- Email: [email protected]
Classical capacitance of few-electron dielectric spheres
ARTICLE IN PRESS
Microelectronics Journal 37 (2006) 1293–1296 www.elsevier.com/locate/mejo
Classical capacitance of few-electron dielectric spheres Jinwen Zhua, Tim LaFave Jr.b,, Raphael Tsub a
Department of Engineering Technology, Missouri Western State University, St. Joseph, MO 64507, USA b Department of Electrical Engineering, University of North Carolina, Charlotte, NC 28223, USA Received 19 June 2006; accepted 10 July 2006 Available online 28 August 2006
Abstract The capacitance of few-electron dielectric spheres differs from the many-electron Gauss model of infinitesimally-divisible charge as a result of the electrostatic interaction of discrete electrons. Minimization of the total classical interaction energy for nanometer-size devices without quantum effects for up to 12 electrons is obtained. Unlike the Gauss model, capacitance is non-constant. The variation of capacitance with N, thus, with voltage, in a non-magnetic, classical domain, opens a new field of discrete charge nanometer-size devices and applies to the general chemistry of nanoparticles. r 2006 Elsevier Ltd. All rights reserved. Keywords: Dielectric materials; Capacitance; Discrete electrons
1. Introduction In the nanometer regime, the dielectric constant [1], capacitance [2], and doping [3] have been shown to be quite different from conventional models. Confined electrons acquire kinetic energy which may dominate over the electrostatic potential energy. The present study involves generalization from two electrons in the original approach [2] to twelve electrons in the classical case when the device size avoids the need for a quantum mechanical treatment. Electrons are positioned inside a sphere of uniform dielectric constant with a minimum total interaction energy configuration. Our classical treatment assumes that the N-electron system is dielectrically confined to a spherical volume characterized by a uniform dielectric constant embedded in a uniform dielectric medium. The Gauss model assumes that all electron charge may be lumped together as a single charge at the center of the sphere or as a uniformly distributed, continuous surface charge. Here the distribution of induced charge is assumed to be continuous due to the denseness of atoms in a solid. This is consistent with the assumption of a uniform and continuous dielectric function. When the sphere’s dielectric Corresponding author. Tel.: +1 704 687 3447; fax: +1 704 687 4762.
E-mail address: [email protected] (T. LaFave Jr.). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.07.013
constant, e, is greater than the dielectric constant of the surrounding medium, e0 , each electron induces a negative surface charge distribution at the polarized dielectric boundary. Thus, for a one-electron system, a minimized energy configuration is obtained when the electron is located at the center of the sphere due to interaction with its self-induced surface charge. For a two-electron system, mutual coulomb repulsion pushes each electron toward the surface while induced surface charges push them away. The electrons are allowed to slowly move to an equilibrium configuration of equal radii on opposite sides of the origin. For three electrons, the minimum energy configuration is a two-dimensional equilateral triangle centered about the origin, and for four electrons, a three-dimensional tetrahedron is obtained. Beyond four, we have observed minima electron configurations coincident with geometric solutions of the well-established Thomson problem [4,5] of N electrons at a fixed radius. An expression for the capacitance of dielectric spheres is derived from the assembly of a collection of discrete charges as opposed to the Gauss model approximation due to an infinitesimally assembled charge. In the case of N ¼ 1, the capacitance is twice the value resulting from the Gauss model. As N grows very large the capacitance approaches an asymptotic constant value as anticipated. The varying capacitance of few N systems has far-reaching
ARTICLE IN PRESS J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
1294
consequences for discrete charge devices (DCDs) in the nanometer regime. 2. Theory and computation A spherical geometry is chosen for the purpose of demonstration and mathematical simplicity. A classical evaluation of the total electrostatic energy of N electrons inside a sphere requires solutions [6,7] of Poisson’s equation for charges in a dielectric sphere ( ¼ 12) of radius a ¼ 10 nm, greater than the mean-free-path of electrons in the dielectric, embedded in a more insulating dielectric medium (0 ¼ 4). The model dielectric interface is assumed to be abrupt. The self-polarization interaction energy of an electron at ri is given by ei fðpi Þ 2 1 r 2l e eð 0 Þ X ðl þ 1Þ i ¼ ; 2 4p0 a l¼0 ½0 þ lð þ 0 Þ a
