Two Dimensional Magnetopolaritons and the Associated Landau Quantized Magnetoconductivity Tensor

Abstract

We address the magnetopolariton spectrum and magnetoconductivity tensor of a two dimensional plasma of nonrelativistic charge carriers subject to Landau quantization in a normal magnetic field. The analysis is carried out in the random phase approximation for linear electromagnetic response. Various regimes of magnetic field strength and wave number are considered. An exact integral representation of the magnetoconductivity tensor is derived and several expansions in terms of modified Bessel functions and Laguerre polynomials are obtained, as well as a low wave number power expansion. These results encompass the nondegenerate and degenerate statistical regimes as well as intermediate field and quantum strong field strengths. The two-dimensional magnetopolariton dispersion relation is formulated and solved for the electromagnetic normal modes, and local and nonlocal magnetopolaritons/plasmons are discussed, including nonlocal Bernstein modes, all subject to Landau quantization. The leading nonlocal corrections to the local modes are shown to exhibit de Haas-van Alphen oscillatory quantum structure in the degenerate statistical regime.

Appendices

15.4 Appendix 1: Alternative Formulation of the 2D Magnetoplasma Dispersion Relation

We consider a 2D plasma in a uniform, static magnetic field perpendicular to the plane of the plasma. The electrons are assumed mobile while the positive charges are taken to be fixed. From Gauss’ and Stokes’ Laws, one has the boundary conditions

$$\begin{aligned} E_\perp (0) = \frac{2\pi n^s}{\epsilon _m} \qquad \text {and} \qquad \widehat{z} \times \vec {H} (0) = \frac{2\pi \vec {J}^s}{c}, \end{aligned}$$

(15.4.1)

where $\perp , \, \parallel$ refer to field components perpendicular, parallel to the plasma surface at $z = 0$ and $n^s, \, \vec {J}^s$ are the surface charge and current densities respectively. For a monochromatic wave with frequency $\Omega$ and wavenumber $\vec {k}$, Maxwell’s curl equation yields

$$\begin{aligned} \vec {k} \times \vec {E}(0) = \frac{\Omega }{c} \vec {H}(0), \, \therefore \, \widehat{z} \times (\vec {k} \times \vec {E}(0)) = \frac{2\pi \Omega }{c^2} \vec {J}^s, \end{aligned}$$

(15.4.2)

and one has

$$\begin{aligned} E_\perp (0) \vec {k}_\parallel - \vec {E}_\parallel (0) k_\perp = \frac{2\pi \Omega }{c^2} \vec {J}^s. \end{aligned}$$

(15.4.3)

Eliminating $E_\perp (0)$ in terms of $n^s$ using (15.4.1) and employing current continuity $\Omega n^s - \vec {k}_\parallel \cdot \vec {J}^s = 0$, one obtains

(15.4.4)

where $k^2 = k_\perp ^2 + k_\parallel ^2 = \frac{\epsilon _m \Omega ^2}{c^2}$. One can then use Ohm’s Law to obtain

(15.4.5)

The dispersion relation for 2D plasma waves is obtained by setting the determinant of the bracketed expression in (15.4.5) to zero (compare with (15.1.16) in the text). One can also define a 2D dielectric tensor :

(15.4.6)

where $\epsilon _m$ is the surrounding bulk medium permittivity. From (15.4.5), the 2D plasma dispersion relations are

(15.4.7)

and with some simple manipulation, this can also be expressed as

(15.4.8)

15.5 Appendix 2: The Time-Ordered Exponential Time Development Operator

An operator $U(t, t_0)$ defining the time development of a state $\Phi (t)$ from an initial state $\Phi (t_0)$,

$$\begin{aligned} \Phi (t) = U(t, t_0) \Phi (t_0), \end{aligned}$$

subject to the initial condition $U(t_0, t_0) = 1$, satisfies the Schrödinger equation $(\hbar \rightarrow 1)$

$$\begin{aligned} i(\partial / \partial t) U(t, t_0) = H(t) U(t, t_0). \end{aligned}$$

