Inter-dimensional effects in nano-structures
© Dick; licensee Springer. 2012
Received: 12 July 2012
Accepted: 3 October 2012
Published: 23 October 2012
We report on two extensions of the traditional analysis of low-dimensional structures in terms of low-dimensional quantum mechanics. On one hand, we discuss the impact of thermodynamics in one or two dimensions on the behavior of fermions in low-dimensional systems. On the other hand, we use both quantum wells and interfaces with different effective electron or hole mass to study the question when charge carriers in interfaces or layers exhibit two-dimensional or three-dimensional behavior. We find in particular that systems with different effective masses in the bulk and in the interface exhibit separation of two-dimensional and three-dimensional behavior on different length scales, whereas quantum wells exhibit linear combination of two-dimensional and three-dimensional behavior on short length scales while the behavior on large length scales cannot be associated with either two-dimensional or three-dimensional behavior.
KeywordsDensity of states Coulomb interactions Exchange interactions Scattering in nano-structures Thermal properties in nano-structures Fermi energy in nano-structures
Nano-structures traditionally provide approximate realizations of low-dimensional systems through confined electron states in one dimension (thin films, interfaces or quantum wells), two dimensions (quantum wires or nano-wires), or three dimensions (quantum dots or color centers). We emphasize electron states rather than electrons in the following discussions, because for conduction bands with large filling factors or p-doped semiconductors, we usually think of the unoccupied electron states as holes, which can also be confined[1–3].
Here, m is the (effective) mass, and q is the charge of the particle, q = ±e for holes or electrons, respectively. is the angular momentum of the particle. Low-dimensional systems in spintronics are therefore directly linked to confined electron states because electrons or holes do not only provide the lightest movable charge carriers, but also interact stronger with magnetic fields than any other readily available (quasi-)particle in materials. Furthermore, energy gaps between different spin configurations are determined by exchange integrals, and the spatial extent of the wave function of a bound particle of mass m typically scales with 1/m. Exchange interactions between electrons or holes can therefore align or anti-align spins, whereas inter-nuclear exchange interactions are negligible in materials science.
A well-known primary effect of a reduced number d of dimensions in nano-systems is the significant change in the energy dependence of the density of states in the energy scale,. This directly affects the thermal and electrical conductivity properties of nano-systems and impacts the use of spins for information storage and processing. Furthermore, even without confinement, particles can exhibit low-dimensional behavior on certain energy and length scales if their propagation properties are affected by the presence of layers, interfaces, or wire-type structures in which the particles propagate with an effective mass which is very different from their effective mass in the adjacent bulk materials[4, 5].
Figure1 illustrates that in lower dimensions, interactions are comparatively stronger at large distances and weaker at short distances. The same effect would apply for any other interaction which would be mediated by confined bosons; for example, it would also apply to phonon-mediated interactions between electrons or holes. The reduced interaction strength at smaller distance in lower dimensions is a consequence of the weaker singularity of the field near its source, whereas the increase in strength at larger distance intuitively can be attributed to the squeezing of field lines into a smaller number of dimensions. This change in distance behavior directly impacts electric forces between charges and implies the potential emergence of electrical confinement in systems with dimensionally restricted electromagnetic fields. In addition, it also impacts effective spin-spin interactions in spintronics because the d-dimensional electrical potential also appears in the exchange integrals which determine the energy splits between spin configurations.
Low-dimensional quantum mechancis with one or two-dimensional Hamilton operators, or three-dimensional Hamiltonians with confining boundary conditions are widely used to analyze and understand the importance of quantum effects on confined particles in nano-systems. Here, we wish to report on extensions of this analysis in two directions: (1) impacts of low-dimensional thermodynamics on the behavior of charge carriers and (2) quantum mechanical analysis of inter-dimensional behavior in materials with a low-dimensional component. We focus also on a thin interface or layer as the low-dimensional component, but the same methods can be applied, e.g., to analyze dimensional competition in the case of a nano-wire on a surface. Inter-dimensional effects in these systems can be relevant, e.g., for charge transport in nano-wires, which attract a lot of interest, e.g., for its use in photovoltaics. We will use both the method of inter-dimensional Green’s functions[4, 5, 10] and grand canonical ensembles in low-dimensional systems to analyze impact of dimensionality of a system on the behavior of electrons and photons.
