Surface Waves Everywhere

M. I. Dyakonov

Laboratoire Charles Coulomb, Université Montpellier, CNRS, Montpellier, France

1 Introduction

By “surface waves” one means a special kind of waves that propagate at the interface between two different media. There exists a large variety of such waves, which are interesting on their own, and sometimes have also practical importance and technological applications. This article presents a brief and nonexhaustive review of this vast subject, designed as an introduction to the field for nonspecialists. Similarities between surface waves of completely different origin are outlined. I am concerned with the physical picture and avoid math as strongly as possible (no differential equations and no boundary conditions), as well as many interesting details. Sometimes I omit numerical coefficients. My own contributions to this field are also presented.

It should be understood that the material in each section of this article is a subject of many books and hundreds, if not thousands, of journal publications, which the interested reader should address for more information.

2 Water waves

These are certainly the first kind of surface waves that mankind has encountered. However, our prehistoric ancestors did not yet realize that waves are characterized by the frequency ω and the wave vector k = 2π/λ, where λ is the wavelength. The relation between ω and k is called the dispersion law. We now establish the dispersion laws for water waves from considerations of dimensionality only.

However, first we must distinguish between several kinds of water waves, differing in the nature of the return force that drives the oscillatory motion: this can be either gravity or surface tension.

Also we must distinguish between the cases of deep and shallow water. To do this, we must compare the water depth h with some other length. There are two characteristic lengths in our problem: the wavelength λ and the wave amplitude. If we are concerned with the linear regime, the amplitude is irrelevant (so long as it is small compared to the wavelength). Thus, in the linear regime we must compare h and λ, so that one has deep water when and shallow water if .

• Gravitational waves

The return force is provided by gravity. In this case, we have the following material for constructing the link between frequency ω with dimension [s⁻¹] and the wave vector k with dimension [cm⁻¹]: the freefall acceleration g [cm/s²] and water depth h [cm].

Shallow-water waves (h λ)

Now, independently of the wavelength, the wave velocity is v ∼ (gh)^1/2 and the dispersion law is linear: ω = vk, as for sound waves. This formula gives a simple estimate for the time needed for a perturbation of the ocean surface caused by an earthquake to cross the Pacific and cause a tsunami on a distant coast.

The so-called “shallow earthquakes” have their focus up to 70 km deep in the Earth crust (say 50 km), which is much greater than the ocean depth (∼3 km). Thus, the ocean surface will be perturbed on a scale of 50 km and the resulting surface wave will be a shallow water wave, however strange it may sound when applied to the Pacific Ocean. This wave will propagate with a velocity ∼(gh)^1/2, where h ∼ 3 km. This is about the velocity of an aircraft!

Note that gravity waves do not depend on the density of the liquid, because, as Galileo showed by dropping stones from the Pisa tower, the motion in the gravity field does not depend on the mass of the object.

• Capillary waves

Here, the return force is due to surface tension σ. As before, we construct the dispersion law from dimensionality considerations only. This time our ingredients are surface tension σ [g/s²] and water density ρ [g/cm³]. Then the only possible form of the dispersion law is ω ∼ (σk³/ρ)^1/2, which leads to the wave velocity v ∼ (σk/ρ)^1/2 ∼ (σ/ρλ)^1/2. Note that in contrast to gravitational waves, now the wave velocity decreases with increasing wavelength.

Thus, generally the dependence of velocity on wavelength is nonmonotonic: there exists a minimal velocity v_MIN at a wavelength λ_MIN at which the roles of gravity and surface tension are roughly equal. For water, λ_MIN is about 2 cm, and the corresponding velocity v_MIN is about 30 cm/s.

• Excitation of water waves by wind

The wind can excite water waves with phase velocities that are inferior to the wind velocity. The previous analysis shows that if the wind velocity is less than v_MIN, it cannot excite any waves at all. This is the rare case when the water surface is absolutely flat and mirror-like.

On the other hand, when the wind speed exceeds v_MIN, it will excite both the long gravity waves and the short capillary waves; as a consequence, we see long waves covered with capillary ripples.

