Optical control of THz reflectivity with surface waves

S. Vassant*a,b

, F. Pardob, P. Bouchon

b, F. Marquier

a, JJ. Greffet

a and JL. Pelouard

b

aLaboratoire Charles Fabry, UMR 8501, Institut d'Optique, CNRS, Univ Paris Sud 11,

2, Avenue Augustin Fresnel, 91127 Palaiseau, France

bLaboratoire de Photonique et Nanostructures, CNRS-UPR20, Route de Nozay,

91460 Marcoussis, France

ABSTRACT

We propose an experimental demonstration of a THz modulator with a visible optical command. The device is a n-doped

GaAs grating with subwavelength dimensions. The principle of this modulator is the control of the THz resonant

absorption by surface waves supported by the grating. This absorption is modulated with low power visible light, leading

to a modulation of the reflected THz beam. From experimental polarized THz reflectivity measurement of the grating,

we show that a depletion layer at the surface of the doped GaAs has to be taken into account to correctly describe the

observed resonant absorption. From experimental observation and modeling we are able to ascribe this absorption to the

coupling of incident THz light with surface plasmon-phonon polariton mode propagating along each wall of the grating.

Thus, each wall acts as a nano-antenna that resonantly absorbs light. The grating can be viewed as a metamaterial

composed of individual resonators. The theoretical model indicates that the reflectivity dip linked to the surface wave is

sensible to the electronic density in the walls of the grating. We performed an experiment to measure the THz

reflectivity while illuminating the grating with visible photons having energy higher than the bandgap of GaAs. The

created photoelectrons change the effective mode index, leading to a shift of the resonant absorption frequency. This

demonstrates the modulation of THz radiation around 8.5 THz with a visible optical command at room temperature.

Keywords: THz, GaAs, surface plasmon polariton, surface phonon polariton, strong coupling, modulator, grating

1. SURFACES WAVES ON SEMICONDUCTORS

In this section, we will review some basics about surface waves propagating on polar semiconductor. For clarity, we will

only consider the single interface geometry, between a polar semiconductor and a dielectric. We will first study

separately surface phonon and surface plasmon polariton (SPhP and SPP) on semiconductor, and end this section by the

coupling that occurs between these two surface waves.

1.1 Surface phonon polaritons on semiconductors

A crystal supports SPhP as soon as it has a polar behavior. A crystal is polar if it has two atoms or more per unit cell.

This includes most of the semiconductors used in microelectronics (III-V and II-VI compounds), and also IV-IV, I-VI

crystals and their alloys. GaAs for example has two atoms per unit cell, that allows longitudinal and transverse optical

phonons modes. We will refer to them in this manuscript as LO and TO phonons modes respectively. These phonon are

mechanical modes of the crystal where Ga and As atoms moves toward each other. As there is a slight difference in the

electronegativity of the Ga and As atoms in the crystal, their movement results in the formation of a dipole that couples

with the electromagnetic field. They impacts the polarizability of the crystal and are classically described by a

Lorentzian resonance in the dielectric function of the crystal !(") centered on the TO-phonon frequency1 :

!(") = !# 1+"

LO

2 $"TO

2

"TO

2 $" 2 $ i%"

&

' ( (

)

* + + . (1)

"LO and "TO are respectively the longitudinal and transverse optical phonon frequencies and # is a damping factor.

We represent in Figure 1a the real and imaginary part of the dielectric function, near the optical phonon frequencies for

GaAs. The corresponding reflectivity at normal incidence is shown in Figure 1b. The parameters for these calculations

are taken from Ref. 2, and listed in Table 1.

!

Figure 1: Optical properties of GaAs. a) Dielectric function as a function of the wavelength. Data for the parameters are

taken from Ref. 2. b) Reflectivity of a GaAs semi-infinite crystal at normal incidence as a function of the wavelength.

Table 1: Dielectric function parameters for GaAs2.

