# Trend: The quest for dilute ferromagnetism in semiconductors: Guides and misguides by theory

and , National Renewable Energy Laboratory, Golden, CO 80401, USA
, Yokohama National University, Department of Physics, 240-8501 Yokohama, Japan
Published June 28, 2010  |  Physics 3, 53 (2010)  |  DOI: 10.1103/Physics.3.53

## Introduction

Magnetism mandates that the electrons in a material collectively align their spins. In a ferromagnet there is an imbalance in the occupancy of spin-up and spin-down states; such a collective spin alignment leads to a macroscopic spontaneous magnetization that persists up to a Curie temperature $TC$. Many common ferromagnets (such as iron or nickel) are metals, exhibiting a set amount of free electrons that make them conduct electrically. In semiconductors, on the other hand, the concentration and type of electronic carriers is controllable externally, for instance, by doping with select atom types or directly by injecting carriers. Most semiconductors are not magnetic. A material that exhibits both ferromagnetic and semiconductor properties offers the exciting prospect of combining nonvolatile magnetic storage and conventional semiconductor electronics in a single device. Magnetic semiconductors offer a number of interesting possibilities in the pursuit of “spintronics,”a branch of science and technology that exploits the spin dimension of the electron in addition to its charge, for novel electronic devices. These materials combine the properties of a semiconductor and a magnetic material, providing, for instance, a way to create $100%$ spin-polarized currents, and by the same token, the promise of electrical control of magnetic effects. While in some magnetic semiconductors, for example, magnetite, all of the material’s constituent ions are intrinsically magnetic (“concentrated magnets”), the most recent focus has been on nonmagnetic semiconductor host materials that can be doped by a small amount of magnetic transition-metal ions or by defects that promote magnetism (“dilute magnets”).

The first experiments on dilute magnetic semiconductors focused on host semiconductors with narrow band gaps, such as $GaAs$ and $InAs$, which are infrared-absorbing black substances. When doped with manganese, $Mn2+$ substitute for $Ga3+$ ions, thus releasing “holes” (a positive charge resulting from the absence of an electron) that are said to collectively align the $Mn2+$ spins ferromagnetically when the sample is cooled below $TC$. Ferromagnetism in $GaMnAs$ cannot, however, be enhanced indefinitely by adding more $Mn$ ions, as some of the additional manganese will occupy interstitial sites or form some clustered phases that inhibit a global ferromagnetic order. This has limited the ferromagnetism in $GaMnAs$ and other doped III-V semiconductors to experimentally relevant, but not technologically useful, temperatures. For this reason, the possibility of doping common insulators like , and $HfO2$ instead was seized upon with great enthusiasm. These materials have the additional bonus feature of a wide band gap, promising the control of magnetism through charge carriers in an optically transparent medium. Reports on ferromagnetism above room temperature in such wide-gap materials quickly appeared—even in samples doped with elements that were not in themselves magnetic, or not doped at all! The excitement brought back fond memories of the early days of room-temperature superconductivity, when unusual phenomena were discovered in a group of unsuspected oxides and nitrides. What was special about these developments in pushing the $TC$ boundaries in dilute magnets was the role of theory, which in many cases preceded experiments and influenced the type of systems that were studied. The nature of the delicate tango danced by theory and experiment in this field is the subject of this article.

The tools of solid-state theory of magnetic systems have played a central role in this rapidly developing field, offering not only ex post facto explanations of measured phenomena, but very often leading the field by predicting specific combinations of host insulators with impurity atoms that would lead to room-temperature ferromagnetism. In this pursuit, two proactive cultures of predictive theories have emerged. The initial basic philosophy was to use model Hamiltonians, in which certain specific magnetic interactions between ions are postulated (and others are excluded). For this technique, the “user” needs to guess at the outset the type of magnetic interactions that will dominate. This style of model-Hamiltonian theory has had a glorious past in numerous areas of condensed matter physics. The most prominent supposition for the interaction type between spin-polarized $3d$ impurity ions (such as $Mn$, $Co$, and other common dopants from the transition-metal series) in insulators assumed it was similar to the interaction of nuclear spins mediated by the conduction electrons in metals, and the model also assumes that the host crystal is largely unperturbed by the presence of the $3d$ impurity ion. This type of Ruderman-Kittel-Kasuya-Yosida (RKKY) [1] model-Hamiltonian approach [2] went all the way to predict ferromagnetism in various $3d$ impurity-doped compound semiconductors. In particular, the prediction that ferromagnetism persists to a high Curie temperature with $TC$’s well above room temperature in various wide-gap oxides and nitrides [2]—has resulted in thousands of papers in the past decade, trying to capitalize on these exciting predictions offering high $TC$ in common insulators.

