Calculation of the Capacitances of Conductors--Perspectives for The

Calculation of the Capacitances of Conductors Perspectives for the Optimization of Electronic Devices

Thilo Kopp and Jochen Mannhart Center for Electronic Correlations and Magnetism, Institute of Physics, Universität Augsburg, D-86135 Augsburg, Germany (Dated: June 9, 2009) The equation describing the capacitance of capacitors is determined. It is shown that by optimizing the material of the conducting electrodes, the capacitance of capacitors reaching the quantum regime can be substantially enhanced or reduced. Dielectric capacitors with negative total capacitances are suggested and their properties analyzed. Resulting perspectives to enhance the performance of electronic devices are discussed.

PACS numbers: 71.27.+a,73.20.-r,73.21.-b,73.40.-c Keywords: negative capacitance, strongly correlated electrons, multiferroics

I. INTRODUCTION In fact, the capacitance may be even negative. Here 0 is the dielectric constant of the vacuum, r is the dielectric As the characteristic properties of electronic circuits constant of the dielectric material between the two elec- reaches quantum scales, new possibilities emerge for the trodes of area A, and d is the thickness of the dielectric. realization of novel, quantum electronic devices which ex- It is specically shown that capacitors can be fabri- ploit the quantum nature of solids. Here we show that cated that do not follow Eq. (1). By altering the con- by using quantum eects, the capacitance of electronic ducting materials of the electrodes and by varying the devices can be optimized to a great extent, which is a geometry of the capacitor in novel ways the capacitance key to the further miniaturization of electronic compo- of such capacitors can be greatly enhanced or lowered as nents.1,2,3,4 While in MOSFETs, for example, large gate compared to Eq. (1). Electrode materials of interest in- capacitances are needed for operation at small gate volt- clude materials with strong electronic correlations such as ages, their gate insulators are required to have an equiv- transition metal oxides, two-dimensional electron gases alent thickness of at least 1 nm to keep the gate currents as generated at interfaces in oxides or semiconductors, at acceptably small values. To meet both requirements, metals with small carrier densities, organic conductors, and materials such as graphene. We point out that these high-κ gate insulators, consisting for example of HfO2 based compounds, have been introduced into the gate quantum phenomena and their technical implementation stack of MOSFETs used in microprocessors. provide a possible strategy to control large or small ca- As the sizes of the devices approach the nanometer pacitances by the electronic properties of the conducting length scale, quantum mechanical phemonena become electrodes as opposed to the standard search for low or high-κ dielectrics11. increasingly relevant. The well-known classical models 10 of semiconducting devices and the corresponding scaling It is common knowledge that classical electrostatics laws break down. Ballistic transport processes, for exam- determines the capacitances. In the textbook case of a ple, gain importance, as does the statistical distribution capacitor with two electrodes, the capacitance is charac- of the dopant atoms in the small drain-source channels. terized by Quantum eects also alter the capacitance of large, Q = CV (2) macroscopic capacitors that reach the quantum regime because their dielectrics become thin. As we show, for where Q is the charge on a capacitor electrode, C is such devices the capacitance comprises not only the well- the capacitance, and V is the voltage between the elec- known geometric capacitance, but four more capacitance trodes. In more general terms, for multiple conductors terms. These originate from the kinetic energy of the (i = 1, 2, . . . , n), the capacitance is characterized by the corresponding static response relations P , electrons, from their exchange energy, their correlation Qi = j CijVj where and are the potential and the total charge arXiv:0902.4673v2 [cond-mat.mtrl-sci] 9 Jun 2009 energy, and the electron-phonon coupling energy. Vi Qi In this paper, we present the general equation for the of conductor i, respectively. The coecients Cij de- capacitance of capacitors that considers the quantum me- ne the capacitance matrix. According to the classi- chanical energies of the electronic systems involved. This cal approach, the coecients of a capacitor consisting equation applies to two-plate capacitors, also in case their of two innitely extended parallel metallic plates are (Eq. (1)). Here, fringe dielectric is ultrathin. It is shown that for given and C11 = C22 = −C12 = −C21 ≡ C r eld eects have been neglected. In this classical descrip- A the capacitance of capacitors reaching the quantum regime may dier considerably from the well-known value tion, the capacitance is not inuenced by the electronic properties of the conducting plates. For a two-plate capacitor, the geometric capacitance is entirely determined 0rA C = (1) by the eective distance ? . d d = d/r 2

It is apparent that Eq. (1) is not very useful for the lateral dimensions and macroscopic numbers of charge design of ultra-small devices, where, for example, single carriers, if their dielectric layers are suciently thin. It electron eects play an essential role (see, e.g., Ref. 16). is hereby particularly interesting that also correlation ef- The calculation of the electronic properties of such de- fects in the electron systems of the conducting electrodes vices is typically based on the direct calculation of quan- may alter the capacitances in unique ways. We therefore tum mechanical energies. propose to use these quantum phenomena as a possible Quantum mechanical eects, however, are also of im- tool to optimize devices such as MOSFETs. Desired large portance for large devices with macroscopic lateral di- or small capacitances can potentially be attained by tun- mensions. The conductors of electrodes can screen, for ing the electronic properties of the conducting electrodes example, the electric eld only over non-vanishing lengths in addition to the standard approach of using low- or such as the Thomas Fermi screening length. This eect high-κ dielectrics and small or large barrier thicknesses. obviously enhances the eective dielectric thickness d?. The enhancement of d? has rst been observed by Mead5, conrmed by Hebard et al.6 for a known thickness of the II. THE COMPOSITION OF THE 7,8 CAPACITANCE OF MACROSCOPIC dielectric, and deduced from the charge distribution CAPACITORS prole in the electrodes which is controlled by the kinetic energy of the charge carriers (see the detailed discussion and evaluations in Refs. 9). A framework to calculate the To determine the equation that describes the capacitance of capacitors, we start by considering the energy capacitance coecients Cij that considers the eld penetration into the conductor has been formulated by Büt- functional for the electronic ground state of a given ca- tiker.12 He evaluated the space-dependent Lindhard func- pacitor. For clarity, we restrict our evaluation to capaci- tion. The eld penetration into the capacitor electrodes tors with two parallel plates and neglect eects arising introduces corrections to the geometrical capacitance:12 from nite temperatures and from fringe elds. The in the case of space independent density of states (DOS) plates are considered to be so far apart that their electronic wave functions do not overlap. Tunneling and right up to the surface (TF) (TF) A/C = (d/r + λ1 + λ2 )/0 exchange contributions between the plates are therefore where (TF) are the Thomas-Fermi screening lengths 17 λ1,2 negligible. The ground state energy as a functional of of the two metal plates. For a capacitor with two- the electronic densities may be decomposed as:18 dimensional metallic plates, (TF) (TF) is replaced12,44 λ1 +λ2 by a /2 where a = 4π 2/(me2) is the Bohr radius. X X X X B B 0~ E = E + E + E + E + E (3) Lang and Kohn14 introduced the center of mass posi- H kin,i x,i c,i ext,i i i i i tions (1,2) of the induced surface charge density at the x0 The sums are used to describe the energies of the two electrodes and related the distance between the center of possibly dierent metallic plates. In capacitors for which mass positions to the capacitance of a parallel plate ca- surfaces are relevant, the electronic density ni(r) is a pacitor in the vacuum ( ): (1) (2) . r = 1 A/C = (x0 − x0 )/0 function of the space coordinate r. The energy E[ni(r)] They calculated (i) within density-functional theory is then a functional of the densities on the two electrodes. x0 15 (DFT) for a metal with a uniform distribution of pos- The Hartree term EH in Eq. (3) includes all direct itive background charge (jellium model). They deter- Coulomb interactions between the electrons of the con- mined the center of mass positions to be outside of the duction bands; these long-range Coulomb potentials also surface of the electrode (given by the edge of the pos- include the electrostatic interaction between the plates. itive background charge at (1,2)) so that the capaci- The other energies, which are of pure electronic origin, xb tance is increased with respect to its geometric value are not caused by inter-plate interactions, but rather are local attributes of each plate. These energies are the ki- (1) (2) . The information on the electrodes is 0A/(xb − xb ) netic energy of the electrons Ekin,i, the exchange or Fock encoded in the distances δx(1,2) = x(1,2) − x(1,2) which 0 b energy Ex,i, and the correlation energy Ec,i, which rep- generate interface capacitances serially connected to the resents the electron-electron interaction energies beyond geometric capacitance. the Hartree-Fock description. ? For a macroscopic d , the corrections arising from the The external energy Eext,i is the energy of the conduc- Thomas-Fermi lengths or δx(1,2) are tiny, because they tion electrons in an external potential; it most promi- represent small, quantum mechanical length scales. How- nently includes the interaction energy between the elec- ever, exceeding these eects described, quantum phenom- trons and the background charge that is caused by the ena are of fundamental relevance in controlling the capac- nuclei. Because this Coulomb interaction between elec- itances of electronic circuits as soon as the eective thick- trons and nuclei is long-ranged, it diverges with increas- ness of their dielectric layers d? approaches quantum me- ing area A; this divergence is canceled by a similar diverg- chanical lengths such as raB. Current devices like mod- ing term of opposite sign, which results from the electron- ern MOSFETs are close to this transition, and quantum electron interaction in the Hartree contribution.19 Inho- phenomena need to be considered in their design. As mogeneities such as surface potentials or potential wells we show, this even holds for capacitors of macroscopic are also introduced through Eext,i. 3