!" #LO #TO $

292.1 cm-1

268.7 cm-1

2.4 cm-1

11

34.24 µm 37.22 µm 4166.67 µm

From figure 1, we see that the real part of the dielectric function has negative value between #LO and #TO. This frequency

band, is called the Reststrahlen band because, as can be seen in Figure 1b, the crystal has a “metallic” optical behavior :

the reflectivity reaches high values. In this frequency range, GaAs will support SPhP. The derivation of SPhP dispersion

relation is similar to that used for SPP on metal surfaces. Such derivation can be found in Ref. 3 for example, and we

will not be detailed here. However, we will underline some differences between SPP and SPhP.

The first particularity of SPhP is the limited frequency range where they can exist. This frequency range is included in

the Reststrahlen band and mainly depends on the dielectric in contact with the polar crystal. This can be seen from the

expression of the asymptotic frequency of the mode given by Re(!SC(#sup)) = -Re(!d):

!sup ="#!LO

2+ "'

d!TO

2

"# + "'d. (2)

In this expression, !SC is given by equation 1, and !d= !’d + i!’’d. The dispersion relation of the mode is given by:

kSPhP

=!

c

"SC"d

"SC+ "

d

, (3)

where kSPhP is the wavevector of the SPhP, # the pulsation, c the speed of light in vacuum, !SC the semiconductor

dielectric function, and !d the dielectric function of the dielectric on top of the semi-conductor. The dispersion relation

(kSPhP as a function of #) is represented in Figure 2.

Figure 2: Dispersion relation for a SPhP propagating at a GaAs/vacuum surface. The horizontal dashed lines represent the

LO and TO frequencies. The horizontal point-dashed line represents the asymptotic frequency of the surface waves. The

oblique dashed line represent the light line, i.e. kx = !/c. Both abscissa and ordinates axis are normalized respectively by k0

= !/c and the LO phonon frequency.

The SPhP only exists inside the Reststrahlen band, and more specifically between !TO and !sup. It is also important to

notice that the asymptotic frequency, where there is a very large density of states is located in the Reststrahlen band,

between the LO and TO phonon frequencies. For GaAs, it is close to 8.5 THz. For SiC, optical phonons are in the 10

microns range, for GaN, in the 13 microns range.

1.2 Surface plasmon polaritons on semiconductors

Semiconductors can also be doped by the addition of donor or acceptor impurities in the crystal. We will only focus on

the case of donor, leading to a n-type doping. The supplementary electrons are free in the crystal. As in metal, they have

a plasma frequency, and the semiconductor can supports surface plasmon polaritons. A Drude model describes the

electron contribution to the dielectric function. For this section, we take the case of a non-polar material, Si for example.

The interaction with optical phonon in polar material will be discussed in the next section. The Drude model takes the

following form:

!(") = !# 1$" p

2

" 2+ i% e"

&

' (

)

* + , (3)

where !p is the plasma frequency, and "e is a damping term, describing the electronic interactions. The plasma frequency

in the semiconductor takes the form:

! p

2=

ne2

"0"#meff

, (4)

with n is the electronic density, e the electronic charge, meff the effective electronic mass in the semiconductor, and #0 the

static dielectric constant. Note the plasma frequency depends on the electronic density in the semiconductor. The

dispersion relation is also described by equation 2, and is represented in figure 3. The asymptotic frequency of the SPP is

given by:

!sup ="#

"# + "'d. (5)

Figure 3: Dispersion relation of a surface plasmon polariton at a doped semiconductor/vacuum surface. The oblique dashed

line represent the light line, i.e. kx = !/c. Both abscissa and ordinates axis are normalized respectively by k0 = !/c and the

plasma frequency.

In this case, the mode always exists at frequencies lower than the asymptotic frequency. It is important to note here that

the electronic density can be controlled by the amount of doping impurities incorporated in the semiconductor. Thus, the

asymptotic frequency can be tuned in the limit of the doping density range for the semiconductor. For metal, the

electronic density is in the order of 1023

cm-3

electron, giving a plasma and asymptotic frequency in the UV/visible. For

semiconductor, the electronic density is much lower (1017

to 1019

cm-3

). These doping values give a plasma frequency in

the far infrared range (" 230-20 microns). In GaAs for example, the plasma frequency of free electrons can be close to

the optical phonon frequency. The next section will describe what happens in this case.

