**3. Physical models for analyzing plasmonic nanolasers**

In order to analyze a plasmonic nanolaser, we need theoretical tools for describing the carrier-plasmon dynamics in the cavity. In this section, several models are discussed with different precision, but every method has its own limitations and should be used for a specific category of devices or a certain purpose.

#### **3.1 Plasmon cavity quantum electrodynamics (PCQED)**

Interaction of electron-hole pairs and plasmons in a nanocavity is the fundamental mechanism in any plasmonic nanolaser. As an example of this interaction, energy transfer diagram in a quantum well based nanolaser is illustrated in **Figure 3**.

This interaction should be treated similar to light-matter interaction in a laser cavity by the cavity quantum electrodynamics (CQED). However, in a plasmonic nanocavity due to Purcell enhancement of spontaneous emission and nanoscale dimensions and considerable loss and dispersion, there should be considerable differences that lead to a new cavity electrodynamic model for plasmonic cavities or PCQED [14].

The key difference between CQED and PCQED is in the method of controlling the interaction of electromagnetic fields with the medium. One of the most fundamental differences between them is the enhancement of spontaneous emission rate in a plasmonic cavity by the Purcell factor. The physical structure of the cavity affects the spectral characteristics of the plasmonic mode oscillations and results in a difference in the local density of optical states and the Purcell factor based on the designer's will. In other words, CQED controls interaction dynamics by their relationship with the quality factor of the resonator, while in PCQED dynamic of interactions is controlled by the Purcell factor. In dielectric microcavities, the quality factor is very high (even 1010), while modal volume is limited to the refraction limit (few microns in each dimension), and Purcell enhancement does not occur. On the other hand, for plasmonic nanocavities because of intense mode confinement, equivalent mode volume is far smaller than the diffraction limit and results in considerable Purcell factor and density of state manipulation [14].

**Figure 3.** *Energy transfer diagram of a quantum-well based nanolaser.*

Moreover, in PCQED loss and dispersion are critical factors and are necessary for correct modeling. Finding precise quantum mechanical models for these phenomena in plasmonic nanocavities still needs more research. However, we can use the photon/plasmon analogy and developed methods and tools of the photons like the density of state matrix and decay channels for estimating quantum mechanical behavior of plasmonic nanocavities [14].

## **3.2 Quantum mechanical atomic-scale model**

oscillations which have both wave and particle properties, and the corresponding

**3. Physical models for analyzing plasmonic nanolasers**

**3.1 Plasmon cavity quantum electrodynamics (PCQED)**

should be used for a specific category of devices or a certain purpose.

considerable Purcell factor and density of state manipulation [14].

*Energy transfer diagram of a quantum-well based nanolaser.*

Polariton is a joint state of light and matter introduced by Hopefield for providing a quantum model for the polarization field describing the response of matter to light [30]. Based on Hopfield's model, Ritchie and Elson proposed the first quantized description of surface plasma waves called Surface Plasmon Polariton or SPP. However, Hopefield's model did not consider the scattering and loss in the metal and effects of valance electrons and later Huttner and Barnett propose a model based on the Hopefield model including dispersion and loss, and recently a macroscopic quantization model based on Green's functions has also been

In order to analyze a plasmonic nanolaser, we need theoretical tools for describing the carrier-plasmon dynamics in the cavity. In this section, several models are discussed with different precision, but every method has its own limitations and

Interaction of electron-hole pairs and plasmons in a nanocavity is the fundamental mechanism in any plasmonic nanolaser. As an example of this interaction, energy transfer diagram in a quantum well based nanolaser is illustrated in **Figure 3**. This interaction should be treated similar to light-matter interaction in a laser cavity by the cavity quantum electrodynamics (CQED). However, in a plasmonic nanocavity due to Purcell enhancement of spontaneous emission and nanoscale dimensions and considerable loss and dispersion, there should be considerable differences that lead to a new cavity electrodynamic model for plasmonic cavities or

The key difference between CQED and PCQED is in the method of controlling the interaction of electromagnetic fields with the medium. One of the most fundamental differences between them is the enhancement of spontaneous emission rate in a plasmonic cavity by the Purcell factor. The physical structure of the cavity affects the spectral characteristics of the plasmonic mode oscillations and results in a difference in the local density of optical states and the Purcell factor based on the designer's will. In other words, CQED controls interaction dynamics by their relationship with the quality factor of the resonator, while in PCQED dynamic of interactions is controlled by the Purcell factor. In dielectric microcavities, the quality factor is very high (even 1010), while modal volume is limited to the refraction limit (few microns in each dimension), and Purcell enhancement does not occur. On the other hand, for plasmonic nanocavities because of intense mode confinement, equivalent mode volume is far smaller than the diffraction limit and results in

quanta (plasmon) is a boson [27].

published [27].

