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# Journal Club Theme of March 2009: Mechanics Issues in Nanocapacitors and Ramifications for Energy Storage

Next generation advances in energy storage for nanoelectronics, micro and nanosensors among others, require capacitors fabricated at the nanoscale. High dielectric constant materials such as ferroelectrics are important candidates for those. Consider the following: the expected capacitance of a 2.7 nm SrTiO3 thin film is 1600 fFmicro-m-2. What is the likely value in reality? 258 fFmicrom-2! This dramatic drop in capacitance is attributed to the so-called "dead layer" effect.

Energy conversion, storage and transport are rapidly emerging to be focal research topics in physical sciences due to socio-economic imperatives. An under-appreciated fact is that the so-called "energy problem" exists at multiple length scales: at the "global or macroscopic level", affecting individual cars to cities and at the "microscopic level" influencing next generation micro and nano-electronics.

With the relentless increase in the world's energy consumption combined with depleting fossil fuel resources, there is a need, now more than ever before, for clean, renewable and efficient sources of energy. However, many such renewable energy sources are intermittent in nature and require efficient electrical energy storage devices. Two most popular electrical energy storage technologies are batteries, and capacitors. While batteries store energy in chemical reactants that in turn generate charge, capacitors store energy directly in the form of electrical charge.

Batteries as a source of energy have been around since a very long time and are ubiquitous in portable electronic devices. Conventional batteries however suffer from a few drawbacks such as limited lifespan and low power density. Though new technologies such as Li-ion rechargeable batteries are being considered for use in hybrid vehicles (which use a combination of gasoline and batteries to power themselves), some serious issues still remain including slow recharge cycles and low energy storage per unit volume. Capacitors, on the other hand, are projected to circumvent some of the problems that batteries pose and provide an alternative solution to meet tomorrow's energy storage needs. While conventional capacitors are often made from dielectrics, new high capacity electrolytics, also called super/ultra capacitors or electrochemical double layer capacitors, have been in development for use in hybrid vehicles. As can be seen from the so-called Ragone plot (see Figure 1), they offer faster recharge times combined with high power densities.

Figure 1: The Ragone Plot: How do nanocapacitors compare with other energy storage devices? (Graph adapted from "Basic Research Needs for Electrical Energy Storage", the Report of the Basic Energy Sciences Workshop on Electrical Energy Storage, 2007.)

In addition to the need for large capacitors with the capability of storing large amounts of energy at high energy and power densities, say in hybrid vehicles, there is an entirely different length scale at which there is a pressing need for development of high density capacitors: the micro and nanoscale. With the increasing demand for miniaturization of electronics, there is a need for the development of nano-sized capacitors which can store energy at high densities for use in electronic circuits as dynamic and permanent memories. Indeed the discovery of new nanostructured materials and the development of precision thin-film manufacturing nanotechnologies have opened up opportunities for advancements in ferroelectric thin film nanocapacitors. Further, new computational analysis and atomic scale simulations provide an important tool for achieving new breakthroughs in this area. For example, thin film nanocapacitors are expected to store more energy than conventional capacitors with applications in micro and nanoelectronics among others.

In its simplest manifestation, a parallel plate capacitor made up of a dielectric with dielectric permittivity and thickness d has the following capacitance per unit area (as predicted by classical electrostatics): e/d (where e is the dielectric constant and d is the thickness).

Several experiments have documented the dependence of capacitance on the thickness of very thin dielectric films and observed that the plot of the inverse capacitance versus thickness does not have a zero intercept (as predicted by classical electrostatics) but rather a finite one, hinting at the presence of a disruptive dead layer at the metal-dielectric interface. Researchers attribute the presence of the dead-layer to a variety of reasons including a secondary low-permittivity phase at the surface of the films, nearby-surface variation of polarization (field induced or spontaneous), presence of misfit dislocations , electric field penetration into the metal electrodes, among others.

Recent pioneering ab initio calculations by Stengel and Spaldin [1] on SrRuO3/SrTiO3/SrRuO3 and Pt/SrTiO3/Pt thin film capacitors with atomistically smooth interfaces (to exclude effects due to, say, misfit dislocations) have confirmed the intrinsic nature of this effect and that electric field penetration occurs in real metal electrodes giving rise to a passive dead layer at the metal-dielectric interface (see Figure 2). Their results also raise questions about the validity of models traditionally used to model the dead layer at the metal-dielectric interface.