1.3 Coupled surface plasmon-phonon mode

We will first consider a bulk doped polar semiconductor in which both bulk plasmon and bulk phonon can exist. Varga4

has theoretically described the strong coupling between them in 1965. Köhl5 later confirmed this experimentally by

reflectivity measurements on doped InSb with different carrier concentration in 1971. The same year, Chiu and Quinn6

predicted a strong coupling of SPP and SPhP. Only one year later, Bryksin7 performed an experimental study of surface

waves propagating on a flat surface of doped InSb. His results clearly show a strong coupling between surface plasmon

polariton and surface phonon polariton. The same year, Rieder et al.8 also observed such coupled mode on CdO

microcrystals. The mix of SPP and SPhP is called a surface plasmon-phonon polariton (SPPhP).

The coupling between SPP and SPhP can be calculated in the long wavevector limit, by studying the asymptotic

frequency of both surface waves as a function of the doping density. The asymptotic frequency of the coupled mode can

be calculated by finding the solution of Re(#(!))=-Re(#d), with the following model for the dielectric function :

!(") = !# 1+"LO

2 $"TO

2

"TO

2 $" 2 $ i%"$

" p

2

" 2+ i% e"

&

' ( (

)

* + + , (6)

that takes into account both phonon and free electron contribution. We take the example of GaAs and plot in Figure 4 the

asymptotic frequency of the SPPhP as a function of the plasma density in GaAs. For simplicity, we take vacuum as a

dielectric (#d=1). The anti-crossing of the modes is typical of a strong coupling between SPP and SPhP.

Figure 3: Evolution of the asymptotic frequency of SPPhP at a doped GaAs/air surface as a function of the plasma

frequency. The asymptotic frequencies for uncoupled modes are shown in dashed line. The complete model shows a typical

anti-crossing behavior, which results from the strong coupling between the two modes.

Although the nature of the SPPhP is clearly established, the experimental data obtained in Ref 7-9 are not well described

by theory. In particular in Ref. 7 and 8, the influence of a possible depletion layer is invoked to explain the difference

between experiments and theory. In 1974, Cunningham9 et al have theoretically predicted the effect of a depletion layer

for such modes. However, it was found to occur only for large wavevectors. Therefore, most of the experimental and

theoretical studies about depletion and/or accumulation layers on doped semiconductors10,11

consider electron energy loss

spectroscopy. To our knowledge, there is no optical measurement clearly showing the depletion layer effect on SPPhP

modes, with geometrically well-controlled semiconductor structures. In the next section, we optically excite SPPhP with

high wavevectors in a subwavelength grating, and show that a depletion layer has to be modeled to correctly describe the

optical response of the grating.

2. DEPLETION LAYER EFFECT ON SURFACE PLASMON-PHONON POLARITON

In this section, we study the optical response of a subwavelength grating made of doped GaAs. We first describe the

fabrication process and geometry of the grating, then present reflectivity measurement of the grating and the model used

for the simulation.

2.1 Fabrication

We start the process from an n-doped GaAs wafer. Lines are defined by electronic lithography in a PMMA resist

previously spin-coated on the wafer. After metallization and subsequent lift-off process, we obtain thin Ti/Ni lines that

will be used as a mask for Reactive Ion Etching with Inductively Coupled Plasma (RIE-ICP). We use BCl3/CL2/Ar

gases for the plasma. Finally, the rest of the metallic mask is removed in a diluted HF solution. Figure 4 present a

scheme of the fabricated grating.

Figure 4: Scheme of the fabricated grating. The period of the grating is denoted by !, the wall width by W and the grating

height by h. An incident plane wave with a TM (or p) polarization, with an angle of incidence "i is represented.

In figure 5, we show Scanning Electron Microscope (SEM) pictures of the cross-section of the grating.