• Kelvin wake

A duck swimming in a pond leaves a V-shaped trail behind. The same wake exists for a small fishing boat and for an enormous aircraft carrier. The amazing fact is that the wake angle is universal: it is the same for the duck and the aircraft carrier, and it depends neither on the shape and speed of the vessel nor on the density of the liquid (provided that the excited waves are deep water gravitational waves).

Kelvin¹ showed that the wake angle θ is determined by the simple equation sin(θ/2) = 1/3, giving θ = 19.5°. This celebrated result, shown in Fig. 1, is a direct consequence of the dispersion law for deep-water gravitational waves: ω = (gk)^1/2.

• Rogue waves

For many centuries, these waves, also known as freak waves, monster waves, killer waves, and extreme waves, have been considered as mythical. However, they are a real phenomenon, consisting in a sudden appearance, on the background of a mild ∼5 m wave pattern, of a huge ∼30 m single wave (the height of a 10-story building!), looking like a “wall of water” with a deep trough in front.

The first scientific registration of rogue wave is presented in Fig. 2.² With increasing maritime traffic such extreme waves are seen more and more often.

So far, there does not exist any theoretical understanding of rogue waves, and they still remain a mystery. The problem can be formulated as that of the statistics of extremely rare events, which apparently is not Gaussian: large fluctuations occur more frequently than the conventional theory predicts. There are indications^{3 4} that similar phenomena exist in optical and electrical noise.

3 Surface acoustic waves

In contrast to liquids and gases, in solids there are two types of sound: longitudinal and transverse, the speed of the latter being greater because of the contribution of shear stress. In 1885, Rayleigh predicted⁵ that the longitudinal and transverse acoustic waves existing in the bulk of a solid can combine to produce a hybrid surface mode (the Rayleigh wave) with a velocity inferior to the velocities of both longitudinal and transverse sound (this is a necessary condition for their existence; otherwise, they would excite bulk acoustic waves and disappear). This hybridization makes it possible to satisfy the boundary conditions at the surface. The amplitude of the Rayleigh wave decreases exponentially away from the surface.

Such waves are used for nondestructive testing and in some electronic devices.

4 Surface plasma waves and polaritons

These are electromagnetic waves that exist when the dielectric constant is positive on one side of the interface and negative on the other side.⁶ Negative dielectric constant (ω) is a well-known property of plasma below the so-called plasma frequency ω_P = 4πne²/m, where n, e, and m are the electron density, charge, and mass, respectively. The dielectric constant of plasma is given by

1

Electromagnetic waves cannot propagate in a plasma if ω < ω_P. This explains the possibility of radio communication with the opposite side of the Earth, a fact that was puzzling at the dawn of the radio era. Later, it was understood that the ionosphere acts as a mirror for frequencies below ω_P. In contrast, for television one needs much higher frequencies (because of the necessity of having a broad band), which are greater than the characteristic plasma frequency in the ionosphere. This is why, until the invention of cable television, one had to build high towers with emitting antennas and television could be received only in areas within the horizon viewed from the top of the tower.

Surface plasma waves can propagate along the interface between media with positive and negative dielectric constant (e.g., air and metal). Similarly, surface electromagnetic waves (polaritons) may exist at the air/dielectric crystal interface, where < 0 for frequencies between the limiting frequencies of the optical and acoustical phonons.

5 Plasma waves in two-dimensional structures

There exist two distinct situations: an ungated two-dimensional electron gas (2DEG), in a quantum well or heterostructure, and a 2DEG with a metallic gate, like in a field-effect transistors (FETs). The theoretical analysis of electromagnetic waves in such structures was done in Refs 6–8, whereas the first experimental results were obtained in Refs 9–11.

• Ungated electron gas

The relevant properties of the 2D plasma are the electron concentration, n, the electron charge, e, and the electron effective mass, m*. From dimensional considerations one obtains the following dispersion relation:

2

(Obviously, the 2π numerical factor cannot be guessed from dimensionality arguments. One needs some math to obtain this factor correctly.)