Figure 2: This figure is adapted from Stengel-Spaldin [1]. The red curve is the classical electrostatic based prediction of the potential while the blue line the one observed in their ab initio calculations.

Calculations performed by Stengel and Spaldin [1] provide a much deeper understanding of the origins of the dead-layer. In particular, their results show that the electrostatic potential profile in the dielectric part of the capacitor exhibits considerable non-linear behavior (as opposed to the linear variation predicted by classical electrostatics) and that the capacitance of the dielectric layer is subject to some additional size-dependent scaling beyond what is suggested by the simple scaling formula I gave before. Also (as the interested reader will find after reading their paper) the dielectric permittivity is found to vary considerably within the dielectric.

So, where do the mechanics issues come in?

Let me first list the three journal papers I wish to put up for discussion:

[1] M. Stengel, and N. A. Spaldin, "Origin of the dielectric dead layer in nanoscale capacitors", Nature 443, 679 (2006).

[2] K. M. Rabe, "Nanoelectronics: New life for the 'dead layer'", Nature Nanotechnology 1, 171-172 (2006).

[3] R.D. Mindlin, "Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films," International Journal of Solids and Structures, v 5, n 11, Nov. 1969, p 1197-1208

The first paper is by Stengel-Spaldin while the second one (by Rabe) provides a "layperson" summary of Stengel-Spaldin's work (I suggest that Rabe's paper be read first). One of the reasons why the first paper is pioneering is that it is very difficult to perform first principles calculations for a system such as a nanocapacitor under a finite electric field. Overall, metal-insulator-metal supercell behaves metal-like and density functional theory become inapplicable (DFT is for ground state calculations not non-equilibrium problems). In such a non-equilibrium problem, in principle TDDFT may be used but there are computational problems. (Please also see discussion in the previous month's jclub where I enquired about the computational expediency of TDFT and possible recourse by finite elements). Stengel and Spaldin have found a clever way to accomplish such types of calculation.

One of the intriguing results of Stengel-Spaldin [1] is that signficant ionic relaxation is observed (i.e. mechanical strain). Their results under atomic relaxation (strain) are significantly different from the unrelaxed case. What role does the strain play in the drop in the capacitance? It turns out that a very famous mechanician (Mindlin, [3]) looked at this problem nearly 40 years ago when the first set of dead-layer experiments were published! He argued that indeed in thin enough films, inhomogeneous electric fields (of the type observed by Ref 1) would drive inhomogeneous strain fields in the thin film leading to a capacitance drop. This happens due to a phenomenon called flexoelectricity which is essentially a link between strain gradients and electric fields or conversely polarization gradients and strain. This phenomena while small in ordinary dielectrics (e.g. Si, NaCl) is very pronounced in ferroelectrics. (---this has been documented experimentally). The three papers collectively hint at many materials and mechanics issues that still need to be explored further in the context of ferroelectric based nanocapacitors and I hope that this journal club issue will inspire some mechanicians to do so. Personally, I have had a very enjoyable last year or so trying to understand the possible role mechanics can play in such systems.

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## Comments

## Materials challenges facing electrical energy storage

Energy storage is an excellent topic. Here is an excellent review article that gives a broader perspective on energy-storage technologies:

M.S. Whittingham, Materials challenges facing electrical energy storage, MRS Bulletin 33, 411-420 (2008).

## Stengel and Spaldin's approach

Pradeep,

I'm curious about their approach. Can you elaborate on how they're accomplishing such calculations, and what makes it clever?

Thanks in advance,

John

## John, sorry for the delay

John, sorry for the delay in my response. Your question is deceptively simple; the answer I am afraid is not and it took me some time to pen my thoughts. First, I have to provide quite a bit of background as your question touches some of the cutting edge research in this area. I get quite turned with jargon filled technical answers so I tried really hard to come up with a response that is accessible to a mechanics audience and free of discipline specific jargon. I probably have not succeeded in either of the goals but hopefully the answer below more or less answers your question as well as serves to provide some background on this topic.

The concept of macroscopic polarization is central to the study of dielectrics and its text-book definition is the "dipole moment per unit volume". Once polarization is defined, derived or calculated for a dielectric, numerous other properties e.g. dielectric tensor, spontaneous polarization (in case of ferroelectrics) may be established.