To treat the electronic exchange eects, all electron- electron interactions, including those with the electrons of the ionic core states, need to be contained in the Hartree-Fock and correlation terms. In eective models, however, the exchange contribution of ionic core states Cgeom Ckin Cx Cc Cel-ncc with conduction electrons is often neglected, an approximation that allows to include the occupation of these electronic core states in a pseudopotential or in a suit- FIG. 1: Equivalent circuit diagram of a two-plate capacitor according to Eq. (6). The capacitor consists of a serial con- ably dened ionic core charge. In those cases, it is useful nection of the geometric capacitance and the quantum capac- to introduce an eective dielectric constant eﬀ,i into the itances of each of the two plates. The capacitance results electronic interaction terms which accounts for the po- from the Hartree energy (∝ 1/C ), the kinetic energy larizability of the ionic cores. geom (∝ 1/Ckin), the exchange energy (∝ 1/Cx), the correlation In case the electron systems on the two parallel plates energy (∝ 1/Cc), and the short range interaction energy of are homogeneous, i.e. if they don't reside on lattices,20 charge carriers with all non charge-carrier (ncc) degrees of P 2 . In such a situation, is freedom (∝ 1/Cel−ncc). Positive and negative capacitances EH + i Eext,i = Q d/(20rA) d given by the distance between the boundaries of the uni- are drawn in both blue and red, respectively. The capacitance is presented in blue and red, because the sign form positive background charge of the electrodes. This Cel−ncc of the capacitance depends on the microscopic model for the energy term matches the standard equation for the en- coupling of the ncc degrees freedom to the charge carriers; for ergy of the capacitor, phonons it is expected to be negative.

2 cording to the coecients of 2 : ECoul = Q /(2Cgeom) (4) Q /2 X X 1/C = 1/Cgeom + 1/Ckin,i + 1/Cx,i with the geometric capacitance Cgeom = 0rA/d. Yet, also the other three terms in the energy functional gener- i i ate capacitances. These capacitances result from distinct X X + 1/Cc,i + 1/Cel−ncc,i (6) quantum phenomena. i i To derive the total capacitance C for Q → 0, we ex- Eq. (1) is replaced by this general equation, which reveals pand the energy functional to second order in Q. If the two plates dier in their physical properties, also a lin- that the total capacitance is caused by a series connection of the capacitances that result from the additive terms in ear contribution in Q arises, which originates from the dierence in the work functions of the electrode materi- the energy functional (Fig. 1). Although Eq. (1) seems to imply that the total capacitance is always smaller than als. The work function Φ, dened as the minimum energy necessary to extract an electron, is composed of the chem- the geometrical capacitance, the total capacitance can, in fact, well exceed the geometrical one, because and ical potential of the innite system (relative to the elec- Cx may have large negative values. trostatic potential in this system) and the electrostatic Cc Note that the external energy consists of a long potential energy dierence across the respective surface Eext,i of the electrode (for details and the evaluation see, e.g., range Coulomb term, which compensates the diverging Refs. 21 and 15). The contribution to the work func- term of the purely electronic Hartree energy, and a re- tion from the chemical potential is generated by all en- maining short range contribution. The inverse geometric capacitance is generically dened as the 2- ergy terms. The electrostatic potential dierence across 1/Cgeom Q the surface, the so-called electrostatic dipole barrier, is coecient from the Hartree energy and the static long- range part of the external energy. The short range part caused by the external energy. A dierence ∆Φ of the of contribution includes the interaction between work functions of the surfaces of the opposing electrodes Eext,i generates a contact voltage, so that associated surface the charge carriers and all non-charge carrier degrees of freedom (ncc) such as phonons or local potentials. Ex- charges ±Qs are induced. This electronic charge recon- struction compensates the work function mismatch. The amples for the latter are the static potentials of the ionic dierential capacitance can be evaluated by expanding cores and the conning potentials in semiconductor het- erojunctions. These contributions generate the the energy functional around the charge-reconstructed 1/Cel−ncc states of the electrodes. term in Eq. (6). In this paper we will not investigate Cel−ncc in more detail. Focussing rst on the capacitance Eventually, to obtain the voltage dependent capacitance C(V ) beyond the linear response limit of Eqs. (5) C = Q/V (5) and (6), one takes the Legendre transform E of the energy E(Q) with respect to Q and identies the capacitance in linear response, i.e., for V → 0 (or, for nite C(V ) = Q(V )/V from E(V ) = E(Q(V )) − QV . 22 Qs = −C∆Φ/e, on C = (Q − Qs)/V ), we collect the For the sake of clarity, we explicitly note that the value quadratic terms and identify the inverse capacitances ac- of the dierential capacitance for specic values of V , Q, 4 or the density of charge carriers, n, is given by the ratio terlayer Coulomb interaction. The mean value of the dQ/dV , where dQ is the amount of charge that a volt- charge in the SILC state is zero on each of the two lay- age change dV adds to a plate of the capacitor, which is ers. The Coulomb energy ECoul therefore vanishes in characterized by V , Q, or n. We also note that the volt- this state. The possible magnetic, charge separated, and age V between the two plates, i.e., the dierence between SILC states have been categorized in Hartree-Fock theory the electrochemical potentials of the plates, does in gen- at zero magnetic eld, with and without applied bias.35 eral not equal the dierence of the electrical potentials A nite bias may support charge separated states. of the electrodes, which denes the electrical voltage . Ve Spin polarized and SILC states oer exciting possibil- Whereas the voltage V is the voltage that is measured ities to explore and use quantum capacitances. Never- by standard voltmeters and, for example, is given by the theless, in the rest of this article we will focus on un- output of standard voltage sources, the electrical volt- polarized states without interlayer coherence. While the age is the electrical potential dierence between the Ve SILC states will be the true ground states, coherency plates, the gradient of which is the electric eld between will be lost in capacities operating with nite electric the plates. The electrochemical potential dierence con- elds in noisy electronic circuits at room temperature. sists of the electrical voltage and the dierence of the The states with charged electrodes may be generated chemical potentials of the electrodes. In standard capac- through an applied voltage in cases where the dierential itors, the shifts of the chemical potential upon charging capacitance C for Q → 0 is negative (see the following are negligible. However, if the chemical potential dier- subsections). The states with charged electrodes are ex- ence becomes signicant, as is the case when the charging pected to persist up to high temperatures of the order of the capacitors cause a non-negligible shift of the chem- of P P P . In com- |ECoul + i Ekin,i + i Ex,i + i Ec,i|/kB ical potentials, the measured voltage will dier from the pounds with strongly electronic correlations, the correla- electric potential dierence. tions may favor either the SILC states or the states with charged electrodes. We consider the unpolarized homogeneous electron gas on two nanoscopically close plates III. CAPACITORS WITH ELECTRODES as a model capacitor and refer to possibly relevant, cor- COMPRISING ELECTRON SYSTEMS WITH related electron systems in the following chapters. UNIFORM BACKGROUND We conceive that practical capacitors with negative, total capacitance can eventually be realized, in particu- Electron gases with uniform, positively charged back- lar by using strongly correlated systems. The existence of ground (jellium model) serve as model systems which al- devices with negative total capacitance has already been low to evaluate thermodynamic electronic properties in addressed in the literature, concerning electronic circuits 23 28 2D and 3D without the complications arising from (see, e.g, Ref. 39), electron injection through interfaces the coupling to lattice degrees of freedom. In the fol- (see, e.g, Ref. 40), and structures involving ferroelectrics lowing we will calculate the capacitance for 2D or 3D (see, e.g, Refs. 41, 42). Here, we consider a far more con- electron gases in the jellium model to make several of ventional system, standard capacitors, such as plate ca- our statements on the quantum mechanical properties of pacitors comprising conventional dielectric materials and capacitors more lucid. The ground state of the electron operated in steady state. gases in a conguration with two unbiased but electro- statically coupled sheets, each comprising a dilute electron gas, has been investigated intensely. In particular, it has been laid open that the exchange term P can i Ex,i drive the double-layer electron gas into ferromagnetic and A. Electrodes with Two-Dimensional, charge separated states.33,34,35 Homogeneous Electron Systems A spontaneous spin polarization of a dilute electron gas was already suggested by Felix Bloch36 and has been included in several recent investigations of double layer sys- The capacitance C for a capacitor with two- tems. An exchange driven instability towards a state in dimensional electron systems on each plate is determined which all electrons occupy a single layer, i.e., a state with by the inverse capacitances given in Eq. (6). To shed a maximally charged capacitor has been proposed33 but light on the physical origin of the rhs of Eq. (6) and to it has been shown that the ground state in an unbiased present a method to approximately calculate these terms, double layer system with P P we introduce the thermodynamic compressibility of an E = ECoul + i Ekin,i + i Ex,i κ 34 18 is of a dierent nature: it is a coherent quantum me- electron system. As follows from Seitz's theorem , κ−1 chanical superposition of the two states that each carry is given by the rst derivative of the chemical potential the electrons on either of the two layers. This ground µ or by the second derivative of the total energy E, both state has been introduced and predicted for bilayer quan- taken with respect to the electronic density n = N/A: 37,38 tum Hall systems. It is characterized by a spon- κ−1 = n2∂µ/∂n = n2 d2(E/A)/dn2, which is a standard taneous interlayer phase coherence (SILC) of two lay- thermodynamic relation in the limit of zero temperature. ers which are each entirely isolated except for their in- For 2D-systems we rewrite Eq. (6) in terms of the inverse 5