E s ðri Þ ¼
ð1Þ
in which pi is the surface charge induced by the ith electron, ei , and a factor of 1/2 appears as a consequence of interaction with an infinitesimally polarized surface charge. The Gauss model represents only one charge, or electron, involving only its self-energy. Thus, the Gauss energy [2] of a dielectric sphere containing a single charge Q is given by Q2 1 1 E G ðÞ ¼ , (2) 8p0 a 0 with Q ¼ Ne, E G ðÞ ¼ N 2 E S ð1eÞ. In systems consisting of NX2 electrons, the repulsive Coulomb interaction serves to push electrons apart while each electron interacts with the electrostatic potential due both to its self-induced surface charge and surface charges induced by the other N 1 electrons. Individual Coulomb interactions are given by E C ðri ; rj Þ ¼
e2 1 4p0 jri rj j
(3)
and polarization interactions by E P ðri ; rj Þ ¼
l l 1 e2 X ð 0 Þðl þ 1Þ ri rj Pl ½cos ðgij Þ; 4p0 l¼0 ½0 þ lð þ 0 Þ a2lþ1
(4)
with Pl ½cos ðgij Þ the Legendre polynomials for each angle, * * gij , subtended by r i and r j . The total electrostatic potential energy of an N-electron system is then given by EðNÞ ¼
N X i¼1
E s ðri Þ þ
N N 1 N N 1 X X X 1X E C ðri ; rj Þ þ E P ðri ; rj Þ. 2 i¼1 jai i¼1 jai
(5) The total equilibrium energy, Eq. (5), is minimized when all N electrons are equidistant from the center of the sphere using a variational radius, b, given as a fraction of the
dielectric sphere radius, a, and distributed in the corresponding Thomson geometry [5]. As this work is a continuation of previous work [2], a few unfortunate errors in the previous paper merit correction. In Eq. (3.6) of Ref. [2], there is a factor of ðl þ 1Þ missing, the first term in the denominator of the summation of Eq. (4.2b) should be e2 (e0 in the current case), and a factor of 2 should be included in the second term of Eq. (5.3) as there are two polarization interactions of equal magnitude. The latter error affects the last term of Eq. (5.4) in the summation, where the numerator should be 2ð1Þl . These corrections are incorporated here in the total equilibrium energy of N-electron systems in Table 1. The choice of a ¼ 10 nm is dictated by the condition that the size is small but larger than the quantum coherence length of electrons such that classical evaluation is valid. In general, the total equilibrium energy corresponds to an electron distribution with the highest possible spatial symmetry. For the two-electron system, each charge is repelled to opposite poles of equal radius, boa, inside the sphere (Fig. 1). For the three-electron system, each electron is located at the vertices of an equilateral triangle such that all are equidistant from the origin. For four electrons, the equilibrium energy corresponds to electrons at the vertices of a tetrahedron—the first of only five possible Platonic geometries [8] having regular polygon faces. Of the next three Platonic geometries, only N ¼ 6 and 12 yield equilibrium energies with all electrons located at Platonic vertices, while N ¼ 8 yields an equilibrium energy only when one face of the cube is rotated by p/4. Therefore, the complexity of interactions leads to symmetric electron configurations requiring more than casual intuition. Table 1 lists equilibrium energies for 1pNp12 electron systems. The corresponding values of energy, E G ðÞ, obtained with the Gauss model are provided for comparison. The radius of each minimized electron configuration of electrons, b, normalized to a ¼ 10 nm, is also provided. As the number of electrons increases, the electron radius b increases. Fig. 2 compares the equilibrium energy resulting from an evaluation of the total Table 1 Evaluated electrostatic parameters for systems consisting of 1pNp12, with a ¼ 10 nm, ¼ 12 and ¼ 4 N
E G ðÞ (eV)
EðNÞ (eV)
b=a
CðNÞ (e2 )
1 2 3 4 5 6 7 8 9 10 11 12
0.012 0.048 0.108 0.192 0.300 0.432 0.588 0.768 0.972 1.200 1.452 1.728
0.0120 0.0816 0.2079 0.3899 0.6319 0.9287 1.2878 1.7025 2.1760 2.7085 3.3012 3.9474
0.000 0.632 0.727 0.774 0.803 0.824 0.839 0.851 0.860 0.868 0.875 0.881
83.333 49.020 43.290 41.036 39.563 38.764 37.049 35.592 37.224 36.921 36.653 36.480
ARTICLE IN PRESS J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
1295
4
Potential Energy (eV)
3.5 3 2.5 Interactions Model
2 1.5 1
Gauss Model
0.5 0
1
2
3
4
5
6 7 8 N Electrons
9
10
11
12
Fig. 2. The equilibrium interaction energy for N-electron systems exceeds twice the energy provided by the Gauss model. The system parameters are ¼ 12; 0 ¼ 4, and a ¼ 10 nm.
