = \bf{y} = (y_1, \ldots, y_m)$ where $y_i=\bf{m}_i \cdot \bf{h}$ for some $\bf{m}_i \in R^n$ and apply the CS inequality.
2. If we denote by $(a_{k,j})_{1\leq k\leq m,1\leq j\leq n}$ the coefficients of the matrix $M$, we have to notice that $y_i =\sum_{j=1}^na_{ij}h_j$. Now, the question is: what is the vector $m_i$ and what does the Cauchy-Schwarz inequality give?


text:
# Some Tralics obsolete features

## module (environment)

Before 2007, the Raweb preprocessor changed

\begin{module}[3]{}{appAI}{Static Analysis}
...
\end{module}

into

\begin{module}[3] {TEAM} {4} {appAI} {module7} {Static Analysis}
...
\end{module}

As you can see, there is one more argument TEAM, the symbolic section (in the example, the empty group {} was replaced by a default value namely `domaine') has been replaced by its numeric value (here 4), and a second additional argument has been added before the title.

These hacks were removed in 2007. For compatiblity reasons, the preprocessor, when used, generates a modulex environment, translated as dedcribed here.

The translator ignores the first two arguments (that were added by the preprocessor). It uses the next argument as a label name (the real name of the label is mod:appAI; but you should really use \moduleref {TEAM} {section} {appAI}, see \moduleref). The next argument is used for a html attribute.

back to home page © INRIA 2007 Last modified $Date: 2007/08/02 12:26:18$


text:
# Need Help With a Logarithmic Problem

• Jan 2nd 2013, 08:06 PM
jorgon1022
Need Help With a Logarithmic Problem
I dont understand what I am doing wrong.

Here it is
Given log a ^2= .69, log a ^3=1.10 and log a ^5=1.61, use the properties of logarithms to evaluate:

log a 1/square root of 15

Can any of you please help with this problem? I would really appreciate it.
• Jan 2nd 2013, 08:11 PM
ibdutt
Re: Need Help With a Logarithmic Problem
The question is not clear. What do you want to find
log a 1/square root of 15???? or log a^1/square root of 15??
• Jan 2nd 2013, 08:19 PM
jorgon1022
Re: Need Help With a Logarithmic Problem
Attachment 26452

Here is a picture of the problem. Thank you.
• Jan 3rd 2013, 06:02 AM
skeeter
Re: Need Help With a Logarithmic Problem
$\log_a{\frac{1}{\sqrt{15}}} = -\log_a{\sqrt{15}} = -\frac{1}{2}\log_a{15} = -\frac{1}{2}\log_a(3 \cdot 5) = -\frac{1}{2}\left(\log_a{3}+\log_a{5}\right)$


text:
uniform distribution with interval (0,2) and sample 12

Q): Suppose that you wish to sample $$12$$ observations randomly from a uniform distribution on the interval $$(0,2)$$. An approximate value of the probability that the average of your sample will be less than $$0.5$$.

what i did was i found the $$\mathbb E(x)$$ to be $$1$$ and variance to be $$1/3$$ and then i wasnt sure where to go with this so i decided to do the $$z$$-formula of $$(0.5-1)/\sqrt{1/3}/\sqrt{12}$$ to get me $$z = -3$$ then i got $$p= 0.001350$$ from that , i was just wondering if that was correct.

• Please do not use pictures for critical portions of your post. Pictures may not be legible, cannot be searched and are not view-able to some, such as those who use screen readers. – Brian Mar 28 at 12:11
• Your approximation to an answer is correct. – NCh Mar 28 at 13:32
• Simulation in R: a = replicate(10^6, mean(runif(12, 0, 2))); mean(a < .5) returns 0.000979, accurate to about 3 places and agrees with $\approx 0.001$ from good normal approx. – BruceET Mar 28 at 15:57
• @BruceET -but how many simulations would you need to distinguish between $0.001350$ and $0.001007$? – Henry Mar 28 at 16:55
• @Henry. I was just doing a 'reality check' on a tail probability so far out. With a million iterations it's $.001 \pm 0.0006.$ Maybe $10^7$ or $10^8$ would settle it, But the point is that the normal aprx should be good enough--even with an avg of only 12. – BruceET Mar 28 at 18:25

The expectation of a single sample is $$1$$, with variance $$\frac13$$, so the average of twelve samples has an expectation of $$1$$ and variance of $$\frac{1}{36}$$, saying (as you found) that for this average $$0.5$$ would be $$3$$ standard deviations below the mean. For a standard normal distribution, you are correct that the probability of values less than $$3$$ standard deviations below the mean is about $$0.00135$$

The difficult question is whether a normal approximation is a good approximation in this case: in particular normal approximations can sometimes perform relatively poorly in the tails

Your question actually involves a Bates distribution, but it is easier to handle translating it to the Irwing-Hall distribution: you want the probability that the sum of $$12$$ uniform i.i.d. random variables on $$[0,1]$$ is less than $$3$$. This is $$\frac{1}{12!}\left({12 \choose 0}(3-0)^{12} - {12 \choose 1}(3-1)^{12} + {12 \choose 2}(3-2)^{12}- {12 \choose 3}(3-3)^{12}\right) \approx 0.001007$$

So the exact answer is rather less than the normal approximation. Whether the normal approximation is good enough, I leave for you to judge


text:
Browse Questions

# Which of the following biomolecules is insoluble in water?

$\begin {array} {1 1} (A)\;a-Keratin & \quad (B)\;heamoglobin \\ (C)\;ribonucleus & \quad (D)\;adenine \end {array}$