9 compressibility of a 2D conductor: and obviously even more so in those cases in which A/Cx and A/Ckin almost cancel each other. X −1 To estimate the correlation capacitance of the 2D ho- A/C(2D) − AC−1 = κi (7) geom 2 mogeneous electron gas, the parameter is introduced. (eni) rs i It is a dimensionless number that characterizes the inter- The inverse kinetic and exchange capacitances can be particle distance: determined from Eq. (7), using the well-known expres- 1 1 sions for the energy functional of a 2D homogeneous elec- 2 (11) rs[ni] = 2 tron system (see, e.g., Ref. 24): aBπni Tanatar and Ceperley24 have cast the -dependence of 4πd rs 4π0A/Cgeom = (8) the correlation energy into an approximate functional r form which ts their Monte Carlo data. With the en- (2D) πaB 4π0 1 ergy functional of Ref. 24, the inverse capacity is identi- 4π0A/C = = (9) kin,i m?/m e2 (2D) ed by using the correlation contribution to Eq. (7). The i ρi (εF ) 2 1 1 1 exchange and correlation capacitances are found to equal 4π A/C(2D) = − 2 √ (10) 0 x,i π n √ i eff,i (2D) (12) 4π0A/Cx,i = − 2 aB rs/eff,i where 2 2 Å is the bare Bohr ra- aB = 4π0~ /(me ) ' 0.5 and dius, ? is the eective electronic mass and the ef- mi eff,i fective dielectric constant in plate . This constant char- (2D) i 4π A/C = (13) acterizes the screening in the metallic 2D plate that does 0 c,i 2 2 not arise from the screening by the conduction band elec- πaBrs a0 3 3 x x z − u + (1 + u) + (1 + u)( 2 − ) , trons themselves. The band electron self-screening is eff,i y 32 4 y y y included in the correlation term Cc discussed below. It is respectively. Here, the polynomial functions u, x, y, z are notoriously dicult to estimate eff,i. For very dilute 2D electron systems in GaAs/AlGaAs heterostructures that dened as: 11 −2 may have nominal densities of 10 cm , eff is given 1 2 (14a) approximately by the dielectric constant of the AlGaAs u[rs] = a1rs layer.45 The eective band mass ? can be more eas- 1 3 mi y[r ] = 1 + a r 2 + a r + a r 2 (14b) ily determined. Conveniently it is chosen such that the s 1 s 2 s 3 s 1 1 1 3 3 resulting DOS matches the DOS (2D) of the con- 2 2 (14c) ρi (εF ) x[rs] = a1rs + a2rs + a3rs ductor. A summation over bands is used in the DOS 4 2 4 1 3 5 2 3 21 2 in case several conduction bands are present or a band z[rs] = a1rs + a2rs + a3rs (14d) degeneracy exists.43 Such a band (degeneracy) factor de- 32 8 32 creases the inverse kinetic capacitance and, consequently, 24 and the coecients al are increases the total capacitance C. The kinetic capacitance (2D), which has been intro- a = −0.3568, a = 1.1300, a = 0.9052, a = 0.4165 Ckin,i 0 1 2 3 duced by Luryi44 as quantum capacitance, is invariably The inverse capacity of each of the plates contributes a positive. The exchange capacitance A/C , however, is x term (2D) to the total inverse capacity. For large , always negative, irrespective of the dimensionality of the A/Cc,i rs electron gas. The exchange capacitance varies inversely i.e., for dilute, homogeneous electron systems, (2D) √ A/Cc ∝ with the square root of the electronic density. Corre- −rs ∝ −1/ n. This implies that in systems with very spondingly, based on Eq. (6) we recognize that capacitors few carriers, C is driven by Cx and Cc to small, negative with negative capacitances can be built. The negative values (Fig. 2). capacitance can be obtained, for example, by fabricating Eqs. (7) and (9), (11) and (13) reveal that the total capacitors with electrode materials that have a very low capacitance can dier considerably from the geometri- density of states. A negative C implies that the plates cal capacitance. It is only for the case that Cgeom/A of the capacitor self-charge once they are connected. In is much smaller than 2 that the total capaci- |κi| (eni) these systems, the inverse exchange A/Cx plus correla- tance is bound to be given by the geometrical capaci- tion A/Cc capacitances dominate the inverse kinetic ca- tance. The strong dependence of C on rs is illustrated pacitances A/Ckin plus the geometrical capacitance, so in Fig. 2 for a symmetric capacitor. As expected from that the total capacitance C becomes negative. the previous discussion, with increasing rs the capaci- In a homogeneous electron gas, electron-phonon contri- tance diverges and becomes negative when the negative butions are not present by denition. Correspondingly, terms 2(A/Cx + A/Cc) compensate the positive terms A/Cel−ncc in Eq. (6) is disregarded here. (2A/Ckin + A/Cgeom) at rs = r0. The rs value at which The last term to be considered in A/C of Eq. (6) is the the capacitance changes sign is higher than the one at correlation term. If present, Cc is particularly important, which the compressibility becomes negative. We assign to 6