85 80 75 C (N) (e2)
70 65 60 55 50 Dielectric Gauss Model
45
(41.668)
40 35 1
2
3
4
5
6 7 8 N Electrons
9
10 11 12
Fig. 3. The dielectric interactions capacitance, C(N), is not a constant value for all N, contrary to the capacitance provided by the Gauss model. The system parameters are ¼ 12; 0 ¼ 4, and a ¼ 10 nm.
given by Fig. 1. The geometries of minimized electrostatic energy configurations of 1pNp12 electrons inside a dielectric sphere of radius a ¼ 10 nm represent the greatest possible electron symmetries. The size of the internal Thomson sphere, b=a, is not drawn to scale.
electrostatic interaction energy and energies obtained by the dielectric Gauss model. For very small Coulomb repulsions, as may be simply shown by Eq. (5), EðN; ! 1Þ ! 2E G ðÞ for large N. The definition of capacitance [2,9,10], C dQ=df, appropriate for applications of parametric amplification [11], may be explicitly defined in terms of interaction energy. Most importantly, addition of an electron to an N-electron system changes the symmetry to that of N+1 electrons. The capacitance is therefore defined solely within each configuration, allowing treatment of each N-electron system as a unique phase. In the Gauss model capacitance the work necessary to assemble a collection of infinitesimally divisible charges is
Z E G ðÞ ¼
Q
f dq ¼ 0
Q2 2C G
giving a factor of 1/2. However, since for discrete electrons, dq ! Dq ¼ e, the capacitance of a few-electron system is derived from the interaction energy, Eq. (5), CðNÞ ¼
ðNeÞ2 EðNÞ
with Q ¼ Ne. For few-electron systems, the capacitance varies largely (Fig. 3), passes near the dielectric capacitance of the Gauss model [C G ðÞ ¼ 41:668e2 for a ¼ 10 nm] and approaches the familiar metallic Gauss capacitance [C G ð ! 1Þ ¼ 4p0 0 a ¼ 27:778e2 for a ¼ 10 nm] as N tends to infinity. This trend is easily seen as the electron radius approaches the sphere’s surface when N grows very large to mimic a metallic system. The uniformity and infinitesimally divisible characteristics of the Gauss model are not present in few-electron dielectric spheres. Indeed, an equipotential of a metallic sphere coincides with the sphere’s surface, whereas for electrons in a dielectric
ARTICLE IN PRESS 1296
J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
sphere, equipotentials describe complicated three-dimensional surfaces which make the application of the divergence theorem prohibitive. For this reason, there is a lack of proper treatment of the capacitance of dielectric spheres in textbooks which commonly treat only the capacitance of metallic spheres and often incorrectly apply it to dielectric spheres. 3. Discussion and conclusions The approach presented here differs from the problem originally approached by Babic´ et al. [2] for up to two electrons, and the quantum mechanical treatment of capacitance by others [9,10] involving many-electron quantum dots. The primary difference is the exact nature of the present evaluation which is valid when quantum effects are relatively insignificant. However, the approach is not too different from the classical Thomson problem for computing equilibrium configurations of N electrons confined to some fictitious Thomson radius [4,5]. Therefore, a monophasic capacitance is defined solely within a phase characterized by the particular symmetry of each N-electron system. Present MOSFET technology is fast approaching the days of storing few electrons in a gate dielectric. In such systems, the capacitance differs considerably from capacitances obtained by conventional models as demonstrated here. Moreover, since symmetry plays a significant role, any changes therein lead to substantial changes in the stored electrostatic energy and capacitance. A nonconstant capacitance is not unexpected. Conventionally treated as a function of device geometry, the capacitance