This comes as a surprise to many (as it did to me a while back) that only recently, perhaps in the last ten years or so, has a consensus emerged regarding a microscopic interpretation of macroscopic polarization. For an excellent tutorial article, I would refer the reader to Resta and Vanderbilt [1]. I believe Vanderbilt's webpage has a pdf copy of this book chapter (or I can email the interested reader).

What is the issue? The conventional viewpoint (referred to as the Clausius-Mossotti picture) is that basic readily identifiable "polarization centers" can be established in a dielectric (upon external stimuli like an electric field). In a crystal the dipole moments are then added in the cell and upon division by the cell volume, yield the polarization. This type of a viewpoint turns out to be quite inaccurate however. In real materials, such a partition is generally neither possible nor physical. The charge density is often delocalized and certainly in case of ferroelectric oxides (which are most often the materials of interest) the bonding is of mixed ionic/covalent type and indeed the distribution is quite delocalized. If one tries to "force" the "polarization centers" picture on a real material, significant error in estimation of polarization and properties like the dielectric constant can arise.

In general, as the research in the last two decades seems to indicate, defining polarization through charge density distribution is problematic. Another alternative is to assume a macroscopic but finite crystal. Then, using textbook definition, the polarization may be calculated by the integral of the charge density moment (over the whole sample) divided by the finite crystal sample volume. This definition, while on a first glance seems reasonable, causes a problem. We may alter the surface conditions (while maintaining charge neutrality) such that the overall polarization of the sample will change even though the induced periodic charge density interior to the crystal is independent of any such alterations.

A further alternative is to deduce the aforementioned integral over a unit cell in the interior and simply divide by the cell volume. This can be shown to be flawed as well. The results depend on the choice of the unit cell! A simple illustration is given by Bhattacharya and Ravichandran [2] in a review article. Assume a sinusoidal charge density i.e. q(x)=csin(x). Polarization is then the integral of x multiplied with q(x). The limits for this 1-D example are "a" and "a+2*pi". The 2*pi reflects the periodicity. The answer is: -2*pi*c*cos(a). Thus the result depends on the choice of the unit cell.

What is then the resolution to this paradox? King-Smith, Vanderbilt, Resta are some of names who helped the modern theory of polarization and as I alluded to earlier a consensus began to emerge just recently. The central idea is that upon stimuli (e.g. strain in case of piezoelectric crystal), a transient electrical current flows. The rate change of polarization is the current and the change in polarization is obtained as an integral of the current density (despite the flowing current, time-dependence is assumed to be slow enough for adiabatic conditions to apply). It turns out that the definition of macroscopic polarization in this manner gets rid of many of the paradoxes I described before. The key point (from first principles viewpoint) is that these currents are related to the phase of the quantum mechanical wave-function. Recall that the charge density depends solely on the modulus of the wave-function (and phase information is lost)---thus the phase it turns out is not quite the useless thing that elementary quantum mechanics textbooks sometimes tell us! This overall concept is known as the "Berry phase formalism". Calculation of polarization through the Berry phase approach is now a standard option in most quantum mechanical codes. Again, these developments are relatively recent.

In dealing with metal-insulator combinations, a finite electric field must be applied. Even if the metal were not present, this is problematic for DFT type approaches. Under an electric field, the ground state is not well-defined. Recall that the electric potential (linear) in case of uniform potential is unbounded. A few researchers however have suggested remedies; one example (among many) is Nunes and Gonze [3].

I return now to the metal-insulator-metal structure under a finite electric field which is needed to study capacitors. In this case the conditions become non-equilibrium. DFT type approach requires a unique Fermi level. Thus, in the case of metal-insulator-metal configuration, the system must remain insulating. The metal-insulator-metal system has an overall metallic character and the Berry phase approach (which is used to correctly calculate the polarization) is not routinely applicable (recall the adiabatic condition). Of course, time-dependent density functional theory or non-equilibrium Green's function approach are applicable although at a computational cost that can be prohibitive. In particular, I know this to be correct for TDFT. Stengel and Spaldin find a way around this problem. Some years back, Vanderbilt and co-workers, Resta, Umari and Pasquarello [4] among others worked on something called the "Wannier function formalism". In perhaps oversimplified terms, Wannier functions are Fourier-like transforms of the Bloch state i.e. a Bloch state (Psi_n_k) where n refers to a band and k to the wave vector can be defined in terms of localized functions (called Wannier functions), W_n_R where R is the lattice vector of the cell. The macroscopic polarization can be also calculated in terms of Wannier functions. In fact, Wannier functions take a delocalized charge distribution and represent it in a localized fashion (thus ironically getting us closer to the Clausius-Mossotti picture). King-Smith and Vanderbilt [5] in fact showed a formal connection between the sum of the Wannier centers and the Berry phase. Several further advancements were made subsequent to these developments. Stengel and Spaldin first found a way to use Wannier functions to obtain accurate estimates of polarization. They show that due to its real-space nature of the Wannier function approach, metal-insulator combinations can be treated in a straightforward manner and that their approach converges very rapidly (compared to various alternatives). This method development allowed them to carry out the study described in the jclub paper I discuss. The details of the method itself are not in the paper under discussion (my focus being mostly on the physics) but is available on arXiv (and published in PRB)

