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Difficulties in the ab initio description of electron transport through spin filters

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 104203 (7pp)

doi:10.1088/0953-8984/26/10/104203

Difficulties in the ab initio description of electron transport through spin filters Mikaël Kepenekian1 , Jean-Pierre Gauyacq2 and Nicolás Lorente1,3 1

ICN2—Institut Catala de Nanociencia i Nanotecnologia, Campus UAB, E-08193 Bellaterra (Barcelona), Spain 2 Institut des Sciences Moléculaires d’Orsay, ISMO, Unité mixte CNRS-Univ Paris-Sud, UMR 8214, Bâtiment 351, Univ Paris-Sud, F-91405 Orsay CEDEX, France 3 CSIC—Consejo Superior de Investigaciones Cientificas, ICN2 Building, E-08193 Bellaterra (Barcelona), Spain E-mail: [email protected] Received 30 August 2013, revised 30 August 2013 Accepted for publication 26 November 2013 Published 19 February 2014

Abstract

Spin-transport calculations present certain difficulties which are sometimes overlooked when using density-functional theory (DFT) to analyze and predict the behavior of molecular-based devices. We analyze and give examples of some caveats of spintronic calculations using DFT. We first describe how the broken-symmetry problem of DFT can cause serious problems in the evaluation of the spin polarization of electron currents. Next, we signal the low-energy scale of magnetic excitations, which makes them ubiquitous at already rather small biases. The existence of excitations in spin transport has catastrophic consequences in the reliability of the usual transport calculations. Finally, we compare DFT and configuration-interaction calculations of a ferrocene-based double decker that has been heralded as a possible spin-filter, and we cast a word of caution when we show that DFT is qualitatively wrong in the description of both the ground state and the excited states of ferrocene double deckers. Keywords: spintronics, spin-filter, density-functional theory, configuration interaction, broken symmetry, magnetic excitations, electron transport (Some figures may appear in colour only in the online journal)

1. Introduction

with non-equilibrium Green’s function (NEGF) techniques. Examples abound in the recent literature of calculations predicting the behavior of molecular devices as spin filters [6–14], spin valves [15], and memory devices [16], for a recent non-exhaustive list. The DFT–NEGF technique is well established and excellent review articles expound its secrets [17–19]. However, the use of DFT can, to say the least, lead to pitfalls. The first problem is due to the fact that DFT is a ground-state theory that is pushed into the calculation of non-equilibrium electronic properties. Actually, it has been proven that this ‘extension’ of DFT is empirically working, as comparison with time-dependent DFT results shows [20] and theoretical constraints [21] prove. Another problem comes with common DFT implementations because they are based on single-reference methods

Molecular spintronics is a thriving field fueled by the promise of ever shrinking circuitry [1] using the peculiar properties of the spin [2, 3]. Not only are molecules complex enough to attain extraordinary functionalities, but they are cheap to manufacture using chemical synthesis. Additionally, it has been possible to address individual molecules while taking advantage of their hierarchical growth to create structures of increasing complexity [4]. All this progress permits us to foresee a successful molecular-based spintronic technology. Parallel to the experimental evolution in the field, theory has greatly profited from new computational schemes to reveal the main properties of magnetic molecular devices. The workhorse for electronic transport simulations, including spin transport [5], is density-functional theory (DFT) together 0953-8984/14/104203+07$33.00

