Broadband, Polarization-Sensitive Photodetector Based on Optically-Thick Films of
Macroscopically Long, Dense, and Aligned Carbon Nanotubes
Sébastien Nanot, Aron W. Cummings, Cary L. Pint, Akira Ikeuchi, Takafumi Akiho, Kazuhisa
Sueoka, Robert H. Hauge, François Léonard & Junichiro Kono
Supplementary Materials:
Materials and Methods:
A. Modeling.
1. Thermoelectric potential in a thermocouple
We begin by illustrating how the photothermoelectric potential of the nanotube detector comes
from the individual contributions of the metal electrodes and the CNT film. Figure S1 shows the
representation of the CNT photodetector as a thermocouple, with one arm corresponding to the
CNT film and one arm corresponding to the metal. The thermoelectric potential across the CNT
film is given by
TL
R
L
VCNT  VCNT
 VCNT
  SCNT (T )dT ,
(S0)
TR
while the thermoelectric potential across the metal is
Vm  VmR  VmL 
TL
S
m
(T )dT .
(S0)
TR
The total thermoelectric potential is then given by
Vtot  VCNT  Vm 
TL
TL
TR
TR
 SCNT (T )dT   Sm (T )dT .
(S0)
If the Seebeck coefficients of the metal and CNT were independent of temperature and position
we would obtain
(S0)
Vtot   SCNT  Sm TL  TR  .
This is the expression that has been previously used when interpreting the photothermoelectric
effect at contacts. The point is that, while this expression looks like it originates from a junction
effect, it actually originates from the independent contributions of each arm of the thermocouple.
This has important consequences on the photothermoelectric effect measured using local
illumination, as we discuss in the following sections.
Suppl. Fig. S1. Illustration of the photodetector as a thermocouple with one arm
corresponding to the CNT film and one arm corresponding to the metal. Details are
provided in the Supplementary Material section A.1.
2. Calculation of the photothermoelectric potential under local illumination
We consider a CNT film with the x direction along the channel and the y direction parallel to
the edge of the electrodes. The thermoelectric potential is given by the expression

R
L
VCNT
 VCNT



TL

TR

dy  sCNT (T )dT 

xL
dy  sCNT ( x, y )Tdx,
(S1)
xR
where s is the Seebeck coefficient per unit length, and the integral in the y direction arises
because each nanotube in the array contributes individually to the voltage, and thus their
contributions to the photovoltage are summed (Ref. S1). sCNT and the usual Seebeck coefficient
SCNT are related through SCNT  sCNT l , where l is the average spacing between nanotubes in the
array.
For the case where sCNT is constant in the film and where the temperature profile is given by
a constant gradient in the channel equal to (TL  TR ) / ( xL  xR ) , one recovers the usual expression
V  SCNT T as discussed in A1. However, in the case of local illumination, the temperature
profile is given by a peaked, symmetric profile, and if SCNT were constant along the CNT film the
xL
 Tdx  0 .
photovoltage would be exactly zero because
Thus, to obtain a non-zero
xR
photovoltage due to the photothermoelectric effect under local illumination, we necessarily must
have a Seebeck coefficient that depends on position. This can be made explicit by integrating
equation (S5) by parts to obtain
R
CNT
V
V
L
CNT
 S
L
CNT
TL  S
R
CNT
TR  

xL
 dy  T ( x, y)

xR
dsCNT
dx.
dx
(S2)
L
R
 SCNT
For the nanotube film, we can take xR   and xL   ; in that case SCNT
, local
illumination gives TL  TR and the first term vanishes. (Note that this first term is the one usually
included when deriving the expression V  SCNT T , as discussed in section A1.) We are left
with the expression

dSCNT
dx.
dx

In this last equation, we assumed that SCNT is independent of y, and defined
R
CNT
V
V
L
CNT
   T ( x)
(S3)

T ( x)  l 1  T ( x, y )dy .