2D C C0 3 6 10 4 ! " 5 # 0 2 C L 0 eff 2 D Ε

2 r r 0

H c o

C -2

-4 !5

1 -6 1 3 5 7 9 !10 m! m 0 2 4 6 8 10 FIG. 3: Contour plot, showing for an ideal two-dimensional rs symmetric capacitor the lines of constant capacitance. The 2D ? capacitance C is a function! of the eective mass m and FIG. 2: Dependence of the capacitance of an ideal two- of the eective dielectric constant eff of the electrode ma- dimensional capacitor (a parallel plate capacitor with two 2D- terials (assumed to be equal on both electrodes). Here, the 1 electrodes) on the carrier spacing. The capacitance is a func- ` 2 ´ 2 dimensionless interparticle distance rs = 1/aB πn in bare tion of the electron density on the electrodes, parameterized atomic units is taken to be and the eective distance 1 rs = 7 ` 2 ´ 2 ? by the dimensionless interparticle distance rs = 1/aB πn of the capacitor plates to be d ≡ d/r = 1.23 aB , where r is in bare atomic units. In the calculation presented, the eec- the dielectric constant of the dielectric material between the ? ? tive mass is taken to be m /m = 1, the eective dielectric plates. The white line traces the values of m /m versus eff constant in the electrodes to be eff,i = 1, and the eective at which the capacitance diverges (cf. Fig. 2). ? distance to be d = aB . The dotted line shows the capacitance of the conventional, classical capacitor C0 ≡ Cgeom. non-homogeneous electron systems, the phase transition into an electronic state with negative compressibility has the latter the dimensionless length scale rc, and r0 > rc. not been investigated rigorously. It cannot be excluded In fact, the transition into a state of negative electronic that for rs > rc the capacitance behaves dierently than compressibility has already been experimentally observed shown by Fig. 2. In particular, a metal-insulator transi- at interface electron gases in Si-MOSFETs and IIIV het- tion50 with a spatially inhomogeneous insulating phase51 45,46,47 erostructures . The observation of a negative κ has been observed which is probably controlled by strong does not necessarily contradict the thermodynamic sta- disorder. Also, a phase transition into a charge density bility criterium of a positive total compressibility. The wave state seems possible.52 The functional properties latter condition applies to the sum of the compressibil- of the capacitance nevertheless reect the energy depen- ities of all electronic and ionic subsystems. Even in dence of the electronic state on the charge carrier density thermodynamic equilibrium, the compressibility κ of the of the electrodes. electronic subsystem on one or on both of the electrodes For the ideal capacitor with two-dimensional elec- 23 may be negative if it is compensated by other contri- trodes, the kinetic term has a negligible inuence on the butions to the total compressibility. capacitance for large density of states or, equivalently It has to be expected that the electronic states on for large eective mass (see Fig. 3). This applies for (2D) 2 ? the plates change qualitatively at the transition into ρ (εF ) 20r/(e d), i.e., for m /m raB/(2d). 48 the regime of negative electronic compressibility at rc. Leaving the model capacitor, the description of a real- The new ground state is characterized by exponentially istic quasi-2D electronic system has to consider also the damped charge density modulations (cf. [49]). A further nite thickness of the quantum well which hosts the 2D transition into a self-charged state takes place at r0 where electron system. As discussed, the Hartree term con- the capacity becomes negative. tains such corrections which are included in d?. Also As shown by Fig. 2, the capacitance exceeds the clas- the exchange and correlation terms depend on the nite sical capacitance (C > Cgeom) for rc < rs < r0. This thickness which may be considered by a form factor result relies on the assumption that, also at these car- that acts as a prefactor for the respective energies. Such rier densities, the electron system is homogeneous. For a form factor has been introduced by Stern9 for the ex- 7 change term and has been applied by Eisenstein et al.45 To elucidate the behavior of capacitors with metallic in the evaluation of their experiments. This nite thick- electrodes that contain 3D electron gases,28 we consider ness correction does not alter in a qualitative way the in the following a model capacitor. In this capacitor, dependence of the capacitances as given by Eq. 12. the 3D bulk compressibility of the electronic systems is Furthermore, also the inuence of the charge carrier taken to be constant up to the electrode surfaces; con- density on the band energy (band bending) has not been sequently the formation of a surface dipole layer is not considered in the model capacitor model. This eect de- included. With this approximation, the precise distance pends on the details of the electronic structure of the between the mirror planes of the electrodes cannot be electrode material. A detailed analysis is given in Ref. 45. identied, as in Ref. 14 for the jellium-model electrodes separated by vacuum. However a comparison of the qualitative dependence of the eective distance on exchange B. Electrodes with Three-Dimensional, Metallic and correlation eects is of considerable interest and will Electron Systems be presented below. This allows to introduce an eective model which captures an important part of the physics and is much easier to treat than the fully self-consistent Quantum corrections to the capacitance that result evaluation of Ref. 14. It may seem that the approach, from the kinetic energy of the additional carriers of a which is presented below and builds on the evaluation charged 3D capacitor, have been investigated in Refs. 9, of the density-density correlation, cannot account for the 12,55. If the corresponding states are conned to the sur- displacement of the center of mass of the induced charge face, i.e., if they form subbands of surface bound states, density as introduced in Ref. 14. The density-density they are two-dimensional states. Their existence requires correlation in the uncharged state of the electrodes is, that their energies are well separated on the temperature however, related via the uctuation-dissipation theorem scale and that the inelastic scattering between states kBT to the displacement of the induced charge for nite, pos- of these subbands is negligible. In that case the evalua- itive compressibility. tion of Sec. III A applies. In this subsection we consider In three dimensions, the parameter is dened by the case that the electronic states are not bound to the rs surface of the electrode. 3 1 r [n ] = 3 (16) Also in 3D systems, the geometric capacitance is given s i 3 4πniaB by Eq. (8), and presents, like in the 2D systems, the dimensionless length scale of the carrier spacing. We assume that ? (15) 4π0A/Cgeom = 4πd the electrodes in the uncharged state consist of a homogeneous electron system with a uniform positive back- except that the charge does not completely reside in the ground which compensates the electron charge (jellium surface planes of the electrodes. A simple approach for model). Correspondingly, r and the energy E are not 3D systems assumes the charge density to decay into the s functionals of n [x] but just functions of the average elec- bulk with a length scale of the order of the screening i tronic densities ni as in Sec. III A. In a microscopic evalu- length λ. The eective distance between the capacitor ation of the work function of the electrodes, this approx- plates is thereby slightly enhanced by 2λ, because the imation would miss the electrostatic potential across the static Coulomb energy is determined by the charge dis- metal surface, because the homogeneous electron system tribution which is inuenced by the boundary condition 7,9,13,53,54,55 (up to the edge of the positive background) does not in- of the electron wave function at the surface. clude the surface dipole barrier. However, these terms The results of such an approach have to be contrasted should be included phenomenologically in the dierence 14 with those of Lang and Kohn. They include the for- of the work functions for the two electrodes in the capac- mation of a surface dipole barrier that is caused by itance. For electrode plates of the same material and the the spilling out of the mobile electrons into the vac- same crystallographic orientation of their inside parallel uum. Already in the uncharged state this dipole layer faces, these terms cancel. forms a non-homogenous charge prole. This charge With the 3D electronic density n = N/V, the prole modies the distribution of the induced screen- compressibility relation is: −1 = n2∂µ/∂n = ing charge for nite voltage so that the eective distance κ n2 d2(E/V)/dn2. For a capacitor with 3D plates, the between the electrodes is reduced rather than enhanced screening of the charge causes the charge density to be by 2λ, assuming that the dierence between the edge non-uniform perpendicular to the plates on the scale of of the uniform background charges denes the distance the screening lengths λi. With the above introduced d?. Whereas the width of the screening charge density assumption of constant compressibility, the relation be- scales approximately with the Thomas-Fermi screening tween the capacitances and compressibilities has the ap- 56 legnth apart from the characteristic Friedel oscilla- proximate form tions in the interior of the metalthe center of mass position of the induced charge is not easy to determine. −1 X 1 ∂µi X Its calculation requires a fully self-consistent treatment A/C(3D)−AC−1 = = κi (17) that includes the exchange and correlation terms. geom e2λ ∂n λ (en )2 i i i i i i 8

(3D) In case the density of states (DOS) ρ (εF ) (the DOS at the Fermi energy εF for both spin directions) is spatially uniform and if the screening follows the Thomas- −2 2 Fermi screening of a free electron gas with (TF) λi = −1 2 (3D) , Eq. (17) is equivalent to the relation 0 e ρi (εF ) (3D) −1 −1 P (TF) given by Büttiker12. A/C − ACgeom = 0 i λi The relevant surface volume of the electrode is given by i 0 1 the surface area times the screening length . λi C #