Comparing this result with that of Section 2.1, one can notice a striking similarity with the dispersion law for deep-water waves! One of the consequences is that a phenomenon similar to the Kelvin wake should also exist in a 2DEG. Certainly, we cannot sail a boat in our heterostructure; however, we could make some inclusion or defect in the 2D gas and pass a current. Then, a static Kelvin wake with an angle of 19.5° will appear in the downstream direction and produce some measurable electrical disturbance at the lateral boundaries of the sample. So far such an experiment was never done.

• Gated electron gas or FET

In addition to the electron concentration, in a gated 2DEG we now have another important parameter: the gate-to-channel capacitance per unit area C. The local electron charge in the channel ne is related to the local gate-to-channel voltage V_G by the plane capacitor formula: ne = C(V_G − V_T), where V_T is the threshold voltage. For gate voltages below V_T, there are no electrons in the channel. The difference U = V_G − V_T is called the gate voltage swing.

Plasma waves in a FET consist in joint oscillations of local gate voltage and electron concentration in the channel. Dimensionality arguments lead to the existence of a characteristic velocity:

3

and hence the dispersion law for plasma waves in a FET should be ω(k) = vk.

Note again the interesting similarity with shallow water waves (or sound waves) described in Section 2.1. One has only to replace the gravitational energy per unit mass gh for shallow water waves by the electrostatic energy per unit mass eU/m* in the case of plasma waves in a FET!

This analogy has led Shur and me to the prediction of plasma wave instability¹² in a FET for sufficiently high source–drain current, provided the boundary conditions at the source and drain are asymmetric. A similar instability is responsible for the performance of wind musical instruments (the clarinet is asymmetric and no sound is produced unless one blows strongly enough).

6 Electronic surface states in solids

In a crystal, the periodic structure of the lattice results in the band structure of the electron energy spectrum and the states are described by Bloch wavefunctions. However, the termination of the periodic structure at the boundary of the crystal will mix the wavefunctions belonging to different bands and this leads to the formation of specific electron surface states.

• Tamm and Shockley states

In the early days of the quantum theory of solids, Tamm¹³ and later Shockley¹⁴ demonstrated the existence of such states within a model illustrated in Fig. 3. Tamm used the tight-binding approximation, while Shockley used the nearly free electron approximation. The physical nature of Tamm states is identical to that of Shockley states.

While being quite important as a matter of principle, these results do not have predictive value. In particular, the energy of Tamm–Shockley surface states crucially depends on the exact point chosen for termination of the periodic potential, which clearly is unphysical.

• Surface states in HgTe quantum wells

It is well known that the electron spectrum in a quantum well consists of two-dimensional minibands. In a rectangular, infinitely deep quantum well, the textbook energy spectrum is given by E_n(k) = ħ²π²n²/2ma² + ħ²k²/2m, where n = 1, 2, 3, … , a is the well width, m is the electron mass, and ħk is the in-plane electron momentum.

Some 33 years ago, Khaetskii and Dyakonov¹⁵ became interested in understanding the energy spectrum in a quantum well formed by a gapless semiconductor, such as HgTe. The bulk energy spectrum of this peculiar material is presented in Fig. 4(a). The conduction and the valence bands have the same origin as bands of light and heavy holes in Ge, GaAs, and other III–V semiconductors, in which there exist light and heavy holes with a degeneracy at k = 0. The specific feature of HgTe is that the light-hole band is inverted, becoming the conduction band. The hole is ∼20 times heavier than the electron.

One could expect that in a quantum well, the holes and the electrons would be quantized independently according to the above simple formula. This is indeed the case at k = 0 (see Fig. 4(b)). However, at finite k we have a surprise: the first hole subband (h1) is inverted (it becomes electronic-like). The electron subbands (e1, e2, …) as well as other hole subbands (h2, h3, …) are “normal.”

Moreover, while the wavefunction in the h1 band at k = 0 has a maximum in the middle of the well (which is normal), for large k the maxima of the wavefunction appear near the well boundaries. This means that at large k the h1 band is formed by surface states! If the Fermi level is located in this band, but below the e1 band, only the surface states should contribute to conductivity.