Alongside the culture of model-Hamiltonian theories of dilute magnets, first-principles calculations have rapidly penetrated this field. The philosophy here is to describe each host and impurity system by considering its fundamental electronic structure, thereby obtaining the magnetic interaction type as the answer, not the question. Instead of prejudging the preferred type of magnetic interactions between ions, one articulates fundamental electron-ion and electron-electron interactions and lets the magnetic interactions emerge as solutions to the fundamental many-electron Hamiltonian. The availability of density-functional approximations (exchange and correlation functionals for the inter-electronic interactions), accurate pseudopotentials (simplifying the calculation greatly), and the ubiquitous computer packages encoding such developments into friendly interfaces freed the theorist from having to guess at the outset how a particular impurity would affect a particular host crystal. At the same time, this apparent freedom sometimes created the illusion that such modeling can be done on “automatic pilot,” disregarding potential pitfalls. Regrettably, such pitfalls sometimes created “false positive” results (suggesting high $TC$’s when no magnetic ordering should occur), spurring enthusiasm and optimism about high-temperature ferromagnetism even when not warranted. Indeed, it has often been the case that materials synthesized from computational recipes have not behaved in the way that was expected.

In the following we will (i) discuss some of the issues and potential pitfalls that can come with uncritical application of theoretical methodology, (ii) analyze the physics of the artifact, and (iii) suggest specific ways out of these dilemmas. The good news is that the insights gained from such critical evaluation can now be embodied (in different ways!) into practical first-principles calculations, offering a truly predictive design option. Furthermore, such calculations clarify what types of interactions are indeed critical and offer model-Hamiltonian approaches a safer, albeit ex-post-facto, method to construct more realistic Hamiltonians.

## Transition-metal impurities in semiconductors: Interesting physics, no ferromagnetism

The electronic properties of isolated transition-metal impurities in semiconductors were studied in great detail in the 1980s [3, 4], when the growth methods produced ultradilute samples (defined as containing $1015$$1017$ impurities/$cm3$) with virtually no impurity-impurity interaction. The high crystalline perfection of such samples enabled a clear experimental and theoretical understanding of the properties and chemical trends in the periodic table of isolated impurities, including their spin configuration, donor and acceptor transitions, and optical excitations. A number of intriguing physical phenomena were revealed due to coexisting localized $3d$ impurity states. One of these is the “self-regulating response” [5, 6], recently revisited by Raebiger et al. [7], which shows that adding electrons to the $3d$ derived impurity levels does not change the integrated electron density around the $3d$ impurity atom because the orbitals of the surrounding ligand atoms rehybridize to minimize the impact of impurity charging. A second phenomenon is the “vacuum pinning rule,” whereby transition levels of the same impurity atom in different host semiconductors line up in approximately the same way, regardless of the host chosen [8, 9], thus permitting the deduction of band offsets between different pure materials from the knowledge of impurity level positions [10]. Third, another consequence is called “exchange-correlation negative $U$,” [11] whereby the strongly varying exchange splitting of differently charged impurities can lead to effective electron-electron attraction. These systems all show interesting physics, however, such systems revealed no ferromagnetism, as the generally low solubility of dopants precluded collective interactions.

Collective magnetic ordering requires the individual magnetic impurity ions to be close enough to one another so as to interact. In practice, this implies large concentrations, often well above typical thermodynamic solubility limits. The low solubility limit of most $3d$ ions in semiconductors can, however, be overcome by using nonequilibrium growth techniques (such as very low-temperature molecular beam epitaxy). These techniques increased the impurity concentration by a few orders of magnitude to $1020$ dopants/$cm3$. This, however, comes at the price of severely lowering the structural quality of the samples. Such high concentrations of magnetic ions in semiconductors, such as $GaAs:Mn$ [12, 13], leads to ferromagnetism with relatively high Curie temperatures of around $100K$—a temperature range that has opened the era of spintronics [14, 15].