Eq. (17) yields the value of the inverse capacitance. "

The kinetic term in the energy functional generates for D 3 each electrode a capacitance, the inverse value of which ! is: C 0 rc ro 1 4 1 − 3 π 3 a n 4π 1 4π A/C(3D) = B i = 0 0 kin,i ? 2 (3D) 3 λi m /m e λj i ρi (εF ) (18) !1 A straightforward conversion to the Thomas-Fermi approach, which also accounts for the polarizability of the 2 4 6 8 10 underlying lattice and of the ionic cores through a corresponding eective dielectric constant eff,i (see, e.g., rs Ref. 13), identies: FIG. 4: Dependence of the capacitance of a three- 1 dimensional, parallel plate capacitor on the carrier spac- 2 (TF) (19) λi = eff,i λi ing, according to the serial connection of the capacitances where the bulk Thomas Fermi screening length is of Eqs. (15), (18), (23) and (24). The capacitance is a function of the electron density on the surface of the electrodes, parameterized by the dimensionless interparticle dis- 1 1 2π 1 r m √ (TF) 3 1 λi = = ? aB rs tance ` 3 ´ 3 . In the calculation shown, the eec- q (3D) 2 3 m rs = 3/4πnaB e2ρ (ε )/ i ? i F 0 tive mass is mi /m = 1, the eective dielectric constant in the ? (20) plates is eff,i = 1, and their eective distance is d = aB . Eq. (18), jointly with Eqs. (20) and (19), yields the simple The dotted line displays the capacitance of the correspond- 13 relation: ing classical capacitor C0 ≡ Cgeom. The dashed line refers to a capacitor with eff,i = 2, and the dot-dashed line to a (TF) 4πλ capacitor with eff,i = 30. (3D) i (21) 4π0A/Ckin,i = √ eff,i Here we have used the density of states (DOS) relation electronic correlations the capacitance therefore grows to for an electron gas: jump at rs > r0 to negative values (see Fig. 4). 1 ? ? 3 1 4π n 3 m 1 3 2 4π m The exchange contribution to the capacitance results ρ(3D)(ε ) = 3 0 i = 3 0 i i F 4 2 2 2 from the standard exchange energy functional18 (using π e aB m π 2π e aB rs m (22) Eq. (17)): We derived the relation Eq. (21) from the (constant) compressibility of the kinetic energy term, whereas the standard derivation determines rst the induced charge den- (3D) 1 1 4π0A/Cx,i = − 1 2 (9π) 3 3 sity in Thomas-Fermi approximation. Both schemes are λieff,i ni obviously equivalent if the screening length is identied 1 2 16π 3 a (TF) 2 B (23) as . = − rs λi 81 λ The quantum kinetic term (Eq. (18)) depends on the i eff,i DOS and always lowers the capacitance as compared to its classical value. As shown by Eqs. (18) and (20), mul- The correlation contribution is derived from the ap- tiple or degenerate conduction bands in the DOS increase proximate functional form of the 3D energy t to Monte C(3D). Conversely, small eective masses m? of the car- Carlo calculations as given by Ceperley:30 riers reduce C(3D) (cf. Fig. 6). Coulomb interactions beyond the classical approxi- 2 3 2 mation qualitatively alter the properties of the capaci- (3D) 4πaBrs h4 g[rs] 4π0A/Cc,i = 2 g[rs] − s[rs] + tance, because exchange and correlation terms introduce 3λieff,i f[rs] f[rs] in Eq. (6) negative inverse capacitances. With increasing (24) 9

3D C C0 5 6 2 5! " 4 4 # 3

0 1 2 C #

eff 1 " 3 Ε D

3 0 ! !1 C 0 2 !2 !3 !4 !1 1 1 3 5 7 9 !5 m! m 2 4 6 8 10 FIG. 6: Contour lines of equal capacitance for the three- rs dimensional model capacitor. The capacitance C3D is a function of the eective mass ? and of the eective dielectric FIG. 5: Dependence of the capacitance of a three- !m constant eff of the electrode materials. Here, the dimension- dimensional, parallel-plate capacitor as a function of carrier 1 ` 3 ´ 3 spacing. The eective distance between the plates used in the less interparticle distance rs = 3/4πnaB in bare atomic units is taken to be and the eective distance of the calculation is d? = a and m?/m = 5. The eective dielectric rs = 7 B i capacitor plates to be ? . The gray line constants are equal for both electrodes with = 1, 2, and d ≡ d/r = 1.23 aB eff,i traces the values of ? versus at which the capaci- 30 for the solid, dashed, and dot-dashed lines, respectively. m /m eff The dotted line shows the capacitance of the corresponding tance diverges. classical capacitor C0 ≡ Cgeom.

at higher eective mass and larger carrier spacing rs (i.e. The polynomials f, g, s are hereby dened by: lower density) than for the 2D electrodes.

1 We nally address the question, how these results com- 2 f[rs] = 1 + β1rs + β2rs (25a) pare to those of Ref. 14 where eff = 1 and r = 1 have to 1 1 1 be chosen.58 Since we did not include the surface dipole g[r ] = r 2 + β r (25b) s 6 s 3 2 s layers, we cannot expect to nd agreement concerning the 7 1 2 eective capacitance length as dened by the inverse total s[r ] = r 2 + β r (25c) s 72 s 9 2 s capacitance. We therefore compare the dependencies of the kinetic term and of the combined exchange and cor- and the coecients and are taken from Ref. 30 βl h4 relation terms on . We take the centre of mass position (where 57 as in Table IV of Ref. 30): rs h4 ≡ α4 − h1β2 of the induced charge density from a recent publication (26) (Table 1 in Ref. 59) in which the values with and without β1 = 1.15813, β2 = 0.34455, h4 = −0.2942 exchange and correlation are explicitly listed. The total The dependence of the total capacitance C(3D) of a sym- values of the centre of mass position agree with those of Ref. 14 at r = 2 and 4, except for a factor of 2 in Ref. 14 metric capacitor on the electron density parameter rs, on s the eective mass, and on the dielectric constant of the which has been suppressed in all subsequent publications: electrodes is presented in Figs. 4 and 5, for parameter The inverse kinetic capacitance, which they obtain, in- ? sets with m /m = 1 and 5, respectively. The contour creases similarly with rs as the one we are nding. The lines of constant capacitance C(3D) in Fig. 6 are qualita- inverse capacitance associated with exchange and corre- tively dierent from those of the 2D capacitance (Fig. 3), lation energies is also negative in their data. The mag- because the Thomas Fermi screening length with its func- nitude of the capacitance also increase with rs, however ? tional dependence on m and rs enters the expression for at a smaller rate. Our calculation based on a homoge- C(3D). Consequently, the transition to a negative capac- nous electron gas apparently overestimates the exchange itance occurs at higher dielectric constants eff in the 3D eect. The overall trend of the contributions to the ca- electrodes than for the ideal 2D capacitor (cf. the scales pacitance reported in Ref. 59 agrees well with our nd- of eff in Fig. 3 and Fig. 6). For eff = 1, however, the ings, which seems surprising if the crude treatment of the transition of a capacitor with 3D electrodes takes place surface inhomogeneity is considered.60 10

