论文部分内容阅读
1. Escuela Nacional Preparatoria 5, Universidad Nacional Autónoma de México, México
2. Departamento de Estado Sólido, Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
Received: April 28, 2011 / Accepted: May 12, 2011 / Published: February 25, 2012.
Abstract: We model a system of alternating conducting layers arranged periodically and an interface of other conducting material as a system of wells and barriers. Each conducting layer can be made of either a metallic material or a highly doped semiconductor. Within the hydrodynamic model we employ additional boundary conditions (ABC’s) to find an analytical expression of the longitudinal electromagnetic surface modes of the truncated superlattice. These longitudinal surface states are similar to those found in quantum-well structures (Tamm states).
Key words: Surface states, elemental semiconductors, electrical properties, low-dimensional structures (superlattices).
1. Introduction??
Spatial dispersion or non-locality of the dielectric response means that the polarization induced at a given position depends not only on the electric field applied at that point, but also at nearby points. A well-known manifestation of non-local behavior is the existence of longitudinal electromagnetic waves or plasma oscillations in conductors.
Since these longitudinal waves are not as common as the transversal electromagnetic waves, let us briefly discuss their origin and main features. In the presence of a surface, the normal component of an incident p-polarized wave pushes the conduction charge towards or away from the surface creating an excess charge. When the frequency of the electromagnetic wave is above the plasma frequency, this charge propagates as a bulk longitudinal wave (longitudinal plasmon). Thus, at a surface, conductors can support and couple both longitudinal and transverse oscillations.
These bulk longitudinal electromagnetic waves(hereafter referred as bulk longitudinal plasmons, in order to differentiate them from surface plasmons) can propagate in conducting hetero-structures which are obtained by heavily doping the semiconductor layers or by replacing some of them with metallic ones. Each conducting layer thus obtained can sustain collective modes such as surface and bulk plasmons, which yield a very rich structure for the normal modes of the composite system.
On the other hand, when the periodicity of an infinite periodic system is broken, then surface wave states localized at interface are possible in addition to bulk waves. This is a general fact for different kinds of waves. Here we are interested in surface longitudinal plasmons or “longitudinal Tamm plasmon states”. In
quantum mechanics these states were predicted by Tamm [1], and have been experimentally observed, for example, in finite superlattices by Ohno et al [2].
In conducting systems transverse and longitudinal electromagnetic waves can be coupled by inhomogeneties. For instance, coupling of transverse and longitudinal waves is important in both semiconductor plasmons [3] and in cold magnetized plasmas [4], although in the latter case the propagation of intense electromagnetic waves is tackled through a relativistic hydrodynamic approach.
To handle the coupling of coupling of transverse and longitudinal waves in layers, Mochán and del Castillo-Mussot used a 4 × 4 transfer matrix formalism within the hydrodynamic model for multi-layered conducting hetero-structures to find expressions for the electromagnetic bulk states [5] similar to those that described the quantum bulk states in infinite quantum-well structures. However, the electromagnetic case is more complex than the quantum case, in the sense that transverse and longitudinal waves are coupled except for propagation perpendicular to the flat interfaces [5]. A good review of the hydrodynamic model can be found in Ref. [6].
As in the quantum case, if the periodicity is broken by the addition of an interface, as in the case of a semi-infinite super-lattice, then there are surface states localized at the interface. Here we are interested in the bulk- plasmon surface states that lie outside the bulk bands. It is important to note that for propagation perpendicular to the surface, there is not longitudinal-transverse coupling and therefore there exist two uncoupled electromagnetic modes; one longitudinal and one transversal, being both similar to quantum propagation, but with different dispersion relations, of course [5]. Transverse electromagnetic surface states are much more studied than longitudinal electromagnetic waves, since they are easier to excite, and unlikely the longitudinal electromagnetic waves also travel in vacuum.
Among the various collective electronic excitations, the longitudinal electromagnetic surface or Tamm states studied here should not be confused with the more common surface plasmons and surface-plasmon polaritons occurring at metal surfaces. For a review of surface plasmons and surface-plasmon polaritons see reference [7], and for a review of plasmons and magnetoplasmons in semiconductor hetero-structures see reference [8].