The basic driving force of this form of ferromagnetism of $3d$ impurities in III-V semiconductors is now understood [16] as the energy stabilization ensuing from the interaction between partially occupied hybrid orbitals located on different impurity sites, $TM1$ and $TM2$, in the host crystal. Figure 1 (top) illustrates that if two such orbitals interact, there will be a gain in energy, i.e., a stabilization of the FM state (relative to the antiferromagnetic state [16]).

One might ask, how would such individual partially occupied hybrid orbitals come to exist in the first place? The answer to this is illustrated in Fig. 1 (bottom). The formation of a substitutional $3d$ impurity can be thought of as a two step process, involving first the removal of a host $Ga$ atom, and second, the placement of a $3d$ impurity atom in its place. The combined impurity/host orbitals simply result from coupling between the host-crystal cation vacancy state (“dangling bonds” of t(p) symmetry) and the impurity orbital of the $3d$ atom filling this vacancy (having t(d) symmetry as well as nonbonding e(d) symmetry). The interaction between these states leads to a bonding state, and an antibonding state. The partial occupancy which ultimately drives ferromagnetism resides in the higher of these states. For $Mn$ in $GaAs$ the antibonding state, dominated by t(p), is called a “dangling bond hybrid” (DBH) state, and the bonding/nonbonding states, dominated by t(d) and e(d), are called “crystal field resonances” (CFR). As we move to the left from $Mn$ in the periodic table, keeping the $GaAs$ host, we approach the light $3d$ impurity elements such as $V$. For these, the atomic t(d) and e(d) are higher in energy and the roles of DBH and CFR are interchanged. The same interchange occurs in the case where $Mn$ is placed in a host crystal with light anion elements, such as nitrogen in $GaN$, where the t(p) level is deeper and t(d) dominates the hole. It is the interaction between the partially occupied hybrid states, centered on different impurity sites in the crystal, which energetically stabilizes the ferromagnetic configuration.

This simple mechanism implies simple rules [17] for impurity-host material combinations that might produce ferromagnetism: one obtains the electronic configuration by constructing an Aufbau ground state for $n$-$m$ electrons ($n$ is the number of TM valence electrons, and $m$ is the valence of the cation site) occupying the relevant symmetry orbitals. For example, a ferromagnetism-promoting partial occupancy of DBH can be expected when $Mn$ occupies a trivalent $Ga$ site, but not the divalent $Zn$ site. This mechanism was directly verified by first-principles calculations of the energy of impurities in a solid as a function of the occupancy of the DBH, and noting the stabilization (destabilization) of the ferromagnetism for partial (full) occupancy.

## Even higher $TC$ promised in wider-gap semiconductors

First-principles electronic structure theory of $Mn$ doped into III-V nitrides, phosphides, and arsenides [18, 19, 20] have shown that the ferromagnetism-promoting hole (i.e., the orbital with partial occupancy, $t+DBH$ in Fig. 1) resides inside the host band gap. When the host anion changes from $As$ to $P$ and then to $N$, the band gap increases primarily by shifting the valence band maximum (VBM) to lower energies. Thus, as one moves from arsenides to phosphides and then to nitride semiconductors, the hole-carrying impurity level, being roughly pinned, appears farther and farther from the VBM, deeper in the band gap. This makes the hole-carrying impurity wave function progressively more localized, weakening its communication with other magnetic ions; thus we lose ferromagnetism. Thus the conclusion of such microscopic calculations [4, 16, 17] was that $Mn$ will have lower $TC$ as we go from a host arsenide to a host phospide and then to a nitride. While initial estimates of $TC$ derived from directly associating the Weiss field with first-principles calculated magnetic coupling energies predicted optimistically higher $TC$ in the wider gap (i.e., nitride) host crystals [21], it was quickly realized (by the same authors) that a proper statistical treatment of these exchange energies (via Monte Carlo, see Ref. [22]) predicted a much reduced $TC$ as one moves to wider gap host crystals.