*Nanoplasmonics*

PCQED [14].

**Figure 3.**

**66**

Modeling phenomena like quantum fluctuations, spectral narrowing, coherency, threshold behavior, and precise dynamic analysis of plasmon nanolasers need an atomic-scale quantum mechanical model. However, in order to find closedform equations, several simplifications are necessary, and thus this model just provides a theoretical tool for investigating fundamental properties of plasmonic nanolasers.

To do so, consider an N-atom system in a low-quality factor nanocavity in which the decay rate of the cavity (κ) is the fastest decay rate of this system. This condition is called the "bad cavity assumption" [28]. Therefore, resonator mode can be adiabatically eliminated, and the system state is totally determined by "N" active atoms. Considering two energy levels for each atom which are coupled to a cavity with resonance frequency (ω) and plasmon lifetime (1/2κ), one can describe the interactions between the atoms and field by Tavis-Cumming Hamiltonian of Eq. (17) [28].

$$H\_{\rm AF} = i\hbar \mathfrak{g} \left( \mathfrak{a}^{\dagger} \mathfrak{J}\_{-} - \mathfrak{a} \mathfrak{J}\_{+} \right) \tag{17}$$

In which "*g*" is the coupling factor which is identical for all of the atoms, "*a*" and "*a*† " are annihilation and creation operators of plasmons, respectively, and "*Jα*" is the operator of collective atomic oscillations in the "*α*" direction and can be defined by Eq. (18) in which "*σjx*" and "*σjy*" are Pauli matrices [28].

$$J\_a = \sum\_{j=1}^{N} \sigma\_{ja}, \qquad a = \{\mathbf{x}, y, z\}, \qquad \sigma\_{j\pm} = \frac{\sigma\_{j\mathbf{x}} \pm \sigma\_{j\mathbf{y}}}{2} \tag{18}$$

Using atomic density operator "*ρ*" in a quantum system with state vector "*ψ*" and by considering the Hamiltonian of Eq. (17), Schrodinger's equation leads us to the dynamic equation of Eq. (19) in which "*γ*↑" is the pumping rate and "*γ*↓" is the spontaneous emission rate and "*γp*" is the dephasing rate of oscillating atoms [28]. The last term describes interaction of active atoms through the cavity mode [28].

$$\begin{split} \dot{\rho} &= -i\frac{1}{2}\alpha[J\_x,\rho] + \frac{\gamma\_\uparrow}{2} \left( \sum\_{j=1}^N 2\sigma\_{j+}\rho\sigma\_{j-} + \frac{1}{2}I\_x\rho + \frac{1}{2}\rho I\_x - N\rho \right) \\ &+ \frac{\gamma\_\downarrow}{2} \left( \sum\_{j=1}^N 2\sigma\_{j-}\rho\sigma\_{j+} - \frac{1}{2}I\_x\rho - \frac{1}{2}\rho I\_x - N\rho \right) + \frac{\gamma\_p}{2} \left( \sum\_{j=1}^N 2\sigma\_{j\mp}\rho\sigma\_{jx} - N\rho \right) + \frac{g^2}{\kappa} (2I\_-\rho) . \end{split} \tag{19}$$

After several mathematic manipulations on Eq. (19) and by defining "*XN <sup>ξ</sup>*, *<sup>ξ</sup>* <sup>∗</sup> ð Þ¼ , *<sup>η</sup> tr <sup>ρ</sup>ei<sup>ξ</sup>* <sup>∗</sup> *<sup>J</sup>*<sup>þ</sup> *<sup>e</sup><sup>i</sup>ηJz <sup>e</sup><sup>i</sup>ξJ*� � �" and its Fourier transform "*P v* <sup>~</sup> , *<sup>v</sup>* <sup>∗</sup> ð Þ , *<sup>m</sup>* " as the atomic polarization operator, we can conclude Eq. (20) as a closed-form dynamic equation describing the system by collective atomic operators [28].

