Supplementary material
“Antenna-Coupled Microcavities for Enhanced
Infrared Photo-detection”
Yuk Nga Chen1, Yanko Todorov*1, Benjamin Askenazi1, Angela Vasanelli1, Giorgio Biasiol2,
Raffaele Colombelli3 and Carlo Sirtori1
1 Univ. Paris Diderot, Sorbonne Paris Cité, Laboratoire Matériaux et Phénomènes
Quantiques, CNRS-UMR7162, 75013 Paris, France
2 IOM CNR, Laboratorio TASC, Area Science Park, I-34149 Trieste, Italy.
3 Institut d’Electronique Fondamentale, Univ. Paris Sud and CNRS–UMR 8622, F-91405
Orsay, France
1. Experimental characterization
The double-metal structures in Fig.1(a) were fabricated by a standard Gold-Gold (Au-Au)
wafer bonding process on a host GaAs substrate. A PdGeAu (Palladium-Germanium-Gold)
metallic layer was evaporated before bonding the Au layer on the active region. This layer
serves both as the lower mirror of the microcavity and ohmic back-contact.
The photocurrent measurements in Fig.1(b) were obtained by illuminating the devices with a
calibrated black body source at 500 °C, under normal incidence. The devices were connected
in series with a known resistance, which allowed to measure the absolute value of the
photocurrent using a lock-in amplifier by mechanically chopping the optical beam.
The spectra from Figure 2 were obtained using the Globar lamp of a Vertex Bruker FTIR
(Fourier-Transform Interferometer) spectrometer. To obtain the responsivity of the detectors
in Fig.2(a), the 500°C calibrated black body source spectra were compared to those from a
Globar lamp in the same experimental conditions as the detector measurements.
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2. Theory of the photo-response of a microcavity-coupled QWIP
In this section we present the derivation of Eq.(1) describing the photoresponce of a
microcavity-coupled infrared photodetector. Let U0 be the total electromagnetic energy stored
in the cavity, then the photocurrent Iphoto can be expressed as in Ref. 1:
Iphot 
(S1)
eg dU0
NQW dt
isb
Here g is the photoconductive gain for one quantum well, equal to the ratio pe/pc (extraction rate
over capture rate), NQW is the number of quantum wells, and the sign “isb” means that we
consider the electromagnetic energy decay owe to the intersubband absorption in the quantum
wells. Let us then consider the electromagnetic energy conservation in the system. For this,
we consider a unit cell of the microcavity array with a surface , as illustrated in Fig. S1.
Then the energy conservation can be deduced from the Poynting theorem2:
Fig. S1. Electromagnetic energy conservation in the system,considering a unit cell of the microcavity
array. For simplicity, we have drawn a transverse section of the device.
(S2)
dU0
dt

isb
dU0
dt
 (Sin  SR )  side (Sside  S 'side )
metal loss
In Eq.(S2) we have also taken into account the energy loss in the metallic walls and the doped
layers of the microcavity. Furthermore, Sin is the z-component of the incoming Poynting flux,
SR is the reflected component, Sside and S’side are the Poynting fluxes of the sides, and side
indicates the surface of the sidewalls. As the system is periodic, the sidewall contributions of
the Poynting flux cancel each other. The remaining contributions Sinand SR can be expressed
as3,4:
2
(S3)
Sin   0 0 |Ein |2 , SR   0 0 |Ein |2 R
Here Ein is the amplitude of the electric field of the incoming wave, 0=2cos/ is the zcomponent of the wavevector of the incoming wave, the angle of incidence and R is the
reflectivity coefficient of the structure. Since the grating period is sub-wavelength, we have
considered only the 0th reflected order, as all high order diffraction waves are evanescent.
In order to express the left hand side of Eq. (S2) we introduce the quality factor of the cavity
Qcav, and a coefficient Aisb that describes the intersubband absorption:
(S4)
1
1 dU0

Qcav U0 dt
metal loss
,
Aisb 
1 dU0
U0 dt
isb
We are now in a position to express the responsivity of the structure defined as the ratio of the
flux of photo-excited electrons over the flux of photons:
(S5)
Resp() 
Iphot
Sin

eg dU0
NQW dt
1
isb
 0 0 |Ein |2
Combining Eq. S1 - Eq. S5 we arrive at Eq. 1 in the main text:
(S6)