here is shown to be dependent on the unique spatial symmetry of each N-electron phase of the system. Additionally, a variable capacitance may prove useful in the design of few-electron parametric devices [11]. For magnetic devices, this purely classical treatment may require additional inclusion of electron spin interactions [12]. These phenomena derive from the discreteness, rather than the wave nature of electrons. The wave nature is only significant when device size is smaller than the electron mean-free-path. These observations may prove useful before devices truly governed by quantum phenomena become reality. The work presented here may be crucial to the development of a new class of DCDs designed and modeled by the classical electrostatic phenomena of discrete electrons presented in this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
R. Tsu, D. Babic´, L. Ioriatti Jr, J. Appl. Phys. 82 (1997) 1327. D. Babic´, R. Tsu, R.F. Green, Phys. Rev. B 45 (1992) 14150. R. Tsu, D. Babic´, Appl. Phys. Lett. 64 (1994) 1806. J.J. Thomson, Philos. Mag. 7 (1904) 237. E.L. Altschuler, T.J. Williams, E.R. Ratner, F. Dowla, F. W, Phys. Rev. Lett. 72 (1994) 2671–2674. L.E. Brus, J. Chem. Phys. 79 (1983) 5566; L.E. Brus, J. Chem. Phys. 80 (1984) 4403. J.F. Bo¨ttcher, Theory of Electric Polarization. 2nd ed., vol 1, Amsterdam, Elsevier, 1973. R.D. Archer-Hind, The Timaeus of Plato, Arno Press, 1973. G.J. Iafrate, K. Hess, J.B. Krieger, Phys. Rev. B 52 (1995) 10737. M. Macucci, K. Hess, G.J. Iafrate, Phys. Rev. B 28 (1993) 17354. A. Yariv, Quantum Electronics, Wiley, New York, 1989. E. Wigner, Phys. Rev. 46 (1934) 1002.
Microelectronics Journal 37 (2006) 1293–1296 www.elsevier.com/locate/mejo
Classical capacitance of few-electron dielectric spheres Jinwen Zhua, Tim LaFave Jr.b,, Raphael Tsub a
Department of Engineering Technology, Missouri Western State University, St. Joseph, MO 64507, USA b Department of Electrical Engineering, University of North Carolina, Charlotte, NC 28223, USA Received 19 June 2006; accepted 10 July 2006 Available online 28 August 2006
Abstract The capacitance of few-electron dielectric spheres differs from the many-electron Gauss model of infinitesimally-divisible charge as a result of the electrostatic interaction of discrete electrons. Minimization of the total classical interaction energy for nanometer-size devices without quantum effects for up to 12 electrons is obtained. Unlike the Gauss model, capacitance is non-constant. The variation of capacitance with N, thus, with voltage, in a non-magnetic, classical domain, opens a new field of discrete charge nanometer-size devices and applies to the general chemistry of nanoparticles. r 2006 Elsevier Ltd. All rights reserved. Keywords: Dielectric materials; Capacitance; Discrete electrons
1. Introduction In the nanometer regime, the dielectric constant [1], capacitance [2], and doping [3] have been shown to be quite different from conventional models. Confined electrons acquire kinetic energy which may dominate over the electrostatic potential energy. The present study involves generalization from two electrons in the original approach [2] to twelve electrons in the classical case when the device size avoids the need for a quantum mechanical treatment. Electrons are positioned inside a sphere of uniform dielectric constant with a minimum total interaction energy configuration. Our classical treatment assumes that the N-electron system is dielectrically confined to a spherical volume characterized by a uniform dielectric constant embedded in a uniform dielectric medium. The Gauss model assumes that all electron charge may be lumped together as a single charge at the center of the sphere or as a uniformly distributed, continuous surface charge. Here the distribution of induced charge is assumed to be continuous due to the denseness of atoms in a solid. This is consistent with the assumption of a uniform and continuous dielectric function. When the sphere’s dielectric Corresponding author. Tel.: +1 704 687 3447; fax: +1 704 687 4762.