The solutions of Equations (1) and (2) provide us with single particle or mean field Green’s functions, which describe scattering of particles and densities of states, and through particular choices or re-definition of the energy parameter, they also determine electric potentials and exchange interactions. In addition, the single particle Green’s functions also enter into the calculation of electronic configurations for many particle systems through application of multiple scattering theory.
(see Appendix I in for derivations). The functions K ν and are modified Bessel functions and Hankel functions of the first kind, respectively, and we follow the notations and conventions of for special functions.
However, if there are parameter ranges in materials and devices where electrons or photons behave according to the laws of two-dimensional or three-dimensional quantum mechanics and electrodynamics, then there should also exist transition regimes with intermittent dimensional behavior. This is the realm where particles or forces are described by the inter-dimensional or dimensionally hybrid Green’s functions introduced in[4, 5]. We should also point out that another important novel approach to inter-dimensional behavior in systems with low-dimensional components concerns the study of inter-dimensional universality for critical scaling laws. This notion has been introduced and studied for domain wall dynamics in nano-wires.
We will review the basic aspects of physics in various dimensions in the section on “Green’s functions, potentials, and densities of states in d dimensions” and then discuss a lesser known but technologically relevant aspect of physics in lower dimensions, viz., the impact of dimensionality on statistical and thermal physics in low-dimensional systems, in the section on “Thermal properties of the charge carriers in d dimensions”. We will then discuss the construction of dimensionally hybrid Green’s functions for quantum wells in the section on “Inter-dimensional effects in interfaces and thin layers”. This will also allow us to calculate the inter-dimensional density of states ϱ(E) and the relation between Fermi energy and electron density in the quantum well in the section entitled “Density of states for the thin quantum well”. Comparison of the results for the quantum well with the results for layers with different effective mass of charge carriers or different permittivity reveals that a difference in potential energy between a layer and a bulk yields linear combinations of two-dimensional and three-dimensional terms at the same length scales, whereas difference in kinetic terms (viz. effective mass which affects kinetic terms for electrons, holes, or permittivity, which affects the kinetic terms for photons), separates two-dimensional behavior on short length scales from three-dimensional behavior at large length scales.
Results and discussion
We can now enter into the discussion of less known results on the low-dimensional quantum and statistical physics of charge carriers and new results and observations concerning inter-dimensional behavior in the presence of layers or interfaces. We will separate this discussion into subsections on interaction potentials and thermal properties in low-dimensional fermion systems, and a subsection on inter-dimensional effects as inferred from Green’s functions.
Green’s functions, potentials and densities of states in d dimensions
and correspondingly screened exchange interactions. Practical realization of low-dimensional Coulomb or Yukawa potentials (Equations (7) and (9) with d = 1 or d = 2) in devices may be possible with the help of photonic bandgap materials, and the two-dimensional logarithmic behavior should be realized at short distances in high permittivity thin films.
The most interesting feature of this result from a nano-device point of view concerns suppressed high energy scattering and enhanced low energy scattering from impurities in low dimensions roughly according to.
This makes physical sense: In a smaller number of dimensions, we need a larger Fermi sphere in space to accommodate the same electron density in space.
Equations (13) and (14) a priori refer to a free electron gas model. In materials science, this is a useful approximation for semi-conductors and a very good approximation for metals at room temperature. For energy bands with minimal energy E0, corresponding effective mass m∗, and a low filling factor, Equation (13) applies for the electron density of states with the substitutions E → E − E0 and m → m∗. For nearly filled bands with maximal energy E1, the substitutions E → E1− E, m → m h yield the hole density of states.
Densities of states are important for electrical and thermal transport properties of materials and for the optical properties of materials. For example, the photon absorption cross-section for excitation of an electron from a discrete donor or quantum dot state into a continuous energy band is directly proportional to the density of final electron states. Therefore, the densities of states (13) for d = 1, 2, and 3 are common items for information in nano-technology textbooks. However, strict electron confinement to a quantum wire or an interface is apparently a bad approximation in most cases and makes only sense for the subset of low lying energy states in deep quantum well structures. Therefore, we will revisit the density of states in the section on “Density of states for the thin quantum well” in the framework of a solvable quantum well model.