Aisb
Resp( )  1  R()  
 Aisb  1 / Qcav
  eg
 
  NQW



At this point we have to provide an explicit expression for the coefficient Aisb. Note that this
coefficient is different from the absorption quantum efficiency  usually defined for detector
structures. Indeed, the definition of  is defined as the ratio of the number of photons
absorbed by the quantum well over the total number of photons incident on the active
medium1. The coefficient Aisb (Eq.(S4)) is defined as the ratio of the number of photons
absorbed during one cycle of oscillation 2/ over the total number of photons circulating in
the microcavity. The coefficient Aisb is therefore more suitable for the case of microcavity
geometry, whereas is more suitable for the case where light propagates in a waveguide,
such as the multi-pass geometry. In the following we will provide a relationship between Aisb
and  in the limit of small intersubband absorption.
To estimate Aisb we use the general definition of the dielectric absorption (4):
(S7)
dU0
dt
 1 |D|2
  fw Im
V


(

)

isb
 zz  0
Here V is the volume of the microcavity, D is the displacement field of the cavity mode, fw is
the geometric overlap factor between the absorbing region and the mode, and zz() is the z3
component of the QW dielectric tensor. We have used the fact that the cavity mode is
essentially polarized along the z-direction. Next, we use the expression of the electromagnetic
energy density of the cavity:
(S8)
U0  
1
2 0
|D|2dV  
µ0
|H |2dV
2
Here D and H are respectively the electric displacement and magnetic field. We neglect in this
case the dispersion induced by the quantum wells, and use  the dielectric constant of the
barriers. We can now express Aisb as a function of the frequency, geometrical overlap and
dielectric constant of the absorbing quantum wells:
(S9)
  
Aisb  fw Im

  zz () 
2
2
2
Using a dielectric function ofthe form  /  zz ()  1  P /(  21  i) and performing a
rotating wave approximation we arrive at a Lorenzian profile:
(S10)
Aisb ()  fw
P2

421 (  21 )2  2 / 4
In this case the coefficient Aisb can be related to the absorption quantum efficiency  as
defined in the literature1: 
(S11)
P2LQW sin²

 () 
4c
cos (  21 )2  2 / 4
Here c is the speed of light, LQW is the quantum well thickness and  is the internal angle of
propagation in the waveguide in a multipass configuration. Note that in our definitions the
transition oscillator strength is contained in the plasma frequency. The ratio between Eq.(S10)
and Eq.(S11) provides the link between  and Aisb:
(S12)
Aisb ( )   ( )
cos 21
fw
sin² 2 LQW
Here 21  2 c / 21 is the wavelength of the ISB transition.
3. Additional data: Polariton splitting measurements
In Fig. S2(a) we provide reflectivity curves obtained from panels containing 6 identical mesas
(3x2) with a patch size s. The measurements were taken at 77K, normal incidence and light
polarized perpendicular to the 150nm wire. These measurements allow one to map the
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polariton anti-crossing curve of the sample (Fig. S2(b)), and extract the dielectric function as
explained in Ref. 5.
Figure. S2. (a) Reflectivity measurement for samples with various stripwidth s. (b) Polaritonic
dispersion curves obtained from the reflectivity measurements.
4. Additional data: Etching of the semiconductor surface
The semiconductor material around the patches and wires has been etched in an ICP reactor,
to limit the detector dark current as described in the main text. A SEM picture of the etched
sample is provided in Figure S3(a). As indicated in the main text, the etching results in a shift
of the resonant frequency, due to a change in the effective modal index of the cavity, neff. This
is illustrated in the data of Figure S3(b), where we compare the photocurrent spectra of an
s=1.1µm device before and after etch. To compensate for this change, we have fabricated a
device with a larger patch size, s=1.4µm, that becomes resonant with the ISB absorption after
the etch process (Fig. S3(b)).
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Figure S3. (a) SEM picture of the structure after ICP etch of the semiconductor region. The
arrows indicate the etched semiconductor layer. (b) Photocurrent spectra of devices before
and after the etch process. The etch causes a blueshift of the cavity mode, as seen with a
s=1.1µm structure. A device with s=1.4µm becomes resonant with the ISB absorption after
etch.
References
1. H. Schneider and H. C. Liu, Quantum well infrared photodetectors: Physics and
applications (Springer, Berlin, 2007).
2. L. Landau and E. Lifchitz, Electrodynamics of continuous media (Mir, Moscow,
1969).
3. Y. Todorov and C. Minot, J. Opt. Soc. Am. A, 24 3100–3114 (2007).
4. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
5. Y. Todorov, L. Tosetto, A. Delteil, A. Vasanelli, A.M. Andrews, G. Strasser, C.
Sirtori, Phys. Rev. B, 86, 125314 (2012).
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