E-mail address: [email protected] (T. LaFave Jr.). 0026-2692/$ - see front matter r 2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.mejo.2006.07.013
constant, e, is greater than the dielectric constant of the surrounding medium, e0 , each electron induces a negative surface charge distribution at the polarized dielectric boundary. Thus, for a one-electron system, a minimized energy configuration is obtained when the electron is located at the center of the sphere due to interaction with its self-induced surface charge. For a two-electron system, mutual coulomb repulsion pushes each electron toward the surface while induced surface charges push them away. The electrons are allowed to slowly move to an equilibrium configuration of equal radii on opposite sides of the origin. For three electrons, the minimum energy configuration is a two-dimensional equilateral triangle centered about the origin, and for four electrons, a three-dimensional tetrahedron is obtained. Beyond four, we have observed minima electron configurations coincident with geometric solutions of the well-established Thomson problem [4,5] of N electrons at a fixed radius. An expression for the capacitance of dielectric spheres is derived from the assembly of a collection of discrete charges as opposed to the Gauss model approximation due to an infinitesimally assembled charge. In the case of N ¼ 1, the capacitance is twice the value resulting from the Gauss model. As N grows very large the capacitance approaches an asymptotic constant value as anticipated. The varying capacitance of few N systems has far-reaching
ARTICLE IN PRESS J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
1294
consequences for discrete charge devices (DCDs) in the nanometer regime. 2. Theory and computation A spherical geometry is chosen for the purpose of demonstration and mathematical simplicity. A classical evaluation of the total electrostatic energy of N electrons inside a sphere requires solutions [6,7] of Poisson’s equation for charges in a dielectric sphere ( ¼ 12) of radius a ¼ 10 nm, greater than the mean-free-path of electrons in the dielectric, embedded in a more insulating dielectric medium (0 ¼ 4). The model dielectric interface is assumed to be abrupt. The self-polarization interaction energy of an electron at ri is given by ei fðpi Þ 2 1 r 2l e eð 0 Þ X ðl þ 1Þ i ¼ ; 2 4p0 a l¼0 ½0 þ lð þ 0 Þ a
E s ðri Þ ¼
ð1Þ
in which pi is the surface charge induced by the ith electron, ei , and a factor of 1/2 appears as a consequence of interaction with an infinitesimally polarized surface charge. The Gauss model represents only one charge, or electron, involving only its self-energy. Thus, the Gauss energy [2] of a dielectric sphere containing a single charge Q is given by Q2 1 1 E G ðÞ ¼ , (2) 8p0 a 0 with Q ¼ Ne, E G ðÞ ¼ N 2 E S ð1eÞ. In systems consisting of NX2 electrons, the repulsive Coulomb interaction serves to push electrons apart while each electron interacts with the electrostatic potential due both to its self-induced surface charge and surface charges induced by the other N 1 electrons. Individual Coulomb interactions are given by E C ðri ; rj Þ ¼
e2 1 4p0 jri rj j
(3)
and polarization interactions by E P ðri ; rj Þ ¼
l l 1 e2 X ð 0 Þðl þ 1Þ ri rj Pl ½cos ðgij Þ; 4p0 l¼0 ½0 þ lð þ 0 Þ a2lþ1
(4)
with Pl ½cos ðgij Þ the Legendre polynomials for each angle, * * gij , subtended by r i and r j . The total electrostatic potential energy of an N-electron system is then given by EðNÞ ¼
N X i¼1
E s ðri Þ þ
N N 1 N N 1 X X X 1X E C ðri ; rj Þ þ E P ðri ; rj Þ. 2 i¼1 jai i¼1 jai
(5) The total equilibrium energy, Eq. (5), is minimized when all N electrons are equidistant from the center of the sphere using a variational radius, b, given as a fraction of the