In next Sect. we describe the system and the hydrodynamic formula employed here. In Sect. 3, after by applying appropriate boundary conditions, we derive the equation that yields the dispersion relation of the longitudinal Tamm plasmon states. The mathematical procedure we employ here is similar to quantum surface states but with different dispersion relations. In Sect. 4 we present our numerical results and Sect. 5 is devoted to discussion and concluding remarks.
the interface with frequencies outside the bulk bands, as we show here.
In this work, we have studied only the uncoupled longitudinal electromagnetic surface states of semi-infinite multilayered conducting hetero-structures. To our knowledge, our work is the first study of electromagnetic longitudinal waves in truncated periodic superlattices in contact with a conducting slab. A possible reason of the apparent lack of studies of these systems, as mentioned above, is that these waves do not propagate in vacuum, and therefore, they have not been so extensively studied as transversal electromagnetic waves. We modeled a system of conducting alternating layers (made of metallic or highly doped semiconductor) as a system of wells and barriers and used the hydrodynamic model to find a very general dispersion relation for longitudinal electromagnetic modes of the superlattice by employing additional boundary conditions (ABC’s) which imply the continuity of some parameters multiplying the current density and electronic density. The corresponding mathematical procedure we employ here to find longitudinal electromagnetic surface Tamm states is straightforward and it is similar to that employed to find Tamm states in quantum systems, but, of course, with different dispersion relations.
Finally, for some appropriate parameters, we show plots of longitudinal electromagnetic surface Tamm frequencies, together with the corresponding bulk longitudinal electromagnetic modes of the infinite superlattice. For each bulk band, we obtain a surface state. Tamm states can lie above or below the bulk band depending on the relative values of the high-density layers in the superlattice and the density of the slab that truncates the superlattice.
We hope that this work may stimulate further theoretical and experimental research on surface states of hetero-structures and related longitudinal
2. Departamento de Estado Sólido, Instituto de Física, Universidad Nacional Autónoma de México, Apartado Postal 20-364, 01000 México, D.F., México
Received: April 28, 2011 / Accepted: May 12, 2011 / Published: February 25, 2012.
Abstract: We model a system of alternating conducting layers arranged periodically and an interface of other conducting material as a system of wells and barriers. Each conducting layer can be made of either a metallic material or a highly doped semiconductor. Within the hydrodynamic model we employ additional boundary conditions (ABC’s) to find an analytical expression of the longitudinal electromagnetic surface modes of the truncated superlattice. These longitudinal surface states are similar to those found in quantum-well structures (Tamm states).
Key words: Surface states, elemental semiconductors, electrical properties, low-dimensional structures (superlattices).
1. Introduction??
Spatial dispersion or non-locality of the dielectric response means that the polarization induced at a given position depends not only on the electric field applied at that point, but also at nearby points. A well-known manifestation of non-local behavior is the existence of longitudinal electromagnetic waves or plasma oscillations in conductors.
Since these longitudinal waves are not as common as the transversal electromagnetic waves, let us briefly discuss their origin and main features. In the presence of a surface, the normal component of an incident p-polarized wave pushes the conduction charge towards or away from the surface creating an excess charge. When the frequency of the electromagnetic wave is above the plasma frequency, this charge propagates as a bulk longitudinal wave (longitudinal plasmon). Thus, at a surface, conductors can support and couple both longitudinal and transverse oscillations.
These bulk longitudinal electromagnetic waves(hereafter referred as bulk longitudinal plasmons, in order to differentiate them from surface plasmons) can propagate in conducting hetero-structures which are obtained by heavily doping the semiconductor layers or by replacing some of them with metallic ones. Each conducting layer thus obtained can sustain collective modes such as surface and bulk plasmons, which yield a very rich structure for the normal modes of the composite system.
