Quantum theory of plasmons at metallic spherical surface

In the present work the functional integral technique was applied to the study of the collective excitations of electron gas at a spherical surface. The quanta of these collective excitations are the surface plasmons. Starting from the fundamental principles of the quantum theory, the generating functionals of the many-electron system in a spherical surface were introduced in two different cases, when the electron–electron Coulomb interaction is absent or is present. By means of the Hubbard–Stratonovich transformation, the generating functional of the electron system in the presence of the electron–electron interaction was transformed into that of a scalar field, the effective action of this scalar field was established and the plasmon field was introduced. Then the expression of the Hamiltonian of the quantum field of plasmons was derived and the system of algebraic equations determining the energies of plasmons was established.


Introduction
Plasmons are quasiparticles exhibiting elementary excitations in the electron gas. They appear as resonances in the collective oscillations of electron gas under the action of a monochromatic electromagnetic field with a suitable frequency called the plasma frequency ω .
p As a rough approximation we can calculate ε ω ( ) by using the classical equation of motion of electrons in a free electron gas under the action of the electrical field of a monochromatic radiation and derive a simple formula for the plasma frequency ω .
p We use the unit system such that the electrostatic potential V x ( ) created by a charge distribution with the charge density ρ x ( ) satisfies the Poisson equation x ( ) 4 ( ). 2 Then the expression for plasma frequency is determined by the following formula where e and M are the electron charge and mass, N is the electron density in the electron gas [1]. Rigorous derivation of the formula for the dielectric function ε ω k ( , ) p depending not only on the frequency ω, but also on the wave vector k, required the application of the quantum theory of many-body systems [2][3][4][5]. From the condition where E F is Fermi level energy of the electron gas [1,6,7].
In the previous works [34,35] we have proposed to apply the functional integral technique in quantum field theory for deriving the effective Hamiltonian of plasmons from the fundamental expression of the Hamiltonian of the electron gas. As a clear demonstration of the general method we have considered infinite homogeneous three-dimensional electron gas with Coulomb electron-electron interaction and rederived the results which have been known in conventional theories. In the present work we apply this general method to another electron system-that of interacting electrons at a spherical surface. The quantum field theory of collective excitations in this system will be established. Its quanta are the SPRs in metallic spherical nanoparticles.
Section 2 is devoted to the elaboration of the suitable functional integral technique. The canonical quantization procedure is presented in section 3. As the final result, the explicit expression of the canonical quantum field of SPRs will be established. Section 4 is the conclusion.

Functional integral technique
Let us apply the functional integral technique elaborated in the previous works [34,35] to the system of electrons moving on the surface S of a sphere with the center at the point O and the radius R. We chose the center O to be the origin of the coordinate system. The angular coordinates of each point M on the surface S are the same as those of the unit vector and each function f on the surface S can be considered as a function of n and denoted f n ( ). The generating functional of the electron system on the surface S contains the surface integrals over this surface. In the calculation of the integral of some function f n ( ) on the surface S it is convenient to use the following identity x , and to replace each surface integral by a corresponding volume integral over the whole three-dimensional space with the measure Without the loss of generality we can omit the spin index of electron wave function and consider electron as a spinless x , the Grassmann variables describing the electron field on spherical surface S and its Hermitian conjugate. Introducing corresponding Grassmann parameters η ( ) t , where Ĥ 0 is quantum-mechanical Hamiltonian of the free electron on the spherical surface S: The generating functional (1) has following explicit form  In particular, it is easy to derive the Wick theorem expressing the 2n-point Green function ,C onsider now the interacting electron system, the electron-electron interaction being the Coulomb interaction, and denote ′ − V x x ( ) the Coulomb energy potential of the system of two electrons located at two points x and ′ x . This interacting electron system has the following functional integral We can linearize the interaction Hamiltonian with respect to the electron density ψ ψ ,   we transform the functional integral Z of the electron system into the new form Following the method presented in previous work [34,35] and calculating the functional integral over the     where . (17) 00 From formula (7) for ε − ( ) This means that ( ) n x x is the electron density at the spherical surface.
In the second order approximation with respect to the , ; .
The integral transformation with the kernel ′ − ′ A t t x x ( , ; ) in the r.h.s of equation (19) is symmetric in the sense that From the principle of the extreme action 0 0 For this purpose we perform the Fourier transformation In terms of the Fourier transform φ ω does not depend on t, equation (25). It describes the background state of the electron system on the spherical surface. The difference   ) we use also that of i t i t and obtain  Introducing the Fourier transformation of the Coulomb potential

Conclusion
In the present work we have elaborated the quantum theory of plasmons at the metallic spherical surface by means of the functional integral technique. Starting from the principles of electrodynamics and quantum mechanics we have established the effective action of the scalar field describing the quantum fluctuation of the electron density in the harmonic approximation. Then we have derived the system of algebraic equations determining the discrete spectrum of energies (angular frequencies) of the surface plasmons. The solution of this system of equations requires the application of suitable simulation calculations.