Figure 5: SEM Cross-section view of the fabricated grating.

From the SEM pictures, we measure the dimension of the grating. The period ! = 2.5 µm serves as an absolute etalon

for the measurements. The height of the grating is h = 800 nm, and the width is W = 80 nm. The area of the grating is

5x5mm2. Note that we will investigate the reflectivity of the grating around "i = 35µm. This gives ! # "i/14, h # "i/43

and W # "i/437.

2.2 Reflectivity measurements

The grating was placed into a Bruker Vertex 70 Fourier Transformed InfraRed (FTIR) spectrometer, equipped with a

Mylar beamsplitter, an A-513Q reflectivity setup and a wire-grid THz polarizer (Tydex). With this setup we can take

reflectivity spectrum of samples at frequencies ranging from 650 to 100 cm-1

(#15 to 100 µm), and for angle of incidence

($i) from 15 to 60°. We are thus able to measure the optical response of our grating in the Reststrahlen region of GaAs at

different angles with polarized incident light.

First, we measure the flat GaAs wafer used for the grating fabrication. We fit the measured reflectivity (shown in Figure

6) using equation 6 and 4 and the values of table 1. The fit is made with the electronic density n and the electron

damping factor %e as free parameters. We obtain a doping density n=1.466 (± 0.05) &1018

cm'3

and a damping of %e

=69.54 (± 1) cm'1

.

Figure 6: Experimental and theoretical reflectivity of a flat surface of doped GaAs ($i = 15°). The strong dip at 39 µm is due

to the TO-phonon of GaAs.

The reflectivity begins to be high around 25 µm, which is due to the electronic contribution in the dielectric function

(Re(()<0). Around 39 µm, a large dip in reflectivity appears, that can be linked to the TO- phonon of the GaAs : at the

TO phonon frequency, we have Re(!)>0 even with the free electron contribution. For larger wavelength, free electrons

dominate the optical response of the doped semi-conductor and the reflectivity is again very high (Re(!)<0).

The grating is then measured, in TM (or p) polarization. The obtained reflectivity curve are shown in Figure 7:

Figure 7: Reflectivity of the grating as a function of the incident wavelength, for different angles of incidence. A dip at 35

µm appears for large angle of incidence.

The reflectivity of the grating shows a supplementary dip around 35 µm. This dip increases in amplitude when

increasing the angle of incidence, but it’s frequency remains constant. The reflectivity in TE polarization (not shown

here) is similar to the reflectivity of a flat surface and does not exhibits this supplementary dip. Note that the reflectivity

at "i=15° is close to the reflectivity of a flat surface. This dip appears because of the excitation of an SPPhP in the

grating. Further investigations via numerical simulations will confirm this and give us a better understanding of the

mode.

2.3 Modeling

To model the electromagnetic response of the grating, we used a code developed by F. Pardo, based on Rigorous

Maxwell equations with Constitutive Approximation (RMCA12

). We use the previously described dielectric function.

The bottom curvature that can be seen in Figure 5 is taken into account []. We first perform numerical calculation

without any depleted layer. We consider two extreme cases: i) fully doped grating walls and ii) fully depleted grating

walls. In the first case, there is a supplementary dip as in the measured spectrum, but shifted towards smaller

wavelength, and with a much larger full width at half maximum (FWHM). For the second case, there is also a

supplementary dip, but this time at larger wavelength, with a smaller FWHM. This indicates that the real structure is in

between these two extreme cases. We will consider a doped GaAs core, with an outer shell of depleted GaAs. To take

into account this depletion layer, we calculated an approximation of the depleted thickness ds given by:

ds= 2!

stat!"Vs

ne , (7)

with e the electronic charge, Vs the surface potential, !0 the vacuum permittivity, !stat the static dielectric constant, and n

the electronic density. For the measured electronic density and a surface potential of 0.7 eV13

, we obtain a depleted layer

of thickness ds=24 nm. This depleted layer was included in the simulation. A scheme of the simulated geometry is shown

in Figure 8a, and the comparison between experimental and theoretical reflectivity with this model are shown in Figure

8b. There is a very good agreement between the two curves indicating that the “core-shell” model is a good

approximation of the system.