In parallel with such first-principles calculations, more traditional model-Hamiltonian approaches were developed [2, 23] that assume a scenario for the underlying interactions, and then solve the scenario mathematically. The classic $k⋅p$ approach in semiconductor physics, as described in Ref. [24], expands the states of a system in terms of a small number of preselected bands of the host crystal at the Brillouin-zone center. In the field of dilute magnetic semiconductors, the use of such $k⋅p$ concepts assumes that the hole that promotes magnetism can be described by just a few host crystal states, becoming essentially a “hostlike hole” [2, 23]). Such simple concepts differ substantially from the atomistic picture of Fig. 1 (bottom). The model Hamiltonian approach is then able to produce, analytically, the dependence of the ferromagnetism Curie temperature $TC$ on basic material parameters. This lead to the highly influential bar-diagram published by Dietl et al. [1] (cited more than $3200$ times) predicting the highest $TC$ for binary semiconductors having the shortest bond length and thus the largest band gap. This design principle pointed towards wide-gap binary compounds containing a first-row element such as $C$, $N$, or $O$ as the best candidates for high-$TC$ ferromagnets. This prediction started a world-wide quest for high-$TC$ ferromagnets in wide-gap metal oxides (), nitrides ($GaN$), and carbides based on dilute transition metal impurities. As exciting as such predictions were, they did not have the additional virtue of becoming true. Indeed, the assumed scenario that the ferromagnetism-promoting hole has the personality of the host valence band did not seem to be true for wide-gap insulators doped with transition metals such as $Mn$. The simple lesson from microscopic models that the $Mn$ acceptor (hole-carrying) level becomes progressively more localized (i.e., contributing to lower $TC$) as one moves up the group $V$ elements from arsenic to nitrogen was lost in the global enthusiasm to seek high $TC$ in the widest gap host systems.

## The role of theory as a guide: Pitfalls and fixes

Magnetic ions were not the only dilute species in wide-gap insulators that were implicated with high-$TC$ ferromagnetism. Indeed, simple structural defects such as cation vacancies [25, 26, 27, 28] or carbon or nitrogen impurities [29, 30, 31] also became prime candidates for point sources of ferromagnetism behavior in wide-gap insulators. The basic thought here is that a proper combination of defect/impurity and a host crystal will lead to a hole-carrying (partially occupied) level, centered about this defect/impurity site, and that a ferromagnetic spin arrangement will be stabilized when different such centers interact [Fig. 1 (top)]. The search for the particular combination of defect/impurity and host crystal was, in many instances, guided by theory. Indeed, an impressive sign of the maturity of electronic structure theory based on first-principles was the central role that it had occupied in guiding the international quest for ferromagnetism in dilute wide-gap insulators [32, 33, 34, 35].

Unfortunately, a number of overlooked factors contributed to confusion. We next discuss a number of ways in which such calculations can produce unfounded optimism, such as a too high-$TC$ ferromagnetism. Fortunately, recognition of such factors has lead to proposing a fix. Efforts worldwide to improve these fixes promise to create a robust guide for new design principles for such materials.

## If it’s not one thing, it’s another

Guiding experiments towards promising defects or impurities that can promote carrier-induced ferromagnetism in certain wide-gap materials has proven to be difficult. Even if you correctly formulate the underlying electronic structure problem so that false-occupancy is avoided, the LDA failure to localize holes can get you. And if these are fixed, you still need to establish the defect concentration that satisfies percolation, yet is tolerated mechanically by the lattice. And you still need to consider the microstructure, i.e., the microscopic distribution of defects, since ideal single oxide crystals with ideal impurities do not lead to ferromagnetism. Indeed, if it’s not one thing that leads to false FM prediction, it’s another.

Finally, one should ask whether it is appropriate to describe the collective magnetism of a sample in terms of interacting single isolated point defects? Statistics tells us that the probability of two or more defects on one of the sublattices in a zinc blende, wurtzite, or rock salt crystal occupying neighboring sites is $>50%$ for defect concentrations $>5%(∼1020/cm3$) [69]. Once a defect pair has another defect next to it (e.g., with a $50%$ probability), it has a lowered point group symmetry and consequently a different electronic spectrum, and is likely to exhibit completely different magnetic interactions compared to its isolated relatives [48, 70]. This has the detrimental effect that magnetic interactions no longer can be mapped on Dirac-van Vleck-Heisenberg type models ($E=ΣJijsi⋅sj)$ [71]. As expected, this problem can be circumvented by introducing additional correction terms or explicit configuration dependences to the model, as was done in Refs. [49, 51], which, however, mandates the explicit calculation of various additional clustered/microstructured configurations and their energetics. Moreover, the flaws in the underlying spin models seem to be of a fundamental nature, since, as it turns out, magnetic coupling energies, even of the simplest hydrogen molecules, have no mapping thereupon [72]. This failure of Heisenberg’s theory [73] can further be traced back onto the Heitler-London theory [74], of which it is a generalization, and which has been abandoned on several occasions [75, 76, 77].