IV. CAPACITORS COMPRISING face is known to occur in disordered correlated electronic ELECTRODES WITH CORRELATED systems62. Al'tshuler and Aronov identied a square root INHOMOGENEOUS ELECTRON SYSTEMS suppression of the density of states at the Fermi level with decreasing temperature as a consequence of the inter- In a homogeneous electron gas, which resides in a conference between inelastic and multiple, elastic scattering tinuous medium, the compressibility of the interacting processes. Therefore we expect a corresponding decrease electrons is dominated by the kinetic and by the exchange of the capacitance in case the surface of the electrodes contributions. In contrast, an electronic system that re- consists of such a material. sides on a lattice may behave in a rather dierent man- ner, as epitomized by the Hubbard model in which the correlation term dominates the physics close to the Mott V. CAPACITORS WITH MULTIPLE STABLE transition. Inter alia, it is the strong on-site interaction OR METASTABLE STATES of charge carriers in the lattice models which on the one hand makes these models dicult to treat but on the The energy functional of the capacitor is given by other hand generates highly intriguing properties. Such Eq. (3). The energy E as a function of charge per strongly correlated systems may display a multitude of area, Q/A, is identied from the dependence of E on phase transitions, sometimes induced by small variations the charge carrier density. For the model capacitor of the electronic density. But also, by altering tempera- (Sec. III A), for example, the energy per area is plotted as ture, magnetic eld, pressure, or epitaxial strain, phase a function of charge density in Fig. 7. The curvature at transitions may be induced. Intermediate to strong elec- Q/A = 0 represents the inverse dierential capacitance tronic correlations have been identied in systems such C−1(Q = 0) = dV/dQ which was calculated in Sec. III A. as the manganites, which display a colossal magnetoresis- Due to the negative capacitances of this model system, tance eect, or such as the cuprates with their enigmatic all electrons will accumulate on one of the two equivalent pseudogap and superconducting states. parallel plates, leave holes on the other plate, and thereby Although for small variations of thermodynamic vari- ables the compressibility of the electronic system is often a dull thermodynamic quantity, close to phase transitions the compressibility can vary strongly. Correspond- ingly, the capacitance of an electrode built from such D -0.8 3

a material will reect these variations in the electronic B state specically the density dependence of the elec- a tronic energy. It is pointed out that the compressibility 0 -0.9

is essentially proportional to the density of states of the ΠΕ interacting electronic system. Consequently, the capaci- 4 tance depends on spectral weight transfer and formation 2 e of coherence peaks in the momentum integrated spec- @ -1.0 tral function. Both of these phenomena are typical for strongly correlated electron systems and can, to a cer- 4

tain extent, be controlled by temperature and magnetic 10 eld. It is therefore expected that they can produce ca- -1.1 pacitive eects for electrodes. This prediction has to be A investigated further.

Whereas one does not expect the compressibility to E vary appreciably close to the Mott transition in the single -1.2 band Hubbard model, for the multi band case the situ- -1.0 -0.5 0.0 0.5 1.0 ation is markedly dierent: a large interorbital charge 3 2 transfer with negative compressibility is feasible in a Q A 10 @e aB D multiband model with at least one band being close to a Mott transition, as has been suggested by Liebsch for FIG. 7: Energy per area of a two-dimensional, parallel-plate the insulator-metal transition in Sr-doped LaTiO .61 In capacitor plotted as a function of charge per area for several 3 spacings of the plates. The energy has been calculated for such multi-band (bulk) systems, the dierent orbitals act the model capacitor described in Sec. III A. The electronic like the electrodes of a capacitor with negative compress- systems of the electrodes form dilute and homogenous 2D ibility at one electrode and zero distance between the electron gases. The charge carrier density (carriers per area) electrodes. Similar to these multiband correlated mate- −3 2 13 −2 is 10 /aB = 3.6 × 10 cm , the eective mass of the carri- rials, one may expect for mixed or intermediate valence ers is m?/m = 1, and the eective dielectric constant in the systems a stronger dependence of the compressibilty on electrodes is eﬀ = 1 for all lines. The eective distance be- ? control parameters such as doping and temperature. tween the plates d /aB = d/(r aB ) is 10.0, 8.0, 6.0, 5.0, and A suppression of the density of states at the Fermi sur- 3.0 from top to bottom. 11

that only use dielectric and not ferroelectric dielectrics is considered relevant for electronic devices such as memory

D - devices or Qbits.

3 1.03 Capacitors with negative dierential capacitance gen- B

a erate in the charged state electric elds in the dielectric

0 and an electrical voltage between their electrodes. How-

ΠΕ -1.04 ever, the charged state is typically in thermal equilib-

4 rium so that the electro-chemical potential vanishes and C2 2 no charge ows through the leads by which the two elec-

e ¯

@ trodes are connected. It is noted that while such capac- -1.05 itors are charged when being in one of the equilibrium 4 C1 states, these states are obviously stable states. The ca- 10 pacitors do therefore not, as all other capacitors do, loose

¯ their charge with time by discharging. Treated as com- -1.06 ponents in electronic circuits, capacitors with negative A capacitances, such as the one for which the E(Q) char-

E acteristic is sketched by the continuous line of Fig. 8, will be characterized by the impedance Z = −i/(ωC) if -1.0 -0.5 0.0 0.5 1.0 operated close to the energy maximum. Thus, the fre- 3 @ 2D quency dependence of their impedance is the one of a Q A 10 e aB standard capacitor. Because C is negative, however, the induced phase shift has the opposite sign of the phase FIG. 8: Sketch, illustrating several possible scenarios for shift of standard capacitors, so that the phase shift cor- the charge dependence of the energy of the two-dimensional, responds to the phase shift of a standard inductance. In parallel-plate capacitor (compare Fig. 7). The continuous red ? line corresponds to the red line of Fig. 7 with d /aB = 3.0. The dotted and dash-dotted lines are tentative extrapolations for high charge densities of the order of the carrier density −3 2 13 −2 1.0 10 /aB = 3.6 × 10 cm . The latter lines have not been calculated, but rather present a sketch. The capacitor with four minima has dierent capacities in the metastable and stable states, as reected by the dierent curvatures of the D characteristics in the minima. Here, C1 and C2 refer to the 2 0.5 inverse of the curvature at the minima which are marked by B a the vertical arrows. The capacitance of such a capacitor can therefore be switched by current pulses. e @

3 0.0

63 10

charge the capacitor (see the lower two curves in Fig. 7). The capacitor is characterized by multiple stable states, A and may be switched between these by small bias pulses. -0.5

? Q For a capacitor with d /aB = d/(r aB) = 5.0 (the second curve from the buttom, in orange) and an area of A = 100 nm2, the energy dierence to the zero charge state is approximately 0.5 eV, and the states will be stable -1.0 aganst thermal activation. In this model, also capacitors -10 -5 0 5 10 with three stable or metastable states can be realized (the 2 third curve from the buttom, in green), however with an V 10 @e4ΠΕ0 aBD energy dierence of only about 50 meV. The model is not suited to precisely describe the FIG. 9: Charge per area of a two-dimensional, parallel-plate capacitor plotted as a function of voltage between the plates. multiple-state properties of real capacitors, because the The charge per area Q(V ) has been calculated for the model nonlinear response of an inhomogeneous electron system capacitor described in Sec. III A. The electronic systems of will typically stabilize a charged state at charge densi- the electrodes form dilute and homogenous 2D electron gases. ties that are smaller than provided by the model. This −3 2 The charge carrier density (carriers per area) is 10 /aB = scenario is illustrated in Fig. 8 (dotted curve). Phase 3.6 × 1013cm−2, the eective mass of the carriers is m?/m = transitions in electrodes with strongly correlated, inho- 1, and the eective dielectric constant in the electrodes is ? mogeneous electron systems might even produce addi- eﬀ = 2.5. The eective distance between the plates d /aB = tional minima in the charge dependence of the energy d/(r aB ) is 10.0, where r is the dielectric constant of the functional (see Fig. 8, dash-dotted curve). The inves- dielectric. The dotted line shows the Q(V )-characteristic of the conventional, classical capacitor with . tigation of such multistable charge states in capacitors Q = CgeomV 12 analogy, we suggest the possible realization of conductors with negative inductances, the phase shift of which 80 corresponds to the phase shift of standard capacitances. It is also amusing to consider the threshold behavior at a possible electrical breakdown of the dielectric. When approaching breakdown, localized charge carriers in the 40 dielectric start to get released and move to the electrode of opposite charge. This charge redistribution shifts the electro-chemical potential so that an equal number of 0 C charge carriers ows through the leads. The original charge state is hereby stabilized and the release of charges L 0 V does therefore not seem to change the electronic state H in the electrodes. At the same time, the change of the C charge distribution in the dielectric reduces the release of further carriers in the dielectric. The Coulomb-eld of -40 the altered charge distribution, however, aects the energy of the electron systems of the plates and thereby in- directly alters the charge distribution on the electrodes. Should at breakdown a conducting channel be induced -80 in the dielectric, the same argument applies, if the chan- 1.0 D nel is considered as an additional lead that connects the 2 B

electrodes. The electronic state of the electrodes is not a

0.5 dependent on this channel and the electrodes will stay e charged. @ 3 0.0 10 -0.5