dielectric sphere radius, a, and distributed in the corresponding Thomson geometry [5]. As this work is a continuation of previous work [2], a few unfortunate errors in the previous paper merit correction. In Eq. (3.6) of Ref. [2], there is a factor of ðl þ 1Þ missing, the first term in the denominator of the summation of Eq. (4.2b) should be e2 (e0 in the current case), and a factor of 2 should be included in the second term of Eq. (5.3) as there are two polarization interactions of equal magnitude. The latter error affects the last term of Eq. (5.4) in the summation, where the numerator should be 2ð1Þl . These corrections are incorporated here in the total equilibrium energy of N-electron systems in Table 1. The choice of a ¼ 10 nm is dictated by the condition that the size is small but larger than the quantum coherence length of electrons such that classical evaluation is valid. In general, the total equilibrium energy corresponds to an electron distribution with the highest possible spatial symmetry. For the two-electron system, each charge is repelled to opposite poles of equal radius, boa, inside the sphere (Fig. 1). For the three-electron system, each electron is located at the vertices of an equilateral triangle such that all are equidistant from the origin. For four electrons, the equilibrium energy corresponds to electrons at the vertices of a tetrahedron—the first of only five possible Platonic geometries [8] having regular polygon faces. Of the next three Platonic geometries, only N ¼ 6 and 12 yield equilibrium energies with all electrons located at Platonic vertices, while N ¼ 8 yields an equilibrium energy only when one face of the cube is rotated by p/4. Therefore, the complexity of interactions leads to symmetric electron configurations requiring more than casual intuition. Table 1 lists equilibrium energies for 1pNp12 electron systems. The corresponding values of energy, E G ðÞ, obtained with the Gauss model are provided for comparison. The radius of each minimized electron configuration of electrons, b, normalized to a ¼ 10 nm, is also provided. As the number of electrons increases, the electron radius b increases. Fig. 2 compares the equilibrium energy resulting from an evaluation of the total Table 1 Evaluated electrostatic parameters for systems consisting of 1pNp12, with a ¼ 10 nm, ¼ 12 and ¼ 4 N
E G ðÞ (eV)
EðNÞ (eV)
b=a
CðNÞ (e2 )
1 2 3 4 5 6 7 8 9 10 11 12
0.012 0.048 0.108 0.192 0.300 0.432 0.588 0.768 0.972 1.200 1.452 1.728
0.0120 0.0816 0.2079 0.3899 0.6319 0.9287 1.2878 1.7025 2.1760 2.7085 3.3012 3.9474
0.000 0.632 0.727 0.774 0.803 0.824 0.839 0.851 0.860 0.868 0.875 0.881
83.333 49.020 43.290 41.036 39.563 38.764 37.049 35.592 37.224 36.921 36.653 36.480
ARTICLE IN PRESS J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
1295
4
Potential Energy (eV)
3.5 3 2.5 Interactions Model
2 1.5 1
Gauss Model
0.5 0
1
2
3
4
5
6 7 8 N Electrons
9
10
11
12
Fig. 2. The equilibrium interaction energy for N-electron systems exceeds twice the energy provided by the Gauss model. The system parameters are ¼ 12; 0 ¼ 4, and a ¼ 10 nm.
85 80 75 C (N) (e2)
70 65 60 55 50 Dielectric Gauss Model
45
(41.668)
40 35 1
2
3
4
5
6 7 8 N Electrons
9
10 11 12
Fig. 3. The dielectric interactions capacitance, C(N), is not a constant value for all N, contrary to the capacitance provided by the Gauss model. The system parameters are ¼ 12; 0 ¼ 4, and a ¼ 10 nm.
given by Fig. 1. The geometries of minimized electrostatic energy configurations of 1pNp12 electrons inside a dielectric sphere of radius a ¼ 10 nm represent the greatest possible electron symmetries. The size of the internal Thomson sphere, b=a, is not drawn to scale.