On the other hand, when the periodicity of an infinite periodic system is broken, then surface wave states localized at interface are possible in addition to bulk waves. This is a general fact for different kinds of waves. Here we are interested in surface longitudinal plasmons or “longitudinal Tamm plasmon states”. In
quantum mechanics these states were predicted by Tamm [1], and have been experimentally observed, for example, in finite superlattices by Ohno et al [2].
In conducting systems transverse and longitudinal electromagnetic waves can be coupled by inhomogeneties. For instance, coupling of transverse and longitudinal waves is important in both semiconductor plasmons [3] and in cold magnetized plasmas [4], although in the latter case the propagation of intense electromagnetic waves is tackled through a relativistic hydrodynamic approach.
To handle the coupling of coupling of transverse and longitudinal waves in layers, Mochán and del Castillo-Mussot used a 4 × 4 transfer matrix formalism within the hydrodynamic model for multi-layered conducting hetero-structures to find expressions for the electromagnetic bulk states [5] similar to those that described the quantum bulk states in infinite quantum-well structures. However, the electromagnetic case is more complex than the quantum case, in the sense that transverse and longitudinal waves are coupled except for propagation perpendicular to the flat interfaces [5]. A good review of the hydrodynamic model can be found in Ref. [6].
As in the quantum case, if the periodicity is broken by the addition of an interface, as in the case of a semi-infinite super-lattice, then there are surface states localized at the interface. Here we are interested in the bulk- plasmon surface states that lie outside the bulk bands. It is important to note that for propagation perpendicular to the surface, there is not longitudinal-transverse coupling and therefore there exist two uncoupled electromagnetic modes; one longitudinal and one transversal, being both similar to quantum propagation, but with different dispersion relations, of course [5]. Transverse electromagnetic surface states are much more studied than longitudinal electromagnetic waves, since they are easier to excite, and unlikely the longitudinal electromagnetic waves also travel in vacuum.
Among the various collective electronic excitations, the longitudinal electromagnetic surface or Tamm states studied here should not be confused with the more common surface plasmons and surface-plasmon polaritons occurring at metal surfaces. For a review of surface plasmons and surface-plasmon polaritons see reference [7], and for a review of plasmons and magnetoplasmons in semiconductor hetero-structures see reference [8].
In next Sect. we describe the system and the hydrodynamic formula employed here. In Sect. 3, after by applying appropriate boundary conditions, we derive the equation that yields the dispersion relation of the longitudinal Tamm plasmon states. The mathematical procedure we employ here is similar to quantum surface states but with different dispersion relations. In Sect. 4 we present our numerical results and Sect. 5 is devoted to discussion and concluding remarks.
the interface with frequencies outside the bulk bands, as we show here.
In this work, we have studied only the uncoupled longitudinal electromagnetic surface states of semi-infinite multilayered conducting hetero-structures. To our knowledge, our work is the first study of electromagnetic longitudinal waves in truncated periodic superlattices in contact with a conducting slab. A possible reason of the apparent lack of studies of these systems, as mentioned above, is that these waves do not propagate in vacuum, and therefore, they have not been so extensively studied as transversal electromagnetic waves. We modeled a system of conducting alternating layers (made of metallic or highly doped semiconductor) as a system of wells and barriers and used the hydrodynamic model to find a very general dispersion relation for longitudinal electromagnetic modes of the superlattice by employing additional boundary conditions (ABC’s) which imply the continuity of some parameters multiplying the current density and electronic density. The corresponding mathematical procedure we employ here to find longitudinal electromagnetic surface Tamm states is straightforward and it is similar to that employed to find Tamm states in quantum systems, but, of course, with different dispersion relations.
Finally, for some appropriate parameters, we show plots of longitudinal electromagnetic surface Tamm frequencies, together with the corresponding bulk longitudinal electromagnetic modes of the infinite superlattice. For each bulk band, we obtain a surface state. Tamm states can lie above or below the bulk band depending on the relative values of the high-density layers in the superlattice and the density of the slab that truncates the superlattice.
We hope that this work may stimulate further theoretical and experimental research on surface states of hetero-structures and related longitudinal