References

[1] R. Resta and Vanderbilt and D. Vanderbilt, "Theory of Polarization: A Modern Approach", Physics of Ferroelectrics: A Modern Approach, Eds C.H> Ahn, K.M. Rabe, and J.M. Triscone, Springer-Verlag, 2007

[2] K. Bhattacharya and G. Ravichandran, "Ferroelectric Perovskites for Electromechanical Actuation", Acta Materialia, 51, 5491, 2001

[3] R.W. Nunes and X. Gonze, "Berry Phase Treatment of the Homogeneous Electric Field Perturbation in Insulators, Physical Review B, 63, 155107, 2001

[4] P. Umari and A. Pasquarello, "Ab Initio Molecular Dynamics in a Finite Homogeneous Electric Field", Physical Review Letters, 89, 157602, 2002

[5] R.D. King-Smith, D. Vanderbilt, "Theory of Polarization of Crystalline Solids", Physical Review B, 47, 1651, 1993

## macroscopic polarization

Dear Pradeep:

Thank you for the excellent discussion. The unit cell dependence of polarization in a lattice of charges is quite interesting (very much like the conditional convergence of energy per unit cell for an infinite lattice of charges). If quantum mechanics can help to calculate the correct polarization, shouldn't it be able to tell us what "unit cell" corresponds to the "correct" polarization?

In the case of energy per charge, what is known is that the sequence of subbodies considered should be charge free with zero moments up to second order if I remember it correctly. I'm wondering if something like this is known for calculating macroscopic polarization.

It would be great if you could, in simple words, explain what Berry phase is. As far as I understand it, it's a geometric phase (it's like parallel transporting a vector along a closed path in a space with curvature and returning to a different vector. I guess here "phase" is like the difference between those two vectors?).

Regards,

Arash

## Arash, thanks for our

Arash, thanks for your comments. Your question about the unit cell identification since we know the "correct" definition of the polarization, is kind of intriguing. It may be possible that once polarization is correctly calculated, we identify or "force" a Clausium-Mossotti type unit cell that gives the same answer but I don't readily see if there is a unique way to do so. To some extent, using Wannier's function does provide a way to visualize a "unit cell". I will have to think a bit more about this....I will get back to you on this again once I return from the APS conference.

Regarding the notion of Berry phase, I am adding a few more details here. Actually your sentence related to parallel transport captured the essence pretty well. Given your expertise in geometrical physics, you can probably explain this much better than I can...I have jotted down a few thoughts below which are hopefully accessible to those who have not seen this before.

First of all, Berry's phase occurs in a variety of contexts e.g. optics, classical dynamics among others (polarization in solids being simply one of many). In fact, in a historical review, Berry indicated that this was first discovered by Pancharatnam in the fifties in the setting of optics. A simple example, given by Berry himself (and which is also popularly used in relativity) is how we may transport a pencil around a spherical ball or earth. Ensure that the pencil points along one of the longitudinal lines (you may assign the north pole as the starting point but of course you could repeat the exercise starting from any other point on the sphere). While transporting southbound, keep the pencil parallel to its path. At some point you will hit the equator. Stop. Turn east and then return to the north pole along a longitudinal line. The pencil, once it returns to its starting position, won't point in the same direction (i.e. the vector is rotated in reference to its original position). This is a version of parallel transport. Essentially, if a vector parallel is transported around a closed path on a curved surface its state becomes altered. For a flat surface, this does not occur. Thus this rotation is purely a topological effect and rotations of such type are called "Berry's phase". The famous Aharonov-Bohm experiment discussed in most quantum mechanics textbooks is a manifestation of this. The reason it goes by Professor Michael Berry's name is that in a landmark paper, he formalized this concept, particularly in relation to quantum mechanics. In addition to other areas, his work had (as evident from the theme of the jclub) an enormous impact in condensed matter physics as well. Incidentally,a couple of years back, (Sir) Michael Berry became the chief editor of Proceedings of the Royal Society. I mention this since many mechanicians publish in that journal.