## Outlook

Bad is Good? No, this is not an attempt at moral revisionism. But consider this: The functionality of optoelectronic semiconductor devices such as lasers, light-emitting diodes, or photovoltaic solar cells is predicated on the structural quality of the active material. “Good materials,” i.e., those leading to efficient functionality, invariably need to have good crystalinity, purity, and very few defects. Yet if one were to use such near-perfect bulk samples of oxides, no ferromagnetism of the type discussed here would be present. Considered from this viewpoint, structurally imperfect and compositionally nonstoichiometric oxide materials (which would be considered “bad materials” in the semiconductor device culture) are “good materials” in this field of dilute magnetic oxides, as they are the only ones that yield, so far, the desired functionality.

## The future of theory guide

Although much of the article has pointed out the various pitfalls encountered in running LDA on automatic pilot, the conclusions are more optimistic: Electronic structure calculations are a powerful tool to describe magnetic ground states, even in complicated clustered or nanostructured configurations. The ensuing results clarify which types of interactions are indeed critical, offering model-Hamiltonian approaches a safer, albeit ex-post-facto method to construct safe, simpler Hamiltonians. Furthermore, the results may be used as input for statistical models constructed to avoid the obvious pitfalls in earlier Heisenberg-like attempts.

## Acknowledgment

This work was funded by the U.S. Department of Energy, Office of Science, Basic Energy Science, Materials Science and Energy Division, under Contract No. DE-AC36-08GO28308 to NREL. HR is supported by a Grant-in-Aid for Young Scientists (A) grant (No. 21686003) from the Japan Society for the Promotion of Science.

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### About the Author: Alex Zunger

Alex Zunger’s research field is condensed matter theory of real materials. He is the recipient of the 2010 Tomassoni Physics Prize and Science Medal of Scola Physica Romana, the 2001 John Bardeen Award of The Material Society, the Rahman Award of the American Physical Society, the 2009 Gutenberg Award (Germany) and the 2000 Cornell Presidential Medal. Israeli raised and educated, he received his Ph.D. from Tel-Aviv University, Israel. He did his postdoctoral research at Northwestern University (with Art Freeman). He then received an IBM Fellowship, which he spent at the University of California, Berkeley (with Marvin Cohen). Dr. Zunger established the Solid State Theory group at the National Renewable Energy Laboratory (NREL), Golden, Colorado, a position he still holds today. He is an NREL Institute Research Fellow, and Director of the Energy Frontier Research Center on Inverse Design.

### About the Author: Stephan Lany

Stephan Lany received his Ph.D. in physics from the Universität des Saarlandes, Germany, in 2002. He is currently a senior scientist at the National Renewable Energy Laboratory. In his recent research activities, he improved the methodology for accurate prediction of the properties of defects and impurities in semiconductors and insulators from electronic structure theory. He applied such methods to magnetic semiconductors and photovoltaic materials, and used the predicted defect energies for thermodynamic modeling of defects, off-stoichiometry, and doping.

### About the Author: Hannes Raebiger

Hannes Raebiger obtained his doctorate at the Helsinki University of Technology in 2006. In 2005 he was awarded the J. W. Corbett prize at the 23rd International Conference on Defects in Semiconductors for his studies elucidating the magnetic interactions in dilute magnetic semiconductors. After obtaining his doctorate, he joined Alex Zunger’s Solid State Theory group at the National Renewable Energy Laboratory as a post-doc. In 2008 he moved to Japan, becoming the first foreign faculty member at the Yokohama National University Graduate School of Engineering. He now heads his independent research group, focusing on the quantum theory of molecules and solids.