Keratin is a family of fibrous structural proteins. Keratin is the key structural material making up the outer layer of human skin. It is also the key structural component of hair and nails. Keratin monomers assemble into bundles to form intermediate filaments, which are tough and insoluble and form strong unmineralized tissues found in reptiles, birds, amphibians, and mammals. The only other biological matter known to approximate thetoughness of keratinized tissue is chitin.
Ans : (A)


text: Another way to put it: as a point moves around a circle at constant speed, its vertical and horizontal coordinates oscillate.


text:
# zbMATH — the first resource for mathematics

On affine normal forms and a classification of homogeneous surfaces in affine three-space. (English) Zbl 0999.53008
The authors contribute to the homogeneity problem of affine hypersurfaces by establishing a new method based on the Taylor expansion for the defining graph function with respect to the Blaschke/Berwald normal as last axis. Specifically, they classify all the affinely homogeneous surfaces in $$3$$-space, thus confirming the result of B. Doubrov, B. Komrakov and M. Rabinovich [Geometry and Topology of Submanifolds VIII, World Scientific, Singapore, 168-178 (1996; Zbl 0934.53007)]. The main difference is that instaed of normalizing the symmetry algebras now the graph functions are normalized. Actually, this is done by a clever induction procedure on the Taylor polynomials of these functions. The authors believe that their approach is more advantegeous. As they realized after the completion of their work, yet another method for the classification of these surfaces has been proposed by A. V. Loboda [Math. Notes 65, No. 5, 668-672 (1999); translation from Mat. Zametki 65, 793-797 (1999; Zbl 0976.53012) and Proc. ‘Pontrjagin’s Readings-6’, Voronezh-98, 124 (1998)].
Several computations related to the problem of homogeneity are nonlinear and rather complex, including enormous branching phenomena. So machine computing is sometimes inevitable. On the other hand, there doesn’t exist an automatism to attack the problem as a whole. For the next higher dimension (hypersurfaces in $$4$$-space) there are recent works by the same authors in the complex domain [Asian J. Math. 5, 721-740 (2001; Zbl 1022.53043)] and by M. Wermann with respect to real classification [Dissertation Univ. Dortmund 2001, Shaker Verlag (2001; Zbl 1028.53010)]. Codimension two homogeneity problems in $$4$$-space have been solved by normalizing Lie subalgebras in the reviewer’s papers [Geom. Dedicata 71, 129-178 (1998; Zbl 0973.53511), Contrib. Algebra Geom. 41, No. 1, 159-180 (2000; Zbl 0976.53013)], and forthcoming [Banach Center Publications, Proc. Conf. ‘PDEs, submanifolds and affine differential geometry’, Banach Center, Warzaw, Sept. 2000].

##### MSC:
 53A15 Affine differential geometry 53C30 Differential geometry of homogeneous manifolds
Full Text:


text:
What is the difference between homotopy and coherent homotopy?

This question had been bugging me for a while, and I have been unable to find a source that is suited to the beginning topologist. Eric Peterson kindly answered this for me, and I found his explanation so astounding that I wish to share on the off chance that you, dear reader, will similarly benefit from this visually rich narrative. All errors are mine and not his.

Coherence is essentially about the existence of diagram categories. For instance, suppose you have some homotopy class $A \wedge A \to A$, which I’d like you to think of as a multiplication map on $A$. You can think of this as a homotopy class

$$S^0 \to F(A \wedge A, A)$$

in the appropriate function spectrum. Then, given any map of this signature, you can build two maps $A \wedge A \wedge A \to A$ out of it:

$A \wedge A \wedge A = A \wedge (A \wedge A) \to A \wedge (A) = A \wedge A \to A$, and
$A \wedge A \wedge A = (A \wedge A) \wedge A \to (A) \wedge A = A \wedge A \to A$,

corresponding on the level of function spectra to a map

$$F(A \wedge A, A) \to F(A \wedge A \wedge A, A)^{\partial \Delta^0}$$

where the right-hand object has the distinguished property

$S^0 \to F(A \wedge A \wedge A, A)^{\partial \Delta^1} = (\partial \Delta^1_+ \wedge S^0) \to F(A \wedge A \wedge A, A)$
$= (\partial \Delta^1_+) \to F(A \wedge A \wedge A, A) = (S^0 \vee S^0) \to F(A \wedge A \wedge A, A)$

i.e., it selects two homotopy classes.

The assertion that the multiplication is associative is the same as saying that these two homotopy classes lie in the same path component of the function space, and a witness to this fact is a map $h$ in

$$S^0 \xrightarrow{h} F(A \wedge A \wedge A, A)^{\Delta^1} \to F(A \wedge A \wedge A, A)^{\partial \Delta^1}$$

which upon restriction to the endpoints of the 1-simplex recovers the two-different-associations map specified above. Then, given such an $h$, you can follow a similar procedure with $F(A \wedge A \wedge A \wedge A, A)$ to build an element of

$$S^1_+ \to F(A \wedge A \wedge A \wedge A, A)$$

that, naively, could be nonzero. This is distinctly different from the algebraic case, where there are still function objects, but they’re all *sets*, and hence they will never have any $\pi_1$, and so this map is *always* fillable. With spaces or spectra (or anything else homotopical), this is no longer guaranteed, and so you have to assure yourself that you can perform this filling. And then, once you’ve chosen such a filling, you can use it to build an element of $\pi_2$ of the next thing…

There are different ways to bundle this formalism, but all of them rely essentially on the existence of these “diagram categories”

$$C^X$$

where $C$ is some suitable category (like: a category enriched in spaces or an ($\infty$, 1)-category, as is the case for Spectra) and $X$ is some indexing object (like: a simplicial set, as is the case for $\Delta$). The two big names are “quasicategories” (which give you a little more than this but are generally useful) and “derivators” (which give you *exactly* this and are considerably less popular for general use). Continuing the example above, an associative algebra object is then a map

$$\Delta \to C$$

(or a point in $C^\Delta$) which restricts on objects to the tensor powers of some fixed object in $C$ (cf. 4.1.2.14-15 of Higher Algebra for a taste).