Figure 8: a) Scheme of the simulated geometry, using a “core-shell” model, b) corresponding theoretical reflectivity

compared with experimental results (!i=45°).

Now that the reflectivity is well described by out theoretical model, we can calculate the near field distribution of the

mode at its resonant frequency. This is shown in Figure 9:

Figure 9: Near field distribution of the x-component of the electric field at resonance (35µm, !i=45°). The white lines

represent the GaAs/air interface. The field is enhanced along the wall of the grating.

We can see in Figure 9 that the field is enhanced along the wall of the grating. The excited mode is composed of two

SPPhP propagating along each side of the wall. Each wall acts independently as a nano-antenna. This explains the

independence of the resonance frequency towards the angle of incidence and the vanishing absorption at near normal

incidence. We obtain numerically the effective index of this mode: neff " 10.

The numerical simulations tells that the system is dependent of the electronic density in the grating walls. In the next

section we will see the influence of a visible light shined on the grating to create photoelectrons that will disturb the

resonance.

3. MODULATION OF THZ REFLECTIVITY WITH AN OPTICAL CONTROL

3.1 Experimental setup

The objective is to create photoelectrons in the grating walls while measuring the THz reflectivity. To do this, we used a

simple HeNe Laser with a wavelength of 633nm. The corresponding photon energy (1.96 eV) is higher than the band

gap of GaAs (1.42 eV): illuminating the grating creates photoelectrons. Furthermore, as the thickness of the walls is very

thin, we can assume a homogeneous increase of the electron density. We couple the HeNe laser to an optical fiber going

inside the spectrometer. This allows us to shine visible light on the grating while measuring the THz reflectivity. The

power at the fiber output is around 16mW/cm2. A scheme of the measurement setup is shown in Figure 11:

Figure 11: Scheme of the experimental setup. A HeNe laser (633 nm) is coupled to a fiber that enters the FTIR spectrometer

and illuminates the grating while measuring the THz reflectivity of the grating.

3.2 Measurements

The reflectivity spectra are first taken without illumination to check that the optical fiber does not perturb the

measurement. Then, the HeNe laser is turned on, and the THz reflectivity with visible illumination is measured. Results

are shown in Figure 12:

Figure 12: THz reflectivity of the doped-GaAs grating with and without illumination by a 633nm HeNe laser. A shift of

about 250 nm of the SPPhP resonance is seen while the rest of the spectrum remains unchanged.

A frequency shift of the resonance is observed, with a small increase in the FWHM. The rest of the spectrum is globally

unchanged. This demonstrates the modulation of THz reflectivity (35.5 µm ! 8.45 THz), with a low-power optical

control at room temperature.

3.3 Modeling

We perform numerical simulation to estimate the number of created photo-carriers in the walls. We do not try to explain

the details of the laser light/matter interaction, but we try a very simple model with strong assumption to have a

qualitative description of the measured spectrum. We suppose that photoelectrons are created homogeneously in the

walls and neglect holes. We also take the same damping factor for the photoelectrons than for the bulk electron. This is a

strong assumption, because for such small semiconductor structures surface recombination may be the predominant

phenomenon. The theoretical reflectivity with our model is compared to experimental reflectivity in figure 13. We found

a good agreement between theory and experiment for a supplementary photoelectron density of 1.7x1017

cm-3

.

Figure 13: a) Experimental and theoretical reflectivity with and without illumination of the grating with the HeNe laser. b)

Corresponding relative reflectivity modulation ((Rmax-Rmin)/Rmin), experimental and theoretical.

