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Nanoscale Res Lett. 2010; 5(10): 1546–1554.
Published online Jul 2, 2010. doi:  10.1007/s11671-010-9675-1
PMCID: PMC2956029

Dimensional Effects on Densities of States and Interactions in Nanostructures

Abstract

We consider electrons in the presence of interfaces with different effective electron mass, and electromagnetic fields in the presence of a high-permittivity interface in bulk material. The equations of motion for these dimensionally hybrid systems yield analytic expressions for Green’s functions and electromagnetic potentials that interpolate between the two-dimensional logarithmic potential at short distance, and the three-dimensional r−1 potential at large distance. This also yields results for electron densities of states which interpolate between the well-known two-dimensional and three-dimensional formulas. The transition length scales for interfaces of thickness L are found to be of order Lm/2m* for an interface in which electrons move with effective mass m*, and An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i62.gif for a dielectric thin film with permittivity An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i63.gif in a bulk of permittivity An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i64.gif. We can easily test the merits of the formalism by comparing the calculated electromagnetic potential with the infinite series solutions from image charges. This confirms that the dimensionally hybrid models are excellent approximations for distances r [greater, similar] L/2.

Keywords: Density of states, Coulomb and exchange interactions in nanostructures, Dielectric thin films

Introduction

When we suppress motion of particles in certain directions through confining potentials, e.g. in quantum wells or quantum wires, we often model the residual low energy excitations in the system through low-dimensional quantum mechanical systems. Prominent examples of this concern layered heterostructures, and one instance where the number d of spatial dimensions enters in a manner which is of direct relevance to technology is in the density of states. In the standard parabolic band approximation, this takes the form (with two helicity or spin states)

equation image
(1)

These are densities of states per d-dimensional volume and per unit of energy. The corresponding dependence of the relation between the Fermi energy and the density n of electrons on d is

equation image
(2)

Variants of these equations (including summation over subbands) are often used for d = 2 or d = 1 to estimate carrier densities in quasi two-dimensional systems or nanowires, and the density of states plays a crucial role in all transport and optical properties of materials. Indeed, the obvious relevance for electrical conductivity properties in micro and nanotechnology implies that densities of states for d = 1, 2, or 3 are now commonly discussed in engineering textbooks, but there is another reason why I anticipate that variants of Eq. (1) will become ever more prominent in the technical literature. Densities also play a huge role in data storage, but with us still relying on binary logic switching between two stable states (spin up or down, charge or no charge, conductivity or no conductivity), data storage densities are limited by the physical densities of the systems which provide the dual states. We could (and likely will) drive information technology and integration much further if we can find ways to utilize more than just two states of a physical system to store and process information. Then, data storage densities should become proportional to energy integrals An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i38.gif of local densities of states. Equation (1) for d = 1 or d = 2 is certainly applicable for particles which have low energies compared to the confinement energy of a nanowire or a quantum well, but how can we effectively model particles which are weakly confined to a nanowire or quantum well, or which are otherwise affected by the presence of a low-dimensional substructure? In these cases, we can devise dimensionally hybrid models [1,2] which yield e.g. densities of states which interpolate between d = 2 and d = 3 [3,4]. This construction will be reviewed in Sect. 2. Based on the experience gained with dimensionally hybrid Hamiltonians for massive particles, we can also construct inter-dimensional Hamiltonians for photons which should be applicable to photons in the presence of high-permittivity thin films or interfaces. These models can also be solved in terms of infinite series expansions using image charges, and the merits of this approach can easily be tested. The case of high-permittivity thin films and testing the theory against image charge solutions will be discussed in Sect. 3.

Dimensionally Hybrid Hamiltonians and Green’s Functions for Massive Particles in the Presence of Thin Films or Interfaces

We use the connection between Green’s functions and the density of states to generalize Eq. (1) for massive particles in the presence of a thin film or interface.

The energy-dependent Green’s function for a Hamiltonian H with spectrum En and eigenstates |n, νright angle bracket is

equation image
(3)

Here, ν is a degeneracy index and the notation implies that continuous components in the indices (n, ν) are integrated. The first equation simply states the relation between the resolvent An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i37.gif of the Hamiltonian and the Green’s function G(E) which is normalized as limm→0,E→0G(E)|d=3 = (4πr)−1.

The zero-energy Green’s function G(0) determines e.g. 2-particle correlation functions and electromagnetic interaction potentials, and the energy-dependent Green’s function G(E) determines e.g. scattering amplitudes for particles of energy E. Application for resistivity calculations is therefore another technologically relevant application of Green’s functions. However, in the present section we are interested in this function because it also determines the local density of states in a system with Hamiltonian H through the relation

equation image
(4)

Here, we explicitly included a factor 2 for the number of spin or helicity states, because the summation over degeneracy indices in (3,4) usually only involves orbital indices.

For our present investigation, the distinctive feature of the interface is that the particles move in it with an effective mass m*, while their mass in the surrounding bulk is m. We use coordinates An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i56.gif parallel to a plane interface, which is located at z = z0. Bold vector notation is used for quantities parallel to the interface, e.g. An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i39.gif and An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i40.gif.