In any case, “coherent” is meant to express that the process is self-referential: each choice of previous homotopy influences choice of the next homotopy (and, indeed, even the ability to choose one at all). Note also that the process is complicated dramatically when considering commutative algebra objects, since then you have all these shuffle maps to keep track of.

This summary is sort of separate, though, from what coherence is useful for. The low-level summary is that ring spectra in the homotopy category have homotopy classes of maps between them (which homotopy-commute with the ring structure maps), but structured ring spectra have *function spaces* (or spectra) between them (which describe how to slide a maps between these $\Delta$-indexed objects around).

So, if you want to do homotopy theory with ring objects, you don’t have access to that until you start saying structured things.

The high-level summary of the Elmendorf-Kriz-Mandell-May book is that coherent ring spectra are the spectra for which you can build a theory of algebra. Coherent ring spectra have categories of modules (with function spaces between them, so this is a statement about homotopies), those categories have suitably monoidal tensor products (where e.g. the associativity of the tensor product is also controlled by homotopies), and the tensor products are controlled by spectral sequences grounded in homological algebra (i.e., there are machines interchanging doing homotopy with algebraic objects for doing algebra with homotopical objects). So, if you have some algebraic question (about formal group laws) and some homotopical objects (like $MU$ and $BP$), then this is the sort of framework you’d want to design in an effort to repeat algebraic proofs, applied to these fancier objects.


text:
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# Enhanced skyrmion stability due to exchange frustration

### Subjects

An Author Correction to this article was published on 28 May 2019

## Abstract

Skyrmions are localized, topologically non-trivial spin structures which have raised high hopes for future spintronic applications. A key issue is skyrmion stability with respect to annihilation into the ferromagnetic state. Energy barriers for this collapse have been calculated taking only nearest neighbor exchange interactions into account. Here, we demonstrate that exchange frustration can greatly enhance skyrmion stability. We focus on the prototypical film system Pd/Fe/Ir(111) and use an atomistic spin model parametrized from first-principles calculations. We show that energy barriers and critical fields of skyrmion collapse as well as skyrmion lifetimes are drastically enhanced due to frustrated exchange and that antiskyrmions are metastable. In contrast an effective nearest-neighbor exchange model can only account for equilibrium properties of skyrmions such as their magnetic field dependent profile or the zero temperature phase diagram. Our work shows that frustration of long range exchange interactions – a typical feature in itinerant electron magnets – is a route towards enhanced skyrmion stability even in systems with a ferromagnetic ground state.

## Introduction

Skyrmions have been predicted to occur in magnetic materials based on micromagnetic models more than 25 years ago1,2. Experimentally skyrmions were first observed in bulk materials with a chiral crystal structure3,4,5. In these systems skyrmions are stabilized by the Dzyaloshinskii-Moriya interaction (DMI)6,7. The DMI occurs due to spin-orbit coupling (SOC) in systems with a broken structural inversion symmetry. Therefore, it is present also at surfaces or interfaces8,9 and can induce spin spirals9,10, chiral domain walls11,12,13,14,15 and skyrmions16,17. Since transition-metal interfaces and multilayers are at the heart of technologically established spintronic devices such as read heads of hard-disk drives based on the giant magnetoresistance effect18,19 the discovery of skyrmions at interfaces16,17 sparked great interest in using them for novel device concepts20,21.

For potential applications20,21,22,23 the stability of magnetic skyrmions becomes a crucial issue. An experimentally well studied system for nanoscale skyrmions at interfaces is Pd/Fe/Ir(111)17,24,25,26,27, i.e. a single atomic layer of Pd grown on an Fe monolayer on the Ir(111) surface. Based on fitting field dependent experimental skyrmion profiles to the micromagnetic model effective parameters have been obtained24 which were used to discuss properties of isolated skyrmions27. Current-induced skyrmion annihilation has also been studied combining the experiments with Monte-Carlo simulations from which energy barriers of about 50 meV have been estimated26. Energy barriers protecting skyrmion states in ultrathin films have been obtained by calculating minimum energy paths (MEPs) for the skyrmion collapse into the ferromagnetic state28,29,30,31 which due to the huge configuration space is a non-trivial task32,33.

While an atomistic spin model has been used in these approaches, the exchange interaction has been treated within the nearest-neighbor (NN) approximation which corresponds to the exchange stiffness within the micromagnetic model. For Pd/Fe/Ir(111) it has been shown based on density functional theory (DFT) calculations that the exchange interaction is ferromagnetic for NN while it is antiferromagnetic for the 2nd and 3rd NN34,35. The effect of exchange frustration on skyrmion lattice formation has been studied in refs36,37,38,39. Therefore, it is not clear in how far spin models using an effective NN exchange interaction are sufficient to describe properties of isolated skyrmions in these systems.

Here, we demonstrate that frustrated exchange greatly enhances skyrmion stability in ultrathin films and can lead to metastable antiskyrmions. We use atomistic spin dynamics simulations parametrized from DFT for Pd/Fe/Ir(111) and obtain energy barriers for skyrmion annihilation into the ferromagnetic state by applying the geodesic nudged elastic band (GNEB) approach28. We compare these calculations with an effective NN exchange model based on our DFT calculations which yields parameters that are in good agreement with those obtained from experiments24,26. We find that equilibrium skyrmion properties such as field-dependent profiles or the zero temperature phase diagram are well described by the effective NN exchange approach. However, energy barriers of skyrmion collapse and skyrmion lifetimes are greatly underestimated if the frustrated exchange interaction is mapped to an effective NN exchange. Antiskyrmions are not even metastable in the effective NN exchange model.