The relative modulation ((Rmax-Rmin)/Rmin) goes up to 40% experimentally, with a very low optical power (16 mW/cm2)

for the control beam. This is possible thanks to two aspects of the device: i) the very subwavelength dimensions of the

structures leading to strongly confined mode with high wavevectors that are sensible to the depletion zone, and ii) the

strong coupling regime, which gives a mode whose frequency is also strongly dependent on the electron density in the

semiconductor. We do not have the possibility to measure experimentally the bandwidth of our system. We expect the

system to be at least as fast as electron-holes recombination time (nanosecond range). This speed may be increase if the

surface recombination is the predominant relaxation process of the system.

Finally, this grating is not yet optimized. Using SPP, SPhP or SPPhP modes it is possible to achieve total absorption of

the incident light. This can be done in this particular case by fabricating higher walls, which is still technologically

challenging, and under work at the Laboratoire de Photonique et Nanostructures (LPN). Having a total absorption will

lead to very large relative modulation of the reflectivity.

To conclude, we presented a device allowing a relative modulation of the reflectivity up to 40% at 8.45 THz, working at

room temperature. This device is a doped GaAs grating with subwavelength dimension supporting very confined modes

and working with surface mode in the strong coupling regime. We showed that in such small structures, the depletion

layer should be included in the model to have a good description of the experimental measurements. This study can be

usefull for space observation14

and porous material studies15

. We also believe that such modes can be used to enhance

light-matter interaction. They are promising for a new class of mid-far infrared devices.

REFERENCES

[1] Born, M. and Huang, K., [Dynamical Theory of Crystal Lattices], Oxford University Press, New York (1954)

[2] Palik, E.D., [Handbook of Optical Constants of Solids], Harcourt Brace Jovanovich, New York (1985)

[3] Raether, H., [Surface Plasmons], Springer (1988)

[4] Varga, B.B., “Coupling of plasmons to polar phonons in Degenerate Semiconductors”, Physical Review B 137(6A),

A1896 (1965).

[5] Köhl, F., “Infrared reflectivity of InP”, Journal of Physics and Chemistry of Solids 32, 2481 (1971).

[6] Bryksin, V.V., Mirlin D.N. and Reshina, I.I., “Surface Plasmon-Phonon interaction in n-InSb”, Solid State

Communications 11, 695 (1972).

[7] Rieder, K.H., Ishigame, N. and Genzel L., Infrared Absorption by Coupled Surface-Phonon-Surface-Plasmon Modes

in Microcrystals of CdO, Physical Review B 6(10), 3804 (1972).

[8]Yamamoto, K., Kimura, K., Ueda, U., Kasahara, H. and Okada, T., “Infrared absorption by coupled surface-phonon-

surface-plasmon modes in small GaAs crystals”, Journal of Physics C: Solid State Physics 18, 2361 (1985).

[9]Cunningham, S.L., Maradudin, A.A. and Wallis, R.F., “Effect of a charge layer on the surface-plasmon-polariton

dispersion curve”, Physical Review B 10(8), 3342 (1974).

[10] Streight, S.R. and Mills, D.L., “Surface plasmons on doped semiconductor films: Influence of the depletion region”,

Physical Review B 38(12), 8526 (1988).

[11] Inaoka, T., “Evolution of elementary excitations at a doped polar semiconductor surface in a depletion layer

formation process”, Physical Review B 63, 165322 (2001).

[12] Pardo, F., Gottesman, Y. and Pelouard, J.L., “Exact formulation of Maxwell’s equations for lamellar gratings

analysis”, To be published, available on demand at fabrice.pardo@lpn.cnrs.fr

[13] Spicer, W.E., Lindau, I., Skeath, P. and Su, C.Y., “Unified defect model and beyond”, Journal of Vacuum Science

and Technology 17(5), 1019 (1980).

[14] Mutschke, H., Andersen, A.C., Clément, D., Henning, Th., and Peiter, G., “Infrared properties of SiC particles”,

Astronomy and Astrophysics 345, 187 (1999).

[15] Spanier, J.E. and Herman, I.P., “Use of hybrid phenomenological and statistical effective medium theories of

dielectric functions to model infrared reflectance of porous SiC”, Physical Review B 61(15), 10437 (2000).