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which, in a nutshell, means that the electronic structure is assumed to be well described by a single Slater determinant. As a consequence the spin is poorly treated since Slater determinants are not always eigenfunctions of the spin operator, and only the projection of the spin is accounted for. This approximate description of the spin wavefunction is known as the broken-symmetry solution of DFT. In most cases, the multi-reference character of the wavefunction cannot be approximated by single-reference methods [22–24]. This problem occurs both in the ground state of the system and in the intermediate states involved in electron transport in the case of an object sandwiched between two electrodes. Indeed, the picture of an electron temporarily attached to an orbital cannot always be described in a single Slater determinant method; for example multi-state effects have been shown to be involved in the case of charge transfer between an open-shell atom/molecule and a metal surface [25–27]. In such cases, explicitly correlated calculations are particularly appealing since the multi-reference character of the wavefunction gives access to information with respect to the weights of the different configurations. This is particularly important when trying to compute magnetic properties, such as spin transport. In addition, electron transport through a molecular device can induce magnetic transitions in it (spin-flip transitions for the transmitted electron) that are not accounted for in DFT–NEGF calculations. Such excitations alter the transport properties through two kinds of effects. First, the spin of the transmitted electron can be modified during the magnetic excitation of the device and second, the excitation leads to a transient population of excited states in the device that possibly have transmission properties different from those of the device ground state, thus influencing transport in the case of large enough currents (this has been carefully revealed experimentally [28–31] and analyzed theoretically [32–35]; for a review see [36]). Magnetic excitation can simply be due to different molecular magnetic states associated with different electronic configurations. In that case, they involve electronic transitions and thus their excitation energy belongs to the eV scale; these will not influence transport at low bias. At a lower scale, we find excitations due to magnetic anisotropy which is a consequence of the combined action of the molecular ligand field and spin–orbit coupling [37]. Consequently, typical excitation energies are within meV and excitations are ubiquitous in molecular devices. Furthermore, detailed studies of these excitations via inelastic electron tunneling spectroscopy (IETS) showed that they were associated with very large excitation probabilities [28–31, 36]. In this paper, we explore the above problems. We first give a qualitative account of broken-symmetry shortcomings, and outline in what type of calculations they are expected to appear. Second, we address the issue of magnetic excitations in the electronic current and apply it to the calculation of the spin-filtering capacities of model molecular devices. Finally, we perform a complete active space self-consistent field (CASSCF) calculation of ferrocene and diferrocene and compare it with a DFT calculation, revealing some of the intrinsic problems of DFT treating complex open-shell molecular systems.

Figure 1. A broken-symmetry description of an open-shell system

between two electrodes can imply that transport is 100% spin-polarized because one of the two spins dominates the electronic structure at the Fermi energy. Using Hund’s rule for this single-level example, the broken picture claims that the intermediate negatively charged state is given by |↑↓i, leading to a fully polarized outgoing current. However, the true intermediate state is √1 {|↑↓i − |↓↑i}, 2 hence both spin directions will contribute to the outgoing current, thus leading to vanishing polarization.

2. Broken-symmetry problem: DFT versus multi-referential description

A simplistic DFT–NEGF calculation would yield a perfect spin-filter for a mixed-valence magnetic impurity between two non-magnetic electrodes. A broken-symmetry picture of a mixed-valence impurity is shown in figure 1. U is the charging energy of the impurity (or quantum dot) and DFT would predict the majority spin to be totally occupied and the minority spin resonant with one of the electrodes. In that case (upper level very coupled to the left electrode and resonant with the left Fermi level), transmission will only involve minority spin electrons and the minority spin would present an average occupation of 1/2. However, beyond the mean-field treatment of DFT, we would obtain that the charge state of the molecule is fluctuating between one and two electrons. The broken-symmetry picture tells us that it is the minority spin that is partially occupied, implying that the actual fluctuation would take place between a charge state with a single spin up and a charge state with |↑↓i. However, the true fluctuation takes place between a fluctuating spin (either up or down) for the first charge state and √1 {|↑↓i 2 − |↓↑i} for the second charge state, hence any of the two spins will contribute to the current and the spin polarization given by P=

I↑ − I↓ I↑ + I↓

(1)

will be zero. Typically, only spin down electrons will enter from the left and spin up and down electrons will exit to the right with equal probabilities. It thus appears that a description only invoking minority and majority spins is oversimplified for the present problem and the actual spin configurations of the system have to be taken into account. 2

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The problem disappears in the event of ferromagnetic electrodes. In this case, the description in terms of majority and minority spins is correct and the broken-symmetry representation of DFT is a good account of ferromagnetic electronic structure. At this point, one can make a reference to a well-known theorem by Anderson [38] stating that a single-reference wavefunction (a Slater determinant) is asymptotically orthogonal to the correct many-body wavefunction. We should keep then in mind the very approximate, and sometimes incorrect, representation of present-day DFT. Finally, the mean-field character of DFT will generally impede any treatment of fluctuations, such as the spin fluctuations encountered in the Kondo effect [37].