## Results

### Atomistic spin model

We describe the magnetic properties of Pd/Fe/Ir(111) using the spin Hamiltonian given by

$$\begin{array}{c}H=-\sum _{ij}{J}_{ij}({{\bf{m}}}_{i}\cdot {{\bf{m}}}_{j})-\sum _{ij}{{\bf{D}}}_{ij}\cdot ({{\bf{m}}}_{i}\times {{\bf{m}}}_{j})\\ \quad \quad -\sum _{i}K{({m}_{i}^{z})}^{2}-\sum _{i}{\mu }_{s}{\bf{B}}\cdot {{\bf{m}}}_{i}\,,\end{array}$$
(1)

which describes the magnetic interactions between the magnetic moments M i of atoms at sites R i where m i = M i/Mi. The parameters for the exchange interaction (J ij ), the DMI (D ij), the magnetic moments (μ s ) as well as an uniaxial magnetocrystalline anisotropy (K) were obtained from DFT using the FLEUR code34,40,41,42 (see methods for details). Note that J ij and D ij are defined per spin because each pairwise interaction contributes twice in the total energy. We do not explicitly include dipole-dipole interactions. However, for ultrathin films this energy term is very small – on the order of 0.1 meV/atom – and it can be effectively included into the magnetocrystalline anisotropy energy30,43.

While the Fe layer follows the fcc stacking of the Ir(111) surface, both Pd stackings have been observed experimentally44. Figure 1a shows the energy dispersions E(q) of flat homogeneous spin spirals including the effect of SOC calculated from DFT for both hcp and fcc stacking of the Pd overlayer. A spin spiral is characterized by a wave vector q from the two-dimensional Brillouin zone (2D-BZ) and the magnetic moment of an atom at site R i is given by $${{\bf{M}}}_{i}=M(\sin ({\bf{q}}{{\bf{R}}}_{i}),\,\cos ({\bf{q}}{{\bf{R}}}_{i}),\,\mathrm{0)}$$ with the size of the magnetic moment M. We find a value of about 2.7 μ B per Fe-atom in our calculations which is fairly constant for all q vectors. Combined with the magnetic moment of 0.3 μ B due to spin polarization of the Pd-atoms, we obtain a total moment of 3.0 μ B that we use for our simulations. Fitting the energy dispersions along the high symmetry directions of the 2D-BZ using the first two terms of Eq. (1) allows to extract the exchange constants and the DMI (see Table 1). Note that the stacking of the Pd overlayer affects the Ruderman-Kittel-Kasuya-Yosida type exchange mediated by conduction electrons due to the key role played by the hybridization at the Pd/Fe/Ir interface34. Our computed shell resolved J ij agree well with those presented in ref.35. In order to obtain a good fit of the DFT energy dispersion many shells need to be taken into account for the exchange while the DMI has been treated in the NN approximation.

It has been suggested to characterize the behavior of the exchange interaction close to the $$\bar{{\rm{\Gamma }}}$$ point of the 2D-BZ – corresponding to the ferromagnetic (FM) state – by an effective NN exchange constant J eff 45 – an approach which has also been used in previous theoretical studies of skyrmion stability26,27,28.

From our DFT energy dispersion we can obtain J eff by using only the NN terms of the first two terms in the spin Hamiltonian, Eq. (1), which yields along the $$\bar{{\rm{\Gamma }}}-\bar{{\rm{K}}}$$ direction $$E(q)={J}_{{\rm{eff}}}(2\,\cos (q)+4\,\cos (q\mathrm{/2)})$$ (with q in units of 2π/a, see Table 2 for J eff). We obtain an excellent approximation around the local spin spiral energy minimum (cf. inset of Fig. 1a). However, the fit drastically underestimates E(q) from DFT for larger values of q. For fcc-Pd/Fe/Ir(111) we had to fit also the DMI to obtain a good agreement. For spin spirals along $$\bar{{\rm{\Gamma }}}-\bar{{\rm{K}}}$$ we use $$E(q)={J}_{{\rm{eff}}}(2\,\cos (q)+4\,\cos (q\mathrm{/2)})+{D}_{{\rm{eff}}}(2\,\sin (q)+2\,\sin (q\mathrm{/2)})$$ and a fit interval close to the energy minimum (see Table 2 for J eff and D eff).

Effective NN exchange and DMI for Pd/Fe/Ir(111) have previously been extracted by fitting experimental data to micromagnetic and Monte-Carlo simulations24,26 (for values see Table 2). The energy dispersions obtained with these values are displayed in Fig. 1a. These curves are between the DFT dispersions for hcp and fcc Pd stacking around the energy minimum (cf. inset of Fig. 1a). For larger values of q, i.e. larger angles between adjacent spins, they are close to the effective fit in fcc stacking but deviate drastically from the DFT curves.

### Zero temperature phase diagrams

To compare the effective description of the exchange (denoted as J eff) and DMI (for fcc-Pd/Fe/Ir(111)) and the full set of exchange constants from DFT (denoted as J DFT), we have obtained the phase diagram at zero temperature as a function of the applied magnetic field (Fig. 1b–e) using spin-dynamics simulations (see methods). Note that we have used the magnetocrystalline anisotropy obtained from DFT in both cases.

