**2. Nanoplasmonics and quantum treatment of plasmons**

The modern era of plasmonic began with the investigation of wood anomalies in the early twentieth century [21]. Later in 1957 Ritchie published a paper on plasma loss due to the electrons at the interface of a thin metal film [11]. In the next few years, theoretical works on collective oscillations of electrons at the surface of metals led to the introduction of plasmons as the quasiparticle corresponding to these oscillations [11]. However, applications of plasmons as a tool for nanoscale manipulation of light has gained significant attention with the paper by H. Atwater in 2007 named "Promise of plasmonics" [21]. In the past two decades, plasmonics has been developed both in theoretical and experimental aspects, and many different devices like switches [22], detectors [23], routers [24], amplifiers [25], and sources [17] have been introduced.

#### **2.1 Basic principles**

In order to find an appropriate model for surface plasma waves at the surface of a metal, we should deal with a charge density wave in an infinite electron gas which is often modeled by hydrodynamic equations [11]. An electromagnetic wave propagating in a material polarizes it and results in a mechanical excitation in electric charges and their movement. Therefore, oscillations in the electric field and mechanical oscillations are coupled. This coupled oscillation is called polariton. In case of metals, the electromagnetic field causes a longitudinal wave of charge density, and the coupled oscillations are known as plasmon polariton waves [11].

According to **Figure 1** at the interface of metal with a dielectric interaction of an electromagnetic field with the surface electrons, a specific type of plasmon polariton waves called surface plasmon polaritons or SPP modes results. Although

*Nanoscale Plasmon Sources: Physical Principles and Novel Structures DOI: http://dx.doi.org/10.5772/intechopen.90842*

**Figure 1.**

called surface plasmon amplification by stimulated emission of radiation (SPASER)

in 2002 [12]. This device utilizes a plasmonic feedback mechanism in a gain medium for exciting stimulated emission in local plasmon modes of metallic nanoparticles. In the next two decades, various mechanisms and devices have been proposed and fabricated for the realization of a nanoscale coherent plasmon source or plasmonic nanolaser [13]. These devices can be categorized in nanoparticles [12],

waveguide-based nanolasers [15], nanowires [16], nanoresonators [17],

of plasmons focusing on special characteristics of plasmons in metallic

mance in room temperature, and so on can be noted [12, 13].

duced, and this chapter will be concluded in Section 6.

sources [17] have been introduced.

**2.1 Basic principles**

*Nanoplasmonics*

waves [11].

**60**

**2. Nanoplasmonics and quantum treatment of plasmons**

nanopatches [18], nanodisks [19], plasmonic crystals [20], and so on. Although these devices have shown significant potentials, there are still serious problems with the nanoscale coherent sources of surface plasmons. For instance, the difficulty of fabrication, over-heating, low output powers, high loss rates, lack of integration capability with commercial fabrication processes like CMOS, inefficient perfor-

In this chapter and in Section 2, we start with basic principles of nanoplasmonics like the definition of surface plasmon polariton (SPP) modes in classical and quantum mechanical pictures, different sources of plasmon loss, and specific properties

The modern era of plasmonic began with the investigation of wood anomalies in the early twentieth century [21]. Later in 1957 Ritchie published a paper on plasma loss due to the electrons at the interface of a thin metal film [11]. In the next few years, theoretical works on collective oscillations of electrons at the surface of metals led to the introduction of plasmons as the quasiparticle corresponding to these oscillations [11]. However, applications of plasmons as a tool for nanoscale manipulation of light has gained significant attention with the paper by H. Atwater in 2007 named "Promise of plasmonics" [21]. In the past two decades, plasmonics has been developed both in theoretical and experimental aspects, and many different devices like switches [22], detectors [23], routers [24], amplifiers [25], and