electrostatic interaction energy and energies obtained by the dielectric Gauss model. For very small Coulomb repulsions, as may be simply shown by Eq. (5), EðN; ! 1Þ ! 2E G ðÞ for large N. The definition of capacitance [2,9,10], C dQ=df, appropriate for applications of parametric amplification [11], may be explicitly defined in terms of interaction energy. Most importantly, addition of an electron to an N-electron system changes the symmetry to that of N+1 electrons. The capacitance is therefore defined solely within each configuration, allowing treatment of each N-electron system as a unique phase. In the Gauss model capacitance the work necessary to assemble a collection of infinitesimally divisible charges is
Z E G ðÞ ¼
Q
f dq ¼ 0
Q2 2C G
giving a factor of 1/2. However, since for discrete electrons, dq ! Dq ¼ e, the capacitance of a few-electron system is derived from the interaction energy, Eq. (5), CðNÞ ¼
ðNeÞ2 EðNÞ
with Q ¼ Ne. For few-electron systems, the capacitance varies largely (Fig. 3), passes near the dielectric capacitance of the Gauss model [C G ðÞ ¼ 41:668e2 for a ¼ 10 nm] and approaches the familiar metallic Gauss capacitance [C G ð ! 1Þ ¼ 4p0 0 a ¼ 27:778e2 for a ¼ 10 nm] as N tends to infinity. This trend is easily seen as the electron radius approaches the sphere’s surface when N grows very large to mimic a metallic system. The uniformity and infinitesimally divisible characteristics of the Gauss model are not present in few-electron dielectric spheres. Indeed, an equipotential of a metallic sphere coincides with the sphere’s surface, whereas for electrons in a dielectric
ARTICLE IN PRESS 1296
J. Zhu et al. / Microelectronics Journal 37 (2006) 1293–1296
sphere, equipotentials describe complicated three-dimensional surfaces which make the application of the divergence theorem prohibitive. For this reason, there is a lack of proper treatment of the capacitance of dielectric spheres in textbooks which commonly treat only the capacitance of metallic spheres and often incorrectly apply it to dielectric spheres. 3. Discussion and conclusions The approach presented here differs from the problem originally approached by Babic´ et al. [2] for up to two electrons, and the quantum mechanical treatment of capacitance by others [9,10] involving many-electron quantum dots. The primary difference is the exact nature of the present evaluation which is valid when quantum effects are relatively insignificant. However, the approach is not too different from the classical Thomson problem for computing equilibrium configurations of N electrons confined to some fictitious Thomson radius [4,5]. Therefore, a monophasic capacitance is defined solely within a phase characterized by the particular symmetry of each N-electron system. Present MOSFET technology is fast approaching the days of storing few electrons in a gate dielectric. In such systems, the capacitance differs considerably from capacitances obtained by conventional models as demonstrated here. Moreover, since symmetry plays a significant role, any changes therein lead to substantial changes in the stored electrostatic energy and capacitance. A nonconstant capacitance is not unexpected. Conventionally treated as a function of device geometry, the capacitance
here is shown to be dependent on the unique spatial symmetry of each N-electron phase of the system. Additionally, a variable capacitance may prove useful in the design of few-electron parametric devices [11]. For magnetic devices, this purely classical treatment may require additional inclusion of electron spin interactions [12]. These phenomena derive from the discreteness, rather than the wave nature of electrons. The wave nature is only significant when device size is smaller than the electron mean-free-path. These observations may prove useful before devices truly governed by quantum phenomena become reality. The work presented here may be crucial to the development of a new class of DCDs designed and modeled by the classical electrostatic phenomena of discrete electrons presented in this work. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
R. Tsu, D. Babic´, L. Ioriatti Jr, J. Appl. Phys. 82 (1997) 1327. D. Babic´, R. Tsu, R.F. Green, Phys. Rev. B 45 (1992) 14150. R. Tsu, D. Babic´, Appl. Phys. Lett. 64 (1994) 1806. J.J. Thomson, Philos. Mag. 7 (1904) 237. E.L. Altschuler, T.J. Williams, E.R. Ratner, F. Dowla, F. W, Phys. Rev. Lett. 72 (1994) 2671–2674. L.E. Brus, J. Chem. Phys. 79 (1983) 5566; L.E. Brus, J. Chem. Phys. 80 (1984) 4403. J.F. Bo¨ttcher, Theory of Electric Polarization. 2nd ed., vol 1, Amsterdam, Elsevier, 1973. R.D. Archer-Hind, The Timaeus of Plato, Arno Press, 1973. G.J. Iafrate, K. Hess, J.B. Krieger, Phys. Rev. B 52 (1995) 10737. M. Macucci, K. Hess, G.J. Iafrate, Phys. Rev. B 28 (1993) 17354. A. Yariv, Quantum Electronics, Wiley, New York, 1989. E. Wigner, Phys. Rev. 46 (1934) 1002.