Correction notice
Scientific Reports (2013) | doi: 10.1038/srep01335
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
In the version of this Article originally published online, the equations in the Supplementary Information file were numbered incorrectly. This error has now been corrected.
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
(S1)
while the thermoelectric potential across the metal is
(S2)
The total thermoelectric potential is then given by
(S3)
If the Seebeck coefficients of the metal and CNT were independent of temperature and position we would obtain
(S4)
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
(S5)
wheres 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). and the usual Seebeck coefficient are related through where l is the average spacing between nanotubes in the array.
For the case where is constant in the film and where the temperature profile is given by a constant gradient in the channel equal to , one recovers the usual expression as discussed in A1. However, in the case of local illumination, the temperature profile is given by a peaked, symmetric profile, and ifwere constant along the CNT film the photovoltage would be exactly zero because. Thus, to obtain a non-zero 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
(S6)
For the nanotube film, we can take and ; in that case , local illumination gives and the first term vanishes. (Note that this first term is the one usually included when deriving the expression as discussed in section A1.) We are left with the expression
(S7)
In this last equation, we assumed that is independent of y, and defined .
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 two-dimensional in the x-y plane, and we thus consider the heat equation in cylindrical coordinates
(S8)
whereT is the temperature, κ is the thermal conductivity, is the thermal conductance between the film and the metal contacts, is the thermal conductance between the film and the substrate, is the temperature of the metal contacts, andis the temperature of the substrate. In this equation, heating by the laser is modeled as a point heat source of strength pat.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
(S9)
whereand
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 where This gives the same equation as equation (S9), except that
(S10)
and
The solution of equation (S9) for is
(S11)
where is the modified Bessel function of order zero, andλ is the thermal length scale given by
(S12)
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, 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 justifying the assumption of an infinite film in the y direction.
The temperature distribution for a Gaussian light spot centered at and is given by
(S13)
which leads to
(S14)
The integral in the square brackets can be performed to give the final result
(S15)
To reflect the fact thatGeff 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
(S16)
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 / CNT orientation / Sm (μV/K) / Tmax (K) / Photovoltage (V/W)Au / Parallel / 2 / 0.16 / 1.6×10-4
Au / Perpendicular / 2 / 0.29 / 2.9×10-4
Pd / Perpendicular / -10 / 0.19 / -9.5×10-4
Ti / Perpendicular / 9 / 4.6 / 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
(S17)
whereρ is the film mass density and Cpis the heat capacity. From this equation we obtain the time-dependence of the maximum temperature as
(S18)
valid when . The time scale τ is given by
.(S19)
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)
(S20)
with
(S21)
wherekB 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, andEF is the Fermi energy. In the flat band case, the above equations can be integrated analytically to give
(S22)
and
(S23)
whereEC 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 . 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 helpedto 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 B33, 551 (1986).
S3. C. L. Pint, et al.ACS Nano4, 1131 (2010).