3.1. Transmission through a single atom

The system considered in the previous section, figure 1, assumes a unique electronic level that is temporarily filled during the transmission event, thus forming a negative ion A− with a spin ST = 0. Also, we showed that this is not a spin-filter since the current does not present any spin polarization. However, if we allow for the existence of more electronic levels of the negative ion A− , then a transmission channel with a triplet configuration, ST = 1, could be possible. Here, things change with respect to the ST = 0 channel. A simple majority versus minority spin argument would conclude that transport only involves majority spins leading to complete polarization. This is not the case, since all three sublevels of the triplet negative ion configuration have to be taken into account. A simple calculation [36, 39] shows that the current polarization, equation (1), is equal to 2/3 in the case of non-polarized electrodes. It shows then some spin-filtering properties but it is not a perfect spin-filter.

3. Magnetic excitations during transport

Low-energy magnetic excitations are ubiquitous in magnetic molecular devices. This is due to the general combination of some magnetic centers, such as magnetic atoms with sizable spin–orbit couplings, and non-negligible ligand fields. The ligand field of the molecule breaks the spherical symmetry around the magnetic atom and the spin–orbit interaction transfers this spatial anisotropy into the spin space [37]. This generates a magnetic anisotropy energy (MAE) landscape as the spin changes directions in the molecule. Since the MAE is of the order of a few meV, there are magnetic excitations that can be excited at biases of a few milli-volts. The efficiency of these excitations is very high due to the strong coupling of spins in a molecule [36, 39]; transport hence involves both elastic and inelastic currents where the spin of the tunneling electron can be conserved or non-conserved during the transmission event. Actually there exist cases where the spin-changing transmission is dominating over the spin-conserving one [29, 30, 40]. The effect on a molecular-based spin-filter is dramatic: the spin of the transmitted electron may change directions when an excitation threshold is overcome by the external bias, hindering the spin-filter character of the device. A calculation using a model many-body wavefunction for the molecular device, and a rate equation approach to describe transport, shows that indeed the full magnetic structure of the molecular system must be taken into account in order to have a correct treatment of the electronic current [41]. In the present work, we simplify their treatment and evaluate the probability that the electron current changes spin as it flows through some model molecular systems. We use the strong-coupling approach of [36, 39] to evaluate the probability of spin flip of the transmitted electrons and compute the spin polarization of the electron current. In this section, we treat two model systems connected to two electrodes. First, an atom, A. Second, a dimer, AB, made of two identical atoms. To keep the discussion simple, we assume that the magnetic atoms A and B of the dimer are spin one-half. We further assume that the current proceeds via the repeated formation of a negative ion, say A− . This corresponds to the process depicted in figure 1, where the total spin of the transmitting system (the spin of the negative ion) is ST = 0.

3.2. Transmission through a dimer

Let us consider a dimer made out of atom A and atom B such that atom A is connected to a left electrode and to atom B, and atom B is connected to A and the right electrode. We further assume a ferromagnetic coupling between A and B, such that the dimer is in one of the three triplet states of the dimer spin, S = 1. Assuming a ferromagnetic Heisenberg exchange coupling, J , between atoms A and B, there is an excited singlet state, S = 0, at an energy J above the triplet. At low bias, V , the electron transport proceeds from the left electrode to the right one, passing first through atom A and then through atom B. Just to keep the same description as for the previous section, let us assume that once the electron is in A, it forms a negative transient ion, A− . As above, we take the spin of A− as ST = 1. Next the electron passes by B, forming B− with local ST = 1. Due to the triplet nature of the ground state of the dimer, transitions among the three triplet sublevels of the dimer can be induced by the travelling electron leading to a spin flip of the electron. It is easily shown that the current polarization, equation (1), is 4/5 for low bias and non-polarized electrodes. At larger bias, when V is such that eV ≥ J , the singlet channel opens. New inelastic transitions appear in addition to the energy-conserving and spin-changing transitions among the triplet states of the dimer that were already present at low bias. This increases the probability for spin-flip transmission. In this case, the current polarization, equation (1), is reduced to 2/3, compared to the low bias value of 4/5. Far from a perfect spin-filter. We then reach a startling conclusion: at low bias, because of the spin properties of the AB molecule, the system is a reasonably good spin-filter with polarization 4/5, however as we increase the bias, one more spin channel becomes available and the polarization drops to 2/3. For completeness, we can mention the case of an applied external magnetic field that removes the degeneracy of the AB triplet state. In this case, for low enough biases (such that only one of the channels of the triplet manifold is open) the 3