We assume that the interface has a thickness L. If the wavenumber component orthogonal to the interface is small compared to the inverse width, |k[perpendicular]L| [double less-than sign] 1, i.e. if the de Broglie wavelength and the incidence angle satisfy λ [dbl greater-than sign]L|cos[theta]|, we can approximate the kinetic energy of the particles through a second quantized Hamiltonian

equation image
(5)

where μ = m*/L. The corresponding first quantized Hamiltonian is

equation image
(6)

The interesting aspect of the Hamiltonians (5,6) is the linear superposition of two-dimensional and three-dimensional kinetic terms. The formalism presented here could and will certainly be extended to include also kinetic terms which are linear in derivatives, in particular in the interface term. This would be motivated either by a Rashba term arising from perpendicular fields penetrating the interface [5-11] of from the dispersion relation in Graphene [12-15]. However, for the present investigation we will use a parabolic band approximation in the bulk and in the interface.

The energy-dependent Green’s function An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i41.gif describes scattering effects in the presence of the interface but also applies to scattering off perturbations which are not located on the interface. In an axially symmetric mixed representation

equation image
(7)

the first order approximation to scattering of an orthogonally incoming plane wave off an impurity potential

equation image

corresponds to

equation image

Green’s functions for surfaces or interfaces are commonly parametrized in an axially symmetric mixed representation like An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i57.gif. In bra-ket notation, this corresponds for the free Green’s function G0(E), which is also translation invariant in z direction, to

equation image

We will briefly recall the explicit form of the free Green’s function G0(E) in the axially symmetric mixed parametrization for later comparison. The equation

equation image

yields

equation image
(8)

To study how this is modified in the presence of the interface, we observe that the Hamiltonians (5) or (6) yield a Schrödinger equation

equation image

The corresponding equation for the Green’s function or 2-point correlation function is

equation image
(9)

The solution of this equation is described in the Appendix. In particular, we find the representation (see Eq. (27))

equation image
(10)

where the definition [ell][equivalent]m/2μ = Lm/2m* was used. The [ell]-independent terms in (10) correspond to the free Green’s function G0(E) (8).

The interface at z0 breaks translational invariance in z direction, and we have with Eq. (7)

equation image

We will use the result (10) to calculate the density of states An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i42.gif in the interface. Substitution yields

equation image

and after evaluation of the integral

equation image
(11)

This is a more complicated result than the density (1) for d = 2 or d = 3. However, it reduces to either the two-dimensional or three-dimensional density of states in the appropriate limits, see Fig. Fig.1.1. For large energies, i.e. if the states only probe length scales smaller than the transition length scale [ell], we find the two-dimensional density of states properly rescaled by a dimensional factor to reflect that it is a density of states per three-dimensional volume,

equation image
(12)
Figure 1
The red line is the two-dimensional limit (12). The blue line is the three-dimensional density of states. The it black line is the inter-dimensional density of states (11) for [ell] = 50 nm

For small energies, i.e. if the states probe length scales larger than [ell], we find the three-dimensional density of states

equation image
(13)

This limiting behavior for interpolation between two and three dimensions is consistent with what is also observed for the zero-energy Green’s function in the interface, see equations (21–22) below.

Equation (11) also implies interpolating behavior for the relation between electron density and Fermi energy on the interface. The full relation is

equation image

This approximates two-dimensional behavior for An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i54.gif,

equation image

and three-dimensional behavior for An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i55.gif,

equation image

It is intuitively understandable that the presence of a layer reduces the available density of states for given energy, or equivalently increases the Fermi energy for a given density of electrons. The presence of a layer generically implies boundary or matching conditions which reduce the number of available states at a given energy.

A condition for relevance of the inter-dimensional behavior is a large transition scale compared to the layer thickness, [ell] [dbl greater-than sign] L, see also Fig. Fig.2.2. In terms of effective particle mass, this means

equation image
(14)
Figure 2
The upper dotted (blue) line is the three-dimensional Green’s function (4πr)−1 in units of [ell]−1, the continuous line is the Green’s function (19) in units of [ell]−1, and the lower dotted (red) ...

i.e. the energy band in the interface should be more strongly curved than in the bulk matrix for the transition to two-dimensional behavior to be observable.

Electric Fields in the Presence of High-Permittivity Thin Films or Interfaces

The zero-energy Green’s function An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i43.gif determines electrostatic and exchange interactions through the electrostatic potential An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i44.gif. Here, q is an electric charge in a dielectric material of permittivity An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i65.gif. The zero-energy Green’s function in d spatial dimensions is given by

equation image
(15)

We cannot infer from the previous section that the zero energy limit of the inter-dimensional Green’s function calculated there also yields a dimensionally hybrid potential, because we were dealing with solutions of Schrödinger’s equation instead of the Gauss law. However, we can rederive the zero energy limit of that Green’s function from the Gauss law for electromagnetic fields in the presence of a high-permittivity interface.

Suppose we have charge carriers of charge q and mass m in the presence of an interface with permittivity An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i66.gif and permeability μ*, We continue to denote vectors parallel to the interface in bold face notation, An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i45.gif, An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i46.gif, etc.