Thermal properties of the charge carriers in d dimensions
A less widely known and less developed aspect of low-dimensional physics concerns the impact of dimensionality of a system on its thermal and statistical properties. The derivation of the basic Fermi-Dirac or Bose-Einstein distributions from maximal information entropy under the boundary conditions of given energy and particle number (if we use a grand canonical ensemble) does not depend on the number of dimensions. However, the calculation of partition functions and thermodynamic quantities from the Fermi-Dirac or Bose-Einstein distributions involves d-dimensional integrals; therefore, thermal properties of a system will depend on the number of dimensions in which particles can move. I would hope that the introduction in this section can serve as a brief compendium and overview of basic aspects of this dependence of thermal properties on d. We will find that the specific heat in particular is affected by d. Due to the particular relevance of confined fermionic charge and spin carriers for nano-technology, we will focus on low-dimensional implications of Fermi-Dirac statistics.
The approximation of an ideal non-relativistic gas,, is known to yield excellent results for metals. For semiconductors with a low filling factor in the conduction band, we can use if we also calculate μ and E F from the minimum of the energy band. For high filling factor, we should calculate the chemical potential and Fermi energy for the holes downwards from the maximum of the energy band, of course, to use.
Inter-dimensional effects in interfaces and thin layers
We will denote two-dimensional coordinate vectors parallel to the interface with,.
We might expect two-dimensional behavior in the limit a → 0 both from the difference of effective mass in the interface and from the interface potential. Indeed, it has been shown that even without a potential difference, the existence of a layer with different effective mass generates Green’s functions in the interface which interpolate between two-dimensional behavior for small distance and three-dimensional behavior for large distance along the interface[4, 5, 10].
The limit κ → 0 reproduces the corresponding representation of the free retarded Green’s function in three dimensions.
To explore the question of two-dimensional or three-dimensional behavior in the quantum well further, we will look at the density of states in the quantum well.
Density of states for the thin quantum well
Here, ν is a degeneracy index, and we tacitly imply that continuous components in the indices (n,ν) are integrated. We include spin in the set of quantum numbers (n,ν).
where a factor g = 2 was taken into account for spin 1/2 states.
Note that K2 = E + (ℏ2κ2/2m) is the kinetic energy of the particles whose wave functions are exponentially suppressed perpendicular to the quantum well. We find that these particles indeed contribute a term proportional to the two-dimensional density of states ϱd = 2(K2) with their energy K2 of motion along the quantum well, but with a dimensional proportionality constant κ which is the inverse penetration depth of those states. Such a dimensional factor has to be there because densities of states in three dimensions enumerate states per energy and per volume, while ϱd = 2(K2) counts states per energy and per area. Furthermore, the unbound states yield a contribution which approaches the free three-dimensional density of states ϱd = 3(E) in the limit κ → 0.
We can also derive these results directly from the energy eigenstates (19 to 21) and the definition (32). However, the derivation from the Green’s function (24) confirms that this is indeed a correct dimensionally hybrid Green’s function which yields inter-dimensional effects.
Not surprisingly, comparison of the relation between Fermi energy and density of fermions for the quantum well with the corresponding results for a layer of different effective mass confirms again that the effective mass layer exhibits separation of two-dimensional behavior for small lengths/high energies and three-dimensional behavior for large lengths/small energies, whereas the quantum well yields a linear combination of two-dimensional and three-dimensional terms for small lengths/high energies.
The thin quantum well is certainly one of the most important model systems for low-dimensional structures in nano-science and technology. We have found that the Green’s function of this system resembles a linear combination of two-dimensional and three-dimensional terms at small distances but exhibits oscillatory behavior at large distances. Furthermore, the local density of states and the relation between particle density and Fermi energy in the quantum well show two-dimensional behavior for Fermi energies below the threshold for scattering out of the quantum well and a linear combination of two-dimensional and three-dimensional behavior plus correction terms above the threshold. This behavior is very different from the behavior of charge carriers which move with different effective mass in a layer: in that case, the analysis in had shown that the system exhibits two-dimensional behavior at small distances and high energies, and three-dimensional behavior at large distances and low energies. The morale of the combination of the present results with the results of is that if we wish to explicitly see transitions between two-dimensional and three-dimensional behavior in a system, then we should look for systems where the interface primarily affects the kinetic terms of fermions through a difference of effective mass between bulk and layer, or the kinetic terms of photons through a difference of permittivity between bulk and layer.
This research was supported by NSERC Canada.
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