In order to find an appropriate model for surface plasma waves at the surface of a metal, we should deal with a charge density wave in an infinite electron gas which is often modeled by hydrodynamic equations [11]. An electromagnetic wave propagating in a material polarizes it and results in a mechanical excitation in electric charges and their movement. Therefore, oscillations in the electric field and mechanical oscillations are coupled. This coupled oscillation is called polariton. In case of metals, the electromagnetic field causes a longitudinal wave of charge density, and the coupled oscillations are known as plasmon polariton

According to **Figure 1** at the interface of metal with a dielectric interaction of an

electromagnetic field with the surface electrons, a specific type of plasmon polariton waves called surface plasmon polaritons or SPP modes results. Although

nanoresonators. Then, the interaction of plasmons with carriers in a cavity will be briefly discussed according to plasmonic cavity quantum electrodynamics. In Section 3, three different methods for analyzing plasmonic nanolasers are discussed, and in Section 4 several previously introduced nanolaser structures are briefly reviewed. In Section 5, the proposed nanolaser structures by the authors are intro-

*Interaction of an electromagnetic field with surface plasma waves and excitation of surface plasmon polariton waves.*

there are different types of plasmons like bulk plasmons and local surface plasmons (LSPs), SPP and LSP modes have significant roles in many plasmonic devices.

There are several models for plasmons, and we are going to briefly overview them here. The most well-known and simple model is Drude's model which describes the metal as a free electron gas system and models the system using the classical spring-mass model with the external force exerted from the incident field "**E**" equals to "-q**E**" acting on the system. We are not going to derive the equations here and only use the final result as shown in Eq. (1) which can be derived as mentioned in many related references like [22]:

$$\varepsilon\_r(\boldsymbol{\alpha}) = 1 - \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}^2 + i\boldsymbol{\gamma}(\boldsymbol{\alpha})\boldsymbol{\alpha}}, \qquad \boldsymbol{\alpha}\_p^2 = \frac{\boldsymbol{m}^2}{\varepsilon\_0 \boldsymbol{m}}, \qquad \boldsymbol{\gamma} = \frac{1}{\tau} \tag{1}$$

where "*n*" is number of electrons in the unit volume of the metal, "*e*" is the electron charge, "*m*" is the electron mass, "*ε*0" is permittivity of vacuum, "*ωp*" is the plasma frequency of the metal, "*γ*" is the total loss, and "*τ*" is the effective lifetime of the electrons associated with all of the decay processes.

According to Eq. (1), permittivity of a metal *εr*(*ω*) can be used in solving Helmholtz equations and finding the behavior of electromagnetic waves propagating at the metal/dielectric interface which are also known as SPP waves. However, Drude's model suffers from several shortcomings which leads to considerable errors especially near the plasma frequency of the metal. This is because in Drude's model, the effect of electrons in other energy bands (not just free electrons) is not included, and nonlocal effects are also not included [26]. To overcome these problems, Drude-Lorentz's model is introduced for the first problem which can be written in general multi-oscillator form as Eq. (2) and Landau damping correction according to Eq. (3) for the second problem. We are not going to further discuss these models either, and you can find details in [11, 26].

$$\epsilon\_{\text{Drude}-\text{Lorentz}}(o) = \mathbf{1} - \frac{f\_0 o\_{p,0}^2}{o^2 + i\gamma\_0 o} + \sum\_{j=1}^{j\_{\text{max}}} \frac{f\_j o\_{p,j}^2}{o\_j^2 - o^2 - i\gamma\_j o} \tag{2}$$

where the first sentence corresponds to Drude's model; "*fj*" is the power of the *j*'th oscillator; and "*ωp,j*," "*ωj*," and "*γj*" are plasma frequency, resonant frequency, and loss coefficient of the *j*'th oscillator, respectively.