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energy-conserving spin-changing transitions among the triplet sublevels are absent and we obtain a perfect spin-filter. The above analysis has simplified all terms and only considered spin one-half atoms. However, experimental data exist for dimers with higher spin [42]. The above analysis can be extended to these systems if the magnetic anisotropy of each atom in its environment (electrodes + other atoms) is taken into account [36]. In these cases too, changing the applied bias will change the number of inelastic channels that are open as well as the number of possibilities of spin-changing transitions, thus affecting the spin-filter properties of the device. 4. An ab initio dimer spin-filter: comparison between DFT and configuration-interaction calculations

The previous section gives a simple example of how the spin of the ground state and first excited states plays an important role in the behavior of a possible molecular device: a dimer formed of two atoms. Here, we explore an iron dimer in which the two metal atoms are separated by 5-carbon cycles with H passivating remaining bonds (cyclopentadienyl). A ferrocene molecule corresponds to one iron atom sandwiched between two 5-carbon cycles with H passivating remaining bonds: FeC10 H10 . To the ferrocene molecule one can add an Fe ion plus a cyclopentadienyl molecule, and create a double decker, and so forth. These multi-decker structures have received some attention in the literature for their spin-filter character [13, 43, 44]. Here, we propose assessing the qualitative description obtained by DFT-based calculations with respect to the one obtained from a configuration-interaction (CI) method where the wavefunction is formed by a linear combination of Slater determinants. For the DFT calculations, we use the PBE functional [45] and a double-ζ plus polarization basis set [46]. The CI method is performed by means of complete active space self-consistent field (CASSCF) [47, 48] calculations which incorporate the leading electronic configurations distributing n electrons in m molecular orbitals (MOs), defining an active space referenced as CAS[n, m]. Nevertheless, such a procedure fails to reproduce the correct relative energies since dynamical correlations are absent. Dynamical correlation is predominantly an atomic effect that can be incorporated within different frameworks on top of the CASSCF wavefunction. In this respect, complete active space second-order perturbation theory (CASPT2) [49, 50] calculations have proven to be impressive tools to investigate spectroscopy issues [51–53] and we resort to this strategy. In our case, the active space can be anticipated to consist of the 10 d-type orbitals and a set of bonding cyclopentadienyliron orbitals (see figure 2); a second set of d-type metalcentered virtual orbitals with an extra radial node, i.e. 3d0 , is traditionally included in the active space to account for the so-called ‘double-shell effect’ [50]. This leads to a challenging 21 electron–24 orbital active space for the neutral species. The feasibility of such a calculation is achieved thanks to a variation of the CASSCF calculation, the restricted active

Figure 2. Selected set of orbitals from the 21 electron–24 orbital

active space: bonding cyclopentadienyl-iron orbitals (bottom) and d-type orbitals (top).