*∂P*~ *<sup>∂</sup><sup>t</sup>* <sup>¼</sup> *L v*, *<sup>v</sup>* <sup>∗</sup> , *<sup>m</sup>*, *<sup>∂</sup> ∂v* , *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> , *<sup>∂</sup> ∂m* � �*P*<sup>~</sup> *<sup>L</sup>* <sup>¼</sup> *<sup>γ</sup>*<sup>↑</sup> 2 *e* �<sup>2</sup> *<sup>∂</sup> <sup>∂</sup><sup>m</sup>* � 1 � �ð Þþ *<sup>N</sup>* � *<sup>m</sup> ∂*4 *<sup>∂</sup>v*<sup>2</sup>*∂<sup>v</sup>* <sup>∗</sup> <sup>2</sup> *<sup>e</sup>* 2 *∂ <sup>∂</sup><sup>m</sup>*ð Þþ *<sup>N</sup>* <sup>þ</sup> *<sup>m</sup>* <sup>2</sup>*<sup>N</sup> <sup>∂</sup>*<sup>2</sup> *∂v∂v* <sup>∗</sup> � �þ þ *γ*↑ 2 2*e* �<sup>2</sup> *<sup>∂</sup> <sup>∂</sup><sup>m</sup>* � 1 þ 2*∂*<sup>2</sup> *∂v∂v* <sup>∗</sup> � � *∂ ∂v v* þ *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> *<sup>v</sup>* <sup>∗</sup> � �þ þ *γ*↓ <sup>2</sup> *<sup>e</sup>* 2 *∂ <sup>∂</sup><sup>m</sup>* � 1 � �ð Þþ *<sup>N</sup>* <sup>þ</sup> *<sup>m</sup> ∂ ∂v v* þ *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> *<sup>v</sup>* <sup>∗</sup> � �þ þ*γ<sup>p</sup> ∂ ∂v v* þ *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> *<sup>v</sup>* <sup>∗</sup> <sup>þ</sup> *∂*2 *<sup>∂</sup>v∂<sup>v</sup>* <sup>∗</sup> *<sup>e</sup>* 2 *∂ <sup>∂</sup><sup>m</sup>*ð Þ *N* þ *m* � � <sup>þ</sup> *<sup>i</sup><sup>ω</sup> <sup>∂</sup> ∂v v* þ *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> *<sup>v</sup>* <sup>∗</sup> � �þ þ *g*2 *κ* 2 1 � *e* �<sup>2</sup> *<sup>∂</sup> ∂m* � �*vv* <sup>∗</sup> � *<sup>∂</sup> ∂v vm* þ *∂ <sup>∂</sup><sup>v</sup>* <sup>∗</sup> *<sup>v</sup>* <sup>∗</sup> *<sup>m</sup>* � � þ *∂*2 *<sup>∂</sup>v*<sup>2</sup> *<sup>v</sup>*<sup>2</sup> <sup>þ</sup> *∂*2 *<sup>∂</sup><sup>v</sup>* <sup>∗</sup> <sup>2</sup> *<sup>v</sup>* <sup>∗</sup> <sup>2</sup> � � (20)

**3.3 Mean-field atomic-scale model (optical Bloch)**

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

*Nanoscale Plasmon Sources: Physical Principles and Novel Structures*

In this model, the classical wave equation is applied to the electric field, while plasmons are assumed quantized, and using Fermi's golden rule, a kinetic equation describing the behavior of plasmons is derived. This model proved to be consistent with the previous model above the threshold while having the advantage to be used for quantum dot gain mediums. Also, Einstein's spontaneous and stimulated emission coefficients can be calculated from this model. Therefore, it can prove the positive effect of increased spontaneous emission due to the Purcell effect on the stimulated emission of the nanolaser. However, similar to the model described in Section 3.2, it needs some simplifying assumptions and works in atomic scales [12]. In this model, the nanoscale system consists of a metal layer with permittivity "*ε*(*ω*)" over a dielectric with permittivity "*εh*." The classic eigenvalue wave equation

can be written for the plasmonic eigenmodes according to Eq. (23) [12].

!� � � *sn* h i∇*<sup>φ</sup> <sup>r</sup>*

where "*θ*(*r*)" is equal to "1" inside the dielectric and "0" inside the metal. Corresponding eigenvalues to the nth mode can be derived by Eq. (24) where "*Ωn*" is the complex frequency of the nth eigenmode in which the real part is equal to the resonant frequency of nth mode "*ωn*" and the imaginary part corresponds to the

*<sup>s</sup>*ð Þ¼ *<sup>Ω</sup><sup>n</sup> sn*, *<sup>s</sup>*ð Þ� *<sup>ω</sup>* <sup>1</sup> � *ε ω*ð Þ

, *s* 0

For the times shorter than the plasmon lifetime "τ<sup>n</sup> = 1/*γn*," corresponding Hamiltonian to the electric field of the quantized surface plasmons can be expressed by Eq. (26) in which "*T*" is the integration time and should satisfy

> *<sup>d</sup><sup>ω</sup> <sup>E</sup>* ! *r* !, *ω* � �*<sup>E</sup>* ! *r* !, �*ω* � � *dω*

Using the extension of the electric field based on the quantized eigenmodes of the system, one can write Eq. (27), and Hamiltonian of Eq. (26) can be written as Eq. (28) which has the standard form of a quantum mechanical harmonic