*Nanoplasmonics*

$$\epsilon\_r(\boldsymbol{\omega}) = 1 - \frac{\boldsymbol{\alpha}\_p^2}{\boldsymbol{\alpha}^2 + i\gamma(\boldsymbol{\alpha})\boldsymbol{\alpha} - \beta \boldsymbol{k}^2}, \quad \boldsymbol{\alpha}\_p^2 = \frac{n\boldsymbol{e}^2}{\epsilon\_0 m}, \quad \gamma = \frac{1}{\tau} \tag{3}$$

**2.2 Specific properties of surface plasmons**

*DOI: http://dx.doi.org/10.5772/intechopen.90842*

Various applications of plasmonic technology in development of nanoscale devices and systems are all based on the same fundamental properties of plasmons. These specific properties include field confinement, enhancement of local density

Confinement of electromagnetic fields in scales much smaller than the wavelength is the most crucial property of surface plasmon modes and can be defined in both parallel and orthogonal planes. Due to the high rate of loss, propagation length of the surface plasma waves in any direction is inversely related to the imaginary part of the wavenumber "1/Im(ksp)." This length for good plasmonic metals like gold and silver is limited to a few microns and is considered as the upper limit of confinement [11]. The lower limit of confinement is exerted by Fourier transform properties with considering a monochromatic field with frequency "ω" and wavenumber "k = ω/c" in the vacuum with far from any surface. It can be concluded that in the "x" direction "ΔxΔα ≥ 2π" in which α is the x component of the wavenumber. Therefore, the lower limit of field confinement is "2π/αmax = λ" which

of optical states, and ultrawide bandwidth and fast response [11].

*Nanoscale Plasmon Sources: Physical Principles and Novel Structures*

is also known as the diffraction limit. However, for surface plasmons, the

*u*^ � *f* ! *r* ! � � � �

to the mismatch between directions of transition dipole and the field.

*Veff* ¼

Ð *ε r* !� � *<sup>E</sup>* ! *r* ! � � � � �

*Max ε r*

!� � *<sup>E</sup>* ! *r* ! � � � � �

<sup>1</sup> <sup>þ</sup> <sup>4</sup>*Q*<sup>2</sup> *<sup>ω</sup>em*�*ω<sup>c</sup>*

�

tiny dimensions (much smaller than the wavelength) [11].

Γ*cav n*1Γ<sup>0</sup>

**63**

<sup>¼</sup> <sup>3</sup> 4*π*<sup>2</sup>

*Fp* <sup>¼</sup> <sup>3</sup> 4*π*<sup>2</sup>

*λem n*1 � �<sup>3</sup> *Q*

*λem n*1 � �<sup>3</sup> *Q*

*Veff*

*Veff*

wavenumber according to the dispersion relation (see **Figure 1**) can be much higher than "ω/c" which implies that surface plasmon modes can be confined in extremely

Enhancement of local density of optical states (LDOS) for surface plasmons can be investigated both near the metal surface and in a metallic nanoresonator. In a metallic nanoresonator, this effect which is also known as Purcell effect or

enhancement of spontaneous emission is the vital property of plasmonic nanolasers. Purcell factor (*Fp*) is defined by the ratio of decay rate due to the spontaneous emission in a cavity over the decay rate in the free space. It can be calculated by Fermi's golden rule in a two-level atomic system and expressed by Eq. (10) [24].

> � � � 2

*u*^ � *f* ! *r* ! � � � �

<sup>1</sup> <sup>þ</sup> <sup>4</sup>*Q*<sup>2</sup> *<sup>ω</sup>em*�*ω<sup>c</sup>*

� � � 2

*ωc* � �<sup>2</sup>

(10)

�

*ωc* � �<sup>2</sup> <sup>¼</sup> *Fp*

In which "Γ*cav*" is the decay rate in the cavity, "Γ0" is the decay rate in the free space, "*n*1" is the refractive index of the propagation medium, "*λem*" and "*ωem*" are the emission wavelength of the medium, "ωc" is the cavity resonance frequency, "*Q*" is the quality factor of the cavity, "*Veff*" is the effective mode volume of the propagating mode in the cavity, and the dot product of the nominator corresponds

In a dielectric microcavity despite the large quality factor, large mode volume results in infinitesimal Purcell factors, but nanoscale metallic resonators (the building block of a plasmonic nanolaser) provide a very small equivalent mode volume expressed by Eq. (11) which results in a large Purcell factor which is crucial for the nanolaser operation. Moreover, since the emission rate is proportional to the LDOS, the higher Purcell factor means the higher local density of optical states [24].