Integration of this with respect to t from $t_0$ to t and applying the initial condition leads to an integral equation,

$$\begin{aligned} U(t, t_0) = 1 - i \int _{t_0}^{t} dt H(t) U(t, t_0), \end{aligned}$$

which can be solved by repeated iterations as

$$\begin{aligned} U(t, t_0)&= 1 - i \int _{t_0}^{t} dt_1 H(t_1) \left[ 1 - i \int _{t_0}^{t_1} dt_2 H(t_2) U(t_2, t_0) \right] \\&= 1 - i \int _{t_0}^{t} dt_1 H(t_1) + (-i)^2 \int _{t_0}^{t} dt_1 \int _{t_0}^{t_1} dt_2 H(t_1) H(t_2) + \dots \\&+ (-i)^n \int _{t_0}^{t} dt_1 \int _{t_0}^{t_1} dt_2 \dots \int _{t_0}^{t_{n-1}} dt_n H(t_1) H(t_2) \dots H(t_n) + \dots \, . \end{aligned}$$

The n’th integrand involves products of the time dependent Hamiltonian H(t) repeated n times with the highest time argument t, on the left, proceeding to the lower time arguments in succession with the lowest time argument $t_n$ on the right. This is a defacto time ordering of the Hamiltonian products, denoted by the “$+$” subscript. $H(t_1) \dots H(t_n) \equiv (H(t_1) \dots (t_n))_+$. It is important to recognize and respect this time ordering because time dependent Hamiltonians generally fail to commute at different times (ie: $[H(t), H(t')] \ne 0$).

In the special case of a time-independent Hamiltonian, the iteration series for $U(t, t_0)$ is just the usual time development operator $U(t, t_0) = e^{-i H(t - t_0)}$. In view of this, it has become conventional to symbolize the iteration series above (for a time dependent H(t)) as $U(t, t_0) = (\exp [-i \int _{t_0}^{t} H(t) dt])_+$, (a “time-ordered exponential”), but this has meaning only as the iteration series above with the time ordered Hamiltonian products under multiple time integrations.

Having identified the correct time development operator for a time dependent Hamiltonian, it is important to recognize that the upper limit of the time integration involved in (15.2.13) is imaginary, $\tau = -i\beta = -i / \kappa _B T$, so the time displacement involved in $<\vec {J}^s>^{i \tau }$ is also imaginary. This imparts an imaginary time periodicity property to the time t-dependence of functions such as $<\vec {J}^s (t)>^{i \tau }$ of (15.2.13) and associated nonequilibrium Green’s functions, with imaginary period $\tau = -i\beta$: This is discussed in detail in [21], sections 7.4 and 9.1. It should be borne in mind that these considerations are in the context of the many-body-problem, which involves a constant particle number such that $[N, H(t)] \equiv 0$ for all times t: This means that the appearance of N in the grand canonical ensemble averaging process is essentially “inert” with respect to the time development process in our present considerations. (If N were not a fixed constant in time, $[N, H(t)] \ne 0$, the situation would be more complicated.)

15.6 Appendix 3: Semiclassical Model

Although our principal interest is in quantum magnetic field effects due to Landau quantization in the 2D magnetopolariton spectrum, it is of interest to explore the precise meaning of the semiclassical model as defined by Chiu and Quinn [30]: That definition neglects the dHvA oscillatory terms embedded in the quantity $\delta$ given by ($r =$ positive integers $\ne 0$)

$$\begin{aligned} \delta \equiv \frac{\pi }{\beta \zeta } \sum _{r \ne 0} (-1)^r \frac{\cos (2\pi r \zeta / \hbar \omega _c)}{\sinh (2\pi ^2 r / \hbar \omega _c \beta )}. \end{aligned}$$

(15.6.1)

These oscillatory terms have maximum amplitude at zero temperature $(\hbar \omega _c \beta>> 1)$, taking the form

$$\begin{aligned} \delta _{T=0} = \frac{\hbar \omega _c}{2\pi \zeta } \sum _{r \ne 0} \frac{(-1)^r}{r} \cos (2\pi r \zeta / \hbar \omega _c), \end{aligned}$$

(15.6.2)

so that its vanishing in the degenerate limit further requires that $\hbar \omega _c / \zeta \rightarrow 0$. In the nondegenerate statistical regime, $\hbar \omega _c \beta<< 1$, (15.6.1) reduces to

$$\begin{aligned} \delta _{T \rightarrow \infty } = \frac{2\pi }{\beta \zeta } \sum _{r \ne 0} (-1)^r \cos (2\pi r \zeta / \hbar \omega _c) e^{-2\pi ^2 r / \hbar \omega _c \beta }, \end{aligned}$$

(15.6.3)

which vanishes exponentially with temperature. While this “semiclassical” model of the magnetopolariton spectrum is of some interest, it is antithetical to the arduous task of a fully quantum mechanical analysis of Landau quantization effects in a magnetic field, and it ignores valuable physical information embedded in the structure of dHvA oscillations.