For hcp-Pd/Fe/Ir(111) the phase diagrams are very similar for both ways of treating the exchange [Fig. 1b,c]. The FM state is the ground state over the whole range of magnetic field values. Homogeneous spin spirals relax into a domain wall structure which gains energy due to DMI11. Isolated skyrmions are metastable for both sets of parameters when an external field of at least ≈0.5 T is applied. For smaller fields, the skyrmions become large bubbles with a fixed rotational sense of the domain walls surrounding the FM core.

For fcc-Pd/Fe/Ir(111) [Fig. 1d,e] the ground state is a spin spiral consistent with the energy minimum in the spin spiral dispersion curve (cf. inset of Fig. 1a). For both J eff and J DFT a skyrmion lattice is energetically favorable at a critical field and at even larger fields there is a transition to the FM phase. However, the critical fields at which these transitions occur are slightly different ($${B}_{{\rm{c}}}^{{\rm{eff}}}\approx 3.6$$ T and $${B}_{{\rm{c}}}^{{\rm{DFT}}}\approx 3.2$$ T) due to the larger DMI obtained by the fit. Using the full set of DFT parameters we were also able to metastabilize antiskyrmions within the skyrmion lattice and FM phase.

In the FM phase isolated skyrmions are metastable in our spin dynamics simulations up to a critical field for both hcp-Pd/Fe/Ir(111) and fcc-Pd/Fe/Ir(111). The skyrmion profiles were obtained by imposing the theoretical profile46 and relaxing this spin structure within our spin dynamics simulation. The radii were extracted as in ref.46. The obtained relaxed skyrmion profiles for both descriptions of the exchange are nearly indistinguishable as exemplified in the insets of Fig. 2. The resulting radii decrease rapidly with applied magnetic field (cf. Fig. 2) consistent with experiments24 and micromagnetic models46. Note that the range of metastable skyrmions stretches beyond B = 8 T for some cases as shown below.

For hcp stacking of Pd the radii are nearly the same using either J eff or J DFT. However, isolated skyrmions are metastable for much larger fields if we use the full set of DFT parameters. For fcc stacking the radii are larger than for hcp stacking and they decrease faster with field considering J eff leading to a crossing of the two curves. The radii obtained from experiments on Pd/Fe/Ir(111)24 are very close to those from J DFT for fcc stacking of Pd. Antiskyrmions can be stabilized only if frustrated exchange is present as in fcc-Pd/Fe/Ir(111). Their radii are smaller than those of skyrmions in the same system. Below B = 3 T they become asymmetric, stretched objects and vanish for fields above B = 5.5 T.

Overall we find a very good agreement between simulations using the full set of exchange constants, the effective parameters and experimental data. We conclude that equilibrium properties of skyrmions in Pd/Fe/Ir(111) such as field dependent profiles or zero temperature phase diagrams can be accurately described by an effective (or micromagnetic) model for the exchange interactions.

### Skyrmion stability

Now we turn to the collapse of isolated skyrmions into the ferromagnetic state. The minimum energy path for this process is obtained using the GNEB method (see methods). Figure 3 shows the energy barriers calculated for both Pd stackings at selected field values and both types of describing the exchange. The initial skyrmion state and the mechanism of collapse are independent of how the exchange interaction is treated. The skyrmion shrinks along the path until at the saddle point the inner spins turn into the film plane after which the topological charge vanishes (see Supplementary Figs 1 and 2) and the skyrmion can collapse to the ferromagnetic state (see Supplementary Movies 1 to 4). This mechanism has already been described based on the effective exchange model28,29,30,32,33. However, the energy barriers are drastically enhanced if the frustrated exchange is explicitly treated in the simulations (see Supplementary Figs 1 and 2 for interaction resolved energy contributions along the reaction path). For the chosen magnetic field values we find an enhancement from about 34 to 59 meV for hcp-Pd/Fe/Ir(111) and from 71 to 143 meV for fcc-Pd/Fe/Ir(111).

The spin structure at the saddle point is more extended if the full set of exchange constants is taken into account for fcc-Pd/Fe/Ir(111) as shown in the insets of Fig. 4. The main contribution to the energy at the saddle point with respect to the FM state stems from the exchange interactions while all other interactions are small. The inner shells of the spin structure give the largest contribution since the angles between adjacent spins are largest. This energy is much enhanced if the full set of exchange constants is considered since using an effective exchange J eff underestimates this term. The role of the frustrated exchange is apparent also from its energy contribution along the reaction path (see supplementary Fig. S1) which exhibits a large barrier – an effect which is absent for an effective exchange. For hcp stacking of the Pd layer a similar picture is obtained (see Supplementary Figs 2 and 3).

Frustrated exchange interactions allow the appearance of metastable antiskyrmions38,39,47,48. We find non-vanishing energy barriers only in our simulations of fcc-Pd/Fe/Ir(111) using the full set of DFT parameters. In this case the exchange interactions alone already lead to a shallow energy minimum in the dispersion of spin spirals. The mechanism of antiskyrmion collapse is similar to that for skyrmions with a shrinking of diameter and a saddle point configuration with inner spins pointing in the film plane (see insets of Fig. 5 and Supplementary Movie 5). We obtain a small energy barrier of about 15 meV at a magnetic field of B = 4 T as shown in Fig. 5 (see Supplementary Fig. 4 for interaction resolved energy contributions along the path).