> � � � 2 *dr*

> > � � �

<sup>2</sup> � � (11)

where "*β*" is the Landau nonlocal parameter which becomes important for large values of wavenumber.

More precise treatment of surface plasmons can be done using the hydrodynamic model which includes solving Bloch equations, i.e., continuity, and Bernoulli and Poisson's equations simultaneously. According to Eqs. (4–6), one can describe collective oscillations of electrons in an arbitrary system using electron density (n) and hydrodynamic velocity (*v r*ð Þ¼� , *t* ∇*ψ*ð Þ *r*, *t* ) [23].

$$\frac{d}{dt}n(r,t) = \nabla n(r,t) = \nabla.[n(r,t)\nabla\psi(r,t)] \tag{4}$$

$$\frac{d}{dt}\varphi(r,t) = \frac{1}{2}\left|\nabla\varphi(r,t)\right|^2 + \frac{\delta G[n]}{\delta n} + \phi(r,t) \tag{5}$$

$$G[n] = \frac{3}{10} \left( 3\pi^2 \right)^{\frac{3}{5}} [n(r, t)]^{\frac{5}{3}}$$

$$
\nabla^2 \phi(r, t) = 4\pi n(r, t) \tag{6}
$$

In the general form, Bloch equations are nonlinear and quite difficult to solve. However, using the perturbation theory, one can find linearized equations of Eq. (7) which helped Ritchie and his team to find plasmon dispersion equation in Eq. (8) for the first time [23].

$$\begin{aligned} n(r,t) &= n\_0(r) + n\_1(r,t) + \dots \\ \boldsymbol{\nu}(\boldsymbol{r},t) &= \mathbf{0} + \boldsymbol{\nu}\_1(r,t) + \dots \\ \left\{ \begin{aligned} &\frac{d}{dt} n\_1(r,t) = \nabla. [n\_0(r) \nabla \boldsymbol{\nu}\_1(r,t)] \\ &\frac{d}{dt} \boldsymbol{\nu}\_1(r,t) = [\boldsymbol{\beta}(r)]^2 \frac{n\_1(r,t)}{n\_0(r)} + \boldsymbol{\phi}(r,t) \end{aligned} \right. \\ \boldsymbol{\Gamma}^2 \phi\_1(r,t) &= 4\pi n\_1(r,t) \\ \boldsymbol{\alpha}^2 &= \frac{1}{2} \left[ \boldsymbol{\alpha}\_p^2 + \boldsymbol{\beta}^2 k^2 + \beta k \sqrt{2\boldsymbol{\alpha}\_p^2 + \boldsymbol{\beta}^2 k^2} \right] \\ \frac{\partial k}{\partial \boldsymbol{\rho}\_p} &\ll 1 \to \boldsymbol{\rho} = \frac{\boldsymbol{\alpha}\_p}{\sqrt{2}} + \frac{\beta k}{2} \end{aligned} \tag{8}$$

The most accurate model for dealing with surface plasmons in atomic scales is solving Schrodinger's equation and calculating dynamical structure factor in Eq. (9) which is related to the oscillations of particle density in a many-particle system [23]:

$$S(r, r'; \alpha) = \sum\_{n} \delta \hat{\rho}\_{0n}(r\_1) \delta \hat{\rho}\_{n0}(r\_2) \delta(\alpha - E\_n + E\_0) \tag{9}$$

where the first two terms are elements of the operator "*ρ*(*r*)-*n*0(*r*)" relating the ground state "*ψ0*" with energy "*E0*" and "*δ*" is the Dirac function, "*n*0(*r*)" represents ground state density of particles, and "*ρ*(*r*)" is the particle density operator.

Using this model one can precisely calculate electron density profile in a manyelectron system like a metal. However, solving the required equations is not easy, and most often approximations like random phase approximation or timedependent density functional theory is used [23].