VI. VOLTAGE DEPENDENCE OF THE A CAPACITANCE

Q -1.0 -4 -2 0 2 4 In many applications it is the voltage and not the 3 charge, with which the state of a capacitor is controlled. V 10 @e4ΠΕ0 aBD To calculate the C(V ) characteristic we consider the Leg- FIG. 11: Capacitance and charge of the two-dimensional, endre transform E of the energy E(Q) with respect to Q, and identify the total capacitance as parallel-plate capacitor of the model capacitor described in Sec. III A plotted as a function of the voltage between the capacitor plates. The charge carrier density (carriers per area) Q(V ) is −3 2 13 −2, the eective mass of the carriers C(V ) = (27) 10 /aB = 3.6×10 cm V is m?/m = 1, and the eective dielectric constant in the electrodes is eﬀ = 1. The eective distance between the plates ? d /aB = d/(r aB ) is 5.6. Top panel: voltage dependence of the capacitance C(V ) = Q(V )/V . Lower panel: voltage dependence of the charge per area Q(V ). In both panels, 1.3 blue and red lines refer to the ranges of positive and negative capacitances, respectively. The dotted line shows the 0

C characteristics of the conventional, classical capacitor with

1.2 L Q = CgeomV . V H

C 1.1 As example we take again the model system of Sec. III A 1.0 with parallel, two-dimensional plates comprising the ca- -10 -5 0 5 10 pacitor. 2 V 10 @e4ΠΕ0 aBD The calculations reveal that already in the regime of moderate eective distance of the capacitor plates, devia- FIG. 10: Capacitance C(V ) = Q(V )/V of a two-dimensional, tions from the conventional characteristics of a textbook parallel-plate capacitor plotted as a function of voltage. The capacitor are manifest: in Fig. 9 we display Q(V ) which capacitance has been calculated for the model capacitor C(V ) is expected to be linear with Q = CgeomV in the text- described in Sec. III A. The parameters are the same as in ? book case or for suciently large d = d/r (green dot- Fig. 9. The capacitance is normalized to , the capacitance C0 ted line). For d? = 10a and = 2.5 the slope at zero of a classical capacitor of the same geometry. B eﬀ voltage is steeper and Q(V ) deviates from a linear behav- 13 ior, most prominently for the strongly charged capacitor. netic phase transitions. Applied electric elds, on the The corresponding voltage dependent capacitance C(V ) one hand, change the carrier density of the magnetic lay- is shown in Fig. 10. With increasing voltage the capac- ers and therefore their magnetic properties. Application itance increases because the capacitance is a function of of magnetic elds, on the other hand, alters the correla- the charge in the plates. The higher the charge in the tion energies and therefore the capacitance and electric plates the higher the capacitance which results in a no- elds of the system. Thus, such articial materials are ticeable upturn of C(V ). candidates for strongly coupled, room temperature mul- This feedback of increasing capacitance with increasing tiferroics. charge may be so strong for nanoscopic distances that the conventional Q(V ) and C(V ) characteristic is completely ? modied (see Fig. 11). For example, for d = 5.6aB (and A. Electrodes with 2D Electron Systems in Transition Metal Oxide Heterostructures eﬀ = 1) we operate in a regime where the capacitor self-charges once a suciently high charge has been ac- cumulated. In that case charge is transfered at vanishing The possibility to control the capacitance of capaci- energy expense (cf. Fig. 7, where V (Q) = dE/dQ = 0 tors by adding a thin, conducting sheet to an electrode for a distinct, nite Q in the green, middle curve). The opens the question how large a capacitance of a capacitor voltage is not zero in the fully charged state (Fig. 11) is expected that uses as an electrode a two-dimensional because this is not a thermodynamic state with a well- electron gas generated at an oxide interface72,73. Indeed, dened minimum in the energy E(Q) as compared to capacitors based on two-dimensional electron gases in ox- the tentative capacitors of Fig. 8 which display minima ide heterostructures are expected to have unique and pos- in E(Q). sibly technically useful properties. The capacitance C(V ) of the model capacitor (top The current model systems for such interfaces are panel of Fig. 11) may have multiple values, depending systems like the n-doped LAO/STO or LVO/STO het- on the charge state. It is conceivable to switch between erostructures. The following discussion will refer to such a positive value of C(V ) (blue curve in Fig. 11) and a interfaces, although future multilayers that do not imply negative value (red curve in Fig. 11), using charge pulses sophisticated epitaxial growth may be preferable for ap- to switch the capacitor in and out of the self-charged plications. For simplicity, we neglect strong correlation state. eects possibly present at such interfaces. As shown by Eqs. (8,9,12,13), the capacitance (Eq. (6)) of a capacitor that uses such an electron gas is a function

VII. SPECIFIC EXAMPLES of rs, of the eective mass of the interface charge carriers ? m , and of the eective dielectric constant eff in the Magnetocapacitive eects have been investigated for interface sheet. several decades. A capacitive measurement, in which Up to now, such oxide interfaces have been prepared 12 Landau levels in a 2D electron gas of a MOSFET were with sheet carrier densities of the order of ni ∼ 5×10 − 14 2 observed, was reported in Refs. 64,65,66,68,69. More re- 5 × 10 /cm , corresponding to bare rs-values ranging cently, the density of states in the fractional quantum from 50 to 5. As Fig. 3 shows for the example of a Hall regime has been investigated by magnetocapacitive parallel-plate capacitor, that comprises a 1.5 nm thick 45,67 measurements. A charge controlled transition of an HfO2 layer as dielectric (with r = 23) and an electron electrode into a magnetic state would also yield large gas with rs = 7.0 in each of the electrodes, the capaci- ∗ magnetocapacitive eects. Such a transition is conceiv- tances can be negative for realistic sets of m and eff val- able in correlated electronic systems, for example, in the ues. Moreover, the capacitance is enhanced for a broad ? manganites. A positive magnetocapacitance has indeed range of eff and m . 70 been observed in La0.7Sr0.3MnO3−δ-titanate junctions which contrasts to the negative magnetocapacitance of 71 B. Other Systems Pd-AlOx-Al thin-lm structures. Whereas the latter is the predicted behavior for a paramagnetic metal in which the Zeeman-split narrow d-bands cause the magnetoca- 1/Ckin may be strongly reduced at van Hove singular- pacitance, the positive magnetoresistance has been re- ities. Conversely, the capacitance of condensators built lated to strong electronic correlations. with electrode materials such as graphene or other ma- Articial materials that, for example, consist of lay- terials with a small density of states, will be reduced be- ered structures such as superlattices, of which the ca- cause then 1/Ckin dominates 1/Cgeom. In that case, the pacitance is inuenced by non-geometrical capacitances small density of states determines the charging of the ca- are predicted to be characterized by unusual properties. pacitance, and the charge is smaller than expected from

Of particular relevance are superlattices with a geometry Cgeom. chosen that the capacitances between the layers are neg- Recently, the quantum capacitance of coaxially gated ative (see Sec. III) so that the layers self-charge, and in carbon nanotubes74 has been calculated using a Green's which a part of the layers consist of materials with mag- function density functional tight-binding approach which 14 is based on a local density approximation (LDA) of the material of the electrodes. It suces to introduce thin, exchange correlation functional. This quantum capaci- conducting sheets of a compound (see Fig. 12b) with the tance includes the terms Ckin and Cx which are evaluated appropriate electron energies to drastically alter the ca- for the quasi one-dimensional nanotube. The behavior pacitance. of this capacitance (see Fig. 3 in Ref. 74) is consistent Therefore, besides optimizing the thickness and the di- with our results for the capacitance of 2D or 3D capaci- electric properties of the dielectric layer, choosing the tors. Specically, in the regime of small carrier densities appropriate materials of the electrode surfaces provides overscreening of the gate eld was observed which corre- an independent access to optimizing the capacitance of sponds to a negative capacitance of the carbon nanotube. capacitors. Materials of interest include materials with The capacitance of carbon nanotube has been measured small electron densities, such as interfaces in oxide het- by a group from Cornell University75 and, in fact, the erostructures, low-electron density metals, materials with measurements suggest the existence of a negative contri- strong electronic correlations such as transition metal ox- bution to the total capacitance. ides, or graphene-based systems. A negative electronic compressibility is also expected Because the capacitance of such capacitors is controlled close to structural transitions. Commonly, this signies by the energy of the electron system of the plates rather that the electronic system is in a phase separated state than by the eld energy, the standard parallel-plate con- or undergoes charge disproportionation. In many cases, guration is not needed to obtain large capacitances or these situations are associated with a metal-insulator eld strengths (for example, in Fig. 12a all blue surfaces transition. If structural transitions can achieve a func- contribute to the capacitance 1/C − 1/Cgeom in Eq. (6)). tional quality for a controlled electronic transition in the Rather, the surface areas of the electrodes are important. electrodes of a capacitor has to be investigated. We note that the desired modication of the capacitance does not require that the electrodes consist entirely of

