Surface Plasmon Polaritons at Terahertz Frequencies on Metal and Semiconductor Surfaces

Surface Plasmon Polaritons at Terahertz
Frequencies on Metal and Semiconductor Surfaces

by
Jörg Saxler

Diploma Thesis in PHYSICS
presented to the
Faculty of Mathematics, Computer Sciences, and Natural Sciences of Aachen University, Aachen, Germany

November 2003

performed in the Institute of Semiconductor Electronics Faculty of Electrical Engineering

Contents

1 Introduction ... 1
1.1 Electromagnetic interface excitations ... 1
1.2 Surface plasmon polariton ... 3
1.3 Terahertz frequency range ... 5
1.4 Conceptual formulation of this thesis ... 6

2 Theory of Surface Plasmon Polaritons ... 9
2.1 Classical eld equations ... 10
2.2 Surface plasmon polaritons at a single interface system ... 11
2.3 Surface plasmon polaritons at a two interface system ... 17
2.4 The Drude model ... 20
2.4.1 A classical model for metals ... 20
2.4.2 Modi ed Drude model for doped semiconductors ... 22
2.4.3 Optical properties of investigated materials ... 24
2.4.4 Limitations of the Drude model ... 25
2.5 Numerical results for single and two interface systems ... 26
2.5.1 SPPs at gold and semiconductor surfaces ... 26
2.5.2 SPPs at dielectric covered gold surfaces ... 29
2.6 Coupling between free electromagnetic waves and surface plasmon polaritons ... 33
2.6.1 Aperture coupling ... 34
2.6.2 Prism coupling ... 35
2.6.3 Grating coupling ... 37

3 Experimental Setup ... 39
3.1 Terahertz time-domain SPP spectroscopy ... 39
3.2 Sample fabrication ... 43

4 Experimental Results 45
4.1 Coupling of Free Electromagnetic Radiation to Surface Plasmon Polaritons ... 45
4.1.1 Aperture coupling ... 45
4.1.2 Prism coupling ... 50
4.2 Study of dielectric coated gold surfaces ... 52
4.3 Study of semiconductor surfaces ... 56
4.3.1 SPP decay perpendicular to the semiconductor surface into air ... 57
4.3.2 SPP damping along the propagation direction ... 61

Bibliography ... 67

Summary and outlook ... 71

Acknowledgements ... 73

Chapter 1
Introduction
This thesis presents the rst experimental study of the propagation characteristics and eld distribution of surface plasmon polaritons (SPPs) at terahertz (THz) frequencies. A measurement setup has been designed which allows the generation, demonstration and systematic investigation of SPPs at at surfaces of various materials. In this chapter a general introduction is given which comprises the di erent concepts that are involved in the presented experiments. After giving a general introduction to electromagnetic interface excitations in section 1.1, a qualitative account of the SPP is found in 1.2. In section 1.3 we discuss the peculiarity of the THz frequency range in which experiments have been carried out, followed by the conceptual formulation of this thesis, including a short summary of the subsequent chapters (section 1.4).

1.1 Electromagnetic interface excitations
A very simple example for an interface excitation is a water wave traveling across the glassy surface of a lake on a calm day. In this case, an interface is de ned by the water surface and the excitation consists of a mechanical up and down movement of water molecules. This results in a non-zero value of the water molecules′ kinetic energy which can therefore defy the earth′s gravitational force and perform an oscillatory motion.

Similarly, the periodic movement of ions or electrons close to the surface of a solid (typically an ionic crystal or a metal) can create an interface polarization. As a consequence, ions or free charge carriers in the solid are subject to restoring Coulomb forces that play the same role as the gravitational force does for the water molecules. Similarly, the Coulomb forces lead to an acceleration of charge and hence to an oscillatory motion. To complete the analogy, the periodic movement of polarization constitutes the source for an electromagnetic eld, i.e., an electromagnetic wave instead of a water wave, which is bound to the solid′s surface and propagating along it.

More general, an electromagnetic surface excitation has its physical origin in a mechanical displacement of charge carriers, atoms or molecules. This displacement leads to the formation of a time-dependent polarization P or magnetization M and the creation of an associated time-dependent electromagnetic eld close to the interface. Hence, mechanical and electromagnetic excitation are not independent from but coupled to each other. This coupled state is often referred to as an interface polariton where the expression polariton is supposed to emphasize the presence of the electromagnetic eld outside the solid although, of course, there must also be a non-vanishing eld inside the solid. There exist many di erent kinds of interface polaritons depending on the nature of the excitation inside the solid, e.g., magnonpolaritons, phonon-polaritons or surface plasmon polaritons [1].

Regarding the solid, a complete treatment of the problem requires the solution of a set of coupled eld equations (Maxwell′s equations) with an appropriate choice of boundary conditions. In this thesis, only systems are considered in which the boundary conditions allow a straightforward mathematical solution. In particular, we will only focus on the case of a perfectly at and parallel set of maximal two interfaces. Such a system, if formally treated as a single, inhomogeneous medium, is sometimes called a strati ed medium. The permittivity " of strati ed media can hence depend only on one space variable. Due to the boundary conditions, the possible solutions of Maxwell′s equations may be grouped into two di erent families namely s and p polarized electromagnetic waves, commonly referred to as electromagnetic surface modes. The electric eld of s-polarized electromagnetic waves is parallel to the interface whereas in the p-polarized case it lies within the plane of incidence. As will be discussed in further detail in chapter 2, in the special case of a surface plasmon polariton only p-polarized solutions will yield a propagating surface wave because only then a charge accumulation corresponding to an interface polarization can occur.

Surprisingly, solid state systems supporting surface polaritons were undergone rigorous electrodynamic treatment only over half a century after Maxwell published his Treatise on Electricity and Magnetism in 1873. This did not, however, prevent scientists like Zenneck and Sommerfeld from elaborating practical concepts which made use of the|at the time already noticed|existence of electromagnetic surface excitations without seeing into their actual physical nature.¹

This changed when Lifshitz and Rosenzweig [3] successfully predicted in 1948 interface excitations at the surface of ionic crystals. The surface plasmon polariton entered the stage about a decade later and shall be presented in the following section.

1.2 Surface plasmon polariton
Surface plasmon polaritons are of interest to a large variety of scientists such as physicists, biologists and chemists. They were recognized in the scienti c community thanks the theoretical work of Ritchie [4] who was the rst to show that SPPs arise as a formal solution of Maxwell′s equations under speci c conditions. A SPP is a collective charge excitation at an interface formed by a conductor and a dielectric as schematically depicted in Fig. 1.1. This excitation can be interpreted as an electromagnetic wave which is trapped to the interface because of the presence of free charge carriers provided by the conductor. Therefore, inside the conductor this excitation has a plasma-like character while inside the dielectric it resembles more a free electromagnetic wave. The term surface plasmon polariton intends to reect this double sided trait.

[...]

1In his work About the propagation of waves in wireless telegraphy [2] Sommerfeld approached the problem of surface waves from a rather phenomenological point of view. He found out that for radio waves with a wavelength λ = 2km propagating along the earth′s surface a relatively undisturbed propagation \of no less than 400km" occurs for undergrounds such as sea water and wet soil.