We also predicted that similar surface states should exist at the boundary of vacuum with a bulk HgTe crystal.¹⁶ However, in this case the band of surface states is superimposed on the bulk band spectrum.

The subject was further explored in Refs 17 and 18. In particular, Ref. 17 considered a quantum well of HgTe sandwiched between thick CdTe layers and made the first prediction of “time-reversal symmetry protected edge states,” which were confirmed experimentally exactly 20 years later.¹⁹

• Topological insulators

The theoretical work of Pankratov et al.¹⁷ was the first prediction belonging to the now very popular field of topological insulators. The exciting property is that in such materials the surface states (in 3D materials) or the edge states in 2D sandwich-like quantum wells are time-reversal symmetry protected, which means that they are immune to potential scattering and spin–orbit interaction (but can be destroyed by scattering by magnetic impurities, i.e., by spin scattering).

The 3D topological insulator was predicted by Fu and Kane²⁰ for binary compounds involving bismuth. The predicted symmetry-protected surface states were discovered in BiSb.²¹ The surface states of a 3D topological insulator is a new type of a 2DEG, where the electron's spin is locked to its linear momentum.

There is a huge recent literature on this subject so we will not dwell on it further.

7 Dyakonov surface waves (DSWs)

Inspired by Rayleigh results for surface acoustic waves,⁵ some time ago²² I started a search for surface waves in another situation, where there exist two kinds of waves in the bulk: the ordinary and extraordinary electromagnetic waves in a birefringent crystal with eigenvalues of the dielectric tensor _|| and _⊥. (It is interesting to note that all the important facts about optics of anisotropic crystals were established by Fresnel²³ long before Maxwell's equations were written.) The simplest configuration in which these new surface waves can exist is presented in Fig. 5.

In order to describe waves decaying away from the interface (x = 0), the x-components of their wave vectors should be imaginary: ik₁, −ik₂, and −ik₃ (for the wave in the isotropic media at x > 0, the extraordinary wave in the crystal at x < 0, and for the ordinary wave in the crystal, respectively). The dispersion laws for the three waves are

4

where all the wave vectors are dimensionless (written in units of ω/c).

These equations should be complemented by the boundary conditions requiring the continuity of the tangential components of the electric and magnetic fields at the interface. After some tedious algebra, one can obtain the fourth equation²³:

5

Now, we can determine the four unknowns: k₁, k₂, k₃, and q, which can be done only numerically. However, from the above equation, one can see that a solution can exist only if _|| > > _⊥ or _⊥ > > _||. Further analysis shows that only the first case is acceptable; thus, the condition for existence of DSW is

6

which means that the birefringent crystal must be positive (_|| > _⊥).

Another possibility was considered by Averkiev and Dyakonov.²⁴ Consider two identical positive birefringent crystals with optical axes parallel to their surfaces. Put one upon the other and make some angle α between their optical axes C₁ and C₂, as shown in Fig. 6(a). For such a configuration, we found that surface waves can exist within two sectors around the bisectrixes of the two angles formed by C₁ and C₂, as shown in Fig. 6(b).

The previously known electromagnetic surface waves (surface plasmons discussed in Section 4) exist under the condition that the permittivity of one of the materials forming the interface is negative, while the other one is positive. In contrast, the DSWs can propagate when both materials are transparent; hence they are virtually lossless, which is their most attractive property. A large number of theoretical works appeared discussing these, or similar, waves under various conditions, in particular, in metamaterials²⁵ (see the review in Ref. 26).

The first experimental observation of DSW was reported only in 2009 by Takayama and coworkers,^{27 28} using the Otto–Kretschmann configuration and an index-matching fluid as an isotropic medium satisfying the condition _|| > > _⊥. The cutoff propagation angle was measured as a function of the permittivity of the fluid (see Fig. 7).

It is believed that the extreme sensitivity of DSWs to anisotropy, and thereby to stress, along with their low-loss (long-range) character render them particularly attractive for enabling high sensitivity tactile and ultrasonic sensing for next-generation, high-speed transduction and read-out technologies.

This subject continues to be of considerable interest, recent work can be found in Refs 29–34.

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