VIII. SUMMARY, CONCLUSIONS AND PERSPECTIVES

Considering the complete energy of capacitors, which includes the Coulomb energy and the energy of the elec- a) tron systems of the capacitor plates, we have evaluated the capacitance of capacitors. We nd that in general the capacitance is given by Eq. (6) and not by Eq. (1). As shown, the capacitance of a capacitor is described by the series connection of the Coulomb capacitance, the kinetic capacitance of the electrons, the exchange capacitance, the correlation capacitance, and the capacitance which results from the interaction of electrons with non charge-carrier degrees of freedom. Because for a free elec- b) tron gas the exchange capacitance and the correlation capacitance are negative and the electron-phonon inter- plate 1 action may also result in a negative value, the total ca- booster layer pacitance may be smaller or larger than the Coulomb capacitance, and may even reach negative values. dielectric Our analysis shows that the capacitance is dominated plate 2 by the eld capacitance for the case that the eld energy exceeds the energies of the electron system, but that the electron energies are signicant if the thickness of the dielectric layer divided by its dielectric constant is com- parable to the Bohr radius. This applies independent of FIG. 12: Sketch, illustrating options to integrate materials the lateral size of the capacitor. The gate capacitances into capacitors to achieve capacitances that deviate strongly of modern MOSFETs are reaching this regime. from the geometric capacitance. Panel (a) sketches a situation where electrodes may cause a strong variaton of the Eq. (6) is of general importance to calculate the capac- capacitance due to their contributions in −1, −1, −1 or Ckin Cx Cc itances of capacitors. Due to their broad applicability, we −1 so that a large total capacitance is obtained with- Cel−ncc have specically provided the detailed calculations of the out using a parallel-plate conguration.. Panel (b) sketches capacitance of parallel-plate capacitors with electrodes a thin, conducting sheet (booster layer) of a compound with that contain two-dimensional and three-dimensional, ho- appropriate electron energies integrated into a standard ca- mogenous electron systems. pacitor. This sheet can considerably enhance or reduce the It is concluded that the capacitance of capacitors can capacitance. The tickness of the sheet may be as small as the be signicantly enhanced or decreased by altering the electric screening length in the compound. 15

case the compressibility of at least one electron system is negative. In a capacitor that self charges, for example, into a positively and negatively charged plate, the electrons and the holes in the two plates interact across the dielectric by their Coulomb eld. It seems possible that electron-hole pairs result that form a coherent state, and, for example modify pairing in a superconducting electrode. Such eects may be particularly useful in case the plate capacitor is designed as a transmission line. We propose that such a superconducting state may even ex- ist if none of the electrodes is superconducting by itself. FIG. 13: Sketch to illustrate the possibility of electric eld Even for electron systems with positive compressibilities, enhancement due to the integration of a material with high the Coulomb energies caused by the presence of a second capacitance into a plate of a capacitor. The material is in- plate will lower or enhance the of a superconducting troduced in a partial coating (blue stripes) of the capacitor Tc electrode. Therefore, modifying Coulomb energies by us- plates (dark gray rectangles). ing capacitor-type congurations, the electrodes of which may even consist of the same materials, oers new pos- strongly correlated compounds. It is sucient to coat sibilities to realize superconductors. the active surface of an electrode with a thin layer of a Also, because the capacitance can be controlled by the correlated material, because the net charges are repelled electronic system of the plates, which may be strongly from the interiors of the electrodes. correlated electron systems that are sensitive to magnetic elds, materials may be designed from capacitor- We emphasize that eff is a property of the electrode materials and, specically, it relates to the properties of derived multilayer congurations that yield large multi- ferroic coupling at room temperature. the materials at their surfaces. eff therefore can be tuned by coating the electrodes with thin layers of desired ma- Capacitors for which the non-geometrical capacitances terials and then is given by the dielectric constant of the are important, and in which one plate has a non- coating material. It accounts for the polarizability of the homogeneous composition, are characterized by inhomo- ionic cores and also includes the short-range screening geneous and unusual electric eld distributions. The processes from excitations in the adjacent material lay- charge distribution of such an electrode is modied and ers, but not the contributions from the electron gas itself. diers from the charge distribution generated in the corresponding standard conguration. Consequently, in the The eective dielectric constant eff is, beyond linear response, also dependent on the charge density. Such a presence of an electric eld, the distribution of the eld dependence induces feedback eects which alter the ca- lines is accordingly altered. As illustrated in Fig. 13, electric eld lines can, for example, be focused onto de- pacitance. While one may consider to estimate eff from an LDA evaluation for the specic heterostructure, there sired areas to extract charges. Or, where desirable, areas can be shielded from electric elds, which may be impor- is no simple scheme to calculate eff . Lacking such a simple scheme, we calculated the capacitance of a simple tant to prevent electric breakthrough phenomena. Thus, inhomogeneous electrodes allow to generate desirable dis- model capacitor for several values of eff . Using optimized capacitor congurations, it is in prin- tributions of charge densities and electric elds without ciple possible to build capacitors that are characterized altering the overall geometrical conguration of a device. by a negative capacitance for zero applied voltage. Such capacitors charge themselves to generate nite electric elds and, although they do not contain any ferroelectric Acknowledgments component, show parallels to ferroelectric devices. It is obvious, that for device applications novel possibilities We would like to thank Yu. S. Barash for his careful are opened by adding truly ferroelectric layers to struc- reading of the manuscript and for bringing several impor- tures with unconventional capacities. Intriguing charac- tant references to our attention. It has been very help- teristics arise if the capacitances are negative. ful that S. Graser kindly double-checked relations and In the case of negative compressibility, adding or re- plots. Illuminating discussions with B. Batlogg, M. Breit- moving an electron from an electrode involves less energy schaft, P. Chaudhari, V. Eyert, P. J. Hirschfeld, R. Jany, than the standard work function or electron anity. It A. Kampf, C. Richter, D. Scalapino, D. G. Schlom, is intriguing to regard the behavior of the electrons in a J. Schubert, S. Thiel, and P. Wöle are gratefully ac- single electrode or on the two electrodes of a capacitor, in knowledged.

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λi attains a complex value for even larger rs. However, λi does not characterize the width of the layer of induced charge at the surface of the electrodes. A more appro-

priate quantity λs/rsaB was dened by Schakel (Eq. (20) in Ref. 49). This length coincides with the Thomas-Fermi screening length (TF) for small and decreases monoton- λi rs −1/2 ically in the entire range 0 < rs < 15 roughly with rs . We approximate the width by (TF) in order to discuss the λi evolution of the capacitance with increasing rs at least in a qualitative way. 61 A. Liebsch, Phys. Rev. B 77, 115115 (2008). 62 B. L. Al'tshuler and A. G. Aronov, Zh. Eksp. Teor. Fiz. 77, 2028 (1979) [Sov. Phys. JETP 50, 968 (1979)]. 63 As discussed in the introductory paragraphs of Section III, the ground state for a symmetric capacitor is characterized by a spontaneous interlayer phase coherence (SILC) in the absence of self-charging. However, the coherence may be lost, if the symmetry is broken, for example, by a nite bias or asymmetric congurations. Here, we consider the unpolarized homogeneous electron gas on a parallel plate capacitor in the absence of SILC as a model system in the expectation that it shares some general features with capacitors comprising strongly correlated electron systems in the presence of a bias. 64 M. Kaplit and J. N. Zemel, Phys. Rev. Lett. 21, 212 (1968).