3. Calculation of the temperature distribution due to local illumination
We consider the temperature distribution in a CNT film of thickness h under local heating
due to a circular Gaussian laser beam of width  . The x direction is along the device channel,
the y direction is in the plane of the film perpendicular to the channel direction, and the z
direction is normal to the film.
We assume that the temperature profile decays over a length scale that is much smaller than
the width of the film in the y direction, and thus consider a film infinitely wide in the y direction.
In addition, we assume that the temperature does not vary appreciably in the vertical z direction.
Both assumptions will be justified later. Under these assumptions, the problem becomes twodimensional in the x-y plane, and we thus consider the heat equation in cylindrical coordinates
h2T  Gm T  Tm   Gs T  Ts    pr 1  r  r ' ,
(S4)
where T is the temperature, κ is the thermal conductivity, Gm is the thermal conductance between
the film and the metal contacts, Gs is the thermal conductance between the film and the substrate,
Tm is the temperature of the metal contacts, and Ts is the temperature of the substrate. In this
equation, heating by the laser is modeled as a point heat source of strength p at r ' . Heating due to
a Gaussian beam is obtained from the solution of equation (S8) as described below. When the
laser is in the channel, optical absorption occurs in the CNT film only, and we assume that the
metals and the substrate remain at the same temperature. In this case, equation (S8) can be
simplified to
h 2T  Geff T  Teff    pr 1  r  r ' ,
where Geff  Gm  Gs and Teff  Ts .
(S5)
When the laser is over the electrode, optical absorption by the metal will cause heating of the
metal as well. This temperature increase is directly related to the CNT heating underneath the
metal, and thus we use a simple relation Tm  T where   1. This gives the same equation as
equation (S9), except that
Geff  1    Gm  Gs
(S6)
and Teff  TsGs / Gs  1    Gm .
The solution of equation (S9) for T  T ( x)  Teff is
T (r , r ') 
p
K0  r  r ' /   ,
2 2Geff
(S7)
where K0 is the modified Bessel function of order zero, and λ is the thermal length scale given
by
h

Geff
(S8)
.
This thermal length scale sets a condition for the validity of the assumptions of uniform
temperature distribution in the z direction, and of an infinite film in the y direction. As discussed
in the main text,  is found to be on the order of several microns even when transport is
perpendicular to the nanotubes, while the film thickness in the z direction is 600 nm. Thus,
 h and the temperature can be taken as uniform in the z direction. Furthermore, the width of
the film W in the y direction is several hundred microns, and thus  W justifying the
assumption of an infinite film in the y direction.
The temperature distribution for a Gaussian light spot centered at x  x0 and y  0 is given
by
p
T ( x, y; x0 ) 
2 2Geff
 

1
  2
2
 x '  x0 2  y '2
e
2 2
K0
 
  x  x '   y  y ' /   dx ' dy ',
2
2
(S9)
which leads to
p
T ( x; x0 ) 
2
2 lGeff
 
1
  2
 

e
2
 x '  x0 2  y '2
2 2

  K0
 
  x  x '   y  y ' /   dy dx ' dy '. (S10)
2
2
The integral in the square brackets can be performed to give the final result
T ( x; x0 ) 
p
4 2lGeff

e

 x '  x0 2
2 2
e
 x x ' /
dx '.
(S11)

To reflect the fact that Geff is different in the channel and under the electrodes, we use two
different values for  when integrating under the electrodes and in the channel. Equation (S15)
combined with equation (S7) is used for the calculations presented in the main part of the paper.
4. Photo-thermoelectric voltage from the electrodes
We estimate the photothermoelectric voltage due to the electrode by considering the
maximum thermoelectric potential that could be generated based on the maximum temperature
reached in the system. With the same sign convention as for the CNT film discussed above we
have for the left electrode
max(VmR  VmL )  S mTmax .
(S12)
The maximum temperatures for the three different metals can be found in Table 1 of the main
text. These can be combined with typical values for the metal Seebeck coefficients to obtain the
maximum expected photothermoelectric voltages, as shown in Table S1. These values, when
normalized by the laser power of 2 mW, are at least an order of magnitude less than those
observed experimentally.
Metal
Au
Au
Pd
Ti
CNT orientation
Sm (μV/K)
Tmax (K)
Parallel
Perpendicular
Perpendicular
Perpendicular
2
2
-10
9
0.16
0.29
0.19
4.6
Photovoltage
(V/W)
1.6×10-4
2.9×10-4
-9.5×10-4
2×10-2
Table S1. Maximum photothermoelectric voltage expected from heating of the electrodes
under local illumination.
5. Calculation of the time scale for temperature decay
To estimate the time scale for the signal, we consider decay of the temperature profile under
the time-dependent heat equation
h C p
dT
 h 2T  Geff T ,
dt
(S13)
where ρ is the film mass density and Cp is the heat capacity. From this equation we obtain the
time-dependence of the maximum temperature as
Tmax (t )  Tmax  t  0 et / ,
(S14)
valid when t
t . The time scale τ is given by