## Discussion

The energy barriers for skyrmion collapse exhibit a nonlinear dependence on the external magnetic field as shown in Fig. 6. For fcc-Pd/Fe/Ir(111) the barriers above the critical field of $${B}_{{\rm{c}}}^{{\rm{eff}}}\mathrm{=3.6}$$ T are of the same order if we use J eff as given by Hagemeister et al.26 based on Monte-Carlo simulations with an effective NN exchange. However, for all field values the barriers are drastically enhanced if the frustrated exchange in this system is treated properly. As a consequence isolated skyrmions are metastable up to much higher magnetic fields. Furthermore, antiskyrmions are only metastable if the full exchange interaction is taken into account with barriers on the order of 10 to 20 meV.

Even for hcp-Pd/Fe/Ir(111) which shows a ferromagnetic ground state at all field values there is a significant difference between the barriers for J eff vs. J DFT. Despite the very similar zero temperature phase diagram (cf. Fig. 1b,c) isolated skyrmions are metastable at much larger fields for J DFT.

Within transition-state theory49 the lifetime τ of skyrmions depends exponentially on the energy barrier ΔE, i.e. $$\tau ={\tau }_{0}\,\exp ({\rm{\Delta }}E/{k}_{{\rm{B}}}T)$$ where T is the temperature and 1/τ 0 is the attempt frequency. The enhancement of the lifetime due to taking exchange beyond NN into account for the barrier is $${\tau }_{{\rm{DFT}}}/{\tau }_{{\rm{eff}}}=\exp ([{\rm{\Delta }}{E}_{{\rm{D}}FT}-{\rm{\Delta }}{E}_{{\rm{eff}}}]/{k}_{{\rm{B}}}T)$$. For both hcp and fcc stacking of Pd there is a huge enhancement by one or three orders of magnitude at 100 K, respectively (see insets of Fig. 6). Even at room temperature there is a large increase of skyrmion lifetime.

The key observation is that at the saddle point which defines the height of the energy barrier the angles between adjacent spins are large and the energy of this configuration cannot be accurately described here by the effective NN exchange or micromagnetic model. From the energy dispersion of spin spirals (cf. Fig. 1a) this is obvious since the effective NN exchange can only describe E(q) well close to the ferromagnetic state for small canting angles between adjacent spins while it breaks down at large q corresponding to large NN angles. The same logic dictates that the NN-exchange model should work better for the skyrmion annihilation at the boundary of a finite-size system since this process does not involve formation of singularities50,51. More detailed analysis of the applicability of the effective NN-exchange description of the skyrmion collapse at the boundary of a finite Pd/Fe/Ir(111) system as well as other itinerant electron magnets is, however, out of the scope of the present study and will be published elsewhere.

The effective exchange model even with parameters obtained from experiment24,26,27 is limited to describe skyrmion properties such as skyrmion profiles or zero temperature phase diagrams which rely on states close to the energy minimum. For all properties which require the treatment of configurations with large angles between adjacent spins such as skyrmion stability it is not sufficient. Therefore, properly treating exchange is also essential to simulate thermal fluctuations in order to obtain phase diagrams at finite temperature and phase transition temperatures52.

Our work shows the importance of accounting for the neighbor resolved exchange interaction even in systems with a ferromagnetic ground state and that engineering exchange frustration45 is a promising route towards enhanced skyrmion stability. Although we focus here on a particular system, the conclusions we arrive at are rather general and should be valid in other itinerant electron magnets where frustrated, long-range exchange is a typical feature.

## Methods

### First-principles calculations

We have explored the ultrathin film system Pd/Fe/Ir(111) from first-principles based on the full-potential linearized augmented plane wave method as implemented in the FLEUR code (www.flapw.de). Within this approach we can calculate the total energy of non-collinear magnetic structures such as spin spirals40 including the DMI in first order perturbation theory with respect to the spin-orbit coupling41. We have used a two-dimensional hexagonal p(1 × 1) unit cell within each layer and the in plane-lattice parameter of the Ir(111) surface as obtained from DFT in ref.34. The relaxed interlayer distances were also taken from ref.34. The magnetic properties were obtained within the local density approximation53 as previously10,16,34. All calculations have been carried out with a plane wave cutoff of k max = 4.3 a.u.−1. The muffin tin radii were set to 2.23 a.u. for Fe and to 2.31 a.u. for Pd and Ir. We have used a 44 × 44 k-point mesh in the full two-dimensional Brillouin zone. In order to obtain quantitative parameters for the exchange interactions, we have converged the small energy minimum of the dispersion curve (Fig. 1(a)) with respect to the number of Ir substrate layers. With 5 layers of Ir substrate34 the energy minimum of fcc-Pd/Fe/Ir(111) is E min = −1.1 meV/Fe (without spin-orbit coupling). Here, we have used asymmetric films with one layer Pd, one layer Fe and 15 layers of Ir substrate which leads to an energy minimum of E min = −0.06 meV/Fe. To determine the exchange constants, we have used the energy dispersion curve of spin spirals E(q) for films with 15 Ir substrate layers for the spiral vector q in [0,1/4] and with 5 Ir layers for q in [1/4,2/3] (in units of 2π/a) along the $$\bar{{\rm{\Gamma }}}\bar{{\rm{K}}}$$ direction. The DM interaction has been taken from the calculations with 5 Ir substrate layers34 as it did not change with substrate thickness.