I return to quantum mechanics. A quantum state is described by a vector in Hilbert space. Imagine that the state is adiabatically changed to another. If the state initially is governed by eigenstate "n" then the final eigenstate is also "n". The change can be large but must be slow. For example, a potential well width "a" may be changed to "2*a". If the particle in the potential well is initially in its ground state, it will remain in its ground state (of course with the state now depending on the new width). Stated differently, let the width, ‘a" or any general parameter, "p", be a function of time, t and imagine that the parameter. Then the particle starting out in the state n(p(0)) will remain in that state at a later time n(p(t)). changes slowly with time. Now imagine that the parameters of the Hamiltonian of a system) are changed adiabatically around a closed path (analogous to the pencil on the sphere example). It can be shown that like the pencil, the Hamiltonian reverts to its original form but acquires a geometrical phase (i.e. the wavefunctions in the two Hamiltonians will differ by a factor of the type, exp(i*theta)). The general viewpoint, at least the one advocated in elementary textbooks, is that observables (which depend on the modulus of the wavefunctions) are not related to the phase information the wavefunction carries. Berry's insight, which allows one to compute the phase acquired when parameters of a Hamiltonian are cyclically transported, implied that the relative phase between initial and final conditions is

measurableand cannot be eliminated by transformations (recall that any instantaneous state can be defined up to an arbitrary phase factor). The connection with condensed matter physics and the topic at hand (polarization) come in since charge density (which is based on the square of the wavefunction) contains no phase information. Traditional DFT approaches calculated charge density. King-smith and Vanderbilt, in their work on polarization, related the concept of tying measuring transient current in dielectric (the now agreed upon correct way to look at bulk polarization) to the Berry phase.## non gauge-invariant?

Hi Pradeep and Arash,

Reading your discussion, does this Berry phase business have to do with non gauge invariance of the basic theory, i. e. the relative phase is no longer a gauge field as it makes a a difference to the observables?

If so, I share with you a connection in a vastly different context. It is related to mechanics of deformation and defects. In the theory of continuously distributed dislocations once one goes beyond the static question of predicting internal stress due to a prescribed dislocation distribution, i.e. goes to the transient theory, a physical equation specifying the compatible part of the plastic distortion has to be specified; this has a direct influence on what displacement, and consequently shape change, is predicted by the model, and this in plasticity is a physically measurable effect. On the other hand, in the absence of dispalcement boundary conditions, this compatible part of the plastic distortion field does not affect the predicted stresses.

Thus if we consider just the stress as the observable of the theory (say static theory of continuously distributed dislocations), then the compatible part of the plastic distortion is a guage field, not affecting the observable physics.

Now add displacement to the set of observables, i.e.e stress and dispalcement, and the field in question is no longer a physically irrelevant transformation.

To illustrate through a physical example: consider a single dislocation in a body; do a thought "anneal" by raising temperature; the dislocation runs out of the body. At the end of the process, the body should have no stress but localized permanent strain, measured from the original reference, along the path of the dislocation. This is a measurable effect and cannot be predicted if the theory does not contain a statement defining the compatible part of the plastic distortion.

- Amit

## Dear Amit, yes, Berry's

Dear Amit, yes, Berry's phase is in fact gauge invariant. In other words, we cannot come up with a clever transformation and make it "go away". It is also measurable. The last insight is what makes Berry's discovery and paper special. On the face of it, your example related to dislocations does seem to bear a close analogy to the Berry phase concept. Since the Berry phase is topological in nature it makes sense that it would occur in the context of dislocations in solids as well. I have not see any formal work that connects these two together though. Perhaps the closest work is due to Edelen and Lagoudas but I don't believe they discuss this particular aspect.

## Thanks, Pradeep. Up to some

Thanks, Pradeep. Up to some some mismatches in terminology, I think we are talking about similar ideas.

As for Edelen and Lagoudas (and all the other gauge theory of dislocations), from what I could understand, it seems they do not bother about predicting permanent deformation, but since my upbringing is in engineering plasticity, this was very important to me.

thanks for a great j-club issue - I have been exposed to interesting new facts...

- Amit