If the photon wavelengths and incidence angles satisfy the condition λ [dbl greater-than sign]L|cos[theta]|, we can approximate the system with an action

equation image

Variation with respect to the electrostatic potential, An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i47.gif, yields the Gauss law in the form

equation image
(16)

and the continuity condition Ez(z0 − 0) = Ez(z0 + 0).

We solve Eq. (16) in Coulomb gauge,

equation image
(17)

where the Green’s function has to satisfy

equation image
(18)

This equation is the zero energy limit of Eq. (9) with the substitution

equation image

We can therefore read off the solution from the results of the previous section with E = 0 and now An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i67.gif.

Equation (10) yields in particular

equation image

with An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i58.gif. Fourier transformation yields

equation image
(19)

The zero-energy Green’s function in the interface is given in terms of a Struve function and a Neumann function1,

equation image
(20)

This yields logarithmic behavior of interaction potentials at small distances r [double less-than sign] [ell] and 1/r behavior for large separation r [dbl greater-than sign] [ell] of charges in high-permittivity thin films,

equation image
(21)
equation image
(22)

see also Fig. Fig.22.

For the comparison with image charges, we set z0 = 0 and recall that the solution for the potential of a charge q at An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i59.gif, z = 0 proceeds through the ansatz

equation image

and symmetric continuation to z < −L/2.

This yields electric fields

equation image

and the junction conditions at z = L/2 yield for n ≥ 0 from the continuity of Er,

equation image

and from the continuity of Dz,

equation image

These conditions can be solved through

equation image

In particular, the potential at z = 0 is

equation image
(23)

We have

equation image

and therefore for An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i68.gif

equation image

The solution from image charges is in very good agreement with the analytic model for distances r [greater, similar] L/2, where both the image charge solution and the analytic model show strong deviations from the bulk r−1 behavior. This is illustrated in Fig. Fig.33 by plotting the reduced electrostatic potential for a charge q, An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i48.gif in the interface.

Figure 3
Different reduced electrostatic potentials are plotted for An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i72.gif. The upper dotted (green) line is the three-dimensional reduced potential L/(4πr). The central dotted (blue) line is the reduced potential following from the image charge solution (22). ...

It is also instructive to plot the relative deviation An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i49.gif between the dimensionally hybrid potential An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i50.gif which follows from (20) and the potential An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i51.gif (23) from image charges.

Figure Figure44 shows that for r [greater, similar] L/2, the dimensionally hybrid model is a very good approximation to the potential from image charges with accuracy better than 10−2 if An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i69.gif. For An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i70.gif, the accuracy is still better than 4 × 10−2.

Figure 4
The relative deviation An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i53.gif between the dimensionally hybrid potential from (19) and the potential (22) from image charges for An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i73.gif

Summary

An analysis of models for particles in the presence of a low effective mass interface, and for electromagnetic fields in the presence of a high-permittivity thin film, yields dimensionally hybrid densities of states (11) and electrostatic potentials (17,20) which interpolate between two-dimensional behavior and three-dimensional behavior. The analytic model for the electromagnetic fields is in very good agreement with the infinite series solution already for small distance scales r [greater, similar] L/2, where the potential strongly deviates from the standard bulk r−1 potential. At distance scales smaller than L/2, r−1, behavior seems to dominate again for the electrostatic potential, in agreement with expectations that for distances which are small compared to the lateral extension of a dielectric slab, bulk behavior should be restored. However, note that neither the inter-dimensional analytic model nor the solution from image charges is trustworthy for very small distances, because both models rely on a continuum approximation through the use of effective permittivities, but the continuum approximation should break down at sub-nanometer scales.

The most important finding is that interfaces and thin films of width L should exhibit transitions between two-dimensional and three-dimensional distance laws for physical quantities at length scales of order Lm/2m* or An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i71.gif, respectively. Interfaces with strong band curvature or high permittivity should provide good samples for experimental study of the transition between two-dimensional and three-dimensional behavior.

Appendix: Solution of Eq. 9

Substitution of the fourier transform

equation image

into Eq. 9 yields

equation image
(24)

This yields with (7) the condition

equation image

Fournier transformation with respect to z yields

equation image
(25)

This result implies that An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i60.gif has the form

equation image

with the yet to be determined An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i61.gif satisfying

equation image

For the treatment of the integrals, we should be consistent with the calculation of the free retarded Green's function (8),

equation image

This yields

equation image

And therefore

equation image
(26)

where the definition [ell][hyphen in circle]m/2μ = Lm/2m*. Fourier transformation of Eq (26) with respect to k[perpendicular] yields finally

equation image
(27)

The Green's function with only k space variables An external file that holds a picture, illustration, etc.
Object name is 1556-276X-5-1546-i29.gif is found from the Fournier transform of Eq. 25,

equation image

and the ensuing equations

equation image

This yields

equation image

It is easily verified that Fournier transformation yields again the result (26).

Footnotes

1Our notations for special functions follow the conventions of Abramowitz and Stegun [16].

Acknowledgements

This research was supported by NSERC Canada.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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