space SCF (RASSCF) method [54], which is used to limit the occupation of the second d-shell, thus reducing the number of configurations in the CI treatment. All of our RASSCF and RASPT2 calculations were performed with the Molcas7.6 package [55] including atomic natural orbitals (ANO-RCC) as basis sets [56]. These basis sets were contracted into [7s, 6p, 5d, 3f, 2g, 1h], [3s, 2p, 1d] and [2s1p] for iron, carbon and hydrogen atoms, respectively [57]. We start by the study of the neutral species. From the DFT inspection, we find that the ground state is an intermediate spin S = 3/2, with a first excited state S = 5/2 lying at 330 meV, then a doublet at 400 meV and the octet at 540 meV (see figure 3(a)). In contrast with these results, the RASPT2 calculations describe the ground state as a doublet followed by the expected spin-ladder formed by the states of spins S = 3/2 (80 meV), S = 5/2 (360 meV) and S = 7/2 (750 meV). The reading of the corresponding wavefunctions allows us to describe the neutral molecule as two iron centers with 6 and 7 d-electrons, respectively. Both are in a high-spin configuration, and antiferromagnetically coupled. The three cyclopentadienyl ligands are negatively charged and do not hold any spin. Let us notice that the wavefunctions show extreme complexity. As an example, the ground state counts 30 configurations with a weight of more than 1% and no configuration with a weight bigger than 5%. Most of them exhibit charge transfer from the cyclopentadienyl ligand to the iron centers or between iron centers. Such an extreme multi-reference character obviously makes the description of the electronic structure extremely challenging for the single-reference DFT. The discrepancy is even more pronounced in the cases of the cation and the anion (see figures 3(b) and (c)). In the 4

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Figure 3. Low-energy spectrum for the iron double decker calculated with DFT and RASPT2 methods for (a) the neutral species, (b) the cation and (c) the anion.

charged ions, DFT finds quintet (cation) or triplet (anion) ground states, while the RAS calculations obtain singlet ground states for both cation and anion. In the neutral case, the main error of DFT is to misplace the doublet as an excited state. If we remove the S = 1/2 state from the calculations, then the DFT is qualitatively giving the same type of spectra, due to the larger weight of one of the references in the RAS calculation for the excited states. However, it is in the excited states that DFT should be unreliable. The ground-state failure is, as said above, due to the truly multi-configurational character of the RAS wavefunction. For the ionic species, this multi-configurational character is preserved all over the spectra, causing a serious disagreement with DFT. The present molecule is made of several parts: two Fe atoms and three cyclopentadienyl planes. If the spin configuration of all these parts corresponds to the maximum spin possible, then the spin of the molecule is large and it can be described by a single determinant, appropriately handled in DFT; in contrast, low-spin states (ground states of all charge states) will require several determinants to be represented correctly, leading to DFT problems. The consequences of these findings are far reaching for transport calculations. In a simple-minded picture of sequential electron hopping (be it incoherent or not) between electrodes and molecule, transport through the ferrocene double-decker device cannot be accounted for by a DFT calculation since the description of the electron flow as transitions among different ionic states will not be based on an electronic structure that can retrieve the static solutions in the limit of very weak electrode–molecule couplings.

yields a perfect spin-filter. This wrong result is due to the use of a simplified majority versus minority spin description that cannot handle all spin configurations, in particular those of open-shell systems. One more problem is the intrinsically elastic nature of DFT–NEGF calculations: they do not include excitation processes and excited-state populations in the electron transport description. We have shown two simple examples and described how much the actual excitation spectra of the system matter in the correct description of transport properties such as current polarization. Finally, we have examined the DFT description of a popular device: a ferrocene-based double decker. We have compared DFT results with involved multi-referential calculations using CI. In this case, DFT fails in predicting the spin characteristics of the ground state, and the spectra of low-energy excitations, for the possible charge states that should be relevant in the prediction of a sequential electron transport process. This casts doubts on the actual capability of DFT–NEGF in reproducing the magnetic properties of an electron transport based molecular device. Nevertheless, for magnetic systems where a single reference is a good approximation (such as maximum-spin molecular systems), the many-body wavefunctions will have few components, and we expect that DFT–NEGF will do a good job under the proviso that the broken-symmetry description is not masking more complex mixed-valence like behavior. Given the difficulty of carrying out static CI calculations, DFT-based methods are certainly needed, although caution is required, and careful initial studies must be undertaken of the molecular devices in order to avoid the above caveats.

5. Concluding remarks References

In this work, we summarized the principal drawbacks of DFT-based transport calculations when applied to the description of spintronic devices. We have shown that the DFT broken-symmetry description of magnetic systems can yield qualitatively wrong results. For this we have chosen a simple device connected to two electrodes that is in a mixed-valence state. As a result, the spin polarization of the current is strictly zero, but the wrong broken-symmetry description of DFT

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