### Spin dynamics simulations

In order to calculate the energy differences between the FM, SkX and SS phases for the low temperature phase diagrams and to relax the isolated skyrmions and antiskyrmions, we used the Landau-Lifshitz equation of spin dynamics:

$$\hslash \frac{d{{\bf{m}}}_{i}}{dt}=\frac{\partial H}{\partial {{\bf{m}}}_{i}}\times {{\bf{m}}}_{i}-\alpha (\frac{\partial H}{\partial {{\bf{m}}}_{i}}\times {{\bf{m}}}_{i})\times {{\bf{m}}}_{i}$$
(2)

where α is the damping parameter, m i is a single spin and H is the Hamiltonian given in Eq. (1). We have used damping parameters from α = 0.5 to α = 0.05 and time steps ranging from 0.05 fs to 0.1 fs to carefully relax the structures. The simulations are carried out on a timescale of 100 to 500 pico seconds and the equation of motion were solved with the semi-implicit integrator as proposed by Mentink et al.54.

### Geodesic nudged elastic band method

Minimum energy paths (MEPs) for skyrmion and antiskyrmion annihilation processes were identified using the geodesic nudged elastic band (GNEB) method28. The GNEB method involves taking some initially generated path between the energy minima, and systematically bringing that to the MEP. A path is represented by a discrete chain of states, or ‘images’, of the system, where the first one corresponds to the skyrmion or antiskyrmion configuration while the last one corresponds to the ferromagnetic state. At each image, the effective field is calculated and its component along a local tangent to the path is substituted by an artificial spring force between the images which ensures uniform distribution of the images along the path. This modified effective field is substituted in Eq. (2) in the overdamped regime and the whole chain of images evolved till convergence. The final, relaxed position of images gives discrete representation of the MEP. The energy maximum along the MEP corresponds to the saddle point (SP) on the multidimensional energy surface of the system and defines the energy barrier separating the stable states. In order to determine the maximum energy accurately, the highest energy image can be treated separately during the iterative optimization and made to move uphill in energy along the path. The effective field on this climbing image (CI) is calculated by deactivating the spring force acting on it and inverting the parallel component of the gradient field. After the CI-GNEB calculation has converged, the position of the CI coincides with the SP along the MEP and gives an accurate value of the SP energy.

### Data availability

The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files.

## Change history

• ### 28 May 2019

A correction to this article has been published and is linked from the HTML and PDF versions of this paper. The error has not been fixed in the paper.

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## Acknowledgements

We gratefully acknowledge computing time at the supercomputer of the North-German Supercomputing Alliance (HLRN). This project has received funding from the European Unions Horizon 2020 research and innovation programme under grant agreement No. 665095 (FET-Open project MAGicSky). P.F.B. acknowledges support from the Icelandic Research Fund (Grant No. 163048-052) and the mega-grant of the Ministry of Education and Science of the Russian Federation (grant no. 14.Y26.31.0015). B.D. thanks the Deutsche Forschungsgemeinschaft for funding within the project DU1489/2-1. A.D. acknowledges financial support from Swedish e-science Research Centre (SeRC), Vetenskapsrådet (grant numbers VR 2015-04608 and VR 2016-05980), and Swedish Energy Agency (grant number STEM P40147-1). Some of the computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at the National Supercomputer Center (NSC), Linköping University, the PDC Centre for High Performance Computing (PDC-HPC), KTH, and the High Performance Computing Center North (HPC2N), Umeå University.

## Author information

Authors

### Contributions

S.v.M. performed the spin dynamics simulations. S.v.M. and P.F.B. performed the GNEB calculations. B.D. performed the first-principles calculations. B.D., P.F.B., A.D., and S.H. conceived the project. S.v.M. and S.H. wrote the manuscript. All authors analyzed and discussed the data and contributed to the writing of the paper.

### Corresponding author

Correspondence to S. Heinze.

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von Malottki, S., Dupé, B., Bessarab, P.F. et al. Enhanced skyrmion stability due to exchange frustration. Sci Rep 7, 12299 (2017). https://doi.org/10.1038/s41598-017-12525-x

• Accepted:

• Published:

• ### Experimental identification of two distinct skyrmion collapse mechanisms

• Florian Muckel
• , Stephan von Malottki
• , Christian Holl
• , Benjamin Pestka
• , Marco Pratzer
• , Pavel F. Bessarab
• , Stefan Heinze
•  & Markus Morgenstern

Nature Physics (2021)

• ### Dense skyrmion crystal stabilized through interfacial exchange coupling: Role of in-plane anisotropy

• Ming-Xiu Sui
• , Zi-Bo Zhang
• , Xiao-Dan Chi
• , Jia-Yu Zhang
•  & Yong Hu

Frontiers of Physics (2021)

• ### Toward room-temperature nanoscale skyrmions in ultrathin films

• Anastasiia S. Varentcova
• , Stephan von Malottki
• , Maria N. Potkina
• , Grzegorz Kwiatkowski
• , Stefan Heinze
•  & Pavel F. Bessarab

npj Computational Materials (2020)

• ### Role of higher-order exchange interactions for skyrmion stability

• Souvik Paul
• , Soumyajyoti Haldar
• , Stephan von Malottki
•  & Stefan Heinze

Nature Communications (2020)

• ### Isolated zero field sub-10 nm skyrmions in ultrathin Co films

• Sebastian Meyer
• , Marco Perini
• , Stephan von Malottki
• , André Kubetzka
• , Roland Wiesendanger
• , Kirsten von Bergmann
•  & Stefan Heinze

Nature Communications (2019)


text:
Projectile Motion.

1. Apr 15, 2005

amcavoy

In rectangular coordinates you have:

$$y=-\frac{1}{2}gt^2+v_{0}\sin{(\theta)}t+h$$

$$x=v_{0}\cos{(\theta)}t$$

Is there any way to use vectors to determine the postion of a projectile? If so, how would you convert rectangular coordinates to vectors?

Thanks.

2. Apr 15, 2005

James R

The position vector r of the object at time t is just:

$$\vec{r} = x(t)\hat{i} + y(t)\hat{j}$$

where the i and j are unit vectors in the x and y directions.