*n*

ℏ*ω<sup>n</sup> a*†

ffiffiffiffiffiffiffiffiffiffiffi 2*π*ℏ*sn εhs*<sup>0</sup> *n*

*nan* þ 1 2

*φ<sup>n</sup> r* !� �*<sup>e</sup>*

s

*εh* � ��<sup>1</sup>

> � � � � *ω*¼*ω<sup>n</sup>*

> > <sup>2</sup>*<sup>π</sup> <sup>d</sup>*<sup>3</sup>

�*γnt ane*

� � (28)

*r* (26)

�*iωnt* <sup>þ</sup> *<sup>a</sup>*† *ne <sup>i</sup>ωnt* � �

(27)

*<sup>n</sup>* � *<sup>d</sup>* Re ½ � *<sup>s</sup>*ð Þ *<sup>ω</sup><sup>n</sup> dω*

!� � <sup>¼</sup> <sup>0</sup> (23)

(24)

(25)

∇*: θ r*

Assuming "*γ<sup>n</sup>* << *ωn*" we can write Eq. (25).

*<sup>H</sup>* <sup>¼</sup> <sup>1</sup> 4*πT*

!, *t*

*<sup>γ</sup><sup>n</sup>* <sup>¼</sup> Im½ � *<sup>s</sup>*ð Þ *<sup>ω</sup><sup>n</sup> s*0 *n*

�ð∞

*d ωε r* !,*ω* h i � �

> !, *t* � � <sup>¼</sup> <sup>X</sup>

*<sup>H</sup>* <sup>¼</sup> <sup>X</sup> *n*

�∞

� �, *<sup>ϕ</sup> <sup>r</sup>*

plasmon decay rate "*γn*" [12].

"*τn*>> *T*>> 1/*ωn*."

oscillator.

� � � �∇*<sup>ϕ</sup> <sup>r</sup>*

*E* ! *r* !, *t*

**69**

It should be noticed that Eq. (20) has not an analytical solution in this form and should be linearized or be solved numerically. In order to find a more familiar form of the dynamic rate equations, we should use linearization. By defining plasmon mode using dimensionless polarization "*σ*" and number of carriers by normalized population inversion "*n*," one can write Eq. (21) for a plasmonic nanolaser with N active atoms and the mentioned assumptions [28]:

$$\begin{aligned} d\left(\frac{\sigma}{n\_{\star}}\right) &= -\left(1-\xi\rho\frac{n}{n\_{\star}}\right)\frac{\sigma}{n\_{\star}}, \sigma = tr\left[\rho I\_{-}e^{i\alpha t}\right]/N, n = tr[\rho I\_{z}]/N\\ d\left(\frac{n}{n\_{\star}}\right) &= \frac{n}{\Gamma T\_{1}} - 1 \quad 4\wp\left|\frac{\sigma}{n\_{\star}}\right|^{2}\\ T\_{1} &= \frac{1}{\chi\_{\uparrow} + \chi\_{\downarrow}}, n\_{i} = \frac{\chi\_{\uparrow} - \chi\_{\downarrow}}{\chi\_{\uparrow} + \chi\_{\downarrow}}, \Gamma = \chi\_{p} + \frac{1}{2T\_{1}}\\ \xi \otimes = \xi \rho\_{0} n\_{i}, \quad \xi \rho\_{0} = \frac{N\mathfrak{g}^{2}}{\kappa\Gamma} \end{aligned} \tag{21}$$

where for threshold parameter, "℘>1" stimulated oscillations are dominant and for "℘<1" nanolaser is working in the subthreshold region, and generated plasmons are not coherent. Using this method, quantum fluctuations of plasmon and carrier numbers in the cavity even for a few numbers of plasmons can be estimated. Eventually, you can find first- and second-order correlation functions of the generated plasmons over time for above threshold pumping and the resulting linewidth "D" in Eq. (22). Significant linewidth narrowing with respect to the natural broadening "Γ" for "℘>1" implies proper laser operation, and time damping quantum fluctuations can be seen from the second-order correlation function [28].

$$\begin{aligned} \mathcal{g}\_{>}^{(1)}(\tau) &= e^{-(i\alpha+D)\tau} \left[ 1 - DT\_1 + DT\_1 e^{\frac{\tau}{T\_1}} \cos \left( \frac{\sqrt{2\Gamma T\_1 (\wp - 1)\tau}}{T\_1} \right) \right] \\ \mathcal{g}\_{>}^{(2)} &= 1 + 4DT\_1 e^{\frac{\tau}{T\_1}} \cos \left( \sqrt{2\Gamma T\_1 (\wp - 1)} \frac{\tau}{T\_1} \right) \\ \mathcal{D} &= \gamma\_p \frac{\Gamma T\_1}{N} \frac{\wp\_0 (\wp\_0 + 1)}{\wp - 1} \end{aligned} \tag{22}$$