h C p
Geff
.
(S15)
6. Calculation of the Seebeck coefficient
To obtain the Seebeck coefficient of semiconducting SWCNTs as a function of the position
of the Fermi level, we use the expression (Ref. S2)
k I
S  B 1,
(S16)
q I0
with

j
 E  EF 
 f 
Ij   
 T E 
 dE ,
k BT 
 E 
 
(S17)
where kB is the Boltzmann constant, T is the temperature, q is the electron charge, T(E) is the
electronic transmission through the SWCNT, f is the Fermi function, and EF is the Fermi energy.
In the flat band case, the above equations can be integrated analytically to give
I0  1  f  EC   f  EV  ,
(S18)
and
 E  EF 
 EV  EF 
I1   C
 f  EC   
f
k
T
k
T

B


B


 EC  EF
1  exp  
k BT

 EV   ln 
 E  EF
1  exp   V
k BT




,


 
(S19)
where EC and EV are the energies of the conduction and valence band edges, respectively. In
Eqs. (S21) - (S23), all energies are measured with respect to the middle of the nanotube band
gap. The position dependence of the Seebeck coefficient can be calculated using the above
equations with a position-dependent value of EF.
7. Impact of nanotube density and composition on the photothermoelectric potential
As mentioned in the manuscript, the CNT film was modeled as an array of uniformly separated
CNTs with a wall-to-wall spacing of 10.8 nm, where 2/3 of the CNTs are (32,0) CNTs and 1/3
are (33,0) CNTs. However, the SEM images show that the density of CNTs varies since the
CNTs are not perfectly straight and sometimes come in contact with other CNTs. In addition, the
samples also contain a distribution of different diameter CNTs. We considered this
heterogeneous nature of the sample by simulating a broad range of nanotube spacings and
compositions. Figure S2 shows the calculated photothermoelectric voltage for these very
different sample conditions for the illustrative case of   4  m . This figure shows that the
qualitative shape of the profile is maintained for both the Au and Ti electrodes; furthermore, the
spacing and nanotube composition chosen for the results in the main text are a good
representation of the average behavior.
Suppl. Fig. S2. Impact of CNT density and composition on the calculated
photothermoelectric voltage. Details are provided in the Supplementary Material section A.6.
The solid line is for the system used for the results presented in the main text.
B. Experimental details.
1. Scanning Photocurrent Microscopy
The setup is illustrated in Fig. S3 and detailed in the Method Summary section.
Suppl. Fig. S3. Schematic of the Scanning Photocurrent Microscopy setup used for this
work. Details are provided in the Supplementary Material section B.1.
2. Device fabrication
Vertical lines of SWCNTs were grown by chemical vapor deposition (CVD) as described in
detail in a previous publicationS3. Lines were transferred individually using tweezers and placed
directly onto SiO2 substrates (see Fig. S4). As these films were directly deposited on the wafer,
no exposure to liquids or processing steps were used in the transfer that could damage the
alignment or chemically alter the pristine SWCNTs. Following transfer, we deposited different
metallic electrodes (Au, Pd and Ti) with a thickness of 50 nm by shadow masking. We did not
use adhesion layers in the case of Au or Pd to clearly determine the role of the metal used to
fabricate the electrodes. This also helped to avoid further accidental doping by the photoresist
and wet processing which would make the nanotube films denser and less aligned.
Figure S5 shows scanning electron microscope images of a device contacted in perpendicular
using Ti and Pd electrodes. The channel length of the device is 300 μm and the photoresponse
under global illumination corresponds to Figs. 1D and 2B. The alignment is preserved under
different types of electrodes.
Suppl. Fig. S4. Diagram of sample fabrication: Vertically-grown carbon nanotube lines are
individually transferred onto silicon oxide substrates and top contacted using e-beam
evaporation and shadow masking.
Fig. S5. Scanning electron microscope images of the photodetector corresponding to the
results presented in Figs. 1D and 2B. Top image: A film is top-contacted with titanium on the left
and palladium on the right with the current flowing perpendicular to the nanotubes. Central
image: Zoom at x2,500 magnification at the Pd contact edge showing very good alignment at
the macroscopic scale in the channel and under the electrode. Bottom image: High
magnification image (13,000x) at the Ti edge showing that the high quality of alignment is fully
preserved at the spot size and wavelength scales.
References:
S1. A. D. Wilson and H. B. Holmes, Rev. Sci. Instrum. 39, 346 (1968).
S2. U. Sivan and Y. Imry, Phys. Rev B 33, 551 (1986).
S3. C. L. Pint, et al. ACS Nano 4, 1131 (2010).