3. Apr 16, 2005

marlon

Alex,

What you are asking about is the very base of Newtonian mechanics. It works like this : Suppose we work in two dimensions denoted by a x-axis and an y-axis. You can work in as many dimensions as you want because all you have to do is add a unit vector to the formula's, as you will see.

Starting from the acceleration ,one can calculate the velocity and position by using integrals at any time : r_0 is initial position at t=0, v_0 is initial velocity

$$\vec{F} = F_x \vec{e_x} + F_y \vec{e_y} = m(a_x \vec{e_x} + a_y \vec{e_y})$$

Where the e_x and e_y denoted the x and y-direction (ie the unit vectors)

Now, integrating will yield
$$\vec{v} = \vec{v_0} + \vec{a}t$$
$$\vec{r} = \vec{r_0} + \vec{v_0}t+ \vec{a} \frac{t^2}{2}$$

Now, the trick really is (and that's the essential part) to apply the same procedure in each direction. The procedure i am talking about is projecting each vector along a direction using the triangle-equalities.

For example : in the x-direction you will have :

$$F_x = ma_x$$
$$v_x = v_{0x} + a_xt$$ and
$$x = r_{0x} + v_{0x}t+ a_x \frac{t^2}{2}$$

$$v_{0x} = ||\vec{v_0}||cos( \theta)$$
$$r_{0x} = ||\vec{r_0}||cos( \theta)$$

the ||.|| denotes the MAGNITUDE of the vector

You see ? the clue is that each vector can be written as a sum of an x and y component $$\vec{A} = A_x \vec{e_x} + A_y \vec{e_y}$$

$$A_x = ||\vec{A}||cos( \theta)$$
$$A_y = ||\vec{A}||sin( \theta)$$

You will need to be carefull with the signs of the x and y components because those depend of the direction of each component with respect to the actual x and y axis.

So, when a force is given, like : F = m(2e_x - 9.81e_y) and the initial position has components r_0 = 2e_x + 6e_y and the initial velocity v_0 = 6e_y, can you write down the equations for both velocity and position in each direction ???

In the end , you must realize that this system is very easy because, nomatter how complicated the force may look, the procedure to determin both position and velocity as a function of time is always the same: Projecting the vectors along the given directions. Once, you have done that, you can do almost anything with these formula's...

regards
marlon


text:
# Question #4dc8a

Apr 11, 2017

$1 , 5351 \cdot {10}^{24}$

#### Explanation:

$0.850 \cdot 6.02 \cdot {10}^{23} = 5.117 \cdot {10}^{23}$
Multiply the answer from above by 3 as there are three atoms in $C {O}_{2}$:
$3 \cdot 5.117 \cdot {10}^{23} = 1.5351 \cdot {10}^{24}$


text:
# Writing statements into symbols Discrete Math

The variable $x$ represents stduents, $F(x)$ means "$x$ is a freshman", and $M(x)$ means "$x$ is a math major"

a) some freshme are math majors? $\exists x:F(x) \implies M(x)$

b) Every math major is a freshman? $\forall x:M(x) \implies M(x)$

c) No math major is a freshman? $\neg\forall x:M(x) \implies F(x)$

-

$(a)$ Some freshmen are math majors
$\sim$ There exist x such that (x is a Freshman and x is a math major):

$$\exists F(F(x) \land M(x))$$

$(b)$ Every math major is a freshman.
$\sim$ For all x (if x is a math major, then x is a Freshman.) $$\forall x (M(x) \rightarrow F(x))$$

$(c):$ No math major is a freshman.
$\sim$ There does not exist an x such that (x is a math major and x is a freshman). \begin{align} \lnot \exists x(M(x) \land F(x)) & \equiv \forall x(\lnot M(x) \lor \lnot F(x))\\ \\ &\equiv \forall x(M(x) \rightarrow \lnot F(x))\end{align}

-
Right up your alley +1 –  Amzoti Dec 7 '13 at 0:20


text:
# position images in several columns

I'm trying to put to images in two columns on a single beamer slide. This is my code

\documentclass{beamer}

\usepackage{multicol}
\usepackage{graphicx}

\begin{document}

\begin{frame}{In short}
\begin{multicols}{2}
\begin{itemize}
\item How to get from here..
\begin{figure}
\includegraphics[width=0.45\textwidth]{figures/demo0-obs.png}
\end{figure}
\vfill\null
\columnbreak
\item ...to here
\begin{figure}
\includegraphics[width=0.45\textwidth]{figures/demo0.png}
\end{figure}
\end{itemize}
\end{multicols}
\end{frame}

\end{document}

And this is the result What am I doing wrong? It seems to me that I have the exact same code for every column. Why are results different? I tried adding different image placement options to the figure environment like t,h p, b but none of that has helped. Could you please offer some hints? Thanks!

beamer has own "multicols" environment, so the best is to use it:

\documentclass[demo]{beamer}

\begin{document}

\begin{frame}{In short}
\begin{column}{.5\linewidth}
\begin{itemize}
\item How to get from here \dots
\begin{figure}\centering
\includegraphics[width=0.9\linewidth]{figures/demo0-obs.png}
\end{figure}
\end{itemize}
\end{column}
\begin{column}{.5\linewidth}
\begin{itemize}
\item ...to here
\begin{figure}\centering
\includegraphics[width=0.9\linewidth]{figures/demo0.png}
\end{figure}
\end{itemize}
\end{column}
\end{columns}
\end{frame}
\end{document}

• thanks for letting me know about this column environment. Works swimmingly! – marcin_j Jul 29 '17 at 20:29